Results 1  10
of
53
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
Abstract

Cited by 47 (10 self)
 Add to MetaCart
(Show Context)
In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.
Proving and Disproving Termination of HigherOrder Functions
 IN: PROC. 5TH FROCOS
, 2005
"... The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higherorder functions using dependency pairs. To this end, the dependency ..."
Abstract

Cited by 46 (19 self)
 Add to MetaCart
(Show Context)
The dependency pair technique is a powerful modular method for automated termination proofs of term rewrite systems (TRSs). We present two important extensions of this technique: First, we show how to prove termination of higherorder functions using dependency pairs. To this end, the dependency pair technique is extended to handle (untyped) applicative TRSs. Second, we introduce a method to prove nontermination with dependency pairs, while up to now dependency pairs were only used to verify termination. Our results lead to a framework for combining termination and nontermination techniques for firstand higherorder functions in a very flexible way. We implemented and evaluated our results in the automated termination prover AProVE.
Definitions by Rewriting in the Calculus of Constructions
, 2001
"... The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. ..."
Abstract

Cited by 44 (6 self)
 Add to MetaCart
The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules.
Complete Monotonic Semantic Path Orderings
 In Proc. 17th CADE, LNAI 1831
, 2000
"... Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination ..."
Abstract

Cited by 31 (8 self)
 Add to MetaCart
Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and generalpurpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several nontrivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO. 1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest po...
HigherOrder Rewriting
 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051
, 1999
"... This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag. ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.
An extension of dependency pair method for proving termination of higherorder rewrite systems
 IEICE Trans. on Information and Systems
, 2001
"... Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the t ..."
Abstract

Cited by 22 (2 self)
 Add to MetaCart
(Show Context)
Abstract. This paper explores how to extend the dependency pair technique for proving termination of higherorder rewrite systems. In the first order case, the termination of term rewriting systems are proved by showing the nonexistence of an infinite Rchain of the dependency pairs. However, the termination and the nonexistence of an infinite Rchain do not coincide in the higherorder case. We introduce a new notion of dependency forest that characterize infinite reductions and infinite Rchains, and show that the termination property of higherorder rewrite systems R can be checked by showing the nonexistence of an infinite Rchain, if R is strongly linear or nonnested. 1
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 ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
(Show Context)
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
Termination and Reduction Checking in the Logical Framework
 IN WORKSHOP OF AUTOMATION OF PROOFS BY MATHEMATICAL INDUCTION
, 2000
"... ..."
Termination and Reduction Checking for HigherOrder Logic Programs
 In First International Joint Conference on Automated Reasoning (IJCAR
"... . In this paper, we present a syntaxdirected termination and reduction checker for higherorder logic programs. The reduction checker verifies parametric higherorder subterm orderings describing input and output relations. These reduction orderings are exploited during termination checking to ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
. In this paper, we present a syntaxdirected termination and reduction checker for higherorder logic programs. The reduction checker verifies parametric higherorder subterm orderings describing input and output relations. These reduction orderings are exploited during termination checking to infer that a specified termination order holds. To reason about parametric higherorder subterm orderings, we introduce a deductive system as a logical foundation for proving termination. This allows the study of prooftheoretical properties, such as consistency, local soundness and completeness and decidability. We concentrate here on proving consistency of the presented inference system. The termination and reduction checker are implemented as part of the Twelf system and enables us to verify proofs by complete induction. 1 Introduction One of the central problems in verifying specifications and checking proofs about them is the need to prove termination. Several automated methods...