Introducing a Resolvable Network-Based SAT Solver Using Monotone CNF–DNF Dualization and Resolution

Abstract

This paper is a theoretical contribution that introduces a new reasoning framework for SAT solving based on resolvable networks (RNs). RNs provide a graph-based representation of propositional satisfiability in which clauses are interpreted as directed reaches between disjoint subsets of Boolean variables (nodes). Building on this framework, we introduce a novel RN-based SAT solver, called RN-Solver, which replaces local assignment-driven branching by global reasoning over token distributions. Token distributions, interpreted as truth assignments, are generated by monotone CNF–DNF dualization applied to white (all-positive) clauses. New white clauses are derived via resolution along private-pivot chains, and the solver’s progression is governed by a taxonomy of token distributions (black-blocked, terminal, active, resolved, and non-resolved). The main results establish the soundness and completeness of the RN-Solver. Experimentally, the solver performs very well on pigeonhole formulas, where the separation between white and black clauses enables effective global reasoning. In contrast, its current implementation performs poorly on random 3-SAT instances, highlighting both practical limitations and significant opportunities for optimization and theoretical refinement. The presented RN-Solver implementation is a proof-of-concept which validates the underlying theory rather than a state-of-the-art competitive solver. One promising direction is the generalization of strongly connected components from directed graphs to resolvable networks. Finally, the token-based perspective naturally suggests a connection to token-superposition Petri net models.

1. Introduction

We assume that the reader is familiar with the SAT problem, so we do not introduce basic notions such as clause, clause set, or CNF, or if we do then we do so only in a very concise way. A detailed description of the field can be found in the Handbook of Satisfiability [1].

The notion of resolvable networks (RN) was introduced by Kusper et al. in 2021 [2]. A resolvable network is a graphical representation of a SAT problem. An RN consists of subnetworks, which are sets of Boolean variables—or, in this context, nodes. Thus, throughout this paper, the terms “Boolean variable” and “node” are used interchangeably.

Subnetworks may overlap; however, directed edges, called reaches, are defined only between subnetworks that are disjointed. This restriction is essential, as we will see later.

A reach $(A,B)$, where $A,B$ are distinct sets of nodes (called the head and the tail of the reach, respectively), represents a logical implication

$$\bigwedge \neg A\Rightarrow \bigvee B,$$

that is, it corresponds to the clause

$$(\neg a_1\vee \cdots \vee \neg a_k\vee b_1\vee \cdots \vee b_m),$$

where $A=\{a_1,\dots ,a_k\}$ and $B=\{b_1,\dots ,b_m\}$.

Special cases are the so-called white clauses, which consist only of positive literals. These clauses are represented by white reaches of the form $(\mathrm{Source},B)$, where $\mathrm{Source}=\bigwedge \varnothing =\top$. Similarly, black clauses consist only of negative literals and are represented by black reaches of the form $(A,\mathrm{Sink})$, where $\mathrm{Sink}=\bigvee \varnothing =\perp$. We say that the reach $(A,B)$ is mixed if the head, A, and the tail, B, are not empty. Finally, the empty clause is represented by the reach $(\mathrm{Source},\mathrm{Sink})$.

This means that there is a one-to-one correspondence between clauses and reaches. Hence, there is a one-to-one correspondence between the field of SAT and the framework of resolvable networks.

RN is not a new logical system, but rather a tool—a viewpoint from which the SAT problem can be observed from a different angle.

The goal of the present work is not to propose a performance-competitive SAT solver, but to introduce and analyze a new reasoning framework for satisfiability, in which global structural properties of clause sets are exploited through graph-based representations and monotone CNF–DNF dualization. In this sense, resolvable networks are used as a unifying abstraction that connects logical inference and graph structures.

Since an RN is a directed graph, and since graph theory is sometimes humorously referred to as the “Hungarian branch of mathematics”, this perspective might provide new insights for solving SAT—at least for Hungarians. To be more serious, we hope that the framework of resolvable networks will inspire novel ways of thinking about SAT and its structure. In this paper, we present one such idea.

Accordingly, the research design of this paper follows a theory-driven approach: we first develop the formal foundations of resolvable networks and token distributions, then derive a sound and complete reasoning procedure (RN-Solver) from these principles, and finally present an implementation whose purpose is to validate and illustrate the theoretical mechanisms rather than to achieve state-of-the-art empirical performance.

From this perspective, RN-Solver should be understood as a foundational framework that explores an alternative reasoning paradigm for SAT, complementary to assignment-driven solvers such as CDCL, rather than as a direct competitor to highly optimized industrial solvers.

The proposed framework builds on classical results from monotone CNF–DNF dualization and hypergraph transversal enumeration, which provide a combinatorial basis for generating and reasoning about token distributions. By integrating these well-studied techniques into the RN setting, the paper aims to bridge established theoretical results with a novel graph-based view of satisfiability.

The rest of this paper is organized as follows. Section 2 recalls several related approaches that represent SAT instances as graphs or networks. Section 3 provides an intuitive overview of resolvable networks using a token-based interpretation. Section 4 introduces the formal definitions used throughout the paper. Section 5 contains the technical lemmas required for our main results. Section 6 presents the main theorems relating token distributions and satisfiability. Section 7 describes the novel RN-Solver algorithm for solving SAT and discusses its properties. Section 8 introduces the Token-Superposition Petri Net as a quantum-inspired perspective on token distributions in resolvable networks. Finally, Section 9 concludes the paper and outlines directions for future work.

2. Related Work

Boolean satisfiability (SAT) is one of the best-known NP-complete problems [3], and consequently it is discussed extensively in the literature; see, for example, [1]. Many other problems have been shown to be NP-complete via reductions to SAT. As a result, a (relatively) efficient solution to SAT can be leveraged to solve a wide range of other computational problems. This is one of the main reasons why SAT and algorithms for solving it play a central role in computer science, both in theory and in practice.

In this paper, we present and use a graph-based representation of the SAT problem; accordingly, we briefly review well-known results in this direction.

2.1. Graph- and Network-Based Representations for SAT and Logical Formulas

A long line of work has investigated how SAT instances, that is logical formulas in general or in specific, e.g., CNF form, can be represented as graphs or networks. Implication graphs are perhaps the best-known example: they encode implications between literals as edges in a skew-symmetric directed graph and provide a natural representation for 2-SAT [4]. Special truth-teller–liar puzzles, where each atomic statement represents the type of the person who says, are also represented in a similar way: the edge representing the statement of person “Y” that “person X is a truth-teller” connects “X” and “Y”, moreover, either both of them are truth-tellers (let us say evaluated to true) or both are liars (evaluated to be false, respectively), revealing a symmetric relation between them [5]. A statement by “Y” that “person X is a liar”, however, means that the types (values) of “X” and “Y” are in opposition. In this way, some self-citation statements are impossible, causing a paradox known as the Liar paradox. To deal with such issues, a third type of person is introduced in [6]. However, in this paper, we concentrate on and use the classical two-valued logic.

For general SAT, implication graphs are used locally inside modern CDCL solvers, introduced in [7], to implement unit propagation and conflict analysis rather than serving as a complete global representation.

Several decision-diagram- and circuit-based formalisms—see, e.g., [8]—also offer graphical representations of propositional formulas. And-inverter graphs (AIGs) are directed acyclic graphs whose internal nodes are AND gates and whose edges may be marked as inverted; they are widely used in hardware verification and logic synthesis as a compact representation of Boolean circuits [9].

Reduced ordered binary decision diagrams (BDDs) represent Boolean functions as directed acyclic graphs whose internal nodes are labeled by variables and whose edges correspond to assigning a variable to 0 or 1. For a fixed variable order, BDDs are canonical up to isomorphism [10]. Zero-suppressed BDDs (ZDDs) are a variant optimized for sparse sets of solutions, where nodes whose 1-edge leads directly to the 0-terminal are eliminated [11]. In both BDDs and ZDDs, edges represent choices in a variable assignment rather than logical consequences.

Beyond these canonical formalisms, a wide range of ad hoc graph-encoding solutions have been proposed for modeling logical puzzles and games. Representative examples of such graph-based representations can be found in [12,13,14]. In [13], there are also some puzzles, where instead of an edge, a bunch of edges have a direct meaning, e.g., representing several atomic statements connected by conjunction. This idea can be thought as an edge that does not connect to a node, but to a subset of nodes (in a somewhat similar way as hyperedges in hypergraphs). In some truth-teller–liar type puzzles, there are also normal people, and the puzzle may be solved “on the fly” based on the information that is already available and when a new piece of information (e.g., a new statement by a person in the puzzle) is arriving, the entire picture may change: in [14], the information flow was also taken into account.

There are also graph forms that originate from the study of minimal unsatisfiable formulas. In particular, minimal unsatisfiable SAT instances can be generated from minimal strongly connected digraphs by interpreting edges as implications and adding suitable monotone clauses, as shown by Abbasizanjani and Kullmann [15].

2.2. Communication Graphs, Weak Models, and the Black-and-White SAT Problem

The present work grew out of the “Graph to SAT and SAT to Graph” project, which studies correspondences between directed graphs and SAT instances [16,17]. In particular, the weak and strong models of communication graphs associate SAT formulas with directed graphs that model, for example, wireless sensor networks, where nodes correspond to sensors and edges represent communication links. The weak model has the appealing property that strongly connected communication graphs give rise to so-called Black-and-White SAT problems with exactly two satisfying assignments; however, the construction is computationally expensive, as it requires enumerating all cycles of the underlying digraph.

In most of these approaches, however, the translation is fundamentally one-way: either a SAT instance is encoded into a graph for analysis or visualization, or a graph is translated into a SAT instance for solving, but the two representations do not coexist as equivalent views of the same object. As a result, logical inference typically happens outside the graph structure itself.

2.3. Resolvable Networks as a Bidirectional Representation

Resolvable networks (RNs) were introduced in [2] precisely to address the asymmetry present in many earlier encodings. In an RN, nodes correspond to Boolean variables, subnetworks are sets of nodes, and directed edges (called reaches) connect non-overlapping subnetworks. Each reach $(A,B)$ is interpreted as the clause $\left({\bigvee }_{x\in A}\neg x\right)\vee \left({\bigvee }_{x\in B}x\right)$, and Kusper et al. show that every SAT instance can be represented as a resolvable network and, conversely, that every resolvable network has a natural CNF interpretation. Thus, there is a tight, bidirectional correspondence between clause sets and resolvable networks. The earlier work also introduces two interpretations of subnetworks (pessimistic and optimistic) and establishes a close connection between resolution and the transitivity of reaches in the pessimistic case: basic forms of path composition in the network correspond exactly to resolution steps on the associated clauses.

In that work, resolvable networks are used primarily as a graphical representation of SAT instances and as a tool for visual reasoning of concrete examples such as pigeonhole formulas and proxy-style networking scenarios. The present paper takes the next step and uses the RN formalism as the foundation of a new satisfiability algorithm: instead of DPLL-style branching on assignments, we perform global reasoning using monotone CNF–DNF dualization.

2.4. Hypergraph Transversals and Monotone Dualization

A central technical ingredient of our approach is the computation of minimal hitting sets (minimal transversals) of the family of white clauses $B_1,\dots ,B_k$. This task is exactly the classical monotone CNF–DNF dualization problem: given a monotone CNF ${\bigwedge }_i\left({\bigvee }_{x\in B_i}x\right)$, compute the minimal terms of an equivalent monotone DNF. The complexity of this dualization problem has been studied extensively; it is known to be solvable in quasi-polynomial time $n^{O(logn)}$ [18], and a number of output-sensitive algorithms have been developed for generating all minimal transversals [19,20]. On the algorithmic side, the MMCS and RS algorithms of Murakami and Uno, together with several related methods, currently form the practical state-of-the-art for hypergraph transversal enumeration [21,22]. Our RN-Solver builds directly on this line of work: it treats the set of white clauses as a hypergraph and uses a customized MMCS-based engine to compute minimal token distributions (DNF terms) as the main global reasoning step.

Resolvable networks thus sit at the intersection of three lines of research: graph-based representations of SAT, communication-graph models such as weak and strong models, and the theory and practice of hypergraph transversal generation. Compared with existing graph encodings, RNs have the unique property that they provide a bidirectional equivalence with CNF formulas and support resolution directly at the graph level. The present paper exploits this structure algorithmically by combining RN resolution with monotone CNF–DNF dualization.

Beyond their static graph-theoretic interpretation, resolvable networks also admit a natural dynamic reading in terms of token distributions and their evolution. This perspective is closely related to classical models of concurrent and distributed computation, most notably Petri nets.

Petri nets are widely used modeling tools for systems that allow parallel or concurrent events [23,24]. They are represented by bipartite graphs and are explicitly designed around token games: a system state is given by a token distribution, transitions define how tokens may move, and well-defined initial and terminal conditions determine the outcome of the computation.

Motivated by this analogy, the present paper enriches the resolvable network framework with a controlled form of token-based dynamics.

3. The Token Distribution Intuition

In this section, we present informal, token-based intuition for resolvable networks. The aim is only to build intuition; the formal definitions and semantics will be introduced in Section 4, Section 5 and Section 6.

3.1. Tokens and the Role of $\mathrm{Source}$ and $\mathrm{Sink}$

We imagine a single abstract token that represents “truth”. A node (variable) is considered to be true if it carries a copy of this token. The distinguished subnetwork $\mathrm{Source}$ is the ultimate origin of truth: it is the unique place where the token is initially present. The distinguished subnetwork $\mathrm{Sink}$ plays the opposite role: reaching $\mathrm{Sink}$ with the token corresponds to a logical contradiction (the empty clause), as we will formalize later.

Crucially, the token does not move like a physical particle; instead, it is copied along logical implications. Once a node has received the token, it remains true; the token is never “consumed” or removed from a node.

3.2. Reach as Both Noun and Verb

In a resolvable network, the word reach has two complementary meanings.

As a noun, a reach $(A,B)$ represents a directed edge or logical implication

$$\underset{a\in A}{\bigwedge }a\Rightarrow \underset{b\in B}{\bigvee }b,$$

equivalent to the clause $\left({\bigvee }_{x\in A}\neg x\right)\vee \left({\bigvee }_{x\in B}x\right)$. Intuitively, A is the set of nodes that must already be true, and B is the set of nodes that can be made true as a consequence.

