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12
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 34 (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.
HigherOrder Unification via Combinators
 Theoretical Computer Science
, 1993
"... We present an algorithm for unification in the simply typed lambda calculus which enumerates complete sets of unifiers using a finitely branching search space. In fact, the types of terms may contain typevariables, so that a solution may involve typesubstitution as well as termsubstitution. the ..."
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Cited by 9 (1 self)
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We present an algorithm for unification in the simply typed lambda calculus which enumerates complete sets of unifiers using a finitely branching search space. In fact, the types of terms may contain typevariables, so that a solution may involve typesubstitution as well as termsubstitution. the problem is first translated into the problem of unification with respect to extensional equality in combinatory logic, and the algorithm is defined in terms of transformations on systems of combinatory terms. These transformations are based on a new method (itself based on systems) for deciding extensional equality between typed combinatory logic terms. 1 Introduction This paper develops a new algorithm for higherorder unification. A higherorder unification problem is specified by two terms F and G of the explicitly simply typed lambda calculus LC; a solution is a substitution oe such that oeF = fij oeG. We will always assume the extensionality axiom j in this paper. In fact we tre...
Sense and the Computation of Reference
 Linguistics and Philosophy
"... The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends ..."
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Cited by 7 (2 self)
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The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of this relation must be given in the form of a logic program. The resulting system does not only throw light on the properties of sense and their relation to computation, but also shows circular behaviour if some ingredients of the Liar Paradox are added. The connection is natural, as algorithms can be inherently circular and the Liar is explained as expressing one of those. Many ideas in the present paper are closely related to those in Moschovakis (1994), but receive a considerably lighter formalization. 1
Compositional Logic Programming
 In Proceedings of the JICSLP'96 postconference workshop: Multiparadigm logic programming, Report 9628. Technische Universitat
, 2000
"... Relational program derivation has gathered momentum over the last decade with the development of many specification logics. However, before such relational specifications can be executed in existing programming languages, they must be carefully phrased to respect the evaluation order of the langu ..."
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Relational program derivation has gathered momentum over the last decade with the development of many specification logics. However, before such relational specifications can be executed in existing programming languages, they must be carefully phrased to respect the evaluation order of the language. In turn, this requirement inhibits the rapid prototyping of specifications in a relational notation. The aim of this thesis is to bridge the gap between the methodology and practice of relational program derivation by realising a compositional style of logic programming that permits specifications to be phrased naturally and executed declaratively.
Towards the uniform implementation of declarative languages
, 1997
"... Current implementation techniques for functional languages differ considerably from those for logic languages. This complicates the development of flexible and efficient abstract machines that can be used for the compilation of declarative languages combining concepts of functional and logic program ..."
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Cited by 5 (0 self)
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Current implementation techniques for functional languages differ considerably from those for logic languages. This complicates the development of flexible and efficient abstract machines that can be used for the compilation of declarative languages combining concepts of functional and logic programming. We propose an abstract machine, called the JUMPmachine, which systematically integrates the operational concepts needed to implement the functional and logic programming paradigm. The use of a tagless representation for heap objects, which originates from the Spineless Tagless Gmachine, supports the integration of different concepts. In this paper, we provide a functional logic kernel language and show how to translate it into the abstract machine language of the JUMPmachine. Furthermore, we define the operational semantics of the machine language formally and discuss the mapping of the abstract machine to concrete machine architectures. We tested the approach by writing a compiler for the functional logic language GTML. The obtained performance results indicate that the proposed method allows to implement functional logic languages efficiently.
Unification in an Extensional Lambda Calculus with Ordered Function Sorts and Constant Overloading
, 1994
"... We develop an ordersorted higherorder calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higherorder ordered sort concept and constant overloading. Huet's wellknown techniques for unifying simply typed lambda ter ..."
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We develop an ordersorted higherorder calculus suitable for automatic theorem proving applications by extending the extensional simply typed lambda calculus with a higherorder ordered sort concept and constant overloading. Huet's wellknown techniques for unifying simply typed lambda terms are generalized to arrive at a complete transformationbased unification algorithm for this sorted calculus. Consideration of an ordersorted logic with functional base sorts and arbitrary term declarations was originally proposed by the second author in a 1991 paper; we give here a corrected calculus which supports constant rather than arbitrary term declarations, as well as a corrected unification algorithm, and prove in this setting results corresponding to those claimed there.
