... this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each second-order variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size <n, (ii) the arity of all second-order variables is <n, (iii) the number of occurrences of second-order variables is #5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables.

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite 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 −−−→

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 DEXPTIME-complete. 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.

Abstract. There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order 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 E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification 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 second-order unification problems with variable occurrence restrictions. 1

Formal calculi of record structures have recently been a focus of active research. However, scarcely anyone has studied formally the dual notion---i.e., argument-passing to functions by keywords, and its harmonization with currying. We have. Recently, we introduced the label-selective -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 ML-like 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 label-selective -calculus is a conservative extension of -calculus by adding numeric labels to stand for argument positions, its...

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 first-order variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are second-order variables that are restricted to be instantiated by linear terms (a linear term is a λ-expression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rank-bound 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

. Theory reasoning is a kind of two-level 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 super-calculus 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. Contents 1 Introduction 2 2 The Tour: Literal level -- Term level -- Variable level 5 2.1 Literal Level Theory Reasoning : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Term Level Theory Reasoning : : : : : : : : : : : : : : : : : : : : : ...

. In the framework of labelled proof nets the task of parsing in categorial grammar can be reduced to the problem of first-order 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 higher-order 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...

. Higher-order E-unification, 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 resource-bounded decidable unification problem (with arbitrary high bounds), providing a complete higher-order E-unification procedure. The techniques are inspired from Gallier's rigid E-unification and from Dougherty and Johann's use of combinatory logic to solve higher-order E-unification problems. We improve their results by using general equational theories, and by defining optimizations such as higherorder rigid E-preunification, where flexible terms are used, gaining much efficiency, as in the non-equational case due to Huet. 1 Introduction Higher-order E-unification is the problem of finding complete sets of unifiers of two simply typed -terms modulo fi or fij-equivalence, and modulo an equational theory E . This problem has applications in higher-order a...

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 inter-variable 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 semi-linear , 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 NP-hard, 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...