As a verb, to reach means that if every node in the head A of a reach $(A,B)$ already holds the token, it is sufficient to copy the token into at least one node of the tail B. The token does not leave A; it is replicated in (at least) one element of B. Once a node has received the token, it stays true, and the propagation may continue through other reaches.

3.3. Comparison with Petri Nets

This firing behavior is similar to the firing rule in Petri nets: a transition may fire only if all its input places contain a token. There are, however, two important differences.

First, in a classical Petri net, when a transition fires, tokens are consumed from all input places and produced on all output places. In contrast, for a reach $(A,B)$ in a resolvable network

• All nodes in A must already carry the token (like input places);
• The token is not removed from any node in A;
• The token is copied to one or more nodes in B, but not necessarily to all of them, because the tail B is interpreted as a logical disjunction.

Thus, a reach $(A,B)$ behaves like a Petri net transition with conjunctive enabling on the inputs and disjunctive, non-consuming propagation to the outputs.

3.4. Time vs. No Time

Logically, a resolvable network is a static object: it is just a finite family of implications (reaches) over a set of variables. There is no built-in notion of time in the semantics. A token distribution is simply a set of nodes that currently hold a copy of the token, and hence are interpreted as true.

When we speak about a reach “firing”, or about the network “evolving”, this refers not to an intrinsic temporal behavior of the RN, but to the actions of an RN solver algorithm. Such a solver (whether based on firing reaches or on global monotone CNF–DNF dualization steps) will manipulate token distributions step by step; this algorithmic progression induces a notion of time in the computation, not in the underlying logical model.

3.5. Token Distributions

Because a reach $(A,B)$ has a disjunctive tail, $\bigwedge A\Rightarrow \bigvee B$, several different token distributions may be compatible with the same network. Whenever all nodes in A already hold the token, we only know that at least one node in B must receive the token; we do not know which one.

Starting from $\mathrm{Source}$, each white reach $(\mathrm{Source},B)$ may fire and activate one node from its tail. As different choices are made along the way, we obtain different token distributions, i.e., different sets of nodes that carry the token.

Example 1.

Consider two white reaches $(\mathrm{Source},\{a,b\left\}\right)$ and $(\mathrm{Source},\{c,d\left\}\right)$. Each reach says that at least one node on its right-hand side must receive the token. Thus, picking exactly one node from each set yields four possible token distributions:

$$\{a,c\},\{a,d\},\{b,c\},\{b,d\}.$$

Later, in the formal development, these will arise as minimal terms of a monotone CNF–DNF dualization of the white reaches.

In summary, the token distribution intuition views resolvable networks as supporting many possible “worlds” (token distributions) that satisfy all white reaches. The RN-Solver introduced in Section 7 will explore these distributions in a global, structured way, and connect their behavior to the SAT/UNSAT classification of the underlying CNF formula via the formal results in Section 6.

4. Formal Definitions

In this section, we collect the basic notions and notational conventions used throughout the paper. We start with standard SAT terminology and then introduce resolvable networks (RNs), token distributions, subsumption, DNF expansions, and the so-called token distribution taxonomy.

Throughout, let V be a finite set of Boolean variables. In the RN setting, we also refer to elements of V as nodes.

4.1. Literals, Clauses, and CNF

A literal is either a variable $x\in V$ or its negation $\neg x$. For a literal ℓ, we denote its complement by $\neg \ell$ and $\neg \neg \ell =\ell$.

A clause is a finite set of literals that contains no complementary pair, i.e., there is no variable x, such that both x and $\neg x$ belong to the clause. A clause is interpreted as the disjunction of its literals. A CNF formula is a finite set F of clauses, interpreted as their conjunction. We use the usual satisfaction relation $F\vDash C$ and the standard notion of (un)satisfiability.

If C is a clause, then the variable set of C, denoted $Var\left(C\right)$, is the set of all variables that occur in C, either positively or negatively.

For a set of literals L, we write $\neg L=\{\neg \ell \mid \ell \in L\}$ for the set obtained by complementing each literal in L. For a clause C, we also write $\neg C$ for the clause obtained by complementing every literal in C.

Resolution is one of the oldest and most widely used inference techniques in propositional satisfiability, dating back to the classical Davis–Putnam procedure [25].

Definition 1

(Resolution on Clauses). Let $\Gamma =L\vee x$ and $\Delta =M\vee \neg x$ be clauses. The resolvent of Γ and Δ on pivot x is

$$Res(\Gamma ,\Delta ;x)=L\vee M,$$

provided that the resolvent is not a tautology, i.e.,

$$L\cap \neg M=\varnothing .$$

Note that the resolution is sound and refutation-complete for CNF UNSAT.

Definition 2

(Left- and right-hand sides of a clause). Let C be a clause over V. We associate with C two disjoint sets of variables:

$$\begin{array}{cc}\hfill C^{-}& =\{x\in V\mid \neg x\in C\},\hfill \\ \hfill C^{+}& =\{x\in V\mid x\in C\}.\hfill \end{array}$$

We call $C^{-}$ the left-hand side (LHS) and $C^{+}$ the right-hand side (RHS) of C. Equivalently,

$$C\equiv \left(\underset{x\in C^{-}}{\bigvee }\neg x\right)\vee \left(\underset{x\in C^{+}}{\bigvee }x\right).$$

Thus, every clause can be seen as an implication

$$\underset{x\in C^{-}}{\bigwedge }x\Rightarrow \underset{y\in C^{+}}{\bigvee }y.$$

We adopt the standard conventions that an empty conjunction equals ⊤ and an empty disjunction equals ⊥.

Definition 3

(White and black clauses). A clause C is called white if $C^{-}=\varnothing$ and $C^{+}\ne \varnothing$, i.e., it contains only positive literals.

A clause C is called black if $C^{+}=\varnothing$ and $C^{-}\ne \varnothing$, i.e., it contains only negative literals.

The empty clause (with $C^{-}=C^{+}=\varnothing$) is neither white nor black.

Definition 4

(Partial valuation and CNF simplification). A partial valuation is a partial function $\alpha :V\rightharpoonup \{0,1\}$, where the value 1 represents true and 0 represents false.

Given a CNF formula F, we write $Simp(F,\alpha )$ for the standard simplification of F under α (as in unit propagation): every clause satisfied by α is deleted, and from the remaining clauses, all literals falsified by α are removed.

We say that α is a satisfying (partial) valuation of F if and only if $Simp(F,\alpha )$ is empty.

Let $X\subseteq V$ and let $C\subseteq X$. We write $V_C^X$ for the total valuation on X, defined by $V_C^X\left(x\right)=1$ for $x\in C$ and $V_C^X\left(x\right)=0$ for $x\in X\setminus C$.

Moreover, for partial valuations α and β with $dom\left(\alpha \right)\cap dom\left(\beta \right)=\varnothing$, their union $\alpha \cup \beta$ denotes the partial valuation with domain $dom\left(\alpha \right)\cup dom\left(\beta \right)$ defined by

$$(\alpha \cup \beta )\left(x\right)=\left\{\begin{array}{cc}\alpha \left(x\right),\hfill & x\in dom\left(\alpha \right),\hfill \\ \beta \left(x\right),\hfill & x\in dom\left(\beta \right).\hfill \end{array}\right.$$

4.2. Resolvable Networks and Reaches

Resolvable networks provide a directed graph representation of CNF formulas.

Definition 5

(Resolvable Network). Let V be a finite set of nodes (i.e., Boolean variables). A resolvable network over V is a directed graph

$$\mathcal{N}=(\mathcal{SN}\cup \{\mathrm{Source},\mathrm{Sink}\},\mathcal{R}),$$

where

• $\mathcal{SN}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}$ is a set of (ordinary) subnetworks , each being a nonempty set of nodes.
• $\mathrm{Source}$ and $\mathrm{Sink}$ are two distinguished symbols. For notational convenience, both are associated with the empty set, but are interpreted semantically as $\bigwedge \varnothing =\top$ and $\bigvee \varnothing =\perp$, respectively. They are not elements of $\mathcal{SN}$ and do not correspond to ordinary subnetworks.
• $\mathcal{R}$ is a set of reaches. A reach is an ordered pair $(A,B)$ with $A,B\in \mathcal{SN}\cup \{\mathrm{Source},\mathrm{Sink}\}$; for such a reach, we call A its head and B its tail. Each reach belongs to exactly one of the following four types:

  −

  Mixed reaches: $(A,B)$ with $A,B\in \mathcal{SN}$, $A\cap B=\varnothing$ and both A and B nonempty; these correspond to clauses containing at least one negative and at least one positive literal;

  −

  White reaches: $(\mathrm{Source},B)$ with $B\in \mathcal{SN}$; these correspond to clauses containing only positive literals;

  −

  Black reaches: $(A,\mathrm{Sink})$ with $A\in \mathcal{SN}$; these correspond to clauses containing only negative literals;

  −

  Empty reach: the special reach $(\mathrm{Source},\mathrm{Sink})$, which represents a direct edge from $\mathrm{Source}$ to $\mathrm{Sink}$ and corresponds to the empty clause.

  No other reaches are allowed: in particular, reaches of the form $(A,\mathrm{Source})$ or $(\mathrm{Sink},B)$ never occur, and $\mathrm{Source}$ can appear only as the head, while $\mathrm{Sink}$ can appear only as the tail of a reach.

We write $U=\bigcup \mathcal{SN}$ for the universe of nodes occurring in ordinary subnetworks.

Note that $U\subseteq V$, since not every node in V must appear in the network.

Note that there is a one-to-one correspondence between reaches of $\mathcal{N}$ and the clauses of ${\mathcal{N}}^{\wedge }$: each reach $(A,B)\in \mathcal{R}$ corresponds to the clause $\left({\bigvee }_{x\in A}\neg x\right)\vee \left({\bigvee }_{x\in B}x\right)$, and conversely, every clause occurring in ${\mathcal{N}}^{\wedge }$ arises from exactly one reach in $\mathcal{R}$.

More generally, given any CNF clause C over V, it induces the pair $(C^{-},C^{+})$; this pair is a valid reach in the sense of Definition 5 precisely when the head and tail are formed by the following convention: the head is $\mathrm{Source}$ if $C^{-}=\varnothing$, and otherwise it is the subnetwork $C^{-}\in \mathcal{SN}$; similarly, the tail is $\mathrm{Sink}$ if $C^{+}=\varnothing$, and otherwise it is the subnetwork $C^{+}\in \mathcal{SN}$. In all cases, the RN side-condition $C^{-}\cap C^{+}=\varnothing$ holds, which is automatic for clauses.

If not stated otherwise, throughout this paper, $\mathcal{N}$ denotes a resolvable network, $\mathcal{SN}\cup \{\mathrm{Source},\mathrm{Sink}\}$ its subnetworks and $\mathcal{R}$ its reaches: $\mathcal{N}=(\mathcal{SN}\cup \{\mathrm{Source},\mathrm{Sink}\},\mathcal{R})$.

A more readable definition of resolvable network is given in Section 1. In our earlier paper [2], several interpretations of resolvable networks and their reaches were allowed. In the present work, we restrict ourselves to the $N^{\wedge }$ and ${(A,B)}^{\vee }$ semantics.

Definition 6

(Semantics and satisfiability of a resolvable network). Let $\mathcal{N}=(\mathcal{SN}\cup \{\mathrm{Source},\mathrm{Sink}\},\mathcal{R})$ be a resolvable network. Each reach $(A,B)\in \mathcal{R}$ abbreviates the logical implication

$$\underset{a\in A}{\bigwedge }a\Rightarrow \underset{b\in B}{\bigvee }b,$$

equivalently represented by the clause $\left({\bigvee }_{x\in A}\neg x\right)\vee \left({\bigvee }_{x\in B}x\right)$. The CNF formula associated with $\mathcal{N}$ is

$${\mathcal{N}}^{\wedge }=\underset{(A,B)\in \mathcal{R}}{\bigwedge }(\left(\underset{x\in A}{\bigvee }\neg x\right)\vee \left(\underset{x\in B}{\bigvee }x\right)).$$

We say that $\mathcal{N}$ is satisfiable (SAT) iff ${\mathcal{N}}^{\wedge }$ is satisfiable as a CNF formula, and unsatisfiable (UNSAT) otherwise.

In this sense, RNs do not introduce a new logic; they provide a structured, directed-graph view of classical propositional CNF formulas.

4.3. Token Distributions and Forward Propagation

Token-based intuition views a truth assignment as a token distribution: a node (i.e., a variable) is considered to be true if it holds the token (or a copy of it). A reach $(A,B)$fires whenever all nodes in A are true, in which case the token propagates forward (i.e., is copied) to one or more nodes in B.

Definition 7

(Token distribution and forward propagation). A token distribution is a set $T\subseteq U$ of nodes. A node $x\in U$ is said to be true under T if $x\in T$; intuitively, such nodes are regarded as currently carrying a copy of the token.

Let $(A,B)\in \mathcal{R}$ be a reach. We say that $(A,B)$ can fire from T if $A\subseteq T$. A single forward-propagation step may then enlarge the token distribution by adding one or more nodes from B; that is, we move from T to some $T^{\prime }$ satisfying

$$T\subset T^{\prime }\subseteq T\cup B.$$

Note that we only count a firing as a propagation step if it strictly increases the token distribution (i.e., if at least one new node from B is added).

Forward propagation therefore induces a monotone evolution of token distributions. Since any subset of B may be added at each step, different choices yield different possible evolutionary paths.

A more readable, intuitive description of these notions can be found in Section 3.

We could define a forward-closure operator and build a resolvable-network solver based solely on forward propagation. However, this would result in nothing more than a DPLL-style SAT solver [26], e.g., a branching search over evolving truth assignments. As we will see later on, we instead rely on monotone CNF–DNF dualization, which updates token distributions globally rather than through purely local forward propagation.

4.4. Definitions on Monotone CNF–DNF Dualization

Note that all subsequent definitions (monotone CNF–DNF dualization, token distribution taxonomy) are intended only for networks in which $(\mathrm{Source},\mathrm{Sink})\notin \mathcal{R}$.

