2.1. Computational Complexity
We assume that the reader is familiar with basic notions from the theory of computational complexity, such as the complexity classes P and NP. For more details, we refer to textbooks on the topic (see, e.g., [
11,
12]).
There are many natural decision problems that are not contained in the classical complexity classes P and NP (under some common complexitytheoretic assumptions). The
Polynomial Hierarchy [
2,
3,
12,
13] contains a hierarchy of increasing complexity classes
${\Sigma}_{i}^{\mathrm{p}}$, for all
$i\ge 0$. We give a characterization of these classes based on the satisfiability problem of various classes of quantified Boolean formulas. A
quantified Boolean formula is a formula of the form
${Q}_{1}{X}_{1}{Q}_{2}{X}_{2}\cdots {Q}_{m}{X}_{m}\psi $, where each
${Q}_{i}$ is either ∀ or ∃, the
${X}_{i}$ are disjoint sets of propositional variables, and
$\psi $ is a Boolean formula over the variables in
${\bigcup}_{i=1}^{m}{X}_{i}$. The quantifierfree part of such formulas is called the
matrix of the formula. Truth of such formulas is defined in the usual way. Let
$\gamma =\{{x}_{1}\mapsto {d}_{1},\cdots ,{x}_{n}\mapsto {d}_{n}\}$ be a function that maps some variables of a formula
$\phi $ to other variables or to truth values. We let
$\phi \left[\gamma \right]$ denote the application of such a substitution
$\gamma $ to the formula
$\phi $. We also write
$\phi [{x}_{1}\mapsto {d}_{1},\cdots ,{x}_{n}\mapsto {d}_{n}]$ to denote
$\phi \left[\gamma \right]$. For each
$i\ge 1$, the decision problem
QSat${}_{i}$ is defined as follows.
QSat${}_{i}$ Instance: A quantified Boolean formula $\phi =\exists {X}_{1}\forall {X}_{2}\exists {X}_{3}\cdots {Q}_{i}{X}_{i}\psi $, where ${Q}_{i}$ is a universal quantifier if i is even and an existential quantifier if i is odd. Question: Is $\phi $ true?

Input formulas to the problem
QSat${}_{i}$ are called
${\Sigma}_{i}^{\mathrm{p}}$formulas. For each nonnegative integer
$i\ge 0$, the complexity class
${\Sigma}_{i}^{\mathrm{p}}$ can be characterized as the closure of the problem
QSat${}_{i}$ under polynomialtime reductions [
2,
3]—that is, all decision problems that are polynomialtime reducible to
QSat${}_{i}$. The
${\Sigma}_{i}^{\mathrm{p}}$hardness of
QSat${}_{i}$ holds already when the matrix of the input formula is restricted to
$3\mathrm{CNF}$ for odd
i, and restricted to
$3\mathrm{DNF}$ for even
i. Note that the class
${\Sigma}_{0}^{\mathrm{p}}$ coincides with P, and the class
${\Sigma}_{1}^{\mathrm{p}}$ coincides with NP. For each
$i\ge 1$, the class
${\Pi}_{i}^{\mathrm{p}}$ is defined as co
${\Sigma}_{i}^{\mathrm{p}}$—that is,
${\Pi}_{i}^{\mathrm{p}}=\{\phantom{\rule{0.166667em}{0ex}}{\{0,1\}}^{\ast}\backslash L\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}L\in {\Sigma}_{i}^{\mathrm{p}}\phantom{\rule{0.166667em}{0ex}}\}$.
