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The HigherOrder Recursive Path Ordering
 FOURTEENTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1999
"... This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by adapting the recursive path ordering definition to terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. The obtained ordering is wellfoun ..."
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Cited by 58 (12 self)
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This paper extends the termination proof techniques based on reduction orderings to a higherorder setting, by adapting the recursive path ordering definition to terms of a typed lambdacalculus generated by a signature of polymorphic higherorder function symbols. The obtained ordering is wellfounded, compatible with fireductions and with polymorphic typing, monotonic with respect to the function symbols, and stable under substitution. It can therefore be used to prove the strong normalizationproperty of higherorder calculi in which constants can be defined by higherorder rewrite rules. For example, the polymorphic version of Gödel's recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Several other nontrivial examples are given which examplify the expressive power of the ordering.
On Computational Interpretations of the Modal Logic S4 IIIa. Termination, Confluence, Conservativity of λevQ
 INSTITUT FUR LOGIK, KOMPLEXITAT UND DEDUKTIONSSYSTEME, UNIVERSITAT
, 1996
"... A language of constructions for minimal logic is the calculus, where cutelimination is encoded as fireduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cutelimination for the corresponding sequent system. It turns o ..."
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Cited by 8 (4 self)
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A language of constructions for minimal logic is the calculus, where cutelimination is encoded as fireduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cutelimination for the corresponding sequent system. It turns out that a natural interpretation of the latter constructions is a calculus extended by an idealized version of Lisp's eval and quote constructs. In this Part IIIa, we examine the termination and confluence properties of the evQ and evQ H calculi. Most results are negative: the typed calculi do not terminate, the subsystems \Sigma and \Sigma H that propagate substitutions, quotations and evaluations downwards do not terminate either in the untyped case, and the untyped evQ H calculus is not confluent. However, the typed versions of \Sigma and \Sigma H do terminate, so the typed evQcalculus is confluent. It follows that the typed evQcalculus is a conservative extension of the typed S4cal...
An LPObased Termination Ordering for HigherOrder Terms without lambdaabstraction
 In Proc. TPHOLs ’98, LNCS 1479
, 1998
"... . We present a new precedencebased termination ordering for (polymorphic) higherorder terms without abstraction. The ordering has been designed to strictly generalize the lexicographic path ordering (on firstorder terms). It is relatively simple, but can be used to prove termination of many high ..."
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Cited by 2 (0 self)
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. We present a new precedencebased termination ordering for (polymorphic) higherorder terms without abstraction. The ordering has been designed to strictly generalize the lexicographic path ordering (on firstorder terms). It is relatively simple, but can be used to prove termination of many higherorder rewrite systems, especially those corresponding to typical functional programs. We establish the relevant properties of the ordering, include a number of examples, and also discuss certain limitations of the ordering and possible extensions. 1 Introduction Specification and interactive reasoning systems, such as HOL [10], Isabelle [12], or PVS [11], typically employ some expressive, higherorder logic as their basis specification formalism. Rewrite techniques, in this context, provide a computational mechanism for the simplification or evaluation of expressions and form an essential part of the (equational) reasoning component of such systems. Termination is one of the key properti...
An Adaptation of Paramodulation and RUEResolution to HigherOrder Logic
, 1998
"... This techreport presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and a calculus ERUE adapting FO RUEResolution [Dig79] to HO logic (based on Church's simply typed ..."
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This techreport presents two approaches to primitive equality treatment in higherorder (HO) automated theorem proving: a calculus EP adapting traditional firstorder (FO) paramodulation [RW69] , and a calculus ERUE adapting FO RUEResolution [Dig79] to HO logic (based on Church's simply typed calculus). EP and ERUE extend the extensional HO resolution approach ER [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by ER that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable differencereducing character the HO paramodulation approach looses its pure termrewriting character. On the other hand examples demonstrate that the extensionality rules harmonise rather well with the differencereducing HO RUEresolution idea.
To My Parents
"... Written under the supervision of Professor JeanPierre Jouannaud and Professor Jerzy Tiuryn ..."
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Written under the supervision of Professor JeanPierre Jouannaud and Professor Jerzy Tiuryn
Specification and Proof in Membership Equational Logic1
"... Abstract: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic foundation on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techni ..."
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Abstract: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic foundation on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been efficiently implemented. This effort started in the late seventies, led by the ADJ group, who promoted equational logic and universal algebra as the semantic basis of program specification languages. An important later milestone was the work around ordersorted algebras and the OBJ family of languages developed at SRIInternational in the eighties. This effort has been substantially advanced in the midnineties with the development of Maude, a language based on membership equational logic. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and sort membership assertions, and its sentences are Horn clauses. It extends in a conservative way both (a version of) ordersorted equational logic and partial algebra approaches, while Horn logic with equality can be very easily encoded.