30,016 research outputs found

    Metamathematics in Coq

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    Chapter 1: Automated Proof Construction in Type Theory using Resolution. We describe techniques to integrate resolution logic in type theory. Refutation proofs obtained by resolution are translated into lambda-terms, using reflection and an encoding of resolution proofs in minimal logic. Thereby we obtain a verification procedure for resolution proofs, and, more importantly, we add the power of resolution theorem provers to interactive proof construction systems based on type theory. We introduce a novel representation of clauses in minimal logic such that the lambda-representation of resolution steps is linear in the size of the premisses. A clausification algorithm, equipped with a correctness proof, is encoded in Coq. Chapter 2: Proof Reflection in Coq. Natural deduction for first-order logic is formalised in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgement is of the form G |- d [:] p, stating that d is a proof term of formula p under hypotheses G; it can be viewed as a typing relation by the Curry-Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, I define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction. Chapter 3: Adbmal To make the notion of scope in the lambda-calculus explicit, we extend the syntax of the lambda-calculus with an end-of-scope operator adbmal. The idea is that an adbmal x ends the scope of the matching lambda x above it (in the term tree). Accordingly, beta-reduction is extended to the set of scoped lambda-terms by performing minimal scope extrusion before performing replication as usual. We show confluence of the resulting scoped β\beta-reduction. Confluence of beta-reduction for the ordinary lambda-calculus is obtained as a corollary, by extruding scopes maximally before forgetting them altogether. Only in this final forgetful step, alpha-equivalence is needed. All our proofs have been verified in Coq

    The Challenge of Unifying Semantic and Syntactic Inference Restrictions

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    While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable

    Higher-order Linear Logic Programming of Categorial Deduction

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    We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.Comment: 8 pages LaTeX, uses eaclap.sty, to appear EACL9

    A language for multiplicative-additive linear logic

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    A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive categories with additives. It is also shown that proof equivalence is decidable by showing that the cut elimination rewrites supply a confluent rewriting system modulo equations.Comment: 16 pages without appendices, 30 with appendice
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