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Binary Function Symbol

The Experts below are selected from a list of 51 Experts worldwide ranked by ideXlab platform

Mateu Villaret – 1st expert on this subject based on the ideXlab platform

  • Simplifying the signature in second-order unification
    Applicable Algebra in Engineering Communication and Computing, 2009
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all Symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one Binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one Binary Function Symbol and constants. The reduction is based on partially currying the equations by using the Binary Function Symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.

  • RTA – Currying Second-Order Unification Problems
    Rewriting Techniques and Applications, 2002
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    The Curry form of a term, like f(a, b), allows us to write it, using just a single Binary Function Symbol, as @(@(f, a), b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform second-order terms into first-order terms, but we have to add beta-reduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one Binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one Binary Function Symbol.We also discuss about the difficulties ofapplying the same ideas to third or higher order unification.

  • Currying second-order unification problems
    Lecture Notes in Computer Science, 2002
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    The Curry form of a term, like f(a,b), allows us to write it, using just a single Binary Function Symbol, as @(@(f,a),b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform second-order terms into first-order terms, but we have to add betareduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one Binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one Binary Function Symbol. We also discuss about the difficulties of applying the same ideas to third or higher order unification.

Jordi Levy – 2nd expert on this subject based on the ideXlab platform

  • Simplifying the signature in second-order unification
    Applicable Algebra in Engineering Communication and Computing, 2009
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all Symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one Binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one Binary Function Symbol and constants. The reduction is based on partially currying the equations by using the Binary Function Symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.

  • RTA – Currying Second-Order Unification Problems
    Rewriting Techniques and Applications, 2002
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    The Curry form of a term, like f(a, b), allows us to write it, using just a single Binary Function Symbol, as @(@(f, a), b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform second-order terms into first-order terms, but we have to add beta-reduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one Binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one Binary Function Symbol.We also discuss about the difficulties ofapplying the same ideas to third or higher order unification.

  • Currying second-order unification problems
    Lecture Notes in Computer Science, 2002
    Co-Authors: Jordi Levy, Mateu Villaret

    Abstract:

    The Curry form of a term, like f(a,b), allows us to write it, using just a single Binary Function Symbol, as @(@(f,a),b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform second-order terms into first-order terms, but we have to add betareduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one Binary Function Symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one Binary Function Symbol. We also discuss about the difficulties of applying the same ideas to third or higher order unification.

Mateu Villaret Auselle – 3rd expert on this subject based on the ideXlab platform

  • On some variants of second-order unification
    , 2004
    Co-Authors: Mateu Villaret Auselle

    Abstract:

    In this thesis we present several results about Second-Order
    Unification. It is well known that the Second-Order Unification
    Problem is in general undecidable; the frontier between its decidable and undecidable subclasses is thin and it still has not been completely defined.
    Our purpose is to shed some light on the Second-Order
    Unification problem and study some of its variants. We have mainly focused our attention on Context Unification and Linear Second-Order
    Unification. Roughly speaking, these problems are variants of
    Second-Order Unification where second-order variables are required to be linear. Context Unification was defined more than ten years ago and its decidability has been an open question since then. Here we make relevant contributions to the study of this question.
    The first result that we present is a simplification on these problems thanks to “curryfication”. We show that the Context
    Unification problem can be NP-reduced to the Context Unification problem where, apart from variables, just a single Binary Function Symbol, and first-order constants, are used. We also show that a similar result also holds for Second-Order Unification.
    The main result of this thesis is the definition of a non-trivial sufficient and necessary condition on the unifiers, for the decidability of Context Unification. The condition is called rank-bound conjecture in order to enforce our belief about its truthness. It lies on a non-trivial measure of terms, the rank, and claims that, whenever an instance of the Context
    Unification problem is satisfiable, there exists a unifier with a rank not exceeding a certain bound depending on the size of the problem. Under the assumption of this conjecture, we give a reduction of the satisfiability problem for Context Unification to the (decidable) satisfiability problem of Word Unification with regular constraints.
    Finally, in the same spirit of the extension of Word Unification with reg