Generating the token distribution from white reaches $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$ is equivalent to converting the monotone CNF formula $\{B_1,\dots ,B_k\}$ into its minimal DNF. This is the classical monotone CNF–DNF dualization (also called hypergraph transversal generation), which has been extensively studied in the literature [18,19,20].

Definition 8

(Minimal DNF via monotone CNF–DNF dualization). Let $\mathcal{W}$ be a family of white clauses of the form $\left({\bigvee }_{x\in B_i}x\right)$ (equivalently, white reaches $(\mathrm{Source},B_i)$) with $B_i\subseteq U$ for $i=1,\dots ,k$. The associated monotone CNF formula is $F_{\mathcal{W}}={\bigwedge }_{i=1}^k\left({\bigvee }_{x\in B_i}x\right).$ The DNF dual of $F_{\mathcal{W}}$, obtained by monotone CNF–DNF dualization, is denoted by $DNF(B_1,\dots ,B_k)$ and defined as

$$DNF(B_1,\dots ,B_k)=\{C\subseteq U\mid C\mathit{is}\mathit{inclusion}-\mathit{minimal}\mathit{and}C\cap B_i\ne \varnothing \mathit{for}\mathit{all}i=1,\dots ,k\}.$$

Each such set C is interpreted as the (partial) assignment ${\bigwedge }_{b\in C}b$, i.e., as a token distribution. Equivalently, the minimal assignments are exactly the minimal hitting sets (i.e., minimal transversals) of the hypergraph with edge set $\{B_1,\dots ,B_k\}$.

Each $C\in DNF(B_1,\dots ,B_k)$ represents a possible initial token distribution obtained by selecting at least one node from each white clause.

The classical theory of monotone CNF–DNF dualization provides the following well-known properties:

• $DNF(B_1,\dots ,B_k)=\varnothing$ if there exists $i\in \{1,\dots ,k\}$, such that $B_i=\varnothing$.
• $DNF(B_1,\dots ,B_k)$ forms an antichain: no C properly contains another.
• $|DNF(B_1,\dots ,B_k\left)\right|$ may be exponentially large in general, even when each set $B_i$ is small.
• The monotone CNF dualization problem—computing $DNF(B_1,\dots ,B_k)$—is solvable in quasi-polynomial time $n^{O(logn)}$, due to Fredman and Khachiyan [18].
• Private-element property; see Lemma 7.

4.5. Subsumption and Resolution on Reaches

In this subsection, we define subsumption and resolution on reaches.

Definition 9

(Resolution on reaches). Let $(A,B)$ and $(C,D)$ be reaches. A node x is a pivot for these reaches if $x\in B$ and $x\in C$. The reach-resolvent on pivot x is

$$Res\left((A,B),(C,D);x\right)=\left(A\cup (C\setminus \{x\left\}\right),(B\setminus \{x\left\}\right)\cup D\right),$$

provided that the head and the tail of the resolvent are non-overlapping, i.e.,

$$\left(A\cup (C\setminus \{x\left\}\right)\right)\cap \left((B\setminus \{x\left\}\right)\cup D\right)=\varnothing .$$

We recall the definition of clause subsumption and generalize it for reaches.

Definition 10

(Clause subsumption). Let F be a clause set. A clause C is subsumed by F if there exists a clause $C^{\prime }\in F$ such that $C^{\prime }\subseteq C$.

Note that if C is subsumed by F, then $F\vDash C$. If $C\in F$, then C is trivially subsumed by F; therefore, in several lemmas, we explicitly examine whether C is subsumed by $F\setminus \left\{C\right\}$ or not.

Definition 11

(Reach subsumption). A reach $(A,B)$ is subsumed by $\mathcal{N}$ if there exist $A^{\prime }\subseteq A$ and $B^{\prime }\subseteq B$ with $(A^{\prime },B^{\prime })\in \mathcal{R}$.

If $(A,B)$ is subsumed by $\mathcal{N}$, then ${\mathcal{N}}^{\wedge }\vDash {(A,B)}^{\vee }$.

4.6. Taxonomy of Token Distributions

Using the notions of black and white reaches and the notion of subsumption, we classify token distributions.

Definition 12

(Black-blocked token distribution). A token distribution $C\subseteq U$ is black-blocked if there exists a black reach $(C^{\prime },\mathrm{Sink})$ such that $C^{\prime }\subseteq C$.

Intuitively, such “branches” are “UNSAT”, i.e., if all token distributions, which are generated from white reaches, are black-blocked, then the resolvable network is UNSAT; see Lemma 13.

Furthermore, if C is black-blocked, then the reach $(C,U\setminus C)$ is subsumed by $\mathcal{N}$; see Lemma 3.

Definition 13

(Terminal token distribution). A token distribution $C\subseteq U$ is terminal if the reach $(C,U\setminus C)$ is not subsumed by $\mathcal{N}$.

Intuitively, if C is a terminal token distribution, then this “branch” is “SAT”, i.e., the whole resolvable network is SAT; see Lemma 17.

Note that a token distribution cannot be black-blocked and at the same time terminal; see Lemma 5.

Definition 14

(Active token distribution). A token distribution C is active if it is not black-blocked, it is not terminal, and there exists a reach $(C^{\prime },D)$ such that $C^{\prime }\subseteq C$ and $D\ne \mathrm{Sink}$.

Intuitively, if C is an active token distribution, then we can add a “new” white clause on this branch; see Lemma 9. Whether this “new” clause is “really new” or not depends on the information carried by the clause. If it is not subsumed, then it is “really new”, otherwise not; see the next definition and the attached lemmas.

Definition 15

(Resolved and non-resolved token distribution). Let C be an active token distribution. We say that C is resolved if for every reach $(C^{\prime },D)$ with $C^{\prime }\subseteq C$, the white reach generated from $(C^{\prime },D)$ via the private-pivot chain of Lemma 9 is already subsumed by $\mathcal{N}$. Otherwise, C is non-resolved.

Note that $D\ne \mathrm{Sink}$, because C is active.

The existence of an appropriate private-pivot chain for any token distribution generated by a monotone CNF–DNF dualization step is guaranteed by Lemma 9.

Intuitively, if C is a non-resolved token distribution, then we should process it and add new white clauses based on it, see Lemma 10. Otherwise, this branch is a kind of “dead” branch, which means the following: If all non-black-blocked token distributions are resolved, then this instance is SAT; see Theorem 1.

We will see that non-resolved branches can be used to extend our knowledge on white reaches, which will result in different token distributions, which will eventually be either black-blocked, terminal or resolved; see Theorems 2 and 3.

Note that a non-terminal token distribution must be either black-blocked, or active, but not both; see Lemma 4. Furthermore, our token distribution taxonomy is complete and distinct; see Lemma 5.

In summary, these four definitions yield the following taxonomy of token distributions. A token distribution is

• Black-blocked.
• Terminal but not black-blocked.
• Active, i.e., neither black-blocked nor terminal.

Every active token distribution is either

• Resolved;
• Non-resolved.

5. Technical Lemmas

After the definitions, we present the theory based on those notions. In this section, we collect a series of technical lemmas that provide the foundational tools used later in the main results. These lemmas formalize several intuitions introduced earlier.

5.1. Unit Propagation

Unit propagation is the simplest form of reasoning both in SAT solving and in resolvable networks. Unit propagation behaves exactly as described in classical CNF.

Lemma 1

(Unit propagation with positive unit). If $(\mathrm{Source},\{b\left\}\right)$ is a reach, then b must be true in every satisfying assignment of ${\mathcal{N}}^{\wedge }$. Consequently, we delete every reach whose tail contains b and remove b from the head of every remaining reach. We also delete b from every subnetwork in $\mathcal{SN}$, and therefore from the universe U. If some subnetwork becomes empty as a result, we delete it from $\mathcal{SN}$. We also delete every subnetwork from $\mathcal{SN}$ that does not occur in any remaining reach. All these simplifications preserve satisfiability.

Proof. 

This is exactly the analogue of the positive unit propagation in SAT: the reach $(\mathrm{Source},\{b\left\}\right)$ corresponds to the unit clause $\left\{b\right\}$, which forces b to be true.    □

Lemma 2

(Unit propagation with negative unit). If $\left(\right\{a\},\mathrm{Sink})$ is a reach, then a must be false in every satisfying assignment of ${\mathcal{N}}^{\wedge }$. Consequently, we delete every reach whose head contains a and remove a from the tail of every remaining reach. We also delete a from every subnetwork in $\mathcal{SN}$, and therefore from the universe U. If some subnetwork becomes empty as a result, we delete it from $\mathcal{SN}$. We also delete every subnetwork from $\mathcal{SN}$ that does not occur in any remaining reach. These simplifications preserve satisfiability.

Proof. 

This is exactly the analogue of the negative unit propagation in SAT: the reach $\left(\right\{a\},\mathrm{Sink})$ corresponds to the unit clause $\{\neg a\}$, which forces a to be false.    □

5.2. Properties of the Token Distribution Taxonomy

The behavior of token distributions plays a central role in the global reasoning steps of the resolvable-network solver. We introduced the intuitive categories: terminal, black-blocked, and active, and now we establish their fundamental structural properties. The following lemmas show why they form a full, i.e., complete and distinct taxonomy.

Lemma 3.

Let $C\subseteq U$ be a black-blocked token distribution. Then, $(C,U\setminus C)$ is subsumed by $\mathcal{N}$.

Proof. 

Since C is black-blocked, by Definition 12 there exists a black reach $(C^{\prime },\mathrm{Sink})$ with $C^{\prime }\subseteq C$. Because $\mathrm{Sink}=\varnothing$ and $\varnothing \subseteq U\setminus C$, we determine, by Definition 11, that $(C,U\setminus C)$ is subsumed by $\mathcal{N}$.    □

The previous lemma isolates the essential property of black-blocked token distributions: their outgoing “maximal” reach is already entailed by the network due to the presence of $\mathrm{Sink}$. While technically simple, this lemma allows us to streamline the proof of the main result of this subsection, Lemma 5, considerably.

Lemma 4

(Non-terminal token distributions are either black-blocked or active). Let $C\subseteq U$ be a token distribution. Assume that C is non-terminal. Then, C is either black-blocked or active, but not both.

Proof. 

Since C is a non-terminal token distribution, we know, by Definition 13, that the reach $(C,U\setminus C)$ is subsumed by $\mathcal{N}$. Thus, by Definition 11, there exists a reach $(A^{\prime },B^{\prime })\in \mathcal{R}$ such that $A^{\prime }\subseteq C$ and $B^{\prime }\subseteq U\setminus C$.

If there exists a black reach $(C^{\prime },\mathrm{Sink})\in \mathcal{R}$ with $C^{\prime }\subseteq C$, then, by Definition 12, C is black-blocked. Moreover, by Definition 14, an active token distribution must not be black-blocked, so in this case C cannot be active.

Otherwise, assume that no such black reach exists, i.e., there is no $(C^{\prime },\mathrm{Sink})$ with $C^{\prime }\subseteq C$. In particular, for every reach $(A^{″},B^{″})\in \mathcal{R}$ with $A^{″}\subseteq C$, we have $B^{″}\ne \mathrm{Sink}$. Since we already know that there exists at least one reach $(A^{\prime },B^{\prime })$ with $A^{\prime }\subseteq C$ and $B^{\prime }\subseteq U\setminus C$, we determine that there exists a reach $(A^{\prime },B^{\prime })$ with $A^{\prime }\subseteq C$ and $B^{\prime }\ne \mathrm{Sink}$, and C is not black-blocked. Together with the assumption that C is non-terminal, all conditions of Definition 14 are satisfied; hence, C is active.

In summary, either C is black-blocked (and then not active), or C is active (and then not black-blocked). Thus, C is either black-blocked or active, but not both.    □

The previous lemma formalizes an intuitive point: every non-terminal token distribution must fall into exactly one of the two remaining categories—black-blocked or active. This observation is used to establish the full taxonomy in the next lemma.

Lemma 5

(The token distribution taxonomy is complete and distinct). Let $C\subseteq U$ be a token distribution. Then, exactly one of the following holds:

1.

C is terminal;

2.

C is black-blocked;

3.

C is active.

In particular, every token distribution falls into one of these three categories (completeness), and no token distribution belongs to more than one of them (distinctness).

Proof. 

Let $C\subseteq U$ be an arbitrary but fixed token distribution.

First, we show the completeness.

We know that either $(C,U\setminus C)$ is subsumed by $\mathcal{N}$; $(C,U\setminus C)$ is not subsumed by $\mathcal{N}$. If $(C,U\setminus C)$ is not subsumed, then C is terminal by Definition 13, and we are in case (1).

If $(C,U\setminus C)$ is subsumed, then C is non-terminal. By Lemma 4, C is either black-blocked or active (but not both), so we are in case (2) or (3). Thus, every token distribution is terminal, black-blocked, or active.

Second, we show the disjoint property.

Assume first that C is terminal. Then, by Definition 13, $(C,U\setminus C)$ is not subsumed. If C were black-blocked, then by Definition 12 there would exist a black reach $(C^{\prime },\mathrm{Sink})\in \mathcal{R}$ with $C^{\prime }\subseteq C$. Since $\mathrm{Sink}=\varnothing$ and $\varnothing \subseteq U\setminus C$, this reach $(C^{\prime },\mathrm{Sink})$ would subsume $(C,U\setminus C)$ in the sense of Definition 11, a contradiction. Hence, a terminal C cannot be black-blocked. Moreover, by Definition 14, an active token distribution must be not terminal, so a terminal C cannot be active either.

Assume next that C is black-blocked. Then, based on Lemma 3, we know that $(C,U\setminus C)$ is subsumed by $\mathcal{N}$, so C is non-terminal. Furthermore, by Definition 14, an active token distribution must be not black-blocked. So C cannot be active.

Finally, assume that C is active. By Definition 14, an active token distribution is neither terminal nor black-blocked. Thus, an active C cannot be terminal and cannot be black-blocked.

Therefore, no token distribution belongs to more than one of the three categories, which proves the distinctness.    □

This is the main result of this subsection. It shows that every token distribution falls uniquely into one of the three categories, i.e., it is a complete and distinct taxonomy. Our RN-Solver, Algorithm 1, relies heavily on this result.