A Combinatorbased Ordersorted Higherorder Unification Algorithm
, 1993
"... This paper develops a sound and complete transformationbased algorithm for unification in an extensional ordersorted combinatory logic supporting constant overloading and a higherorder sort concept. Appropriate notions of ordersorted weak equality and extensionality  reflecting ordersorted f ..."
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Cited by 1 (0 self)
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This paper develops a sound and complete transformationbased algorithm for unification in an extensional ordersorted combinatory logic supporting constant overloading and a higherorder sort concept. Appropriate notions of ordersorted weak equality and extensionality  reflecting ordersorted fijequality in the corresponding lambda calculus given by Johann and Kohlhase  are defined, and the typed combinatorbased higherorder unification techniques of Dougherty are modified to accommodate unification with respect to the theory they generate. The algorithm presented here can thus be viewed as a combinatory logic counterpart to that of Johann and Kohlhase, as well as a refinement of that of Dougherty, and provides evidence that combinatory logic is wellsuited to serve as a framework for incorporating ordersorted higherorder reasoning into deduction systems aiming to capitalize on both the expressiveness of extensional higherorder logic and the efficiency of ordersorted calculi.
RECONSTRUCTION OF EXTENDED POLYNOMIALS FROM THE FINITE NUMBER OF EXAMPLES
"... Abstract The extended polynomials are considered the class of all functions definable in the simply typed *calculus with one basic type. The goal of the thesis was to decide, whether for every extended polynomial there exists a finite set of examples determining that polynomial, and to find an alg ..."
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Abstract The extended polynomials are considered the class of all functions definable in the simply typed *calculus with one basic type. The goal of the thesis was to decide, whether for every extended polynomial there exists a finite set of examples determining that polynomial, and to find an algorithm for constructing such a set for a given polynomial. It is proved that such a finite set exists for every extended polynomial and the cardinality of such a set depends only on the polynomial's arity. An algorithm constructing such a set is also presented.
Towards a Relational Programming Language
, 1995
"... Until recently, the art of deriving computer programs from specification has been performed equationally in a functional calculus. However, it has become evident that a relational calculus affords us a greater degree of expression and flexibility in both specification and proof since a relational ca ..."
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Until recently, the art of deriving computer programs from specification has been performed equationally in a functional calculus. However, it has become evident that a relational calculus affords us a greater degree of expression and flexibility in both specification and proof since a relational calculus naturally captures the notions of nondeterminism along with function converses. Unfortunately, no language exists in which we can directly execute relational expressions. Moreover, those candidate languages most likely to support an implementation of a relational calculus suffer from an inelegant or complicated evaluation model. This dissertation examines the suitability of these languages for supporting an implementation of relations and proposes a novel computational strategy for a relational language based on the category theory of relations. CONTENTS i Contents 1 Introduction 1 2 A Calculus of Relations 2 2.1 Basic Relational Operators \Delta \Delta \Delta \Delta \Delta \Del...
ThirdOrder Matching in lambda>Curry is Undecidable
, 1997
"... Given closed untyped terms x 1 : : : x k :s and t, which can be assigned some types oe 1 ! : : : ! oe k ! and respectively in the Currystyle systems of type assignment (essentially due to R. Hindley) ! Curry [Bar92], ! t [Mit96], TA [Hin97], it is undecidable whether there exist closed term ..."
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Given closed untyped terms x 1 : : : x k :s and t, which can be assigned some types oe 1 ! : : : ! oe k ! and respectively in the Currystyle systems of type assignment (essentially due to R. Hindley) ! Curry [Bar92], ! t [Mit96], TA [Hin97], it is undecidable whether there exist closed terms s 1 ; : : : ; s k of types oe 1 ; : : : ; oe k such that s[s 1 =x 1 ; \Delta \Delta \Delta ; s k =x k ] = fij t, even if the orders of oe i 's do not exceed 3. This undecidability result should be contrasted to the decidability of the thirdorder matching in the Churchstyle simply typed lambda calculus with a single constant base type [Dow94]. The proof is by reduction from the recursively inseparable sets of unsatisfiable and finitely satisfiable sentences of the firstorder theory of binary relation [Tra53, Vau60]. Keywords Typed lambda calculus, Higherorder matching problem, Decidability, Recursive inseparability, Firstorder theory of binary relation, Full type hierarchy Contents...