Induction and Co-induction in Sequent Calculus

Momigliano, Alberto; Tiu, Alwen

doi:10.1007/978-3-540-24849-1_19

Cited by 28 publications

(36 citation statements)

References 46 publications

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“…Several other logics from the programming languages literature formulate the notion of induction similar to IID 0 : the proofs of consistency for FOλ∆ N [MM00], Linc [MT03], and G [GMN08] all rely on logical relations; the proof of consistency for LKID [Bro06] uses model theory. We are optimistic that at least some of these systems can be proven consistent using lexicographic path induction.…”

Section: Resultsmentioning

confidence: 99%

“…We give a novel consistency proof by lexicographic path induction of an intuitionistic theory of inductive definitions, based on the system by Martin-Löf [ML71] that inspired the definition of inductive types in type theory [Dyb91,PM93], and several sequent calculi in the programming languages literature [MM00,MT03,Bro06,GMN08]. The consistency of Heyting Arithmetic follows as a simple corollary, and the weak normalization of Gödel's T follows via a structural logical relation [SS08].…”

Section: Introductionmentioning

confidence: 99%

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Lexicographic Path Induction

Sarnat

Schürmann

2009

Lecture Notes in Computer Science

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Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of Martin-Löf's intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel's T follows indirectly; both have been formalized in a prototypical extension of Twelf.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lexicographic Path Induction

Sarnat

Schürmann

2009

Lecture Notes in Computer Science

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“…The logic G allows for the direct specification of recursive predicate definitions and to interpret them either as a least and or greatest fixed point in the sense of [2,5,22,31]. The rules for induction and co-induction use higher-order predicate schema variables in their premises in order to range over possible pre-and post-fixed points.…”

Section: The Reasoning Logicmentioning

confidence: 99%

Reasoning about Computations Using Two-Levels of Logic

Miller

2010

Programming Languages and Systems

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Abstract. We describe an approach to using one logic to reason about specifications written in a second logic. One level of logic, called the "reasoning logic", is used to state theorems about computational specifications. This logic is classical or intuitionistic and should contain strong proof principles such as induction and co-induction. The second level of logic, called the "specification logic", is used to specify computation. While computation can be specified using a number of formal techniques-e.g., Petri nets, process calculus, and state machines-we shall illustrate the merits and challenges of using logic programming-like specifications of computation.

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“…As a matter of fact, Abella supports two different meta-logics, LG ω [Tiu 2007] and a superset, G . Both support (co)inductive partial definitions [Momigliano and Tiu 2003] and the ∇ quantifier, in the stronger (w.r.t. F Oλ ∆∇ [Miller and Tiu 2005]) "nominal-ish" version.…”

Section: Tu In Abellamentioning

confidence: 99%

Reasoning with hypothetical judgments and open terms in hybrid

Felty

Momigliano

2009

Proceedings of the 11th ACM SIGPLAN Conference on Principles and Practice of Declarative Programming

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Hybrid is a system developed to specify and reason about logics, programming languages, and other formal systems expressed in higher-order abstract syntax (HOAS). An important goal of Hybrid is to exploit the advantages of HOAS within the well-understood setting of higher-order logic as implemented by systems such as Isabelle and Coq. In this paper, we add new capabilities for reasoning by induction on encodings of object-level inference rules. Elegant and succinct specifications of such inference rules can often be given using hypothetical and parametric judgments, which are represented by embedded implication and universal quantification. Induction over such judgments is well-known to be problematic. In previous work, we showed how to express this kind of judgment using a two-level approach, but reasoning by induction on such judgments was restricted to closed terms. The new capabilities we add include techniques for adding arbitrary "new" variables to contexts and inductively reasoning about open terms. Very little overhead is required, namely a small library of definitions and lemmas, yet the reasoning power of the system and the class of properties that can be proved is significantly increased. We illustrate the approach using PCF, a simple programming language that serves as the core of a variety of functional languages. We encode the typing judgment, and prove by induction on this judgment that well-typed PCF terms have unique types.

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Induction and Co-induction in Sequent Calculus

Cited by 28 publications

References 46 publications

Lexicographic Path Induction

Lexicographic Path Induction

Reasoning about Computations Using Two-Levels of Logic

Reasoning with hypothetical judgments and open terms in hybrid

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