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15
Combinatory Reduction Systems: introduction and survey
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simpl ..."
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Cited by 84 (9 self)
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Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and leftlinear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the wellknown confluence proof for λcalculus by Tait and MartinLof. There is a wellknown connection between the para...
Programming in an Integrated Functional and Logic Language
, 1999
"... Escher is a generalpurpose, declarative programming language that integrates the best features of both functional and logic programming languages. It has types and modules, higherorder and metaprogramming facilities, concurrency, and declarative input/output. The main design aim is to combine in ..."
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Cited by 65 (14 self)
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Escher is a generalpurpose, declarative programming language that integrates the best features of both functional and logic programming languages. It has types and modules, higherorder and metaprogramming facilities, concurrency, and declarative input/output. The main design aim is to combine in a practical and comprehensive way the best ideas of existing functional and logic languages, such as Haskell and Godel. In fact, Escher uses the Haskell syntax and is most straightforwardly understood as an extension of Haskell. Consequently, this paper discusses Escher from this perspective. It provides an introduction to the Escher language, concentrating largely on the issue of programming style and the Escher programming idioms not provided by Haskell. Also the extra mechanisms needed to support these idioms are discussed.
Third Order Matching is Decidable
 Annals of Pure and Applied Logic
, 1999
"... The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus, i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. The decidability of this problem is still open. We prove the decidability of ..."
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Cited by 48 (0 self)
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The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus, i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. The decidability of this problem is still open. We prove the decidability of the particular case in which the variables occurring in the problem are at most third order. Introduction The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed calculus i.e. to solve the equation a = b where a and b are simply typed terms and b is ground. Pattern matching algorithms are used to check if a proposition can be deduced from another by elimination of universal quantifiers or by introduction of existential quantifiers. In automated theorem proving, elimination of universal quantifiers and introduction of existential quantifiers are mixed and full unification is required, but in proofchecking and semiaut...
Hierarchical Contextual Reasoning
, 2003
"... VII Zusammenfassung IX Extended Abstract XI Acknowledgements XIII I ..."
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Cited by 18 (9 self)
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VII Zusammenfassung IX Extended Abstract XI Acknowledgements XIII I
Termination and confluence of higherorder rewrite systems
 In Proc. RTA ’00, volume 1833 of LNCS
, 2000
"... Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last on ..."
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Cited by 14 (8 self)
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Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higherorder patternmatching mechanism, resulting in simplytyped CRSs. Then, we show how the termination criterion developed for IDTSs with firstorder patternmatching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higherorder patternmatching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow’s higherorder critical pair analysis technique for proving local confluence can be applied to IDTSs. 1
Equality and Extensionality in Automated HigherOrder Theorem Proving
, 1999
"... Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existenc ..."
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Cited by 13 (10 self)
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Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 Hintikka Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Primitive Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Model Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Extensional HigherOrder Resolution: ER 42 4.1 A Review of HORES and ER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Lifting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5 Theorem Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Extensional HigherOrder Paramodulation: EP 57 5.1 A Naive and Incomplete Adaptation...
The "Hardest" Natural Decidable Theory
 12th Annual IEEE Symp. on Logic in Computer Science (LICS'97)', IEEE
, 1997
"... We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linea ..."
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Cited by 10 (4 self)
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We prove that any decision procedure for a modest fragment of L. Henkin's theory of pure propositional types [7, 12, 15, 11] requires time exceeding a tower of 2's of height exponential in the length of input. Until now the highest known lower bounds for natural decidable theories were at most linearly high towers of 2's and since midseventies it was an open problem whether natural decidable theories requiring more than that exist [12, 2]. We give the affirmative answer. As an application of this today's strongest lower bound we improve known and settle new lower bounds for several problems in the simply typed lambda calculus. 1. Introduction In his survey paper [12] A. Meyer mentioned (p. 479), as a curious empirical observation, that all known natural decidable nonelementary problems require at most (upper bound) F (1; n) = exp 1 (n) = 2 2 \Delta \Delta \Delta 2 oe n Turing machine steps on inputs of length n to decide 1 . Until now the highest known lower bounds for natu...
On statman's finite completeness theorem
, 1992
"... Abstract We give a complete selfcontained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the *definability conjecture implies the higherorder matching conjecture. ..."
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Cited by 4 (0 self)
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Abstract We give a complete selfcontained proof of Statman's finite completeness theorem and of a corollary of this theorem stating that the *definability conjecture implies the higherorder matching conjecture.
Linear Interpolation for the Higher Order Matching Problem
, 1996
"... We present here a particular case of the higher order matching problem  the linear interpolation problem. The problem consists in solving a collection of higher order matching equations of the shape xM 1 : : : M k = N , where x is the only unknown quantity. We prove recursive equivalence of t ..."
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Cited by 3 (0 self)
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We present here a particular case of the higher order matching problem  the linear interpolation problem. The problem consists in solving a collection of higher order matching equations of the shape xM 1 : : : M k = N , where x is the only unknown quantity. We prove recursive equivalence of the higher order matching problem and the linear interpolation problem. We also investigate decidability of a special case of the fifth order linear interpolation problem. The restriction we consider consists in that arguments of variables from the main abstraction in terms M 1 ; : : : ; M k cannot contain variables from the main abstraction. 1 Preface The higherorder matching problem for simply typed calculus has been considered since 1976 ([5]). There were proposed several partial solutions of the problem (second order matching  [4]; correct, but without a proof of completeness, algorithm  [7]; third order matching  [3]; fourth order matching  [6]). In this paper, we pr...
A Lost Proof
 IN TPHOLS: WORK IN PROGRESS PAPERS
, 2001
"... We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which t ..."
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Cited by 2 (1 self)
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We reinvestigate a proof example presented by George Boolos which perspicuously illustrates Gödel’s argument for the potentially drastic increase of prooflengths in formal systems when carrying through the argument at too low a level. More concretely, restricting the order of the logic in which the proof is carried through to the order of the logic in which the problem is formulated in the first place can result in unmanageable long proofs, although there are short proofs in a logic of higher order. Our motivation in this paper is of practical nature and its aim is to sketch the implications of this example to current technology in automated theorem proving, to point to related questions about the foundational character of type theory (without explicit comprehension axioms) for mathematics, and to work out some challenging aspects with regard to the automation of this proof – which, as we belief, nicely illustrates the discrepancy between the creativity and intuition required in mathematics and the limitations of state of the art theorem provers.