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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 secondorder unification
Applicable Algebra in Engineering Communication and Computing, 2009CoAuthors: Jordi Levy, Mateu VillaretAbstract:SecondOrder 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 NPcomplete, whereas it is enough to have just one Binary constant to lose decidability. In this work we reduce SecondOrder Unification to SecondOrder 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 SecondOrder Unification and some of its variants, like Context Unification and Bounded SecondOrder Unification.

RTA – Currying SecondOrder Unification Problems
Rewriting Techniques and Applications, 2002CoAuthors: Jordi Levy, Mateu VillaretAbstract: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 secondorder unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform secondorder terms into firstorder 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 secondorder unification to secondorder 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 secondorder unification problems
Lecture Notes in Computer Science, 2002CoAuthors: Jordi Levy, Mateu VillaretAbstract: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 secondorder unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform secondorder terms into firstorder 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 secondorder unification to secondorder 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 secondorder unification
Applicable Algebra in Engineering Communication and Computing, 2009CoAuthors: Jordi Levy, Mateu VillaretAbstract:SecondOrder 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 NPcomplete, whereas it is enough to have just one Binary constant to lose decidability. In this work we reduce SecondOrder Unification to SecondOrder 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 SecondOrder Unification and some of its variants, like Context Unification and Bounded SecondOrder Unification.

RTA – Currying SecondOrder Unification Problems
Rewriting Techniques and Applications, 2002CoAuthors: Jordi Levy, Mateu VillaretAbstract: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 secondorder unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform secondorder terms into firstorder 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 secondorder unification to secondorder 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 secondorder unification problems
Lecture Notes in Computer Science, 2002CoAuthors: Jordi Levy, Mateu VillaretAbstract: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 secondorder unification, and conclude that one Binary Symbol is enough. By currying variable applications, like X(a), as @(X,a), we can transform secondorder terms into firstorder 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 secondorder unification to secondorder 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 secondorder unification
, 2004CoAuthors: Mateu Villaret AuselleAbstract:In this thesis we present several results about SecondOrder
Unification. It is well known that the SecondOrder 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 SecondOrder
Unification problem and study some of its variants. We have mainly focused our attention on Context Unification and Linear SecondOrder
Unification. Roughly speaking, these problems are variants of
SecondOrder Unification where secondorder 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 NPreduced to the Context Unification problem where, apart from variables, just a single Binary Function Symbol, and firstorder constants, are used. We also show that a similar result also holds for SecondOrder Unification.
The main result of this thesis is the definition of a nontrivial sufficient and necessary condition on the unifiers, for the decidability of Context Unification. The condition is called rankbound conjecture in order to enforce our belief about its truthness. It lies on a nontrivial 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