Algorithm 1 RN-Solver
Input: A resolvable network $\mathcal{N}=(\mathcal{SN},\mathcal{R})$.
Output: SAT if the corresponding SAT problem is satisfiable; UNSAT otherwise.
1: function RN-Solver($\mathcal{N}=(\mathcal{SN},\mathcal{R})$)
2:     Exhaustively apply unit propagation to $\mathcal{N}$.	▹ Store units in A, Lemmas 1 and 2
3:     if $\mathcal{R}$ is empty then
4:         return SAT	▹ Empty network, satisfied by A, Lemma 14
5:     if $(\mathrm{Source},\mathrm{Sink})\in \mathcal{R}$ then
6:         return UNSAT	▹ direct contradiction, Lemma 11
7:     if no white reach $(\mathrm{Source},B)$ with $B\ne \varnothing$ exists then
8:         return SAT	▹A ∪ all-false satisfies the network, Lemma 15
9:     Let the current white reaches be $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$.
10:     Let $\mathcal{T}$ be the set of token distributions $\mathcal{T}=DNF(B_1,\dots ,B_k)$.
11:     for each token distribution $C\in \mathcal{T}$ do
12:         if C is terminal then
13:            return SAT	▹ SAT witness, $A\cup V_C$ satisfies the network, Lemma 17
14:         if C is black-blocked then
15:            mark C as black-blocked
16:         if C is not black-blocked then
17:            for each reach $(C^{\prime },D)\in \mathcal{R}$ with $C^{\prime }\subseteq C$ and $D\ne \mathrm{Sink}$ do
18:                Let $B_{\mathrm{new}}=(\left({\bigcup }_{c\in C^{\prime }}{B}_{j\left(c\right)}\right)\cup D)\setminus C^{\prime }$ and $B_{j\left(c\right)}\cap C=\left\{c\right\}$	▹ Lemma 9
19:                if $(\mathrm{Source},B_{\mathrm{new}})$ is new and not subsumed by $(\mathcal{SN},\mathcal{R})$ then
20:                    add $(\mathrm{Source},B_{\mathrm{new}})$ to $\mathcal{R}$, and add $B_{\mathrm{new}}$ to $\mathcal{SN}$
21:                    mark C as non-resolved
22:            if C is not marked as non-resolved then
23:                mark C as resolved
24:     if every $C\in \mathcal{T}$ is black-blocked then
25:         return UNSAT	▹ Lemma 13
26:     else if every $C\in \mathcal{T}$ is black-blocked or resolved then
27:         Let F be a resolved token distribution
28:         return SAT	▹ White-fixpoint, $A\cup V_F$ satisfies the network, Theorem 1
29:     else
30:         return RN-Solver($(\mathcal{SN},\mathcal{R})$)	▹ recursive call

5.3. Lemmas on Monotone CNF–DNF Dualization

Monotone CNF–DNF dualization plays a central role in our global reasoning framework. In this subsection, we recall the fundamental properties of the dualization process that we rely on later. These results are classical and well known in the literature, and therefore we omit their proofs. Note that the Minimal Term Property is built directly into our Definition 8, which guarantees that the dualization produces only minimal terms.

Lemma 6

(DNF dual is equivalent to CNF). Let $F=\{B_1,\dots ,B_k\}$ be a set of white clauses with a positive value of k. Then $\bigwedge F\equiv \bigvee DNF\left(F\right)$.

Lemma 7

(Private-element property). If $C\in DNF(B_1,\dots ,B_k)$, then for every element $c\in C$, there exists an index j such that

$$B_j\cap C=\left\{c\right\}.$$

In other words, every $c\in C$ is the unique element of C that hits some hyperedge $B_j$; hence, c is private to $B_j$.

5.4. Pivots and the Private-Pivot Chain

The private-element property of monotone CNF–DNF dualization has a direct counterpart in resolvable networks: each private element of a minimal DNF term acts as a unique pivot for resolution. In this subsection, we lift this observation to the level of reaches, showing how private pivots give rise to a structured sequence of resolutions, which is known as the private-pivot chain. This mechanism will later serve as the key tool for generating new white reaches whenever a token distribution is active but not yet resolved.

Lemma 8

(Private-pivot). Let $(A,B_1),\dots ,(A,B_k)$ be reaches with a positive value of k. Let $C\in DNF(B_1,\dots ,B_k)$ be a token distribution. Suppose that there is a reach $(C^{\prime },D)$ such that $C^{\prime }\subseteq C$ and $A\cap D=\varnothing$. By the private-element property, for every $c\in C^{\prime }$ there exists an index j such that $B_j\cap C^{\prime }=\left\{c\right\}$. Fix such a c and the corresponding j. Then, c is the unique pivot, the private pivot, for resolution between $(A,B_j)$ and $(C^{\prime },D)$, and the resolvent is

$$Res\left((A,B_j),(C^{\prime },D);c\right)=\left((A\cup C^{\prime })\setminus \left\{c\right\},(B_j\cup D)\setminus \left\{c\right\}\right).$$

Proof. 

By Lemma 7, for each $c\in C$ there exists $j\left(c\right)$ with $B_{j\left(c\right)}\cap C=\left\{c\right\}$. For any fixed $c\in C^{\prime }$ we can use the same index $j=j\left(c\right)$ and obtain $B_j\cap C^{\prime }=\left\{c\right\}$ as well (since $C^{\prime }\subseteq C$ and $c\in C^{\prime }$). Therefore, c occurs positively in the clause of $(A,B_j)$ and negatively in the clause of $(C^{\prime },D)$, while $A\cap D=\varnothing$ by assumption. Thus, c is the only variable lying in $(B_j\cap C^{\prime })\cup (A\cap D)$, so it is the unique pivot.

By Definition 9, the resolvent on c equals

$$Res\left((A,B_j),(C^{\prime },D);c\right)=\left((A\cup C^{\prime })\setminus \left\{c\right\},(B_j\cup D)\setminus \left\{c\right\}\right),$$

provided the resolvent’s head and tail are disjointed. Now, $A\cap B_j=\varnothing$ and $C^{\prime }\cap D=\varnothing$ hold for reaches (Definition 5), $A\cap D=\varnothing$ holds by hypothesis, and $(C^{\prime }\setminus \left\{c\right\})\cap (B_j\setminus \left\{c\right\})=\varnothing$ follows from $B_j\cap C^{\prime }=\left\{c\right\}$. Therefore, the side condition in Definition 9 is satisfied. Hence, the claim follows.    □

The private-pivot lemma captures the behavior of a single private element of a minimal DNF term: each such element induces a unique pivot for resolution. In the general case, several private elements may appear simultaneously in $C^{\prime }\subseteq C$. The next lemma shows how these private pivots interact when applied successively, yielding a structured sequence of resolutions called the private-pivot chain.

Lemma 9

(Private-pivot chain). Let $(A,B_1),\dots ,(A,B_k)$ be reaches with a positive value of k. Let $C\in DNF(B_1,\dots ,B_k)$ be a token distribution. Let $C^{\prime }\subseteq C$ and suppose there is a reach $(C^{\prime },D)$ with $A\cap D=\varnothing$. For each $c\in C^{\prime }$ choose $j\left(c\right)$ given by Lemma 7 for C, i.e., $B_{j\left(c\right)}\cap C=\left\{c\right\}$; then $B_{j\left(c\right)}\cap C^{\prime }=\left\{c\right\}$. Resolve the clause of $(C^{\prime },D)$ successively (in any order) with the clauses of the reaches $(A,B_{j\left(c\right)})$ on the pivots $c\in C^{\prime }$. The resulting clause is

$$\left(\underset{x\in A}{\bigvee }\neg x\right)\vee \left(\underset{c\in C^{\prime }}{\bigvee }\underset{b\in B_{j\left(c\right)}\setminus \left\{c\right\}}{\bigvee }b\right)\vee \left(\underset{x\in D}{\bigvee }x\right),$$

equivalently, the reach

$$\left(A,(\left({\bigcup }_{c\in C^{\prime }}{B}_{j\left(c\right)}\right)\cup D)\setminus C^{\prime }\right).$$

In particular, if $A=\mathrm{Source}$, this resolvent is white by Definition 3.

Proof. 

By Lemma 7 applied to C, for each $c\in C^{\prime }$ there exists $j\left(c\right)$ with $B_{j\left(c\right)}\cap C=\left\{c\right\}$; hence, $B_{j\left(c\right)}\cap C^{\prime }=\left\{c\right\}$ since $C^{\prime }\subseteq C$ and $c\in C^{\prime }$. Therefore, in the clause form (Definition 6), each $c\in C^{\prime }$ occurs negatively in the clause of $(C^{\prime },D)$ and positively in that of $(A,B_{j\left(c\right)})$, so it is a valid pivot; the assumption $A\cap D=\varnothing$ excludes any additional pivot from $A\cap D$. By Definition 1, resolving on c removes $\neg c$, keeps $\left({\bigvee }_{x\in A}\neg x\right)$ and $\left({\bigvee }_{x\in D}x\right)$ and adds $B_{j\left(c\right)}\setminus \left\{c\right\}$; repeating this for all $c\in C^{\prime }$ eliminates all $\neg c$ ($c\in C^{\prime }$) and yields the displayed clause. So, the equivalent reach has head A and tail $((\left({\bigcup }_{c\in C^{\prime }}{B}_{j\left(c\right)}\right)\cup D)\setminus C^{\prime })$; RN side-disjointedness holds, since $A\cap B_{j\left(c\right)}=\varnothing$, $C^{\prime }\cap D=\varnothing$, and $(B_{j\left(c\right)}\setminus \left\{c\right\})\cap C^{\prime }=\varnothing$. Hence, the claim holds.    □

In some cases, there are several possible private-pivot chains available. Any of them may be chosen, even if they produce different outputs. A reasonable choice is to select one for which $B_{j\left(c\right)}\setminus D$ is empty, or as small as possible. However, this is not required, as our RN-Solver, Algorithm 1, will correct itself in a later recursive call; see Theorem 3.

Example 2

(Illustration of the private-pivot chain). Consider the following CNF formula in DIMACS format:

p cnf 8 4	comment: 8 variables, 4 clauses
1 2 0	comment: $(\mathrm{Source},\{1,2\left\}\right)$ (white reach)
3 4 0	comment: $(\mathrm{Source},\{3,4\left\}\right)$ (white reach)
5 6 0	comment: $(\mathrm{Source},\{5,6\left\}\right)$ (white reach)
−1 −3 7 8 0	comment: $\left(\right\{1,3\},\{7,8\left\}\right)$

We have three white reaches $(\mathrm{Source},B_1)$ with $B_1=\{1,2\}$, $(\mathrm{Source},B_2)$ with $B_2=\{3,4\}$, and $(\mathrm{Source},B_3)$ with $B_3=\{5,6\}$. A minimal term of $DNF(B_1,B_2,B_3)$ is, for instance, $C=\{1,3,5\}$, obtained by picking one element from each $B_i$.

By the private-element property (Lemma 7), each $c\in C$ is private to exactly one $B_j$: $B_1\cap C=\left\{1\right\}$, $B_2\cap C=\left\{3\right\},B_3\cap C=\left\{5\right\}$. Thus, 1 is private to $B_1$, 3 is private to $B_2$, and 5 is private to $B_3$.

Now, take the proper subset $C^{\prime }=\{1,3\}\subset C$ and the reach $(C^{\prime },D)=(\{1,3\},\{7,8\})$ coming from the last clause, with $A=\mathrm{Source}$ and $D=\{7,8\}$. Clearly $A\cap D=\varnothing$, so the assumptions of Lemma 9 are satisfied. Restricting the private-element witnesses from C to $C^{\prime }$, we have $B_1\cap C^{\prime }=\left\{1\right\}$, $B_2\cap C^{\prime }=\left\{3\right\}$, so the private pivots in $C^{\prime }$ are 1 (for $B_1$) and 3 (for $B_2$). The private-pivot chain now resolves $(C^{\prime },D)$ successively on these pivots with the reaches $(\mathrm{Source},B_1)$ and $(\mathrm{Source},B_2)$ (in any order).

By Lemma 9, the final resolvent reach is

$$\begin{array}{cc}\hfill \left(\mathrm{Source},((B_1\cup B_2)\cup D)\setminus C^{\prime }\right)& =\left(\mathrm{Source},\left(\right\{1,2\}\cup \{3,4\}\cup \{7,8\left\}\right)\setminus \{1,3\}\right)\hfill \\ & =\left(\mathrm{Source},\{2,4,7,8\}\right),\hfill \end{array}$$

which is white by Definition 3. Thus, starting from the proper subset $C^{\prime }=\{1,3\}$ of the minimal term $C=\{1,3,5\}$, the private-pivot chain produces the new white reach $(\mathrm{Source},\{2,4,7,8\left\}\right)$.

The private-pivot chain provides a systematic way of deriving new white reaches from an active token distribution. In order to use these newly generated reaches within the solver, we must ensure that extending the network with such a reach does not change the set of models. The following lemma establishes exactly this: every private-pivot chain reach is a logical consequence of the existing network, and adding it preserves satisfiability.

Lemma 10

(Adding a new private-pivot chain-generated reach preserves satisfiability). Let $\mathcal{N}=(\mathcal{SN},\mathcal{R})$ be a resolvable network. Let $(\mathrm{Source},B_{\mathrm{new}})$ be a white reach obtained by the private-pivot chain of Lemma 9, and assume that $(\mathrm{Source},B_{\mathrm{new}})$ is not subsumed by $\mathcal{N}$.

Define the extended network

$${\mathcal{N}}^{\prime }=\left(\mathcal{SN}\cup \left\{B_{\mathrm{new}}\right\}\cup \{\mathrm{Source},\mathrm{Sink}\},\mathcal{R}\cup \left\{(\mathrm{Source},B_{\mathrm{new}})\right\}\right).$$

Then,

$${\mathcal{N}}^{\wedge }\vDash {\mathcal{N}}^{\prime }\wedge .$$

That is, every model of $\mathcal{N}$ is also a model of ${\mathcal{N}}^{\prime }$.

Proof. 

