5,502 research outputs found

    Third Order Matching is Decidable

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    The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed λ\lambda-calculus, i.e. to solve the equation a = b where a and b are simply typed λ\lambda-terms and b is ground. The decidability of this problem is still open. We prove the decidability of the particular case in which the variables occurring in the problem are at most third order

    The variable containment problem

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    The essentially free variables of a term tt in some λ\lambda-calculus, FV β(t)_{\beta}(t), form the set (xx _{\mid}^{\mid} u.t=βux\forall u.t=_{\beta}u\Rightarrow x ϵ\epsilon FV(u)(u)}. This set is significant once we consider equivalence classes of λ\lambda-terms rather than λ\lambda-terms themselves, as for instance in higher-order rewriting. An important problem for (generalised) higher-order rewrite systems is the variable containment problem: given two terms tt and uu, do we have for all substitutions θ\theta and contexts CC[] that FVβ(C[t]θ)_{\beta}(C[t]^{\theta}) \supseteq FVβ(C[uθ])_{\beta}(C[u^{\theta}])? This property is important when we want to consider tut \to u as a rewrite rule and keep nn-step rewriting decidable. Variable containment is in general not implied by FV β(t)_{\beta} (t)\supseteq FVβ(u)_{\beta}(u). We give a decision procedure for the variable containment problem of the second-order fragment of λ\lambda^{\to}. For full λ\lambda^{\to} we show the equivalence of variable containment to an open problem in the theory of PCF; this equivalence also shows that the problem is decidable in the third-order case

    Can a Turing Player identify itself?

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    We show that the problem of whether two Turing Machines are functionally equivalent is undecidable and explain why this is significant for the theory of repeated play and evolution

    Verifying Recursive Active Documents with Positive Data Tree Rewriting

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    This paper proposes a data tree-rewriting framework for modeling evolving documents. The framework is close to Guarded Active XML, a platform used for handling XML repositories evolving through web services. We focus on automatic verification of properties of evolving documents that can contain data from an infinite domain. We establish the boundaries of decidability, and show that verification of a {\em positive} fragment that can handle recursive service calls is decidable. We also consider bounded model-checking in our data tree-rewriting framework and show that it is \nexptime-complete

    Reachability Analysis of Communicating Pushdown Systems

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    The reachability analysis of recursive programs that communicate asynchronously over reliable FIFO channels calls for restrictions to ensure decidability. Our first result characterizes communication topologies with a decidable reachability problem restricted to eager runs (i.e., runs where messages are either received immediately after being sent, or never received). The problem is EXPTIME-complete in the decidable case. The second result is a doubly exponential time algorithm for bounded context analysis in this setting, together with a matching lower bound. Both results extend and improve previous work from La Torre et al

    The Algebraic Intersection Type Unification Problem

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    The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type unification problem is decidable. We give the first nontrivial lower bound for the problem by showing (our main result) that it is exponential time hard. Furthermore, we show that this holds even under rank 1 solutions (substitutions whose codomains are restricted to contain rank 1 types). In addition, we provide a fixed-parameter intractability result for intersection type matching (one-sided unification), which is known to be NP-complete. We place the algebraic intersection type unification problem in the context of unification theory. The equational theory of intersection types can be presented as an algebraic theory with an ACI (associative, commutative, and idempotent) operator (intersection type) combined with distributivity properties with respect to a second operator (function type). Although the problem is algebraically natural and interesting, it appears to occupy a hitherto unstudied place in the theory of unification, and our investigation of the problem suggests that new methods are required to understand the problem. Thus, for the lower bound proof, we were not able to reduce from known results in ACI-unification theory and use game-theoretic methods for two-player tiling games

    An Introduction to Mechanized Reasoning

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    Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics journals. We introduce mechanized reasoning to economists in three ways. First, we introduce mechanized reasoning in general, describing both the techniques and their successful applications. Second, we explain how mechanized reasoning has been applied to economic problems, concentrating on the two domains that have attracted the most attention: social choice theory and auction theory. Finally, we present a detailed example of mechanized reasoning in practice by means of a proof of Vickrey's familiar theorem on second-price auctions

    Two-variable Logic with Counting and a Linear Order

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    We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C2) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NEXPTIME-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata

    Can A Turing Player Identify Itself?

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    We show that the problem of whether two Turing Machines are functionally equivalent is undecidable and explain why this is significant for the theory of repeated play and evolution.Economic Theory
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