The classes
${\Sigma}_{i}^{\mathrm{p}}$ and
${\Pi}_{i}^{\mathrm{p}}$ can also be defined by means of nondeterministic Turing machines with an oracle. Intuitively, oracles are blackbox machines that can solve a problem in a single time step—for more details, see, e.g., Chapter 3 of [
11]. For any complexity class
C, we let
${\mathrm{NP}}^{C}$ be the set of decision problems that are decided in polynomial time by a nondeterministic Turing machine with an oracle for a problem that is in the class
C. Then, the classes
${\Sigma}_{i}^{\mathrm{p}}$ and
${\Pi}_{i}^{\mathrm{p}}$, for
$i\ge 0$, can be equivalently defined by letting
${\Sigma}_{0}^{\mathrm{p}}={\Pi}_{0}^{\mathrm{p}}=\mathrm{P}$, and for each
$i\ge 1$ letting
${\Sigma}_{i}^{\mathrm{p}}={\mathrm{NP}}^{{\Sigma}_{i1}^{\mathrm{p}}}$ and
${\Pi}_{i}^{\mathrm{p}}={\mathrm{coNP}}^{{\Sigma}_{i1}^{\mathrm{p}}}$.
The Polynomial Hierarchy also includes complexity classes between
${\Sigma}_{i}^{\mathrm{p}}$ and
${\Sigma}_{i+1}^{\mathrm{p}}$—such as the classes
${\Delta}_{i+1}^{\mathrm{p}}$ and
${\Theta}_{i+1}^{\mathrm{p}}$. The class
${\Delta}_{i+1}^{\mathrm{p}}$ consists of all decision problems that are decided in polynomial time by a deterministic Turing machine with an oracle for a problem that is in the class
${\Sigma}_{i}^{\mathrm{p}}$. Similarly, the class
${\Theta}_{i+1}^{\mathrm{p}}$ consists of all decision problems that are decided in polynomial time by a deterministic Turing machine with an oracle for a problem that is in the class
${\Sigma}_{i}^{\mathrm{p}}$, with the restriction that the Turing machine is only allowed to make
$O(logn)$ oracle queries, where
n denotes the input size [
14,
15]. It holds that
${\Sigma}_{i}^{\mathrm{p}}\cup {\Pi}_{i}^{\mathrm{p}}\subseteq {\Theta}_{i+1}^{\mathrm{p}}\subseteq {\Delta}_{i+1}^{\mathrm{p}}\subseteq {\Sigma}_{i+1}^{\mathrm{p}}\cap {\Pi}_{i+1}^{\mathrm{p}}$.
There are also natural decision problems that are located between NP and
${\Theta}_{2}^{\mathrm{p}}$. The
Boolean Hierarchy (BH) [
16,
17,
18] consists of a hierarchy of complexity classes
${\mathrm{BH}}_{i}$, for each
$i\ge 1$, that can be used to classify the complexity of decision problems between NP and
${\Theta}_{2}^{\mathrm{p}}$. Each class
${\mathrm{BH}}_{i}$ can be characterized as the class of problems that can be reduced to the problem
${\mathrm{BH}}_{i}$
Sat, which is defined inductively as follows. The problem
${\mathrm{BH}}_{1}$
Sat consists of all sequences
$\left(\phi \right)$ of length 1, where
$\phi $ is a satisfiable propositional formula. For even
$i\ge 2$, the problem
${\mathrm{BH}}_{i}$
Sat consists of all sequences
$({\phi}_{1},\cdots ,{\phi}_{i})$ of propositional formulas such that both
$({\phi}_{1},\cdots ,{\phi}_{i1})\in $${\mathrm{BH}}_{(i1)}$
Sat and
${\phi}_{i}$ is unsatisfiable. For odd
$i\ge 2$, the problem
${\mathrm{BH}}_{i}$
Sat consists of all sequences
$({\phi}_{1},\cdots ,{\phi}_{i})$ of propositional formulas such that
$({\phi}_{1},\cdots ,{\phi}_{i1})\in $${\mathrm{BH}}_{(i1)}$
Sat or
${\phi}_{i}$ is satisfiable. The class
${\mathrm{BH}}_{2}$ is also denoted by DP, and the problem
${\mathrm{BH}}_{2}$
Sat is also denoted by
$\mathrm{SATUNSAT}$. The class BH is defined as the union of all
${\mathrm{BH}}_{i}$, for
$i\ge 1$. It holds that
$\mathrm{NP}\cup \mathrm{coNP}\subseteq {\mathrm{BH}}_{2}\subseteq {\mathrm{BH}}_{3}\subseteq \cdots \subseteq \mathrm{BH}\subseteq {\Theta}_{2}^{\mathrm{p}}$.