By definition,

$${\mathcal{N}}^{\prime }\wedge ={\mathcal{N}}^{\wedge }\wedge {(\mathrm{Source},B_{\mathrm{new}})}^{\vee }.$$

By Lemma 9, the reach $(\mathrm{Source},B_{\mathrm{new}})$ is obtained from reaches in $\mathcal{R}$ by a finite sequence of reach-resolutions. Since reach-resolution corresponds to clause-resolution and clause-resolution is sound, we have ${\mathcal{N}}^{\wedge }\vDash {(\mathrm{Source},B_{\mathrm{new}})}^{\vee }.$ Therefore, ${\mathcal{N}}^{\wedge }\vDash {\mathcal{N}}^{\prime \wedge },$ i.e., every model of $\mathcal{N}$ is also a model of ${\mathcal{N}}^{\prime }$.    □

6. Main Results

In this section, we connect the token distribution taxonomy with the satisfiability of a resolvable network. Using the notions of black-blocked, terminal, active, non-resolved and resolved token distributions, together with the technical lemmas, we derive a set of conditions under which $\mathcal{N}$ is guaranteed to be UNSAT or SAT.

On the UNSAT side, we first observe that the presence of the reach $(\mathrm{Source},\mathrm{Sink})$ forces the associated CNF to be unsatisfiable. Moreover, we show that if all token distributions generated by the monotone CNF–DNF dualization step on the white reaches are black-blocked, then the instance is UNSAT as well. Intuitively, in this case, every “branch’’ starting from $\mathrm{Source}$ leads to $\mathrm{Sink}$, i.e., to a contradiction.

On the SAT side, we identify the following three cases. If there is no white reach, then the all-false valuation satisfies the network. If there exists a terminal token distribution, then its associated valuation is a satisfying assignment. Finally, one of our main theorems, the white-fixpoint implies SAT theorem (Theorem 1), shows that if all token distributions generated by the monotone CNF–DNF dualization step are either black-blocked or resolved (but not all are black-blocked), then every resolved token distribution yields a model of ${\mathcal{N}}^{\wedge }$. This white-fixpoint condition is exactly what our RN-Solver in Section 7 tries to establish, and it forms the key link between the combinatorics of token distributions and the satisfiability of resolvable networks.

6.1. UNSAT Cases

First, we present the UNSAT cases. We also introduce a useful technical lemma, the Distributive Composition Lemma, which shows the power that the monotone CNF–DNF dualization has to “set in motion” a resolvable network.

We start with the most direct way in which unsatisfiability can appear in a resolvable network.

Lemma 11

(Source–sink implies UNSAT). If $(\mathrm{Source},\mathrm{Sink})\in \mathcal{R}$ then $\mathcal{N}$ is UNSAT.

Proof. 

${(\mathrm{Source},\mathrm{Sink})}^{\vee }\equiv \top \to \perp \equiv \perp$.    □

The previous lemma captures the “built–in contradiction’’: if the network already contains a direct edge from $\mathrm{Source}$ to $\mathrm{Sink}$, then the associated CNF formula contains the empty clause and is therefore unsatisfiable.

Our next step is to derive a more general inference principle that allows us to add new reaches without changing satisfiability. This principle is expressed in the following Distributive Composition Lemma, which essentially says that if every minimal way of following the outgoing tail sets from A already forces E, then we may contract this behavior into a single reach from A to E.

Lemma 12

(Distributive composition from A to E). Let $(A,B_1),\dots ,(A,B_k)$ be all reaches with a positive value of k. If, for each $C\in DNF(B_1,\dots ,B_k)$, the reach $(C,E)$ is subsumed by $\mathcal{N}$, then ${\mathcal{N}}^{\wedge }\vDash (\bigwedge A)\Rightarrow (\bigvee E)$, i.e., we can add the reach $(A,E)$ to $\mathcal{N}$ without changing its satisfiability.

Proof. 

By Definitions 6 and 11, for each $C\in DNF(B_1,\dots ,B_k)$, the fact that $(C,E)$ is subsumed implies ${\mathcal{N}}^{\wedge }\vDash (\bigwedge C)\Rightarrow (\bigvee E)$. Therefore, ${\mathcal{N}}^{\wedge }\vDash \left({\bigvee }_{C\in DNF(B_1,\dots ,B_k)}\bigwedge C\right)\Rightarrow (\bigvee E).$ By Lemma 6 we have ${\bigwedge }_i(\bigvee B_i)\equiv {\bigvee }_{C\in DNF(B_1,\dots ,B_k)}\bigwedge C$; thus, ${\mathcal{N}}^{\wedge }\vDash ({\bigwedge }_i\bigvee B_i)\Rightarrow (\bigvee E)$. Again, according to Definition 6, each reach $(A,B_i)$ yields $\left(\left({\bigvee }_{x\in A}\neg x\right)\right)\vee \left({\bigvee }_{x\in B_i}x\right)$, so ${\mathcal{N}}^{\wedge }\vDash (\bigwedge A)\Rightarrow {\bigwedge }_i(\bigvee B_i)$. By transitivity of semantic consequence, ${\mathcal{N}}^{\wedge }\vDash (\bigwedge A)\Rightarrow (\bigvee E)$. Hence, adding the reach $(A,E)$ to $\mathcal{N}$ preserves satisfiability.    □

The real strength of Lemma 12 becomes apparent when we instantiate A and E with $\mathrm{Source}$ and $\mathrm{Sink}$, respectively. In that case, the minimal DNF terms built from white reaches describe all possible token distributions generated from the source. If each such distribution is already “doomed’’ in the sense that it forces the token into $\mathrm{Sink}$, then the whole network behaves as if there were a direct reach from $\mathrm{Source}$ to $\mathrm{Sink}$. Formally, this yields the following characterization of an important UNSAT case.

Lemma 13

(If all token distributions are black-blocked then the instance is UNSAT). Let $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$ be all the white reaches with a positive value of k. If all $C\in DNF(B_1,\dots ,B_k)$ token distributions are black-blocked, then $\mathcal{N}$ is UNSAT.

Proof. 

This lemma is a straightforward consequence of Lemma 12 with $A=\mathrm{Source}$ and, by Definition 12, $E=\mathrm{Sink}$, because, in this case, Lemma 12 implies the reach $(\mathrm{Source},\mathrm{Sink})$, which implies, via Lemma 11, that $\mathcal{N}$ is UNSAT.    □

6.2. SAT Cases

In contrast to the unsatisfiable configurations discussed earlier, the following lemmas identify structural situations in which a resolvable network is guaranteed to be satisfiable.

In the remainder of the paper, for any token distribution $C\subseteq U$, we write $V_C$ for the valuation on U, which sets all variables in C to true and all variables in $U\setminus C$ to false, i.e., $V_C=V_C^U$.

Lemma 14

(Empty network is SAT). Let ${\mathcal{N}}_0$ be a resolvable network and let A be the partial valuation storing units. Let $\mathcal{N}=(\mathcal{SN},\mathcal{R})$ be $Simp({\mathcal{N}}_0,A)$, as defined in Definition 4. If $\mathcal{R}=\varnothing$, then ${\mathcal{N}}_0$ is SAT; moreover, A is a satisfying valuation of ${\mathcal{N}}_0^{\wedge }$.

Proof. 

If $\mathcal{R}=\varnothing$, then $\mathcal{N}$ is satisfiable. Since unit propagation preserves satisfiability, ${\mathcal{N}}_0$ is satisfiable as well, and A is a satisfying valuation of ${\mathcal{N}}_0^{\wedge }$.    □

Lemma 15

(No-white implies SAT). If $(\mathrm{Source},\mathrm{Sink})\notin \mathcal{R}$ and there is no white $(\mathrm{Source},B)$ with $B\ne \varnothing$, then $\mathcal{N}$ is SAT; in particular, the all-false valuation satisfies ${\mathcal{N}}^{\wedge }$.

Proof. 

If there is no white reach $(\mathrm{Source},B)$ with $B\ne \varnothing$, then every reach $(A,B)\in \mathcal{R}$ has $A\ne \mathrm{Source}$; hence, $A\ne \varnothing$. Thus, every clause of ${\mathcal{N}}^{\wedge }$ contains at least one negative literal. The all-false valuation satisfies all negative literals and, hence, all clauses. Since $(\mathrm{Source},\mathrm{Sink})\notin \mathcal{R}$, the empty clause is absent. Hence, $\mathcal{N}$ is SAT.    □

The previous lemma covers the degenerate case in which no white information is present at all. In this situation, the all-false valuation trivially satisfies the resolvable network.

The next lemma is the adaptation of a well-known fact from classical SAT theory to the setting of resolvable networks: if a full-length clause is not subsumed by a clause set, then the valuation that makes all its literals false (i.e., the negation of the clause) satisfies the entire clause set.

Lemma 16

(Non-subsumed full reach implies SAT). Let $C\subseteq U$ be a token distribution, while C may be empty. If $(C,U\setminus C)$ is not subsumed by $\mathcal{N}$, then $V_C$ satisfies ${\mathcal{N}}^{\wedge }$.

Proof. 

Any falsified reach would have $C^{\prime }\subseteq C$ and $B^{\prime }\subseteq (U\setminus C)$, contradicting the non-subsumption of $(C,U\setminus C)$.    □

The next lemma shows why a terminal token distribution C is truly “terminal’’: in this case, a satisfying valuation can be constructed directly as $V_C$.

Lemma 17

(SAT witness). Let $C\subseteq U$ be a terminal token distribution, where $C\ne \varnothing .$ Then, ${\mathcal{N}}^{\wedge }$ is satisfied by the valuation $V_C$.

Proof. 

Assume that $C\subseteq U$ is a terminal token distribution and $C\ne \varnothing .$ Then, by Definition 13, the reach $(C,U\setminus C)$ is not subsumed by $\mathcal{N}$. By Lemma 16, the valuation $V_C$ satisfies ${\mathcal{N}}^{\wedge }$. Hence, $V_C$ is a satisfying valuation.    □

The next theorem is one of the main contributions of this paper. It shows that when the solver reaches a genuine fixpoint, the following occur:

• No more white reaches can be generated;
• No easy SAT case (such as a terminal token distribution) is available;
• No UNSAT case (such as all token distributions being black-blocked).

In this case, this apparent stalemate is in fact good news. The fixpoint itself guarantees satisfiability. In other words, the absence of further progress is not a failure but a certificate proving that the network is SAT.

Theorem 1

(White-fixpoint implies SAT). Assume that $(\mathrm{Source},\mathrm{Sink})$ is not contained in $\mathcal{R}$. Let $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$ be all the white reaches. Assume that all $C\in DNF(B_1,\dots ,B_k)$ token distributions are black-blocked or resolved, but not all are black-blocked. Then, for every resolved token distribution F, we state that $V_F$ satisfies ${\mathcal{N}}^{\wedge }$, i.e., $\mathcal{N}$ is SAT.

Proof. 

Assume that $(\mathrm{Source},\mathrm{Sink})\notin \mathcal{R}$, so $\mathcal{N}$ is not trivially UNSAT.

Let $F\in DNF(B_1,\dots ,B_k)$ be an arbitrary but fixed resolved token distribution. By assumption, such F exists.

Since F is resolved, it is active, so it is neither black-blocked nor terminal by Definition 15 and Lemma 5.

We show that the valuation $V_F$ (setting all variables in F to true and all variables in $U\setminus F$ to false) satisfies ${\mathcal{N}}^{\wedge }$.

Suppose, as a contradiction, that $V_F$ falsifies some reach $(G,H)\in \mathcal{R}$. By Definition 6, this means

$$G\subseteq F\mathrm{and}H\subseteq U\setminus F.$$

We know that $H\ne \mathrm{Sink}$, because otherwise $(G,\mathrm{Sink})$ would black-block F.

Now, we can apply Lemma 9 with

$$A=\mathrm{Source},(A,B_1),\dots ,(A,B_k)\mathrm{the}\mathrm{white}\mathrm{reaches},C=F,C^{\prime }=G,D=H.$$

Since $F\in DNF(B_1,\dots ,B_k)$, Lemma 9 yields a white reach

$$(\mathrm{Source},B_{\mathrm{new}}),$$

where

$$B_{\mathrm{new}}=\left(\bigcup _{c\in G}{B}_{j\left(c\right)}\cup H\right)\setminus G,$$

and for each $c\in F$, the index $j\left(c\right)$ is chosen so that $B_{j\left(c\right)}\cap F=\left\{c\right\}$ by the private-element property (Lemma 7).

Since $B_{j\left(c\right)}\cap F=\left\{c\right\}$, all elements of $B_{j\left(c\right)}\setminus \left\{c\right\}$ lie in $U\setminus F$. As $H\subseteq U\setminus F$, it follows that

$$B_{\mathrm{new}}\subseteq U\setminus F.$$

But F is resolved. Therefore, by Definition 15, every white reach that can be generated from any $(C^{\prime },D)$ with $C^{\prime }\subseteq F$ and $D\ne \mathrm{Sink}$ is already subsumed by $\mathcal{N}$. In particular, $(\mathrm{Source},B_{\mathrm{new}})$ is subsumed, i.e., there exists an i with

$$(\mathrm{Source},B_i)\in \mathcal{R}\mathrm{and}{B}_i\subseteq B_{\mathrm{new}}.$$

Combining this with $B_{\mathrm{new}}\subseteq U\setminus F$ yields

$$B_i\subseteq U\setminus F.$$

But since $F\in DNF(B_1,\dots ,B_k)$, by Definition 8, we must have

$$F\cap B_i\ne \varnothing .$$

This contradicts $B_i\subseteq U\setminus F$.

Therefore, $V_F$ cannot falsify any reach. Hence, $V_F\vDash {\mathcal{N}}^{\wedge }$, and since F was an arbitrary but fixed resolved token distribution, any resolved token distribution gives a satisfying assignment, and $\mathcal{N}$ is SAT.    □

7. The Resolvable Network Solver

The previous section established the structural conditions under which a resolvable network is guaranteed to be SAT or UNSAT. We now describe a solver that operationalizes these results. Rather than performing a branching search as in DPLL, the solver evolves token distributions globally through

• Monotone CNF–DNF dualization of the white reaches;
• Private-pivot chain resolutions that generate new white reaches.

At each recursive iteration, the solver performs the following high-level steps:

• Exhaustive unit propagation on the current resolvable network;
• Checking the easy SAT and UNSAT cases from Section 6;
• If neither applies, it constructs the monotone CNF–DNF dual of the current white reaches, yielding the next set of token distributions;
• Examining these token distributions: if none of them are terminal and not all of them are black-blocked, then every active token distribution is used to generate additional white reaches via private-pivot chains.

