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In this paper, we analyze axiomatic and constructive issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms and unconventional computations change the algorithmic universe, making it open and allowing increased flexibility and expressive power that augment creativity. At the same time, the greater power of new types of algorithms also results in the greater complexity of the algorithmic universe, transforming it into the algorithmic multiverse and demanding new tools for its study. That is why we analyze new powerful tools brought forth by local mathematics, local logics, logical varieties and the axiomatic theory of algorithms, automata and computation. We demonstrate how these new tools allow efficient navigation in the algorithmic multiverse. Further work includes study of natural computation by unconventional algorithms and constructive approaches.

The development of computer science and information technology brought forth a diversity of novel algorithms and algorithmic schemas, unconventional computations and nature-inspired processes, advanced functionality and conceptualizations. The field of information processing and information sciences encountered the stage of transition from classical subrecursive and recursive algorithms, like finite automata, recursive functions or Turing machines, to new super-recursive algorithms, such as inductive Turing machines or limiting recursive functions.

Super-recursive algorithms controlling and directing unconventional computations exceed the boundary set by Turing machines and other recursive algorithms, resulting in an open algorithmic universe and revealing new levels of creativity. As the growth of possibilities involves a much higher complexity of the new, open world of super-recursive algorithms, unconventional computations, innovative hardware and advanced organization, we discuss means of navigation in this open algorithmic world.

The paper is organized as follows: in

As an advanced knowledge system, mathematics exists as an aggregate of various mathematical fields. If at the beginning there were only two fields―arithmetics and geometry, now there are hundreds of mathematical fields and subfields. However, mathematicians always believed in mathematics as a unified system striving to build common and in some senses absolute foundations for all mathematical fields and subfields. At the end of the 19th century, mathematicians came very close to achieving this goal as the emerging set theory allowed the construction of all mathematical structures using only sets and operations with sets. However, in the 20th century, it was discovered that there are different set theories. This brought some confusion and attempts to find the “true” set theory.

To overcome this confusion, Bell [

It is possible to extend the approach of Bell in three directions. First, we can use an arbitrary category as a framework for developing mathematics. When an internal structure of such a framework is meager, the corresponding mathematics will also be indigent. Second, it is possible to take a theory of some structures instead of the classical universe of sets and develop mathematics within that framework without reference to the universal framework. Third, as we know, there are different axiomatizations of set theory. Developed axiomatics are often incompatible, e.g., axiomatics in which the Continuum Hypothesis is true and axiomatics where it is false. Thus, developing mathematics based on one such axiomatics also results in a local mathematics.

A similar situation emerged in computer science, where mathematics plays a pivotal role. Usually, to study properties of computers, computer networks and computational processes and elaborate more efficient applications, mathematicians and computer scientists use mathematical models. There is a variety of such models: Turing machines of different kinds (with one tape and one head, with several tapes, with several heads, with

This diversity of models is natural and useful because each type is suited to a particular type of problem. In other words, the diversity of problems that are solved by computers gives rise to a corresponding diversity of models. For example, general problems of computability involve such models as Turing machines and partial recursive functions. Finite automata are used for text search, lexical analysis, and construction of semantics for programming languages. In addition, different computing devices demand corresponding mathematical models. For example, universal Turing machines and inductive Turing machines allow one to investigate characteristics of conventional computers [

To utilize some models that are related to a specific type of problem, we need to know their properties. In many cases, different classes of models have the same or similar properties. As a rule, such properties are proved for each class separately. Thus, alike proofs are repeated many times in similar situations involving various models and classes of algorithms.

In contrast to this, the

The projective approach in computer science has its counterpart in mathematics, where systems of unifying properties have been used for building new encompassing structures, proving indispensable properties in these new structures and projecting these properties on the encompassed domains. Such projectivity has been explicitly utilized in category theory, which was developed and utilized with the goal of unification [

Breaking the barrier of the Church-Turing Thesis drastically increased the variety of algorithmic model classes and changed the algorithmic universe of recursive algorithms to the multiverse of super-recursive algorithms [

Local mathematics brings forth local logics because each local mathematical framework has its own logic and it is possible that different frameworks have different local logics.

