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On Termination and Confluence Properties of Disjoint and ConstructorSharing Conditional Rewrite Systems
 Theoretical Computer Science
, 1996
"... We investigate the modularity behaviour of termination and confluence properties of (join) conditional term rewriting systems. We give counterexamples showing that the properties weak termination, weak innermost termination and (strong) innermost termination are not preserved in general under signat ..."
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We investigate the modularity behaviour of termination and confluence properties of (join) conditional term rewriting systems. We give counterexamples showing that the properties weak termination, weak innermost termination and (strong) innermost termination are not preserved in general under signature extensions. Then we develop sufficient conditions for the preservation of these properties under signature extensions, and more generally, for their modularity in the disjoint union case. This leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems. Finally, combining our analysis with recent related results on the preservation of semicompleteness, we show how to cover the (nondisjoint) case of combined conditional rewrite systems with shared constructors, too. 1 Introduction, Motivation and Overview Starting with the seminal work of Toyama [31] the investigation of preservation properties of term rewriting systems (TRSs...
Inductive Theorem Proving for Design Specifications
 J. Symbolic Computation
, 1997
"... We present a number of new results on inductive theorem proving for design specifications based on Horn logic with equality. Induction is explicit here because induction orderings are supposed to be part of the specification. We show how the automatic support for program verification is enhanced i ..."
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Cited by 12 (8 self)
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We present a number of new results on inductive theorem proving for design specifications based on Horn logic with equality. Induction is explicit here because induction orderings are supposed to be part of the specification. We show how the automatic support for program verification is enhanced if the specification satisfies a bunch of rewrite properties, summarized under the notion of canonicity. The enhancement is due to inference rules and corresponding strategies whose soundness is implied by the specification's canonicity. The second main result of the paper provides a method for proving canonicity by using the same rules, which are applied in proofs of conjectures about the specification and the functionallogic programs it contains. Contents 1 Introduction 2 1.1 Expander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Proof by term rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
Rewrite, Rewrite, Rewrite, Rewrite, Rewrite, ...
, 1989
"... .We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories ..."
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Cited by 9 (1 self)
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.We study properties of rewrite systems that are not necessarily terminating, but allow instead for trans#nite derivations that have a limit. In particular, we give conditions for the existence of a limit and for its uniqueness and relate the operational and algebraic semantics of in#nitary theories. We also consider su#cient completeness of hierarchical systems. Is there no limit? Job 16:3 1. Introduction Rewrite systems are sets of directed equations used to compute by repeatedly replacing equal terms in a given formula, as long as possible. For one approach to their use in computing, see #23#. The theory of rewriting is an outgrowth of the study of the lambda calculus and combinatory logic, and # Preliminary versions #6, 7# of ideas in this paper were presented at the Sixteenth ACM Symposium on Principles of Programming Languages, Austin, TX #January 1989# and at the Sixteenth EATCS International Colloquium on Automata, Languages and Programming, Stresa, Italy #July 1989#. ...
Specifying Visual Languages with Conditional Set Rewrite Systems
 in 1993 IEEE Symposium on Visual Languages
, 1993
"... We propose Conditional Set Rewriting as a general mechanism for describing the syntax of multidimensional languages. We compare the approach with other existing methods, and give a number of examples that illustrate its strengths. 1 Introduction This paper introduces the notion of a Conditional Se ..."
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We propose Conditional Set Rewriting as a general mechanism for describing the syntax of multidimensional languages. We compare the approach with other existing methods, and give a number of examples that illustrate its strengths. 1 Introduction This paper introduces the notion of a Conditional Set Rewrite System (CSRS). Conditional Set Rewrite Systems are a generalization of Conditional Term Rewrite Systems [10]. Conditional Term Rewrite Systems deal with rewriting a single term, whereas Conditional Set Rewrite systems deal with rewriting a set of terms. Each conditional set rewrite rule specifies how to replace a subset of the set of terms by another set. We suggest two uses for CSRS: they can be used to describe the set of all legal phrases (pictures) of a multidimensional (visual) language, and they can be used to translate a phrase from one language (e.g. a visual one) into another language (e.g. a textual one). The ability of CSRS to translate visual programs into textual ones ...
Logicality of Conditional Rewrite Systems
, 2000
"... A conditional term rewriting system is called logical if it has the same logical strength as the underlying conditional equational system. In this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term ..."
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A conditional term rewriting system is called logical if it has the same logical strength as the underlying conditional equational system. In this paper we summarize known logicality results and we present new sufficient conditions for logicality of the important class of oriented conditional term rewriting systems.
Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
On the confluence of lambdacalculus with conditional rewriting
, 2009
"... The confluence of untyped λcalculus with unconditional rewriting is now well understood. In this paper, we investigate the confluence of λcalculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Müller ..."
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The confluence of untyped λcalculus with unconditional rewriting is now well understood. In this paper, we investigate the confluence of λcalculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Müller and Dougherty for unconditional rewriting. Two cases are considered, whether betareduction is allowed or not in the evaluation of conditions. Moreover, Dougherty’s result is improved from the assumption of strongly normalizing βreduction to weakly normalizing βreduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional
Lifting of Lnarrowing derivations
, 1996
"... If conditional rewriterules are restricted to the form P = ⇒ f(x1,..., xn) → t where P is a finite set of equations, f is any function symbol, x1,..., xn are variables, and t is any term then the premise P contains in general variables which do not occur in the list x1,..., xn. The rule with premi ..."
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If conditional rewriterules are restricted to the form P = ⇒ f(x1,..., xn) → t where P is a finite set of equations, f is any function symbol, x1,..., xn are variables, and t is any term then the premise P contains in general variables which do not occur in the list x1,..., xn. The rule with premise P can be applied if P is satisfiable. Therefore, we need methods to solve P and narrowing must be combined with rewriting. But, narrowing becomes a special case, called Lnarrowing, closely related to lazynarrowing. Two lifting lemmas are shown which characterize the relationship of Lnarrowing derivations if the goals are modified by substitutions. From these lifting lemmas, soundness and completeness results can be concluded. 1