2.2. Parameterized Complexity
We introduce some core notions from parameterized complexity theory. For an indepth treatment, we refer to other sources [
19,
20,
21,
22,
23]. A
parameterized problem L is a subset of
${\Sigma}^{\ast}\times \mathbb{N}$ for some finite alphabet
$\Sigma $. For an instance
$(x,k)\in {\Sigma}^{\ast}\times \mathbb{N}$, we call
x the
main part and
k the
parameter. The following generalization of polynomialtime computability is commonly regarded as the main tractability notion of parameterized complexity theory. A parameterized problem
L is
fixedparameter tractable if there exists a computable function
f and a constant
c such that there exists an algorithm that decides whether
$(x,k)\in L$ in time
$f\left(k\right)\xb7{\leftx\right}^{c}$, where
$\leftx\right$ denotes the size of
x. Such an algorithm is called an
fptalgorithm, and this amount of time is called
fpttime. FPT is the class of all parameterized problems that are fixedparameter tractable. If the parameter is constant, then fptalgorithms run in polynomial time where the order of the polynomial is independent of the parameter. This provides a good scalability in the parameter in contrast to running times of the form
${\leftx\right}^{k}$, which are also polynomial for fixed
k, but are already impractical for, say,
$k>5$.
Parameterized complexity also generalizes the notion of polynomialtime reductions. Let $L\subseteq {\Sigma}^{\ast}\times \mathbb{N}$ and ${L}^{\prime}\subseteq {\left({\Sigma}^{\prime}\right)}^{\ast}\times \mathbb{N}$ be two parameterized problems. A (manyone) fptreduction from L to ${L}^{\prime}$ is a mapping $R:{\Sigma}^{\ast}\times \mathbb{N}\to {\left({\Sigma}^{\prime}\right)}^{\ast}\times \mathbb{N}$ from instances of L to instances of ${L}^{\prime}$ for which there exist some computable function $g:\mathbb{N}\to \mathbb{N}$ such that for all $(x,k)\in {\Sigma}^{\ast}\times \mathbb{N}$: (i) $(x,k)$ is a yesinstance of L if and only if $({x}^{\prime},{k}^{\prime})=R(x,k)$ is a yesinstance of ${L}^{\prime}$, (ii) ${k}^{\prime}\le g\left(k\right)$, and (iii) R is computable in fpttime. Let K be a parameterized complexity class. A parameterized problem L is Khard if for every ${L}^{\prime}\in \mathrm{K}$ there is an fptreduction from ${L}^{\prime}$ to L. A problem L is Kcomplete if it is both in K and Khard. Reductions that satisfy properties (i) and (ii) but that are computable in time $O\left(\rightx{}^{f\left(k\right)})$, for some fixed computable function f, we call xpreductions.
The parameterized complexity classes $\mathrm{W}\left[\mathrm{t}\right]$, for $t\ge 1$, $\mathrm{W}\left[\mathrm{SAT}\right]$, and $\mathrm{W}\left[\mathrm{P}\right]$ can be used to give evidence that a given parameterized problem is not fixedparameter tractable. These classes are based on the satisfiability problems of Boolean circuits and formulas. We consider Boolean circuits with a single output gate. We call input nodes variables. We distinguish between small gates, with fanin $\le 2$, and large gates, with fanin $>2$. The depth of a circuit is the length of a longest path from any variable to the output gate. The weft of a circuit is the largest number of large gates on any path from a variable to the output gate. We say that a circuit C is in negation normal form if all negation nodes in C have variables as inputs. A Boolean formula can be considered as a Boolean circuit where all gates have fanout $\le 1$. We adopt the usual notions of truth assignments and satisfiability of a Boolean circuit. We say that a truth assignment for a Boolean circuit has weight k if it sets exactly k of the variables of the circuit to true. We denote the class of Boolean circuits with depth u and weft t by ${\mathrm{CIRC}}_{t,u}$. We denote the class of all Boolean circuits by CIRC, and the class of all Boolean formulas by FORM. For any class $\mathcal{C}$ of Boolean circuits, we define the following parameterized problem:
pWSat$\left[\mathcal{C}\right]$ Instance: A Boolean circuit $C\in \mathcal{C}$, and an integer k. Parameter: k. Question: Does there exist an assignment of weight k that satisfies C?

