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13
On the Undecidability of SecondOrder Unification
 INFORMATION AND COMPUTATION
, 2000
"... ... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most t ..."
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Cited by 33 (16 self)
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... this paper, and it is the starting point for proving some novel results about the undecidability of secondorder unification presented in the rest of the paper. We prove that secondorder unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two secondorder variables; (2) there is only one secondorder variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all secondorder variables are ground terms of size <n, (ii) the arity of all secondorder variables is <n, (iii) the number of occurrences of secondorder variables is #5, (iv) there is either a single secondorder variable or there are two secondorder variables and no firstorder variables.
Birewrite systems
, 1996
"... In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations ..."
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Cited by 29 (9 self)
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In this article we propose an extension of term rewriting techniques to automate the deduction in monotone preorder theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a birewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→
Rank 2 Type Systems and Recursive Definitions
, 1995
"... We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We int ..."
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Cited by 26 (1 self)
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We demonstrate an equivalence between the rank 2 fragments of the polymorphic lambda calculus (System F) and the intersection type discipline: exactly the same terms are typable in each system. An immediate consequence is that typability in the rank 2 intersection system is DEXPTIMEcomplete. We introduce a rank 2 system combining intersections and polymorphism, and prove that it types exactly the same terms as the other rank 2 systems. The combined system suggests a new rule for typing recursive definitions. The result is a rank 2 type system with decidable type inference that can type some interesting examples of polymorphic recursion. Finally,we discuss some applications of the type system in data representation optimizations such as unboxing and overloading.
Decidable and undecidable secondorder unification problems
 In Proceedings of the 9th Int. Conf. on Rewriting Techniques and Applications (RTA’98), volume 1379 of LNCS
, 1998
"... Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in ..."
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Cited by 15 (9 self)
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Abstract. There is a close relationship between word unification and secondorder unification. This similarity has been exploited for instance for proving decidability of monadic secondorder unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higherorder unification procedures) when variables are restricted to occur at most twice. Hence a wellknown open question was the decidability of secondorder unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid Eunification to secondorder unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid Eunification is decidable. This happens, for example, for onevariable problems where the variable occurs at most twice (because rigid Eunification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable secondorder unification problems with variable occurrence restrictions. 1
The Typed Polymorphic LabelSelective λCalculus
 IN PROC. ACM SYMPOSIUM ON PRINCIPLES OF PROGRAMMING LANGUAGES
, 1994
"... Formal calculi of record structures have recently been a focus of active research. However, scarcely anyone has studied formally the dual notioni.e., argumentpassing to functions by keywords, and its harmonization with currying. We have. Recently, we introduced the labelselective calculus, a ..."
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Cited by 8 (0 self)
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Formal calculi of record structures have recently been a focus of active research. However, scarcely anyone has studied formally the dual notioni.e., argumentpassing to functions by keywords, and its harmonization with currying. We have. Recently, we introduced the labelselective calculus, a conservative extension of calculus that uses a labeling of abstractions and applications to perform unordered currying. In other words, it enables some form of commutation between arguments. This improves program legibility, thanks to the presence of labels, and efficiency, thanks to argument commuting. In this paper, we propose a simply typed version of the calculus, then extend it to one with MLlike polymorphic types. For the latter calculus, we establish the existence of principal types and we give an algorithm to compute them. Thanks to the fact that labelselective calculus is a conservative extension of calculus by adding numeric labels to stand for argument positions, its...
Context unification and traversal equations
 In: Proc. of the 12th International Conference on Rewriting Techniques and Applications (RTA’01
, 2001
"... Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secon ..."
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Cited by 8 (7 self)
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Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing firstorder variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are secondorder variables that are restricted to be instantiated by linear terms (a linear term is a λexpression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rankbound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints. 1
A Unified Approach to Theory Reasoning
, 1992
"... Theory reasoning is a kind of twolevel reasoning in automated theorem proving, where the knowledge of a given domain or theory is separated and treated by special purpose inference rules. We define a classification for the various approaches for theory reasoning which is based on the syntactic con ..."
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Cited by 6 (1 self)
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Theory reasoning is a kind of twolevel reasoning in automated theorem proving, where the knowledge of a given domain or theory is separated and treated by special purpose inference rules. We define a classification for the various approaches for theory reasoning which is based on the syntactic concepts of literal level  term level  variable level. The main part is a review of theory extensions of common calculi (resolution, model elimination and a connection method). In order to see the relationships among these calculi, we define a supercalculus called theory consolution. Completeness of the various theory calculi is proven. Finally, due to its relevance in automated reasoning, we describe current ways of equality handling.
Generation as Deduction on Labelled Proof Nets
 Logical Aspects of Computational Linguistics, LACL’96, volume 1328 of Lecture Notes in Artificial Intelligence
"... . In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Int ..."
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Cited by 5 (1 self)
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. In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of firstorder matching under theory. Here we shall show how to use the same method of labelled proof nets to reduce the task of generating to the problem of higherorder matching. 1 Introduction Categorial grammar provides a mechanism for the analysis of linguistic expressions on the basis of lexicalism and the parsing as deduction paradigm ([17]). 3 In accordance with lexicalism each lexical entry of the language encapsulates all the information needed to analyse the lexical item, and the grammar itself only needs to know how to manage these resources. In the particular case of categorial grammar, a lexical categorisation is a formula, or type, constructed over some basic types by logical connectives; and the grammar constitutes the connectives' syntactic behaviour (i.e. the laws governing the connectives). Within the parsing as deduction paradigm the problem of analys...
HigherOrder Rigid EUnification
 5th International Conference on Logic Programming and Automated Reasoning', number 822 in `Lecture Notes in Artificial Intelligence
"... . Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bound ..."
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Cited by 1 (0 self)
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. Higherorder Eunification, i.e. the problem of finding substitutions that make two simply typed terms equal modulo fi or fij equivalence and a given equational theory, is undecidable. We propose to rigidify it, to get a resourcebounded decidable unification problem (with arbitrary high bounds), providing a complete higherorder Eunification procedure. The techniques are inspired from Gallier's rigid Eunification and from Dougherty and Johann's use of combinatory logic to solve higherorder Eunification problems. We improve their results by using general equational theories, and by defining optimizations such as higherorder rigid Epreunification, where flexible terms are used, gaining much efficiency, as in the nonequational case due to Huet. 1 Introduction Higherorder Eunification is the problem of finding complete sets of unifiers of two simply typed terms modulo fi or fijequivalence, and modulo an equational theory E . This problem has applications in higherorder a...
Sorted Unification And The Solution Of SemiLinear Membership Constraints
, 1992
"... This thesis describes a new representation for sortal constraints and a unification algorithm for the corresponding constrained terms. Variables range over sets of terms described by systems of set constraints , which can express limited intervariable dependencies. These sets of terms are more gene ..."
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Cited by 1 (1 self)
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This thesis describes a new representation for sortal constraints and a unification algorithm for the corresponding constrained terms. Variables range over sets of terms described by systems of set constraints , which can express limited intervariable dependencies. These sets of terms are more general than regular tree languages, but are still closed under intersection. The new unification algorithm shows sorted unification to be decidable for a broad class of sorted signatures, which we call semilinear , and, more generally, for sort theories with a least Herbrand model that can be represented using the new constraints. Even though the unification problem in question is NPhard, the generality of the algorithm may allow for particular efficient implementations for more restricted theories. This unification algorithm can be integrated into any of a variety of deductive systems, resulting in a hybrid substitutional reasoner. As an example, the soundness and completeness of a resolutio...