Higher-Order Matching, Games and Automata

Automata, Languages, and Programming

2013

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“…A position in a play of G(D(z), C(x)) is a pair n θ where n ∈ N and θ is a look-up table (similar to a closure) defined as follows [11]. Definition 13.…”

Section: Definition 12mentioning

confidence: 99%

Proof Systems for Retracts in Simply Typed Lambda Calculus

Automata, Languages, and Programming

2013

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“…As we saw with the monster type in Example 1, fifth-order terms may (essentially) contain infinitely many different variables. In [14], we overcame the first problem but not the second: relative to a fixed finite alphabet, the set of solutions over that alphabet to a matching problem is tree automata recognizable. The proof relies on a similar technology to that used here (a game-theoretic characterisation of matching).…”

Section: Definition 8 a Matching Problem In Simply Typed Lambda Calcmentioning

confidence: 99%

“…The question is how to capture jumping within a tree automaton. This we do using variable assumptions as in [14]: Ong calls them "variable profiles" in his setting. …”

Section: Fig 1 An Interpolation Treementioning

confidence: 99%

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Dependency Tree Automata

Foundations of Software Science and Computational Structures

2009

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Abstract. We introduce a new kind of tree automaton, a dependency tree automaton, that is suitable for deciding properties of classes of terms with binding. Two kinds of such automaton are defined, nondeterministic and alternating. We show that the nondeterministic automata have a decidable nonemptiness problem and leave as an open question whether this is true for the alternating version. The families of trees that both kinds recognise are closed under intersection and union. To illustrate the utility of the automata, we apply them to terms of simply typed lambda calculus and provide an automata-theoretic characterisation of solutions to the higher-order matching problem.

“…Besides the use of games to understand β-reduction, we also emphasize how tree automata can recognize terms of simply typed lambda calculus as developed in [1,[3][4][5][6]. References 1.…”

mentioning

confidence: 99%

Introduction to Decidability of Higher-Order Matching

Foundations of Software Science and Computational Structures

2010

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Abstract. Higher-order unification is the problem given an equation t = u containing free variables is there a solution substitution θ such that tθ and uθ have the same normal form? The terms t and u are from the simply typed lambda calculus and the same normal form is with respect to βη-equivalence. Higher-order matching is the particular instance when the term u is closed; can t be pattern matched to u? Although higher-order unification is undecidable, higher-order matching was conjectured to be decidable by Huet [2]. Decidability was shown in [7] via a game-theoretic analysis of β-reduction when component terms are in η-long normal form.In the talk we outline the proof of decidability. Besides the use of games to understand β-reduction, we also emphasize how tree automata can recognize terms of simply typed lambda calculus as developed in [1,[3][4][5][6]. References 1. Comon, H., Jurski, Y.: Higher-order matching and tree automata.