We denote closure under fptreductions by ${[\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}}]}^{\mathrm{fpt}}$—that is, for any set $\mathcal{L}$ of parameterized problems, ${\left[\phantom{\rule{4pt}{0ex}}\mathcal{L}\phantom{\rule{4pt}{0ex}}\right]}^{\mathrm{fpt}}$ is the set of all parameterized problems ${L}^{\prime}$ that are fptreducible to some problem $L\in \mathcal{L}$. The classes $\mathrm{W}\left[t\right]$ are defined by letting $\mathrm{W}\left[t\right]=\left[\phantom{\rule{4pt}{0ex}}\right\{\phantom{\rule{0.166667em}{0ex}}$pWSat$\left[{\mathrm{CIRC}}_{t,u}\right]$ $:\phantom{\rule{0.277778em}{0ex}}u\ge {1\phantom{\rule{0.166667em}{0ex}}\left\}\phantom{\rule{4pt}{0ex}}\right]}^{\mathrm{fpt}}$ for all $t\ge 1$. The classes $\mathrm{W}\left[\mathrm{SAT}\right]$ and $\mathrm{W}\left[\mathrm{P}\right]$ are defined by letting $\mathrm{W}\left[\mathrm{SAT}\right]=[\phantom{\rule{4pt}{0ex}}$ pWSat$\left[\mathrm{FORM}\right]$${\phantom{\rule{4pt}{0ex}}]}^{\mathrm{fpt}}$ and $\mathrm{W}\left[\mathrm{P}\right]=[\phantom{\rule{4pt}{0ex}}$ pWSat$\left[\mathrm{CIRC}\right]$ ${]}^{\mathrm{fpt}}$.
Let
K be a classical complexity class, e.g., NP. The parameterized complexity class paraK is defined as the class of all parameterized problems
$L\subseteq {\Sigma}^{\ast}\times \mathbb{N}$, for some finite alphabet
$\Sigma $, for which there exist a computable function
$f:\mathbb{N}\to {\Sigma}^{\ast}$, and a problem
$P\subseteq {\Sigma}^{\ast}$ such that
$P\in K$ and for all instances
$(x,k)\in {\Sigma}^{\ast}\times \mathbb{N}$ of
L we have that
$(x,k)\in L$ if and only if
$(x,f(k\left)\right)\in P$. (Here, we implicitly use a representation of pairs of strings in
${\Sigma}^{\ast}\times {\Sigma}^{\ast}$ as strings in
${\Sigma}^{\ast}$.) Intuitively, the class paraK consists of all problems that are in
K after a precomputation that only involves the parameter. The class paraNP can also be defined via nondeterministic fptalgorithms [
24]. The class paraK can be seen as a direct analogue of the class K in parameterized complexity. In particular, for the case of
$\mathrm{K}=\mathrm{P}$, we have
$\mathrm{FPT}=\mathrm{paraP}$.
