# Unconventional Algorithms: Complementarity of Axiomatics and Construction

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## Abstract

**:**

## 1. Introduction

## 2. Local Mathematics vs. Global Mathematics

## 3. Logical Varieties as a Unification of Local Logics

**T**of all Turing machines and the class

**TT**of all total, i.e., everywhere defined, Turing machines, we find that the first class satisfies the axiom of universality [10], which affirms existence of a universal algorithm, i.e., a universal Turing machine in this class. However, the class

**TT**does not satisfy this axiom [10].

**K**of syntactic logical calculi, a set Q of inference rules (Q ⊆ R), and a class

**F**of partial mappings from L to L.

**M**= (A, H, M), where A and M are sets of expressions that belong to L (A consists of axioms and M consists of theorems) and H is a set of inference rules, which belong to the set R, is called:

- (1)
- a projective syntactic (
**K,F**)-prevariety if there exists a set of logical calculi**C**_{i}= (A_{i}, H_{i}, T_{i}) from**K**and a system of mappings f_{i}: A_{i}→ L and g_{i}: M_{i}→ L (i ∈ I) from**F**in which A_{i}consists of all axioms and M_{i}consists of all theorems of the logical calculus**C**_{i}, and for which the equalities A = ∪_{i∈I}f_{i}(A_{i}), H = ∪_{i∈I}H_{i}and M = ∪_{i∈I}g_{i}(M_{i}) are valid (it is possible that**C**_{i}=**C**_{j}for some i ≠ j). - (2)
- a projective syntactic (
**K,F**)-variety with the depth k if it is a projective syntactic (**K,F**)-quasi-prevariety and for any i_{1}, i_{2}, i_{3}, … , i_{k}∈ I either the intersections ∩_{j}_{=1}^{k}f_{ij}(A_{ij}) and ∩_{j}_{=1}^{k}g_{ij}(T_{ij}) are empty or there exists a calculus**C**= (A, H, T) from**K**and projections f: A → ∩_{j}_{=1}^{k}f_{ij}(A_{ij}) and g: N → ∩_{j}_{=1}^{k}g_{ij}(M_{ij}) from**F**where N ⊆ T; - (3)
- a syntactic
**K**-prevariety if it is a projective syntactic (**K,F**)-prevariety in which M_{i}= T_{i}for all i ∈ I and all mappings f_{i}and g_{i}that define**M**are bijections on the sets A_{i}and M_{i}, correspondingly; - (4)
- a syntactic
**K**-variety if it is a projective syntactic (**K,F**)- variety in which M_{i}= T_{i}for all i ∈ I and all mappings f_{i}and g_{i}that define**M**are bijections on the sets A_{i}and M_{i}, correspondingly.

**C**

_{i}used in the formation of the prevariety (variety)

**M**are called components of

**M**.

_{i}and g

_{i}makes a unified system called a prevariety or quasi-prevariety out of separate logical calculi

**C**

_{i}, while the collection of the intersections ∩

_{j=1}

^{k}f

_{ij}(A

_{ij}) and ∩

_{j=1}

^{k}g

_{ij}(T

_{ij}) makes a unified system called a variety out of separate logical calculi

**C**

_{i}. For instance, mappings f

_{i}and g

_{i}allow one to establish a correspondence between norms/laws that were used in one country during different periods of time or between norms/laws used in different countries. In a similar way, relations between components of logical varieties and prevarieties allow one to establish a correspondence between properties of different models of computation and algorithmic classes.

## 4. Projective Mathematics vs. Reverse Mathematics vs. Classical Mathematics

**A**of algorithms, not to prove these theorems but only to check if the initial axioms are valid in

**A**. If this is the case, then it becomes possible to conclude that all corresponding theorems are true for the class

**A**. As we know, computer scientists and mathematicians study and utilize a huge variety of different classes and types of algorithms, automata, and abstract machines. Consequently, such an axiomatic approach allows them to obtain many properties of studied algorithms and automata in a simple and easy way.

- (1)
- Consistent with some result (theorem) T, i.e., when the theorem T is added as a new axiom, the new system remains consistent, allowing one to, in some cases, deduce (prove) this theorem.
- (2)
- Sufficient for some result (theorem) T, i.e., it is possible to deduce (prove) the theorem T using axioms from A.
- (3)
- Irreducible with respect to some result (theorem) T, i.e., the system A is a minimal set of the axiom that allows one to deduce (prove) the theorem T.

**A**of algorithms or automata, not to prove these theorems but only to check if all axioms from the system U are valid in

**A**. If this is the case, then it is possible to conclude that all corresponding theorems are true for the class

**A**. As we know, computer scientists and mathematicians study and utilize a huge variety of different classes and types of algorithms, automata, and abstract machines. In such a way, the axiom system U provides a definite perspective on different classes and types of algorithms, automata, and abstract machines.

_{i}is a component M

_{i}of M. Then we deduce a theorem T from the statements from C. Then instead of proving the theorem T for each domain D

_{i}, we check whether C ⊆ M

_{i}. When this is true, we conclude that the theorem T belongs to the component M

_{i}because M

_{i}is a calculus and thus, the theorem T is valid for the model D

_{i}. Because C usually consists of relatively simple statements, to check the inclusion C ⊆ M

_{i}is simpler than to prove T in M

_{i}. In addition, this approach provides unification for the whole theory of algorithms, automata and computation as it explicates similarities and common traits in different algorithmic models and abstract automata.

## 5. How To Navigate in the Algorithmic Multiverse

**K**and arbitrary data elements b and x informs whether application of A to x gives b as the result, i.e., whether A(x) = b.

## 6. Conclusions and Future Work

- -
- Classical mathematics, with global axiomatization and classical logic.
- -
- Local mathematics, with local axiomatization, diverse logics and logical varieties.
- -
- Reverse mathematics, with axiomatic properties decomposition and backward inference.
- -
- Projective mathematics, with view axiomatization, logical varieties and properties proliferation.

## Acknowledgments

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Dodig Crnkovic, G.; Burgin, M.
Unconventional Algorithms: Complementarity of Axiomatics and Construction. *Entropy* **2012**, *14*, 2066-2080.
https://doi.org/10.3390/e14112066

**AMA Style**

Dodig Crnkovic G, Burgin M.
Unconventional Algorithms: Complementarity of Axiomatics and Construction. *Entropy*. 2012; 14(11):2066-2080.
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**Chicago/Turabian Style**

Dodig Crnkovic, Gordana, and Mark Burgin.
2012. "Unconventional Algorithms: Complementarity of Axiomatics and Construction" *Entropy* 14, no. 11: 2066-2080.
https://doi.org/10.3390/e14112066