Toshiyasu Arai

Bio: Toshiyasu Arai is an academic researcher from Hiroshima University. The author has contributed to research in topics: Mathematical proof & Iterated function. The author has an hindex of 2, co-authored 2 publications receiving 47 citations.

Some results on cut-elimination, provable well-orderings, induction and reflection

Abstract: We gather the following miscellaneous results in proof theory from the attic. 1. 1. A provably well-founded elementary ordering admits an elementary order preserving map. 2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21(X). 3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic. 4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE. 5. 5. Intuitionistic fixed point theories which are conservative extensions of HA. 6. 6. Proof theoretic strengths of classical fixed points theories. 7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣn. 8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions. Each section can be read separately in principle.

A bounded arithmetic AID for Frege systems

Abstract: In this paper we introduce a system AID (alogtime inductive definitions) of bounded arithmetic The main feature of AID is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems F in Buss (Ann Pure Appl Logic 52 (1991) 3–29) We show that AID proves the soundness of F , and conversely any Σ 0 b -theorem in AID yields boolean sentences of which F has polysize proofs Further we define Σ 1 b -faithful interpretations between AID+Σ 0 b -CA and a quantified theory QALV of an equational system ALV in Clote (Ann Math Art Intell 6 (1992) 57–106) Hence ALV also proves the soundness of F

The Lean Theorem Prover (System Description)

Abstract: Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

Quantified propositional calculus and a second-order theory for NC1

Abstract: Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively.

Chapter 7. Proof Complexity and SAT Solving

Abstract: The satisfiability problem (SAT) — i.e., to determine whether a given a formula in propositional logic has a satisfying assignment or not — is of central importance to both the theory of computer science and the practice of automatic theorem proving and proof search. Proof complexity — i.e., the study of the complexity of proofs and the difficulty of searching for proofs — joins the theoretical and practical aspects of satisfiability. For theoretical computer science, SAT is the canonical NP-complete problem, even for conjunctive normal form (CNF) formulas [Coo71, Lev73]. In fact, SAT is very efficient at expressing problems in NP in that many of the standard NPcomplete problems, including the question of whether a (nondeterministic) Turing machine halts within n steps, have very efficient, almost linear time, reductions to the satisfiability of CNF formulas. A popular hypothesis in the computational complexity community is the Strong Exponential Time Hypothesis (SETH), which says that any algorithm for solving CNF SAT must have worst-case running time (roughly) 2 on instances with n variables [IP01, CIP09]. This hypothesis has been widely studied in recent years, and has served as a basis for proving conditional hardness results for many other problems. In other words, CNF SAT serves as the canonical hard decision problem, and is frequently conjectured to require exponential time to solve. In contrast, for practical theorem proving, CNF SAT is the core method for encoding and solving problems. On one hand, the expressiveness of CNF formulas means that a large variety of problems can be faithfully and straightforwardly translated into CNF SAT problems. On the other hand, the message that SAT is supposed to be hard to solve does not seem to have reached practitioners of SAT solving; instead, there has been enormous improvements in performance in SAT algorithms over the last decades. Amazingly, state-of-the-art algorithms for deciding satisfiability — so-called SAT solvers — can routinely handle real-world

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems

Abstract: This thesis is concerned with investigations into the "complexity of term rewriting systems". Moreover the majority of the presented work deals with the "automation" of such a complexity analysis. The aim of this introduction is to present the main ideas in an easily accessible fashion to make the result presented accessible to the general public. Necessarily some technical points are stated in an over-simplified way.

Derivation lengths and order types of Knuth–Bendix orders

Abstract: The derivation length function of a finite term rewriting system terminating via a Knuth–Bendix order is shown to be bounded by the Ackermann function applied to a single exponential function. This result is essentially optimal as there are rewrite systems with such derivation lengths. In a second part the order types of Knuth–Bendix orders over finite signatures are classified within the ordinals up to ω ω .