Results 1  10
of
28
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.
The Syntax Definition Formalism SDF  Reference Manual
, 2001
"... SDF is a formalism for the definition of syntax which is comparable to BNF in some respect, but has a wider scope in that it also covers the definition... ..."
Abstract

Cited by 166 (28 self)
 Add to MetaCart
SDF is a formalism for the definition of syntax which is comparable to BNF in some respect, but has a wider scope in that it also covers the definition...
Natural termination
 Theoretical Computer Science
"... Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 ..."
Abstract

Cited by 84 (11 self)
 Add to MetaCart
(Show Context)
Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1
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.
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.
Termination of Term Rewriting
, 2000
"... Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicograp ..."
Abstract

Cited by 34 (7 self)
 Add to MetaCart
Contents 1 Introduction 2 2 Semantical methods 3 2.1 Wellfounded monotone algebras . . . . . . . . . . . . . . . . . . . . 3 2.2 Polynomial interpretations . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Polynomial interpretations modulo AC . . . . . . . . . . . . . . . . . 13 2.4 Lexicographic combinations . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 A hierarchy of termination 17 3.1 Simple termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Total termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Syntactical methods 25 4.1 Recursive path order . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Justi cation of recursive path order . . . . . . . . . . . . . . . . . . . 30 4.3 Extensions of recursive path order . . . . . . . . . . . . . . . . . . . 36 4.
Confluence by Decreasing Diagrams
 Theoretical Computer Science
, 1994
"... We present a confluence criterion, local decreasingness, for abstract reduction systems. This criterion is shown to be a considerable generalisation of several wellknown confluence criteria. 1 Introduction An abstract reduction system is a set of objects equipped with some binary `reduction&apos ..."
Abstract

Cited by 24 (6 self)
 Add to MetaCart
(Show Context)
We present a confluence criterion, local decreasingness, for abstract reduction systems. This criterion is shown to be a considerable generalisation of several wellknown confluence criteria. 1 Introduction An abstract reduction system is a set of objects equipped with some binary `reduction' relations. Because they have so little structure, abstract reduction systems can be viewed as abstractions of several kinds of rewriting such as string rewriting, term rewriting and graph rewriting. In the case of term rewriting, the objects model terms and the reduction relations model (nondeterministic) computations. A desirable property in computing is that results of computations are unique (if they exist). In the case that whenever we have two `diverging' computations starting from the same term, a common result can be reached by `converging' computations (the socalled confluence or ChurchRosser property), uniqueness is guaranteed. a c d a b d Confluence In this paper we present a ...
Liveness in Rewriting
 IN PROC. 14TH RTA, LNCS 2706
, 2002
"... In this paper, we show how the problem of verifying liveness properties is related to termination of term rewrite systems (TRSs). We formalize liveness in the framework of rewriting and present a sound and complete transformation to transform liveness problems into TRSs. Then the transformed TRS ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
In this paper, we show how the problem of verifying liveness properties is related to termination of term rewrite systems (TRSs). We formalize liveness in the framework of rewriting and present a sound and complete transformation to transform liveness problems into TRSs. Then the transformed TRS terminates if and only if the original liveness property holds. This shows that liveness and termination are essentially equivalent. To apply our approach in practice, we introduce a simpler sound transformation which only satis es the `only if'part. By re ning existing techniques for proving termination of TRSs we show how liveness properties can be veri ed automatically. As examples, we prove a liveness property of a waiting line protocol for a network of processes and a liveness property of a protocol on a ring of processes.
A Parsing Algorithm for ContextSensitive Graph Grammars
, 1995
"... Sentences of visual languages may often be regarded as assemblies of pictorial objects like "circles", "arrows" or "strings" with spatial relations like "above" or "contains" between them, i.e. their underlying structure is a kind of directed graph. ..."
Abstract

Cited by 21 (2 self)
 Add to MetaCart
Sentences of visual languages may often be regarded as assemblies of pictorial objects like "circles", "arrows" or "strings" with spatial relations like "above" or "contains" between them, i.e. their underlying structure is a kind of directed graph. Therefore, graph grammars are a natural means for defining the syntax of visual languages. Their main drawback until now is the lack of general enough and efficiently working parsing algorithms. All published graph grammar or more general visual language parsing algorithms are only able to deal with contextfree graph grammars, where the lefthand side consists of a single nonterminal vertex only. This makes syntax definitions of visual languages hard to read, prohibits the use of complex pattern matching, and disallows graphgrammars which specify transformation processes. We have developed the first parsing algorithm for contextsensitive graph grammars which allows left and rightsides of productions to be almost arbitrary graphs....