The following recursive procedure implements this exact strategy.

In this section, we show that this algorithm is sound, i.e., it never produces an incorrect output, and complete, i.e., it always terminates on every resolvable network.

Note that the algorithm is designed to return only SAT or UNSAT, although in the SAT cases we also indicate, in comments, a model for $\mathcal{N}$. In order to achieve this, unit propagation must have a side effect: every propagated unit must be stored in the global partial valuation A. Since unit propagation removes the propagated variable from the universe U, we may describe any model returned in a SAT branch as the union of A with a valuation for the remaining variables in U.

The algorithm proceeds in a directional manner from $\mathrm{Source}$ toward $\mathrm{Sink}$. From a symmetry-oriented perspective, one could envision a more balanced approach in which black reaches are also generated, effectively allowing the search to propagate backward from $\mathrm{Sink}$ and potentially converge in the middle. Such bidirectional reasoning schemes are known to improve performance in related frameworks [27]. At the present stage, however, the RN-Solver is intentionally restricted to forward generation of white reaches, as the theoretical foundations required to safely integrate backward propagation of black reaches have not yet been established.

The convergence of Algorithm 1 follows directly from the structure of its return conditions and from the monotone evolution of the resolvable network. At each invocation, the solver either terminates immediately by one of the explicit return statements or strictly strengthens the current network before making a recursive call.

Immediate termination occurs in several base cases. If the set of reaches becomes empty after exhaustive unit propagation, the solver returns SAT. If the contradictory reach $(\mathrm{Source},\mathrm{Sink})$ is present, the solver returns UNSAT. If no non-empty white reach exists, satisfiability follows from assigning all remaining variables to the false category. Furthermore, during the inspection of token distributions, the solver may return SAT as soon as a terminal token distribution is found.

If none of these base cases apply, the solver computes the set of token distributions induced by the current white reaches. For each non-black-blocked token distribution, resolution may produce a new white reach $(\mathrm{Source},B_{\mathrm{new}})$. Crucially, such a reach is added only if it is not subsumed by the current network, ensuring that each recursive call strictly increases the strength of the global constraint system. Black reaches are never generated, removed, or weakened.

After all token distributions have been processed, two additional terminating conditions are checked. If all token distributions are black-blocked, the solver returns UNSAT. If all token distributions are either black-blocked or resolved, a white fixpoint has been reached and the solver returns SAT.

Only in the remaining case does the algorithm perform a recursive call. Since each recursive call is preceded by the addition of at least one new non-subsumed white reach, and since the number of distinct reaches is finite up to subsumption, infinite recursion is impossible. Consequently, the RN-Solver must eventually reach one of its terminating return statements, and the algorithm always terminates.

Let us present two examples of how the RN-Solver works on small but interesting problems.

Example 3

(UNSAT example). Consider the following CNF formula in DIMACS format:

p cnf 3 6	comment: 3 variables, 6 clauses
−1 −2 0	comment: $\left(\right\{1,2\},\mathrm{Sink})$ (black reach)
−1 −3 0	comment: $\left(\right\{1,3\},\mathrm{Sink})$ (black reach)
−2 −3 0	comment: $\left(\right\{2,3\},\mathrm{Sink})$ (black reach)
1 2 0	comment: $(\mathrm{Source},\{1,2\left\}\right)$ (white reach)
2 3 0	comment: $(\mathrm{Source},\{2,3\left\}\right)$ (white reach)
1 −2 3 0	comment: $\left(\right\{2\},\{1,3\left\}\right)$ (mixed reach)

The corresponding resolvable network has two initial white reaches $(\mathrm{Source},B_1)$ and $(\mathrm{Source},B_2)$ with $B_1=\{1,2\}$, $B_2=\{2,3\}$.

First monotone CNF–DNF dualization step: The white reaches induce the hyperedges $B_1$ and $B_2$. The minimal hitting sets of $\{B_1,B_2\}$ are

$$\left\{2\right\},\{1,3\},$$

so the monotone CNF–DNF dualization step yields the token distributions

$$C_1=\left\{2\right\}\mathrm{and}{C}_2=\{1,3\}.$$

The distribution $C_2=\{1,3\}$ is already black-blocked because there is a black reach $\left(\right\{1,3\},\mathrm{Sink})$ with head contained in $C_2$. The distribution $C_1=\left\{2\right\}$ is not terminal and not black-blocked; hence, $C_1$ is active.

For the active token distribution $C_1=\left\{2\right\}$, Algorithm 1 considers the mixed reach $(C^{\prime },D)=(\left\{2\right\},\{1,3\})$ with $C^{\prime }\subseteq C_1$ and $D\ne \mathrm{Sink}$. By Lemma 9, using the white reaches $(\mathrm{Source},B_1)$ and $(\mathrm{Source},B_2)$ and the fact that $B_i\cap C_1=\left\{2\right\}$ for $i=1,2$, the private-pivot chain yields a new white reach

$$(\mathrm{Source},B_{\mathrm{new}})=\left(\mathrm{Source},((B_{j\left(2\right)}\cup D)\setminus \left\{2\right\})\right)=(\mathrm{Source},\{1,3\}),$$

where, for instance, we may choose $j\left(2\right)=1$ so that $B_{j\left(2\right)}=B_1=\{1,2\}$ and hence $B_{\mathrm{new}}=(\{1,2\}\cup \{1,3\})\setminus \left\{2\right\}=\{1,3\}$. This reach is new and not subsumed, so it is added as a third white reach.

Second monotone CNF–DNF dualization step. After adding $(\mathrm{Source},\{1,3\left\}\right)$, the current white reaches are

$$(\mathrm{Source},\{1,2\left\}\right),(\mathrm{Source},\{2,3\left\}\right),(\mathrm{Source},\{1,3\left\}\right).$$

The corresponding hyperedges are

$$B_1=\{1,2\},B_2=\{2,3\},B_3=\{1,3\},$$

whose minimal hitting sets are

$$\{1,2\},\{1,3\},\{2,3\}.$$

Thus, the new token distributions are

$$C_3=\{1,2\},C_4=\{1,3\},C_5=\{2,3\}.$$

Each of these is black-blocked:

• $C_3=\{1,2\}$ is black-blocked by $\left(\right\{1,2\},\mathrm{Sink})$,
• $C_4=\{1,3\}$ is black-blocked by $\left(\right\{1,3\},\mathrm{Sink})$,
• $C_5=\{2,3\}$ is black-blocked by $\left(\right\{2,3\},\mathrm{Sink})$.

Since all token distributions in $DNF(B_1,B_2,B_3)$ are black-blocked, Lemma 13 implies that the instance is UNSAT.

The second example illustrates how the RN-Solver uses an active, non-resolved token distribution to generate stronger white reaches via the private-pivot chain, and how repeated monotone CNF–DNF dualization steps can eventually produce a unit white reach.

Example 4

(Emergence of a unit white reach). Consider the following CNF formula in DIMACS format:

p cnf 4 7	comment: 4 variables, 7 clauses
1 2 0	comment: $(\mathrm{Source},\{1,2\left\}\right)$ (white reach)
3 4 0	comment: $(\mathrm{Source},\{3,4\left\}\right)$ (white reach)
1 −2 3 0	comment: $\left(\right\{2\},\{1,3\left\}\right)$ (mixed reach)
1 −2 4 0	comment: $\left(\right\{2\},\{1,4\left\}\right)$ (mixed reach)
−1 −3 0	comment: $\left(\right\{1,3\},\mathrm{Sink})$ (black reach)
−1 −4 0	comment: $\left(\right\{1,4\},\mathrm{Sink})$ (black reach)
1 −2 −3 −4 0	comment: $\left(\right\{2,3,4\},\{1\left\}\right)$ (mixed reach)

In the first monotone CNF–DNF dualization step, the two initial white reaches are $(\mathrm{Source},\{1,2\left\}\right)$ and $(\mathrm{Source},\{3,4\left\}\right)$. They induce the hyperedges $B_1=\{1,2\}$ and $B_2=\{3,4\}$. The minimal hitting sets of $\{B_1,B_2\}$ are

$$\{1,3\},\{1,4\},\{2,3\},\{2,4\},$$

so among the resulting token distributions we find, for example,

$$C_1=\{2,3\}\mathrm{and}{C}_2=\{2,4\}.$$

Consider $C_1=\{2,3\}$ and the mixed reach $(C^{\prime },D)=(\left\{2\right\},\{1,3\})$. Here, $C^{\prime }=\left\{2\right\}\subseteq C_1$ and $A=\mathrm{Source}$, so the assumptions of Lemma 9 are satisfied. For the private element $c=2$ of $C^{\prime }$ we have $B_1\cap C_1=\left\{2\right\}$; hence, $j\left(2\right)=1$ and the private-pivot chain on $c=2$ produces the new white reach

$$\begin{array}{cc}\hfill (\mathrm{Source},B_{\mathrm{new}})& =\left(\mathrm{Source},((B_{j\left(2\right)}\cup D)\setminus C^{\prime })\right)\hfill \\ & =\left(\mathrm{Source},\left(\right\{1,2\}\cup \{1,3\left\}\right)\setminus \left\{2\right\}\right)\hfill \\ & =(\mathrm{Source},\{1,3\left\}\right).\hfill \end{array}$$

Similarly, starting from $C_2=\{2,4\}$ and the mixed reach $\left(\right\{2\},\{1,4\left\}\right)$, the same construction yields the new white reach $(\mathrm{Source},\{1,4\left\}\right)$.

At the end of this round, the set of white reaches has been enlarged to

$$(\mathrm{Source},\{1,2\left\}\right),(\mathrm{Source},\{3,4\left\}\right),(\mathrm{Source},\{1,3\left\}\right),(\mathrm{Source},\{1,4\left\}\right).$$

In the next monotone CNF–DNF dualization step, these four white reaches correspond to the hyperedges

$$\{1,2\},\{3,4\},\{1,3\},\{1,4\},$$

whose minimal hitting sets are

$$\{1,3\},\{1,4\},\{2,3,4\}.$$

Thus, one of the token distributions generated in this round is

$$C_3=\{2,3,4\}.$$

Now, apply Lemma 9 with $C=C_3$ and the mixed reach $(C^{\prime },D)=(\{2,3,4\},\left\{1\right\})$. For each $c\in C_3$, we choose $j\left(c\right)$, such that $B_{j\left(c\right)}\cap C_3=\left\{c\right\}$. Concretely,

$$B_1\cap C_3=\left\{2\right\},B_3\cap C_3=\left\{3\right\},B_4\cap C_3=\left\{4\right\},$$

so we can take $j\left(2\right)=1$, $j\left(3\right)=3$ and $j\left(4\right)=4$, where $B_3=\{1,3\}$ and $B_4=\{1,4\}$ come from the newly added white reaches. By Lemma 9, the private-pivot chain on $C^{\prime }=C_3$ and $(C^{\prime },D)=(\{2,3,4\},\left\{1\right\})$ yields the white reach

$$(\mathrm{Source},B_{\mathrm{new}})=\left(\mathrm{Source},\left((B_1\cup B_3\cup B_4)\cup \left\{1\right\}\right)\setminus \{2,3,4\}\right)=(\mathrm{Source},\left\{1\right\}).$$

Thus, a unit white reach $(\mathrm{Source},\{1\left\}\right)$ emerges from the interaction of the original white reaches, the mixed reaches, and the monotone CNF–DNF dualization.

This unit white reach immediately triggers further simplification by positive unit propagation (Lemma 1) in Algorithm 1, showing how non-resolved token distributions can lead to strong global consequences.

Our example ends here; however, the algorithm does not stop at this point. We stop because the example is ready, and because we have shown that sometimes the algorithm creates some units.

7.1. Soundness

Now, we show that the RN-Solver, Algorithm 1, never misclassifies any input, i.e., it is sound. By soundness we mean that whenever Algorithm 1 returns SAT (respectively, UNSAT), the original resolvable network is indeed satisfiable (respectively, unsatisfiable).

Theorem 2

(Soundness). Algorithm 1 is sound.

Proof. 

Let the initial resolvable network be $N_0$. During the execution of the algorithm, the current network N is modified only in the following two ways:

(1)

Unit propagation (line 2). By Lemmas 1 and 2, both positive and negative unit propagation preserve satisfiability. Hence, the network after line 2 is equi-satisfiable with $N_0$. Unit propagation has several effects. Most importantly, it deletes the propagated unit from U, and stores it in a global set, called A.

(2)

Adding new white reaches (lines 19–21). Whenever a new reach $(\mathrm{Source},B_{\mathrm{new}})$ is added, it is produced by a finite sequence of reach-resolutions as described in Lemma 9. By Lemma 10, adding such a reach preserves satisfiability. Thus, at all later points, the current network N remains equi-satisfiable with $N_0$.

We now inspect all return points of Algorithm 1.

• Return at line 4 (SAT). The condition checked at line 3 is $\mathcal{R}$ is empty. According to Lemma 14, this implies that N is SAT. Hence, returning SAT is correct.
• Return at line 6 (UNSAT). The condition checked at line 5 is $(\mathrm{Source},\mathrm{Sink})\in \mathcal{R}$. By Lemma 11, this implies that N is UNSAT. Hence, returning UNSAT is correct.
• Return at line 8 (SAT). Line 7 detects the case that no white reach $(\mathrm{Source},B)$ with $B\ne \varnothing$ exists. By Lemma 15, the all-false valuation satisfies $N^{\wedge }$. Thus, returning SAT is correct.
• Return at line 13 (SAT). Line 12 checks whether some C is a terminal token distribution, and Lemma 17 states that in this case, $V_C$ satisfies $N^{\wedge }$. Therefore, returning SAT is correct.
• Return at line 25 (UNSAT). Here, every $C\in \mathcal{T}$ is black-blocked. By Lemma 13, if all token distributions in $DNF(B_1,\dots ,B_k)$ are black-blocked, then N is UNSAT. Thus, returning UNSAT is correct.
• Return at line 28 (SAT). Lines 24–26 detect the case in which all $C\in \mathcal{T}$ are either black-blocked or resolved, and at least one of them is resolved. By Theorem 1, any resolved token distribution F yields a valuation $V_F$ that satisfies $N^{\wedge }$. Hence, returning SAT is correct.
• Recursive call at line 30. This case arises when some token distribution is active and non-resolved. The algorithm does not decide here; it only continues with an equi-satisfiable extended network. Since all transformations preserve satisfiability, any later return value is correct for $N_0$ as well.

