Results 1  10
of
127
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
Abstract

Cited by 821 (23 self)
 Add to MetaCart
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
Term Rewriting Systems
, 1992
"... Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstra ..."
Abstract

Cited by 610 (18 self)
 Add to MetaCart
Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems
Completion Without Failure
, 1989
"... We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational the ..."
Abstract

Cited by 140 (21 self)
 Add to MetaCart
We present an "unfailing" extension of the standard KnuthBendix completion procedure that is guaranteed to produce a desired canonical system, provided certain conditions are met. Weprove that this unfailing completion method is refutationally complete for theorem proving in equational theories. The method can also be applied to Horn clauses with equality, in which case it corresponds to positive unit resolution plus oriented paramodulation, with unrestricted simplification.
What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
Abstract

Cited by 56 (2 self)
 Add to MetaCart
This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
Solving Symbolic Ordering Constraints
, 1990
"... We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a ..."
Abstract

Cited by 50 (10 self)
 Add to MetaCart
We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a total precedence is decidable. Keywords: simplification orderings, ordered strategies, term algebras, constraint solving. 1. Introduction The first order theory of term algebras over a language (or alphabet) with no relational symbol (other than equality) has been shown to be decidable 1;2 . See also Refs 3 and 4. Introducing into the language a binary relational symbol interpreted as the subterm ordering makes the theory undecidable 5 . Venkataraman also shows in the latter paper that the purely existential fragment of the theory, i.e. the subset of sentences whose prenex form does not contain 8, is decidable. Venkataraman was concerned with some applications in functional programm...
Rewrite Techniques for Transitive Relations
 IN PROC., 9TH IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1994
"... We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and al ..."
Abstract

Cited by 39 (6 self)
 Add to MetaCart
We propose inference systems for dealing with transitive relations in the context of resolutiontype theorem proving. These inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods. We establish their refutational completeness and also prove their compatibility with the usual simplification techniques used in rewritebased theorem provers. A key to the practicality of chaining techniques is the extent to which socalled variable chainings can be restricted. We demonstrate that rewrite techniques considerably restrict variable chaining, though we also show that they cannot be completely avoided for transitive relations in general. If the given relation satisfies additional properties, such as symmetry, further restrictions are possible. In particular, we discuss (partial) equivalence relations and congruence relations.
Ordered Chaining Calculi for FirstOrder Theories of Transitive Relations
 Journal of the ACM
, 1998
"... this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994). ..."
Abstract

Cited by 35 (4 self)
 Add to MetaCart
this paper have been presented at the 12th International Conference on Automated Deduction (Nancy, France, June/July 1994) and the 9th IEEE Symposium on Logic in Computer Science (Paris, France, July 1994).