We consider the following (trivial) parameterization of SAT, the satisfiability problem for propositional logic. We let ${\mathrm{SAT}}_{1}=\{\phantom{\rule{0.166667em}{0ex}}(\phi ,1)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\phi \in \mathrm{SAT}\phantom{\rule{0.166667em}{0ex}}\}$. In other words, ${\mathrm{SAT}}_{1}$ is the parameterized variant of SAT where the parameter is the constant value 1. Similarly, we let ${\mathrm{UNSAT}}_{1}=\{\phantom{\rule{0.166667em}{0ex}}(\phi ,1)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\phi \in \mathrm{UNSAT}\phantom{\rule{0.166667em}{0ex}}\}$. The problem ${\mathrm{SAT}}_{1}$ is paraNPcomplete, and the problem ${\mathrm{UNSAT}}_{1}$ is para$\mathrm{coNP}$complete. In other words, the class paraNP consists of all parameterized problems that can be fptreduced to ${\mathrm{SAT}}_{1}$, and para$\mathrm{coNP}$ consists of all parameterized problems that can be fptreduced to ${\mathrm{UNSAT}}_{1}$.
Another analogue to the classical complexity class K is the parameterized complexity class
$\mathrm{X}{\mathrm{K}}^{\mathrm{nu}}$, that is defined as the class of those parameterized problems
Q whose slices
${Q}_{k}$ are in K, i.e., for each positive integer
k the classical problem
${Q}_{k}=\{\phantom{\rule{0.166667em}{0ex}}x\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}(x,k)\in Q\phantom{\rule{0.166667em}{0ex}}\}$ is in K [
20]. For instance, the class
${\mathrm{XP}}^{\mathrm{nu}}$ consists of those parameterized problems whose slices are decidable in polynomial time. Note that this definition is nonuniform, that is, for each positive integer
k, there might be a completely different polynomialtime algorithm that witnesses that
${Q}_{k}$ is polynomialtime solvable. There are also uniform variants
$\mathrm{X}\mathrm{K}$ of these classes
$\mathrm{X}{\mathrm{K}}^{\mathrm{nu}}$. We define XP to be the class of parameterized problems
Q for which there exists a computable function
f and an algorithm
A that decides whether
$(x,k)\in Q$ in time
${\leftx\right}^{f\left(k\right)}$ [
20,
22,
24]. Similarly, we define XNP to be the class of parameterized problems that are decidable in nondeterministic time
${\leftx\right}^{f\left(k\right)}$. Its dual class we denote by Xco−NP. Alternatively, we can view XNP as the class of parameterized problems for which there exists an xpreduction to
${\mathrm{SAT}}_{1}$ and Xco−NP as the class of parameterized problems for which there exists an xpreduction to
${\mathrm{UNSAT}}_{1}$. (For any
$L\in \mathrm{XNP}$, we know that
L can be xpreduced to
${\mathrm{SAT}}_{1}$ by following a suitable variant of the proof of the CookLevin Theorem [
25,
26]. Conversely, any parameterized problem
L that can be xpreduced to
${\mathrm{SAT}}_{1}$ we can solve in nondeterministic time
${\leftx\right}^{f\left(k\right)}$ by first carrying out the xpreduction, and then solving the resulting instance of
${\mathrm{SAT}}_{1}$. The case for Xco−NP and
${\mathrm{UNSAT}}_{1}$ is entirely analogous.)
2.3. FptReductions to SAT and Parameterized Complexity Classes at Higher Levels of the PH
Problems in NP and coNP can be encoded into SAT in such a way that the time required to produce the encoding and consequently also the size of the resulting SAT instance are polynomial in the input (the encoding is a polynomialtime manyone reduction). Typically, the SAT encodings of problems proposed for practical use are of this kind (see, e.g., [
27]). For problems that are “beyond NP”, say for problems on the second level of the PH, such polynomial SAT encodings do not exist, unless the PH collapses. However, for such problems, there still could exist SAT encodings which can be produced in fpttime with respect to some parameter associated with the problem. In fact, such fpttime SAT encodings have been obtained for various problems on the second level of the PH [
28,
29,
30,
31]. The classes paraNP and paracoNP contain exactly those parameterized problems that admit such a manyone fptreduction to
${\mathrm{SAT}}_{1}$ and
${\mathrm{UNSAT}}_{1}$, respectively. Thus, with fpttime encodings, one can go significantly beyond what is possible by conventional polynomialtime SAT encodings.