Since every transformation preserves satisfiability and all return points are justified by the corresponding lemmas and theorems, the algorithm never outputs an incorrect answer. Hence, Algorithm 1 is sound.    □

The proof checks that all transformations performed by the algorithm preserve satisfiability, and that every return point is justified by the lemmas and theorems established in the previous sections, which are also highlighted in the comments.

7.2. Completeness

In this subsection we state the completeness of Algorithm 1. The result shows that the solver is not only sound, i.e., every returned answer is correct, but also complete: the procedure always terminates and eventually reaches one of the decisive cases described in Section 7. The proof relies on the finiteness of the universe U and the fact that the algorithm can introduce only a finite number of non-subsumed white reaches; therefore, it cannot trigger an infinite sequence of white-reach extensions.

Theorem 3

(Completeness). Algorithm 1 is complete: for every resolvable network $\mathcal{N}$, the algorithm terminates and, upon termination, correctly returns either SAT or UNSAT.

Proof. 

Let ${\mathcal{N}}_0$ be the initial resolvable network given as input to Algorithm 1. We prove completeness in two parts.

(1)

Termination. We first show that Algorithm 1 always terminates.

Work performed in a single call. In a call RN-Solver$\left(\mathcal{N}\right)$ the algorithm performs only the following steps:

• Exhaustive unit propagation, which only removes literals and reaches and therefore must terminate, since the underlying CNF instance is finite (Lemmas 1 and 2);
• A finite number of tests for the easy SAT/UNSAT cases after unit propagation;
• Computation of the finite set $\mathcal{T}=DNF(B_1,\dots ,B_k)$ of token distributions induced by the current white reaches $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$;
• A finite outer loop over $\mathcal{T}$, where each $C\in \mathcal{T}$ is processed in finite time (checking whether C is terminal or black-blocked, and possibly generating candidate new white reaches from C via private-pivot chains);
• For each active (not terminal, not black-blocked) C, a finite inner loop over all reaches $(C^{\prime },D)\in \mathcal{R}$ with $C^{\prime }\subseteq C$ and $D\ne \mathrm{Sink}$; every iteration of this loop performs only a finite number of operations.

Thus, in a single invocation of RN-Solver, all loops and case distinctions range over finite sets and terminate. The only possible source of non-termination is, therefore, an infinite sequence of recursive calls.

Finiteness of possible white reaches. Let $U=\bigcup \mathcal{SN}$ be the universe of nodes of the network. Since ${\mathcal{N}}_0$ is finite, the universe U is finite as well, and the algorithm never introduces new nodes. While new subnetworks (i.e., new subsets of U) may be introduced, no new nodes are ever added to the universe. Any white reach has the form $(\mathrm{Source},B)$ with $B\subseteq U$. Consequently, there are at most $2^{\left|U\right|}$ distinct tails B, and hence at most $2^{\left|U\right|}$ distinct white reaches $(\mathrm{Source},B)$ in total.

Moreover, Algorithm 1 adds a new white reach only in the inner loop, only if the candidate $(\mathrm{Source},B_{\mathrm{new}})$ is

• Not already present in $\mathcal{R}$;
• Not subsumed by $(\mathcal{SN},\mathcal{R})$.

In particular, whenever such a candidate is added, the set of distinct white reaches in $\mathcal{R}$ strictly increases.

By the construction of the algorithm, the recursive call at the end is taken only if at least one token distribution $C\in \mathcal{T}$ is marked as non-resolved. This happens when, in the inner loop for that C, a new, non-subsumed white reach $(\mathrm{Source},B_{\mathrm{new}})$ has been generated. Hence, every recursive call is preceded by at least one strict increase in the set of distinct white reaches.

Since there are only a finite number of subsets $B\subseteq U$, there can be only a finite number of distinct white reaches $(\mathrm{Source},B)$. Therefore, the extension step that adds new white reaches, and hence the recursive call, can occur only a finite number of times.

Thus, the recursion depth of RN-Solver is bounded, and each individual call terminates; therefore, Algorithm 1 always terminates.

(2)

Exhaustiveness of the case analysis and correctness. After exhaustive unit propagation we obtain an equi-satisfiable network $\mathcal{N}=(\mathcal{SN},\mathcal{R})$ and a set A of propagated units. At this point, exactly one of the following four mutually exclusive cases holds:

1.

$\mathcal{R}=\varnothing$. Then, the algorithm returns SAT, and by Lemma 14 the (partial) valuation A satisfies ${\mathcal{N}}_0$.

2.

$(\mathrm{Source},\mathrm{Sink})\in \mathcal{R}$. Then, the algorithm returns UNSAT, which is correct by Lemma 11.

3.

$\mathcal{R}\ne \varnothing$ and no white reach $(\mathrm{Source},B)$ with $B\ne \varnothing$ exists. Then, the algorithm returns SAT, and by Lemma 15 the valuation A together with the all-false assignment on the remaining variables satisfies ${\mathcal{N}}_0$.

4.

None of the above: in particular, there exists at least one non-empty white reach $(\mathrm{Source},B)$. In this case, the algorithm constructs $\mathcal{T}=DNF(B_1,\dots ,B_k)$ and enters the main loop.

Thus, if we are not in case (4), the algorithm already terminates in a correct SAT/UNSAT state.

Assume now that we are in case (4) and thus enter the main loop with a non-empty set $\mathcal{T}$ of token distributions. By Lemma 5 every token distribution $C\in \mathcal{T}$ is either terminal, black-blocked, or active, and an active C is, by Definition 15, either resolved or non-resolved. Hence, after the loop has processed all $C\in \mathcal{T}$, exactly one of the following four situations occurs:

(a)

Some $C\in \mathcal{T}$ is terminal.

(b)

Every $C\in \mathcal{T}$ is black-blocked.

(c)

Every $C\in \mathcal{T}$ is black-blocked or resolved, and at least one of them is resolved.

(d)

No $C\in \mathcal{T}$ is terminal, not all of them are black-blocked, and at least one C is active and non-resolved.

The control flow of Algorithm 1 follows this dichotomy:

• In case (a), the algorithm returns SAT, and by Lemma 17 the valuation $A\cup V_C$ satisfies  ${\mathcal{N}}_0$.
• In case (b), the algorithm returns UNSAT, which is correct by Lemma 13.
• In case (c), the algorithm returns SAT. By the White-fixpoint implies SAT Theorem (Theorem 1), any resolved token distribution F yields a model $A\cup V_F$ of ${\mathcal{N}}_0$.
• In case (d), the algorithm does not decide yet: for some active, non-resolved C, it generates at least one new, non-subsumed white reach $(\mathrm{Source},B_{\mathrm{new}})$ via a private-pivot chain (Lemma 9) and extends the network as in Lemma 10. By the same lemma, this extension preserves satisfiability, and then the algorithm makes a recursive call on the extended network.

Because of the finiteness argument in part (1), case (d) can occur only a finite number of times along any recursion branch: every time we are in case (d), the set of non-subsumed white reaches strictly increases, and there are only a finite number of such reaches. Hence, every branch of the recursion eventually falls into one of the decisive cases (1)–(3) above or (a)–(c) in the main loop, where the algorithm returns.

Finally, by Theorem 2, Algorithm 1 is sound: every return point yields the correct classification of ${\mathcal{N}}_0$. Combining soundness with the termination argument and the exhaustive case analysis above, we conclude that for every input resolvable network ${\mathcal{N}}_0$, Algorithm 1 terminates after a finite number of steps and, upon termination, returns the correct answer SAT or UNSAT. Therefore, Algorithm 1 is complete.    □

The strength of our RN-Solver comes from the fact that it treats the white and black reaches as a global object rather than as local constraints to be propagated one by one. In particular, when the input contains only white and black reaches and no mixed ones, a single call of Algorithm 1 suffices to decide satisfiability. Classical hard benchmarks such as the pigeonhole principle can be encoded purely with white and black reaches; in these cases, the solver exploits the global structure of the white clauses (via the monotone CNF–DNF dualization step) and the fixed negative backbone given by the black reaches to derive UNSAT without any recursion or branching.

Whenever recursion does occur, it is driven entirely by the mixed reaches: each recursive step adds a new white reach $(\mathrm{Source},B_{\mathrm{new}})$ that is obtained as a resolution consequence of existing white and mixed reaches (Lemma 9), while the set of black reaches never grows. We could also generate black reaches, but at this early experimental phase of the RN-Solver we decided not to do so.

The mixed reaches act only as a catalyst for generating stronger white constraints; the eventual UNSAT decision is made either by the direct reach $(\mathrm{Source},\mathrm{Sink})$ or by the fact that all token distributions induced by the white reaches are black-blocked (Lemmas 11 and 13). In particularly constrained regions of the search space, this process may even produce white unit reaches. Such situations represent an extreme case for the RN-Solver, since unit clauses typically emerge only in deep recursive calls.

7.3. Complexity Considerations

Although the algorithm is now fully specified, its precise worst-case complexity is not known yet. Preliminary observations suggest that the monotone CNF–DNF dualization step is likely to dominate its running time. It is known that the classical monotone CNF–DNF dualization problem can be solved by a quasi-polynomial-time algorithm, i.e., in $n^{O(logn)}$ time [18].

On the algorithmic side, a long line of work has led to highly optimized enumeration procedures for minimal hitting sets/hypergraph transversals, most notably the MMCS and RS algorithms of Murakami and Uno, which are designed specifically for large-scale instances [21]. A recent comprehensive experimental study by Gainer-Dewar and Vera-Licona reports that these MMCS-type algorithms, together with a small number of related approaches, clearly dominate the practical state-of-the-art on a wide range of benchmarks [22]. Our current implementation of the monotone CNF–DNF dualization step follows this line, using an MMCS-style depth-first search with aggressive pruning.

However, the cost of our main loop, which also contains an inner loop and interleaves dualization with reach-resolution, is not yet characterized. Therefore, establishing the worst-case complexity of the overall solver remains an open problem.

From a broader algorithmic perspective, it is instructive to compare the RN-Solver to the classical Davis–Putnam (DP) procedure [25]. Both approaches are fundamentally resolution-based and operate by progressively deriving new clauses until a contradiction is obtained (the UNSAT case) or a resolution fixpoint is reached (the SAT case). It is well known that the DP procedure may suffer from a severe blow-up in the size of the intermediate clause set: eliminating a single variable can generate a quadratic number of resolvents, since all clauses containing the variable positively must be resolved with all clauses containing it negatively [1]. Repeated applications of such resolution steps may therefore lead to exponential growth of the formula, which is one of the main reasons why DP-style elimination was largely superseded in practice by search-based procedures such as DPLL and CDCL.

Given this close conceptual relationship, it is plausible that the RN-Solver exhibits similar worst-case time complexity characteristics. In particular, whenever recursion is triggered by mixed reaches, the generation of new white reaches via resolution and the subsequent subsumption checks on these newly derived constraints may give rise to behavior analogous to the clause proliferation observed in DP. This interpretation is consistent with the empirical observations reported in the following subsection, where the proof-of-concept implementation reveals that subsumption testing of newly generated resolvents constitutes the dominant computational bottleneck. Such evidence strongly suggests that the practical performance of the RN-Solver is closely tied to controlling the growth of intermediate constraints, much like in classical resolution-based procedures.

7.4. Implementation

We implemented Algorithm 1 in Java. The resolvable network data structure is represented using BitSets for all heads and tails of reaches, and the main solver loop mutates a single in-memory instance of $\mathcal{N}$, performing unit propagation, adding new white reaches, and invoking the monotone CNF–DNF dualization step in each recursive round.

The implementation is intended as a proof-of-concept realization of the RN-Solver framework, focusing on clarity and faithfulness to the theoretical model rather than on low-level performance optimizations. As a consequence, the current solver is not yet suitable for large-scale benchmark evaluation, but it allows us to experimentally validate the core reasoning mechanisms introduced in this paper.

In our final implementation, we use an MMCS-style depth-first search in the spirit of Murakami and Uno [21,22], rather than a Berge-type incremental construction [28].

A key design decision in our implementation is that the MMCS-based dualization algorithm is integrated with the RN taxonomy. In addition to the tails of the white reaches $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$, our MMCSBasedDNFBuilder also receives the heads $A_1,\dots ,A_m$ of all black reaches $(A_j,\mathrm{Sink})$ as input.

Conceptually, the MMCS search enumerates candidate token distributions $C\subseteq U$ that are minimal hitting sets of the white hypergraph, but we augment the search with an additional pruning condition: whenever a partial hitting set H already contains some black head $A_j$ as a subset, i.e., $A_j\subseteq H$, then H and all its extensions would be black-blocked in the sense of our taxonomy, and hence irrelevant for the solver. We therefore cut off that branch of the search immediately and do not generate any full transversal $C\supseteq H$ from it.

This design choice effectively moves the black-blocked test from the recursive solver level into the monotone CNF–DNF dualization process itself, allowing entire regions of the search space to be pruned before any token distribution is materialized.

To the best of our knowledge, this “black-aware” variant of monotone CNF–DNF dualization is novel in the context of hypergraph transversal algorithms.

In extreme cases, this pruning can be very strong, as illustrated by the runtime results in Table 1. In the case of pigeonhole principle instances, the white clauses assert that each pigeon must occupy a hole, while the black clauses forbid any pair of pigeons from sharing a hole. In such formulas, every minimal transversal of the white hypergraph contains the head of some black reach by necessity; that is, every token distribution is black-blocked. Our MMCS-based dualization detects this situation directly at the search level and returns an empty set of token distributions. The solver then identifies this case as the “all black-blocked” scenario of Lemma 13 and correctly returns UNSAT without any token distribution materializing at any point.

Table 1. Runtime comparison on pigeonhole benchmarks from SATLIB. All instances are UNSAT and consist exclusively of black and white clauses. Runtimes are given in milliseconds; for each solver and instance, the best result out of three runs is reported.

