An intuitionistic version of Ramsey's Theorem and its use in Program Termination

Berardi, Stefano; Steila, Silvia

doi:10.1016/j.apal.2015.08.002

Cited by 8 publications

(25 citation statements)

References 16 publications

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“…The claim (1) can be shown in the same way as [3] since that paper did not use classical logic for proving (1). The claim (2) can be easily shown also in intuitionistic logic by using iteration to the least point.…”

Section: Main Ideasmentioning

confidence: 89%

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Equivalence of inductive definitions and cyclic proofs under arithmetic

Berardi¹,

Tatsuta²

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that of the classical system of Martin-Lof's inductive definitions. This paper studies the conjecture for intuitionistic logic. This paper first points out that the countermodel of FOSSACS 2017 paper by the same authors shows the conjecture for intuitionistic logic is false in general. Then this paper shows the conjecture for intuitionistic logic is true under arithmetic, namely, the provability of the intuitionistic cyclic proof system is the same as that of the intuitionistic system of Martin-Lof's inductive definitions when both systems contain Heyting arithmetic HA. For this purpose, this paper also shows that HA proves Podelski-Rybalchenko theorem for induction and Kleene-Brouwer theorem for induction. These results immediately give another proof to the conjecture under arithmetic for classical logic shown in LICS 2017 paper by the same authors. IntroductionAn inductive definition is a way to define a predicate by an expression which may contain the predicate itself. The predicate is interpreted by the least fixed point of the defining equation. Inductive definitions are important in computer science, since they can define useful recursive data structures such as lists and trees. Inductive definitions are important also in mathematical logic, since they increase the proof theoretic strength. Martin-Löf's system of inductive definitions given in [11] is one of the most popular systems of inductive definitions. This system has production rules for an inductive predicate, and the production rule determines the introduction rules and the elimination rules for the predicate.Brotherston and Simpson [5,8] proposed an alternative formalization of inductive definitions, called a cyclic proof system. A proof, called a cyclic proof, is defined by proof search, going upwardly in a proof figure. If we encounter the same sequent (called a bud) as some sequent we already passed (called a companion) we can stop. The induction rule is replaced by a case rule, for this purpose. The soundness is guaranteed by some additional condition, called a global trace condition, which can show the case rule decreases some measure of a bud from that of the companion. In general, for proof search, a cyclic proof system can find an induction formula in a more efficient way than Martin-Löf's system, since a cyclic proof system does not have to choose fixed induction formulas in advance. A cyclic proof system enables us efficient implementation of theorem provers with inductive definitions [4,6,7,9].Brotherston and Simpson [8] investigated Martin-Löf's system LKID of inductive definitions in classical logic for the first-order language, and the cyclic proof system CLKID ω for the same language, showed the provability of CLKID ω includes that of LKID, and conjectured the equivalence.By 2017, the equivalence was le...

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Section: Main Ideasmentioning

confidence: 89%

“…The places we need arithmetic are the proofs of (3) and (4), since they use sequences of numbers. The claims (1) and (2) can be easily shown in almost the same way as [3]. We will show the claim (3) by refining an ordinary proof of Kleene-Brouwer theorem for orders.…”

mentioning

confidence: 85%

Equivalence of inductive definitions and cyclic proofs under arithmetic

Berardi¹,

Tatsuta²

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…Assume that V is a graph whose ancestor relation is included in the complete graph N. We say that V is an Erdős' tree in k colors (e.g. [4,Definition 6.3]) if for all x ∈ V , all i = 1, . .…”

Section: From Omniscience To Homogeneous Setsmentioning

confidence: 99%

RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

Berardi¹,

Steila²

2017

J. symb. log.

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The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of k-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number k ≥ 2, Ramsey's Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for Σ 0 3 formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some i < k and some branch with infinitely many children of index i.

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“…By using this definition they proved their main theorem which states that a program P is terminating if it has a disjunctively well-founded transition invariant. In this section we firstly introduce the Termination Theorem, then we analyse the notion of inductive well-foundedness in order to present the H-closure Theorem [5].…”

Section: Podelski and Rybalchenko's Terminationmentioning

confidence: 99%

“…Although Ramsey's Theorem for pairs is a purely classical result, the Termination Theorem can be intuitionistically proved by using some intuitionistic version of Ramsey providing and providing to consider the intuitionistic notion of well-foundedness. In 2012 Vytiniotis, Coquand and Wahlstedt proved an intuitionistic version of the Termination Theorem by using the Almost-Full Theorem [36], while in 2014 Stefano Berardi and the first author proved it by using the H-closure Theorem [5]. The H-closure Theorem arose by the combinatorial fragment needed to prove the Termination Theorem (see Section 2,3).…”

Section: Introductionmentioning

confidence: 99%

Reverse mathematical bounds for the Termination Theorem

Steila

Yokoyama

2016

Annals of Pure and Applied Logic

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In 2004 Podelski and Rybalchenko expressed the termination of transition-based programs as a property of well-founded relations. The classical proof by Podelski and Rybalchenko requires Ramsey's Theorem for pairs which is a purely classical result, therefore extracting bounds from the original proof is non-trivial task.Our goal is to investigate the termination analysis from the point of view of Reverse Mathematics. By studying the strength of Podelski and Rybalchenko's Termination Theorem we can extract some information about termination bounds. * Dickson's Lemma states that (N k , ≤) (where ≤ is the componentwise order) is a well-quasi order [20,25]; i.e. every infinite sequence σ of elements of N k is such that there exists n < m with σ(n) ≤ σ(m).

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An intuitionistic version of Ramsey's Theorem and its use in Program Termination

Cited by 8 publications

References 16 publications

Equivalence of inductive definitions and cyclic proofs under arithmetic

Equivalence of inductive definitions and cyclic proofs under arithmetic

RAMSEY’S THEOREM FOR PAIRS ANDKCOLORS AS A SUB-CLASSICAL PRINCIPLE OF ARITHMETIC

Reverse mathematical bounds for the Termination Theorem

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