Fpttime encodings to SAT also have their limits. Clearly, para
${\Sigma}_{2}^{\mathrm{p}}$hard and para
${\Pi}_{2}^{\mathrm{p}}$hard parameterized problems do not admit fpttime encodings to SAT, even when the parameter is a fixed constant, unless the PH collapses. There are problems that apparently do not admit fpttime encodings to SAT, but seem not to be para
${\Sigma}_{2}^{\mathrm{p}}$hard nor para
${\Pi}_{2}^{\mathrm{p}}$hard either. Recently, several complexity classes have been introduced to classify such intermediate problems [
7,
8,
30]. These parameterized complexity classes are dubbed the
$k\ast $ class and the
$\ast k$ hierarchy, inspired by their definition, which is based on the following weighted variants of the quantified Boolean satisfiability problem that is canonical for the second level of the PH. The problem
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$
WSat(
$\mathcal{C}$) provides the foundation for the
$k\ast $ class.
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$WSat Instance: A quantified Boolean formula $\exists X.\forall Y.\psi $, and an integer k. Parameter: k. Question: Does there exist a truth assignment $\alpha $ to X with weight k such that for all truth assignments $\beta $ to Y the assignment $\alpha \cup \beta $ satisfies $\psi $?

Similarly, the problem
${\Sigma}_{2}^{\mathrm{p}}[\ast k]$
WSat(
$\mathcal{C}$) provides the foundation for the
$\ast k$ hierarchy—where
$\mathcal{C}$ is a class of Boolean circuits. (The parameterized problems
${\Sigma}_{2}^{\mathrm{p}}[\ast k]$
WSat(
$\mathcal{C}$) seem not to be fptreducible to each other for various classes
$\mathcal{C}$ of Boolean circuits—similarly to the problems
p
WSat$\left[\mathcal{C}\right]$ that are used to define the classes
$\mathrm{W}\left[t\right]$,
$\mathrm{W}\left[\mathrm{SAT}\right]$, and
$\mathrm{W}\left[\mathrm{P}\right]$. This is in contrast to the case of
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$
WSat, where we can use a Tseitin transformation [
32] to reduce arbitrary Boolean circuits to equisatisfiable 3CNF formulas.)
${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat($\mathcal{C}$) Instance: A Boolean circuit $C\in \mathcal{C}$ over two disjoint sets X and Y of variables, and an integer k. Parameter: k. Question: Does there exist a truth assignment $\alpha $ to X such that for all truth assignments $\beta $ to Y with weight k the assignment $\alpha \cup \beta $ satisfies C?

The parameterized complexity class
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$ (also called the
$k\ast .$ class) is then defined as follows:
Similarly, the classes of the $\ast k$ hierarchy are defined as follows:
${\Sigma}_{2}^{\mathrm{p}}[\ast k,t]$ = $\left[\phantom{\rule{4pt}{0ex}}\right\{\phantom{\rule{0.166667em}{0ex}}$ ${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat$\left({\mathrm{CIRC}}_{t,u}\right)\phantom{\rule{0.277778em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}u\ge 1\phantom{\rule{0.166667em}{0ex}}{\left\}\phantom{\rule{4pt}{0ex}}\right]}^{\mathrm{fpt}}$,
${\Sigma}_{2}^{\mathrm{p}}[\ast k,\mathrm{SAT}]$ = $[\phantom{\rule{4pt}{0ex}}$${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat(FORM) ${\phantom{\rule{4pt}{0ex}}]}^{\mathrm{fpt}}$, and
${\Sigma}_{2}^{\mathrm{p}}[\ast k,\mathrm{P}]$ = $[\phantom{\rule{4pt}{0ex}}$${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat(CIRC) ${\phantom{\rule{4pt}{0ex}}]}^{\mathrm{fpt}}$.