Barwise and Seligman [

The basic concept of the information flow theory is a

In a similar way, each class of algorithms from the algorithmic multiverse, as well as a constellation of such classes, forms a local algorithmic universe, which has a corresponding local logic. These logics may be essentially different. For instance, taking two local algorithmic universes formed by such classes as the class

Barwise and Seligman [

Minsky [

There are different types and kinds of logical varieties and prevarieties:

Let us consider a logical language

A triad

a _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i∈I} _{i}_{i}_{i∈I} _{i}_{i∈I} _{i}_{i}_{i}_{j}

a _{1} , _{2} , _{3} , … , _{k}_{j}_{=1}^{k}_{ij}_{ij}_{j}_{=1}^{k}g_{ij}_{ij}_{j}_{=1}^{k}_{ij}_{ij}_{j}_{=1}^{k}_{ij}_{ij}

a _{i}_{i}_{i}_{i}_{i}_{i}

a _{i}_{i}_{i}_{i}_{i}_{i}

The calculi _{i}

Note that different components of deductive logical varieties and prevarieties can not only contain distinct axioms and theorems but also employ distinctive deduction rules. For instance, one component can use classical deduction, while another component of the same variety can be based on a relevant logic.

We see that the collection of mappings _{i}_{i}_{i}_{j=1}^{k} _{ij}_{ij}_{j=1}^{k}_{ij}_{ij}_{i}_{i}_{i}

The main goal of syntactic logical varieties is in presenting sets of formulas as a structured logical system using logical calculi, which have means for inference and other logical operations. Semantically, it allows one to describe a domain of interest, e.g., a database, knowledge of an individual or the text of a novel, by a syntactic logical variety dividing the domain in parts that allow representation by calculi.

In comparison with varieties and prevarieties, logical quasi-varieties and quasi-prevarieties studied in [

While syntactic logical varieties and prevarietis synthesize local logics in a unified system, semantic logical varieties and prevarieties studied in [

In addition, syntactic logical varieties and prevarieties found diverse applications to databases and network technology providing tools for working with inconsistency, imprecision, vagueness, non-monotonic inference, knowledge base unification and database integration (cf., for example, [

According to Suppe [

Mathematics suggests an approach for knowledge unification, namely, it is necessary to find axioms that characterize all theories in a specific area and to develop the theory in an axiomatic context. This approach has worked extremely well in a variety of mathematical fields, providing rigorous tools for mathematical exploration.

Axiomatization has often been used in physics (Hilbert’s sixth problem refers to axiomatization of branches of physics in which mathematics is prevalent and researchers found that, for example, finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics), biology (according to Britannica, [

Since the advent of computers, deductive reasoning and axiomatic exposition have been delegated to computers, which performed theorem-proving, while the axiomatic approach has come to software technology and computer science. Logical tools and axiomatic description have been used in computer science for different purposes. For instance, Manna [

However, in classical mathematics, axiomatization has a global character. Mathematicians tried to build a unique axiomatics for the foundations of mathematics. Logicians working in the theory of algorithms tried to find axioms comprising all models of algorithms.

This is the classical approach – axiomatizing the studied domain and then deducing theorems from axioms. All classical mathematics is based on deduction as a method of logical reasoning and inference.

The goal of deductive mathematics is to deduce theorems from axioms. Deduction of a theorem is also called proving the theorem. When mathematicians cannot prove some interesting and/or important conjecture, researchers with a conventional thinking try to prove that the problem is unsolvable in the existing framework. Creative explorers instead invent new structures and methods, construct new framework, introducing new axioms to solve the problem.

Some consider deductive mathematics as a part of axiomatic mathematics, assuming that deduction (in a strict sense) is possible only in an axiomatic system. Others treat axiomatic mathematics as a part of deductive mathematics, assuming that there are other inference rules besides deduction.

While deductive mathematics is present in and actually dominates all fields of contemporary mathematics, reverse mathematics is the branch of mathematical logic that seeks to determine what are the minimal axioms (formalized conditions) needed to prove a particular theorem [

Reverse mathematics was prefigured by some results in set theory, such as the classical theorem that states that the axiom of choice, well-ordering principle of Zermelo, maximal chain priciple of Hausdorff, Zorn’s lemma [

Projective mathematics is a branch of mathematics similar to reverse mathematics, which aims to determine what simple conditions are needed to prove the particular theorem or to develop a particular theory. However, there are essential differences between these two directions: reverse mathematics is aimed at a logical analysis of mathematical statements, while projective mathematics is directed at making the scope of theoretical statements in general and mathematical statements in particular much larger whilst extending their applications. As a result, instead of proving similar results in various situations, it becomes possible to prove a corresponding general result in the axiomatic setting and to ascertain validity of this result for a particular case by demonstrating that all axioms (conditions) used in the proof are true for this case. In this way the general result is projected on different situations. This direction in mathematics was founded by Burgin [

Projective mathematics has its precursor in such results as the extension of many theorems initially proved for numerical functions to functions in metric spaces [

Here we describe how projective mathematics is used for exploration of computations controlled by algorithms and realized by automata. In this application of projective mathematics, the goal is to find some simple properties of computations, algorithms and automata in general, to present these properties in the form of axioms, and to deduce from these axioms theorems that describe much more profound and sophisticated properties of computations, algorithms and automata. This allows one, taking some class

It is possible to explain goals of classical (deductive) mathematics, reverse mathematics and projective mathematics by means of relations between axioms and theorems.

