Research in mathematical theory of computation

Research progress in the following areas is reviewed: (1) new version of computer program LCF (logic for computable functions) including a facility to search for proofs automatically; (2) the description of the language PASCAL in terms of both LCF and in first order logic; (3) discussion of LISP semantics in LCF and attempt to prove the correctness of the London compilers in a formal way; (4) design of both special purpose and domain independent proving procedures specifically program correctness in mind; (5) design of languages for describing such proof procedures; and (6) the embedding of ideas in the first order checker

Dyson-Schwinger equations in the theory of computation

Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.Comment: 26 pages, LaTeX, final version, in "Feynman Amplitudes, Periods and Motives", Contemporary Mathematics, AMS 201

Complexity vs Energy: Theory of Computation and Theoretical Physics

This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.Comment: 23 pages. Talk at the satellite conference to ECM 2012, "QQQ Algebra, Geometry, Information", Tallinn, July 9-12, 201

Is Classical Mathematics Appropriate for Theory of Computation?

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy

Languages, machines, and classical computation

3rd ed, 2021. A circumscription of the classical theory of computation building up from the Chomsky hierarchy. With the usual topics in formal language and automata theory

Theory of computation and computing machines

At the turn of the century, David Hilbert, a famous mathematician and leader of the formalist school, was convinced of the existence of an algorithm for establishing the consistency or inconsistency of any mathematical system. Kurt Gödel [2] showed in 1931 that the consistency of any system which included the natural numbers could not be established. This result was a corollary to his more startling "incompleteness theorem" which states that if any formal system which contains the natural numbers is consistent, then that system is necessarily incomplete. More directly, there is a statement P in the system such that neither P nor not-P is a theorem of the system. Since either P or not-P must be true, then there is a true statement in the theory which is not provable. Thus the algorithm which Hilbert believed existed, in fact did not exist. The formal notion of algorithm - or "effective" procedure as it is often called - had concerned mathematicians before the result of Godel. How was an algorithm to be defined? When an algorithm was constructed, could it be determined whether or not it was meaningful? These and other questions now appeared more ominous than ever. Logicians turned their efforts toward establishing some type of approach which would enable them to categorize those procedures which were meaningful as opposed to those which were not

Theory of Computation of Multidimensional Entropy with an Application to the Monomer-Dimer Problem

We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.Comment: 35 pages, one pstricks and two eps figures, submitte