Note that these definitions are entirely analogous to those of the parameterized complexity classes of the Whierarchy [
20]. The following inclusion relations hold between the classes of the
$\ast k$ hierarchy:
(See also
Figure 1 for a visual overview of these inclusion relations.)
Dual to the classical complexity class ${\Sigma}_{2}^{\mathrm{p}}$ is its coclass ${\Pi}_{2}^{\mathrm{p}}$, whose canonical complete problem is complementary to the problem QSat${}_{2}$. Similarly, we can define dual classes for the $k\ast $ class and for each of the parameterized complexity classes in the $\ast k$ hierarchy. These coclasses are based on problems complementary to the problems ${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$WSat and ${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat—i.e., these problems have as yesinstances exactly the noinstances of ${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$WSat and ${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat, respectively. Equivalently, these complementary problems can be considered as variants of ${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$WSat and ${\Sigma}_{2}^{\mathrm{p}}[\ast k]$WSat where the existential and universal quantifiers are swapped, and are therefore denoted with ${\Pi}_{2}^{\mathrm{p}}[k\ast ]$WSat and ${\Pi}_{2}^{\mathrm{p}}[\ast k]$WSat. We use a similar notation for the dual complexity classes, e.g., we denote co${\Sigma}_{2}^{\mathrm{p}}[\ast k,t]$ by ${\Pi}_{2}^{\mathrm{p}}[\ast k,t]$.
The class
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$ includes the class para
$\mathrm{coNP}$ as a subset, and is contained in the class Xco−NP as a subset. Similarly, each of the classes
${\Sigma}_{2}^{\mathrm{p}}[\ast k,t]$ include the the class paraNP as a subset, and is contained in the class XNP. Under some common complexitytheoretic assumptions, the class
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$ can be separated from paraNP on the one hand, and para
${\Sigma}_{2}^{\mathrm{p}}$ on the other hand. In particular, assuming that
$\mathrm{NP}\ne \mathrm{coNP}$, it holds that
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]\u2288\mathrm{paraNP}$, that
$\mathrm{paraNP}\u2288{\Sigma}_{2}^{\mathrm{p}}[k\ast ]$ and that
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]\u228a\mathrm{para}{\Sigma}_{2}^{\mathrm{p}}$ [
7,
8]. Similarly, the classes
${\Sigma}_{2}^{\mathrm{p}}[\ast k,t]$ can be separated from para
$\mathrm{coNP}$ and para
${\Sigma}_{2}^{\mathrm{p}}$. Assuming that
$\mathrm{NP}\ne \mathrm{coNP}$, it holds that
${\Sigma}_{2}^{\mathrm{p}}[\ast k,1]\u2288\mathrm{para}\mathrm{coNP}$, that
$\mathrm{para}\mathrm{coNP}\u2288{\Sigma}_{2}^{\mathrm{p}}[\ast k,\mathrm{P}]$ and thus in particular that
$\mathrm{para}\mathrm{coNP}\u2288{\Sigma}_{2}^{\mathrm{p}}[\ast k,1]$, and that
${\Sigma}_{2}^{\mathrm{p}}[\ast k,\mathrm{P}]\u228a\mathrm{para}{\Sigma}_{2}^{\mathrm{p}}$ [
7,
8].