A set

After the discovery of non-Euclidean geometries, the creation of modern algebra and the construction of set theory, classical mathematics’ main interest has been to find whether a statement

In contrast to this, projective mathematics is oriented at the second relation. The goal is to find some simple properties of algorithms or automata in general, to present these properties in the form of a system

It is interesting that Bernays had a similar intuition with respect to axioms in mathematics, regarding them not as a system of statements about a subject matter but as a system of conditions for what might be called a relational structure. He wrote in [

It is possible to formalize the approach of projective mathematics using logical varieties. Indeed, let us take a collection _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

It is possible to see that for a conformist, it is much easier to live in the closed algorithmic universe because all possible and impossible actions, as well as all solvable and insolvable problems can be measured against one of the most powerful and universal classes of algorithms in the algorithmic universe. This has usually been done utilizing Turing machines.

The open world provides many more opportunities for actions and problem solving, but at the same time it demands more work, more effort and even more imagination for solving problems which are insolvable in the closed algorithmic universe. Even the closed algorithmic universe contains many classes and types of algorithms, which have been studied with reference to a universal class of recursive algorithms. In some cases, partial recursive functions have been used. In other cases, unrestricted grammars have been employed. The most popular have been the utilization of Turing machines. A big diversity of new and old classes of algorithms exists that demands specific tools for exploration.

Mathematics has invented such tools and one of the most efficient for dealing with diversity is the axiomatic method. This method was also applied to the theory of algorithms, automata and computation when the axiomatic theory of algorithms, automata and computation was created [

In [

The axiomatic context allows a researcher to explore not only individual algorithms and separate classes of algorithms, computational models and automata but also classes of classes of algorithms, automata, and computational models and processes. As a result, the axiomatic approach goes higher in the hierarchy of computer and network models, thus reducing the complexity of their study. The suggested axiomatic methodology is applied to the evaluation of possibilities of computers, their software and their networks, with the main emphasis on such properties as computability, decidability, and acceptability. In such a way, it becomes possible to derive various characteristics of types of computers and software systems from the initial postulates, axioms and conditions.

It is also worth mentioning that the axiomatic approach allowed researchers to prove the Church-Turing Thesis for an algorithmic class that satisfies very simple initial axioms [

Moreover, the axiomatic approach is efficient in exploring features of innovative hardware and unconventional organization.

It is interesting to remark that algorithms are used in mathematics and beyond as constructive tools of cognition. Algorithms are often opposed to non-constructive, e.g., descriptive, methods used in mathematics. The axiomatic approach is essentially descriptive because axioms describe properties of the studied objects in a formalized way.

Constructive mathematics is distinguished from its traditional counterpart, axiomatic classical mathematics, by the strict interpretation of the expression “there exists” (called in logic the

However, in some situations, descriptive methods can be more efficient and powerful than constructive tools. Language allows one to describe many more objects than it is possible to build by available tools and materials. For instance, sufficiently rich logical languages, according to the first Gödel undecidability theorem, can represent statements that are true but are not provable. That is why descriptive methods in the form of the axiomatic approach came back to the theory of algorithms, automata and computation, becoming efficient tools in computer science.

This paper demonstrates the role of the axiomatic methods for the following paradigms of mathematics and computer science:

Here we have considered only some of the consequences of new trends in the axiomatic approach to mathematical cognition. It would be interesting to study consequences of this approach in other fields such as epistemology and computability theory. Furthermore, inasmuch as computer science is based on mathematics, the new paradigms of mathematics presented in this work form corresponding directions in computer science giving an advantage to unconventional computations and nature-inspired architectures of information processing systems. One of the novel approaches is applying the axiomatic methods of the mathematical theory on information technology [

Another important direction for future work is the study of physical systems as information processing architectures. Computations beyond the Turing model exist not only in the universe of unconventional algorithms but even in the physical universe. The idea of Pancomputationalism (Naturalist computationalism) [

As a consequence, unconventional computing as it appears in natural systems is developing as an important new area of constructive research. It is presented by Stepney [

The authors would like to thank Andree Ehresmann, Hector Zenil and Marcin J. Schroeder for useful and constructive comments on the previous version of this work and three anonymous reviewers for numerous insightful reflections and helpful advices which considerably improved the article.