Instance	#Vars	#Clauses	#White	#Black	Glucose	CSFLOC	CSFLOC-B	RN-Solver
hole6	42	133	7	126	5	35	17	21
hole7	56	204	8	196	108	131	62	50
hole8	72	297	9	288	1554	1141	281	210
hole9	90	415	10	405	10,097	14,097	3245	1664
hole10	110	561	11	550	117,092	189,619	34,163	12,417

Without this black-aware pruning mechanism, the current RN-Solver implementation would already be too slow in pigeonhole instances, demonstrating that the observed performance gains a stem directly from integrating the black-blocked criterion into the dualization step.

In our experiments, we compare three solvers:

• Glucose 4.2.1 (https://github.com/audemard/glucose (accessed on: 9 January 2026)) (C++ implementation) [29,30].
• CSFLOC 19 (https://uni-eszterhazy.hu/fmv/tools (accessed on: 9 January 2026)) (Java implementation) [31].
• our new RN-Solver (Java implementation, version 3) (https://uni-eszterhazy.hu/fmv/tools (accessed on: 9 January 2026)) (Supplementary Materials).

We consider two benchmark families:

• Pigeonhole formulas, where the global white/black view of RNs is expected to be advantageous;
• Random 3-SAT formulas at and around the phase transition [32], where classical CDCL solvers are known to be very strong.

In addition to the instances reported below, the RN-Solver was tested on representative benchmark families from the SATLIB repository. However, in its current form, the implementation does not yet scale to most SATLIB classes within reasonable time limits. For this reason, we have restricted the reported results to small instances that already suffice to expose the strengths and limitations of the proposed approach.

Table 1 summarizes the runtime results for three SAT solvers, Glucose 4.2.1, CSFLOC 19, and the proposed RN-Solver, on classical pigeonhole benchmarks (hole6.cnf–hole10.cnf) obtained from the SATLIB repository (https://www.cs.ubc.ca/~hoos/SATLIB/benchm.html (accessed on: 9 January 2026)). All benchmark instances are unsatisfiable and contain only black and white clauses; no mixed clauses are present.

For each instance, we also report the number of variables, the total number of clauses, and the decomposition into white and black clauses. Each solver was executed three times on each benchmark, and the best runtime (measured in milliseconds) was recorded. All experiments were performed on a standard workstation equipped with a 12th Gen Intel Core i7-1255U processor (1.70 GHz) and 32 GB of RAM.

In addition to its default configuration, CSFLOC 19 was also run, with its specialized variable renaming strategy denoted by B, which assigns low variable indices to variables occurring in black clauses. This strategy is listed as a recommended choice for pigeonhole-style instances [33].

The results show that while Glucose performs very well on the smallest instances, its runtime increases rapidly as the size of the pigeonhole problem grows. CSFLOC benefits substantially from the B strategy, achieving a significant speed-up compared to its default configuration. Notably, the RN-Solver consistently outperforms the default CSFLOC configuration in all instances and remains competitive with CSFLOC (B), clearly outperforming all other solvers on the largest benchmark (hole10).

These results indicate that the RN-based approach can effectively exploit structured unsatisfiable instances containing only black and white clauses, such as pigeonhole formulas.

The favorable results on pigeonhole benchmarks should therefore be interpreted as evidence for the effectiveness of the black-aware dualization strategy. Similar techniques are required to further refine and optimize the RN-Solver.

Table 2 reports runtime results on small random 3-SAT instances from the SATLIB uf20-91 benchmark set (https://www.cs.ubc.ca/~hoos/SATLIB/benchm.html (accessed on: 9 January 2026)). All instances are satisfiable and contain exactly 20 variables and 91 clauses. From the full set of 1000 instances, we selected the first five benchmarks for evaluation, as these already exhibit a number of characteristic behaviors.

Table 2. Runtime comparison on random 3-SAT instances from the SATLIB uf20-91 benchmark set. All instances are satisfiable and contain 20 variables and 91 clauses. Runtimes are given in milliseconds; timeout indicates that the solver did not finish within the 5 min time limit.

Instance	#White	#Black	Glucose	CSFLOC	CSFLOC-IWCR	RN-Solver
uf20-01	10	11	2	6	7	326
uf20-02	11	13	0	7	7	17
uf20-03	8	7	4	5	7	TIMEOUT
uf20-04	11	14	2	4	7	530
uf20-05	12	12	3	5	8	351

Each solver was executed three times on each instance, and the best runtime (measured in milliseconds) was recorded. A timeout of five minutes was imposed; instances exceeding this limit are reported as TIMEOUT. All experiments were performed on the same hardware configuration as that used in the previous experiments.

The results show that Glucose 4.2.1 solves all instances almost instantly, as expected for such small random 3-SAT formulas. CSFLOC 19 also performs well in its default configuration. We also report results for CSFLOC with its random 3-SAT–oriented variable renaming strategy IWCR, which is listed among the recommended strategies for random 3-SAT instances. On these very small instances, however, the overhead of the IWCR strategy outweighs its benefits, resulting in no performance improvement and in some cases a slight slowdown.

In contrast, the RN-Solver performs significantly worse on this benchmark set. Although all instances are small and satisfiable. The solver exhibits high variance in runtime and fails to solve uf20-03.cnf within the given time limit.

Further inspection revealed that this instance has a uniquely satisfying assignment, which appears to be particularly challenging for the current RN-Solver implementation.

These results indicate that, in its present form, the RN-Solver is not well suited for unstructured random SAT instances, even at very small scales. Profiling of the Java implementation shows that approximately 97% of the runtime is spent on checking whether newly generated clause candidates $B_{\mathrm{new}}$ are subsumed or not. This clearly identifies the main computational bottleneck and suggests that substantial performance improvements may be achievable through more efficient subsumption checking or alternative data structures.

Overall, this experiment highlights an important limitation of the current RN-Solver and strongly motivates further work on both implementation-level optimization and potential refinements of the underlying RN-based reasoning principles.

7.5. Further Optimization Idea: Usage of Strongly Connected Components

Resolvable networks can be viewed as a form of directed hypergraph, and it is natural to ask whether classical structural notions from graph theory, in particular strongly connected components (SCCs), admit useful generalizations in this setting. A suitable SCC theory could provide powerful optimization techniques, for example, by allowing the solver to collapse certain subnetworks into single components or to detect forced implications early.

Our current work explores how an SCC-like notion can be formulated for resolvable networks. A preliminary definition is the following.

Definition 16

($\mathrm{Source}$-implied SCC). Let $(\mathrm{Source},B_1),\dots ,(\mathrm{Source},B_k)$ be a subset of white reaches of $\mathcal{N}$ with a positive value of k. A set of nodes $B\subseteq U$ is called a $\mathrm{Source}$-implied strongly connected component (SCC) of $\mathcal{N}$ if and only if for every token distribution $C\in DNF(B_1,\dots ,B_k)$ we have

$${\mathcal{N}}^{\wedge }\wedge \bigwedge C\vDash \bigwedge B.$$

This definition captures the intuition that B behaves as a Strongly Connected Component: every logically feasible partial assignment generated by the white reaches must imply $\bigwedge B$, i.e., the variables in B form a cycle of implications $b_1\Rightarrow b_2,\dots ,b_{z-1}\Rightarrow b_z,$ and $b_z\Rightarrow b_1$, where $B=\{b_1,\dots ,b_z\}$.

Initial lemmas suggest that such components behave monotonically under private-pivot expansions and may serve as contraction points for the solver, allowing the hypergraph to be simplified without affecting satisfiability.

However, the theory is not yet mature: we currently have several partial results but no complete algorithmic toolkit. Developing a full SCC framework for resolvable networks remains a promising direction for future optimization of the solver.

8. Introduction of the Token-Superposition Petri Net

Petri nets are among the most widely used mathematical models for describing distributed, concurrent, and reactive systems. Since their introduction by Carl Adam Petri in 1962 [23], numerous extensions and generalizations have been developed to capture probabilistic, timed, colored, or stochastic behavior (see, e.g., [24,34]). Despite this rich landscape, all classical Petri net variants share two fundamental notions: places, representing conditions or resources, and transitions, representing events. Tokens placed in the places describe the current state of the system, and a transition may fire if all of its input places contain the required number of tokens. Upon firing, the transition consumes tokens from its input places and produces tokens in its output places.

Several authors have proposed “Quantum Petri Nets” (QPNs) in which classical Petri nets are enriched with a quantum state space. Early work by Ito models qubits and a quantum Turing machine by means of Petri nets whose tokens carry quantum states and whose transitions implement unitary steps of the computation [35]. Letia et al. introduce QPNs whose markings live in a Hilbert space: tokens represent superposed quantum states and firing is governed by linear operators corresponding to quantum gates; this framework is then used to model hybrid classical-quantum software architectures and evolutionary systems [36,37]. Schmidt defines QPNs in which the structural part of the net is kept separate from a quantum valuation. He proves that such nets can represent arbitrary quantum circuits and supports a compositional, gate-level view of quantum programs [38]. More recently, Joachim, de Visme, and Haar equip safe Petri nets with local quantum annotations compatible with quantum event-structure semantics, yielding QPNs with a well-defined unfolding semantics and a parallel-composition operator [39]. On the more hardware-oriented side, Shah and Oruç use a simplified QPN model with q-tokens (tokens carrying qubits or multi-qubit states) and quantum-gate-labeled transitions to design and verify quantum buffers and registers [40].

In the present paper, the motivation for a quantum-inspired Petri net extension emerges organically from the study of resolvable networks. A reach $(A,B)$ in an RN represents a logical implication

$$\bigwedge A\Rightarrow \bigvee B,$$

meaning that if all nodes in A are true, then at least one node in B must become true. Crucially, the right-hand side is disjunctive: the token may appear in any element of B, and the model does not (and cannot) specify exactly where. This “token may appear here or there’’ behavior is structurally similar to the idea behind superposition in quantum theory: before observation, a quantum particle may be said to “be’’ in multiple potential states, even though a single outcome is realized when measured.

This observation motivates an interesting analogy. When $\mathrm{Source}$ fires, it does not transfer the token to any particular pre-determined node or set of nodes. Instead, the monotone CNF–DNF dualization step applied to the white reaches produces a family of minimal hitting sets, and each such set is a valid token distribution. Taken together, these token distributions can be seen as the classical components of a single “superposition” that the token may have. This motivates the following informal definition.

Definition 17

(Token-superposition Petri net). A Token-superposition Petri net (TS-PN) is an ordinary Petri net that is extended as follows:

• A token distribution is a set of places that currently carry the token; A superposition is a (non-empty) set of token distributions;
• The current state of the net is given by a superposition of token distributions;
• A transition with input set I is considered to be enabled if there exists a token distribution T in the current superposition such that $I\subseteq T$;
• When such a transition fires, each enabling T may be replaced by one or more extended token distributions $T^{\prime }$, where the token is copied into some of the output places; the new state is the superposition consisting of all unchanged distributions together with all such extensions.

Thus, unlike classical Petri nets where a firing produces a deterministically specified multiset of tokens, in a token-superposition Petri net, a firing step may branch a single token distribution into several successor token distributions. This means that these successors form a new superposition. This is directly analogous to the behavior of token distributions in RN: every family of white reaches gives rise, via the monotone CNF–DNF dualization step, to multiple minimal token distributions, all of which must be considered simultaneously by the solver.

We emphasize that this model is purely inspirational rather than a formal quantum-mechanical construct. We do not attempt to introduce amplitudes, unitary evolution, or measurement; the only quantum-style component is the superposed, non-deterministic appearance of tokens.

The purpose of introducing token-superposition Petri nets here is not to develop a full-blown theory of quantum Petri nets, which would require formal semantics, conflict resolution, and an explicit interplay between superposition, causality, and firing sequences. Instead, our aim is to highlight a conceptual bridge: Resolvable networks naturally exhibit a quantum-like nondeterminism in token propagation, suggesting that Petri net semantics might be fruitfully extended to support superposed token flows.

Nevertheless, we believe that the connection between RN token distributions, hypergraph transversal generation, and quantum-inspired system modeling opens up a new and potentially rich research direction.

9. Conclusions

We presented a novel SAT solving framework based on resolvable networks and token distributions. Instead of assignment-driven branching, the RN-Solver performs global reasoning by combining graph-based clause representations, monotone CNF–DNF dualization of white clauses, and resolution-based refinement via private-pivot chains.

From a theoretical standpoint, we introduced a taxonomy of token distributions and proved the soundness and completeness of the solver. The resulting white-fixpoint criterion provides a new satisfiability condition that differs fundamentally from classical DPLL/CDCL stopping rules and highlights an alternative resolution-centered view of SAT solving.

From an algorithmic perspective, the RN-Solver should be understood as a proof-of-concept realization of this theoretical framework rather than as a state-of-the-art competitive solver. The experimental evaluation is therefore intended to validate the proposed reasoning principles and to expose their practical strengths and limitations, rather than to provide a comprehensive performance comparison against highly optimized CDCL solvers.

The experimental results and the theoretical analysis jointly indicate that the effectiveness of the RN-Solver is closely tied to the structural properties of the underlying resolvable network. In particular, instances containing only white and black reaches, or exhibiting a weak presence of mixed reaches, represent a favorable extreme in which global reasoning can be applied with little or no recursion.

The absence of mixed reaches should not be interpreted as a sharp or quantitative boundary of applicability, but rather as an illustrative edge case that clarifies the underlying mechanism of the solver. More generally, RN-based reasoning appears to benefit from instances in which mixed reaches primarily act as catalysts for strengthening white constraints, instead of inducing extensive recursive branching.

At the current stage, we deliberately refrain from introducing numerical structural metrics or thresholds that would separate “easy” and “hard” instances for RN-based reasoning. Defining such metrics, together with systematic evaluations across structural instance classes, constitutes an important and nontrivial direction for future research.

Several additional directions for future work naturally emerge from this study. On the algorithmic side, improving subsumption checking and exploring more symmetric resolution strategies may substantially enhance practical performance. On the theoretical side, further connections between resolvable networks, resolution-based proof systems, and related formalisms such as Petri nets may deepen our understanding of global reasoning mechanisms for satisfiability. We hope that the RN framework provides a useful conceptual foundation for such developments.