One can also enhance the power of polynomialtime SAT encodings by considering polynomial time algorithms that can query a SAT solver multiple times—that is, polynomialtime Turing reductions. Such an approach has been shown to be quite effective in practice (see, e.g., [
33,
34,
35]) and extends the scope of SAT solvers to problems in the class
${\Delta}_{2}^{\mathrm{p}}$, but not to problems that are
${\Sigma}_{2}^{\mathrm{p}}$hard or
${\Pi}_{2}^{\mathrm{p}}$hard. In addition, here, switching from polynomialtime to fpttime provides a significant increase in power. The class para
${\Delta}_{2}^{\mathrm{p}}$ contains all parameterized problems that can be decided by an fptalgorithm that can query a SAT oracle multiple times—i.e., by an fpttime Turing reduction to SAT. (One can prove this by following the proof of Theorem 4 in [
24] that
$\mathrm{FPT}=\mathrm{para}\mathrm{P}$, with the modification that the algorithms are given access to a SAT oracle.) In addition, one could restrict the number of queries that the algorithm is allowed to make. The class para
${\Theta}_{2}^{\mathrm{p}}$ consists of all parameterized problems that can be decided by an fptalgorithm that can query a SAT oracle at most
$f\left(k\right)logn$ many times, where
k is the parameter value,
n is the input size, and
f is some computable function. (This statement one can prove by following the proof of Theorem 4 in [
24] that
$\mathrm{FPT}=\mathrm{para}\mathrm{P}$, with the modification that the algorithms can query a SAT oracle an amount of times that depends logarithmically on the input size.) Restricting the number of queries even further, we define the parameterized complexity class
${\mathrm{FPT}}^{\mathrm{NP}}\left[\mathrm{few}\right]$ as the class of all parameterized problems that can be decided by an fptalgorithm that can query a SAT oracle at most
$f\left(k\right)$ times, where
k is the parameter value and
f is some computable function [
7,
8].
We get the parameterized analogue paraPSPACE of the class PSPACE by using the definition of paraK for
$\mathrm{K}=\mathrm{PSPACE}$. Similarly, we can define the parameterized complexity class XPSPACE, consisting of all parameterized problems
Q for which there exists a computable function
f and an algorithm
A that decides whether
$(x,k)\in Q$ in space
${\leftx\right}^{f\left(k\right)}$. We also consider another parameterized variant of PSPACE, which is based on parameterizing the number of quantifier alternations in
QSat. An unbounded number of quantifier alternations in this problem results in the class PSPACE, and bounding the number of quantifier alternations by a constant leads to some fixed level of the PH. The parameterized complexity class
PH$\left[\mathrm{level}\right]$ is based on bounding the number of quantifier alternations by the problem parameter [
7,
36]. Formally, we consider the following parameterized problem
QSat(level).
QSat(level) Instance: A quantified Boolean formula $\phi =\exists {X}_{1}\forall {X}_{2}\exists {X}_{3}\cdots {Q}_{k}{X}_{k}\psi $, where ${Q}_{k}$ is a universal quantifier if k is even and an existential quantifier if k is odd, and where $\psi $ is quantifierfree. Parameter: k. Question: Is $\phi $ true?

The parameterized complexity class PH$\left[\mathrm{level}\right]$ is defined to be the class of all parameterized problems that can be fptreduced to QSat(level). We have that $\mathrm{para}{\Sigma}_{2}^{\mathrm{p}}\cup \mathrm{para}{\Pi}_{2}^{\mathrm{p}}\subseteq \mathrm{PH}\left[\mathrm{level}\right]\subseteq \mathrm{paraPSPACE}$.
An overview of the parameterized complexity classes relevant for this paper can be found in
Figure 1.
In the early literature on this topic [
6,
28,
30,
37,
38,
39,
40,
41], the class
${\Sigma}_{2}^{\mathrm{p}}[k\ast ]$ appeared under the names
${\exists}^{k}\forall $ and
${\exists}^{k}{\forall}^{\ast}$. Similarly, the classes
${\Sigma}_{2}^{\mathrm{p}}[\ast k,t]$ appeared under the names
$\exists {\forall}^{k}\mathrm{W}\left[t\right]$ and
${\exists}^{\ast}{\forall}^{k}\mathrm{W}\left[t\right]$. In addition, the class
${\mathrm{FPT}}^{\mathrm{NP}}\left[\mathrm{few}\right]$ appeared under the name
${\mathrm{FPT}}^{\mathrm{NP}}\left[f\left(k\right)\right]$.