Termination of rewriting

Dershowitz, Nachum

doi:10.1016/s0747-7171(87)80022-6

Cited by 602 publications

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“…This can be shown by considering the multiset of "natural" interpretations of all products in a term, letting + and • stand for addition and multiplication, and assigning some fixed value to constants; see [Dershowitz and Manna, 1979] for similar examples. Syntactic "path" orderings (see [Dershowitz, 1987]) work in this case, too. Lipton and Snyder [1977] gave a method for proving termination with interpretations (order-isomorphic to w) for which rules are "value-preserving", as this example is for the natural interpretation.…”

Section: Path Orderingsmentioning

confidence: 93%

“…Nor can we use multisets of the values of the argument of fact, since some rules can multiply occurrences of that symbol. Though path orderings [Dershowitz, 1987] have been successfully applied to many termination proofs, they suffer from the same limitation as do all simplification orderings: they are not useful when a rule embeds as does/aa(s(=)) --, s(x) x fact(p(,(x))).…”

Section: Path Orderingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Tile use, in particular, of polynomial interpretations which map terms into the natural nmnbers, was developed by Lankford [1979]. For a survey of termination methods, see [Dershowitz, 1987].Virtually all orderings used in practice are simplification orderings [Dershowitz, 1982], satisfying the replacement property, that s ~-t implies that any term containing s is not less (under ~) than the same term with that occurrence of s replaced by t, and the snbterm property, that any term containing s is greater or equal to s. Simplification orderings cannot be used to prove termination of "self-embedding" systems, that is, when a term t can be derived in one or more steps from a term t ~, and t t can be obtained by repeatedly replacing subterms of t with subterms of those subterms. Knuth and Bendix [1970] designed a particular class of well-orderings which assigns a weight to a term which is the sum of the weights of its constituent function symbols.…”

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confidence: 99%

“…Another class of simplification orderiugs, tile path orderings [Dershowitz, 1982], is based on the idea that a term u should be bigger than any term that is built from smaller terms, all held together by a structure of function symbols that are smaller in some precedence ordering than the root symbol of u. The notion of path ordering was extended by Kamin and Ldvy [1980] to compare subterms lexicographically and to allow for a semantic component; see [Dershowitz, 1987]. Ilere, we generalize these orderings and the conditions under which they work.…”

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Hierarchical termination

Dershowitz

1995

Conditional and Typed Rewriting Systems

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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. IntroductionIf no infinite sequences of rewrites are possible, a rewrite system is said to have tile termiuation property. In practice, one usually guarantees termination by devising a well-founded (strict partial) ordering ~, such that s :,-t whenever s rewrites to t. As suggested in [Manna and Ness, 1970], it is often convenient to separate reduction orderings into a homomorphism from terms to an algebra with a wellfounded ordering. Tile use, in particular, of polynomial interpretations which map terms into the natural nmnbers, was developed by Lankford [1979]. For a survey of termination methods, see [Dershowitz, 1987].Virtually all orderings used in practice are simplification orderings [Dershowitz, 1982], satisfying the replacement property, that s ~-t implies that any term containing s is not less (under ~) than the same term with that occurrence of s replaced by t, and the snbterm property, that any term containing s is greater or equal to s. Simplification orderings cannot be used to prove termination of "self-embedding" systems, that is, when a term t can be derived in one or more steps from a term t ~, and t t can be obtained by repeatedly replacing subterms of t with subterms of those subterms. Knuth and Bendix [1970] designed a particular class of well-orderings which assigns a weight to a term which is the sum of the weights of its constituent function symbols. Terms of equal weight and headed by the same symbol have their subterms compared lexicographieally. Another class of simplification orderiugs, tile path orderings [Dershowitz, 1982], is based on the idea that a term u should be bigger than any term that is built from smaller terms, all held together by a structure of function symbols that are smaller in some precedence ordering than the root symbol of u. The notion of path ordering was extended by Kamin and Ldvy [1980] to compare subterms lexicographically and to allow for a semantic component; see [Dershowitz, 1987]. Ilere, we generalize these orderings and the conditions under which they work. In the appendix, we describe an implementation of the general ordering.

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Section: Path Orderingsmentioning

confidence: 93%

Section: Path Orderingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Hierarchical termination

Dershowitz

1995

Conditional and Typed Rewriting Systems

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Theorem Proving

Plaisted¹

1999

Wiley Encyclopedia of Electrical and Electronics Engineering

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The sections in this article are History, Status, and Future Prospects Propositional Logic Propositional Proof Procedures First‐order Logic First‐Order Proof Systems Equality Other Logics Current Research Areas Acknowledgment

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Automated Theorem Proving

Pfenning

2008

Wiley Encyclopedia of Computer Science and Engineering

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Automated Theorem Proving is an area in mathematical logic and computer science dedicated to the production of theorem proofs by algorithmical means, and is of great use in many fields such as some areas of mathematics, artificial intelligence, software verification, hardware verification or declarative programming. The aim of the first part of this work is to present Herbrand's theory and the Resolution Method for the first order logic. In the second part we show one of the many applications of the Resolution Method: the Prolog programming language, which is a useful tool for the resolution of problems in the NP class. i 2 Fundamentals of Logic Symbolic logic considers languages whose essential purpose is to symbolize reasoning encountered not only in mathematics, but also in daily life. The aim of this section is to introduce necessary basic concepts for Automated Theorem Proving. 2.1 The Propositional Logic In the propositional logic, the matter of interest are declarative sentences that can be either true or false. Such declarative sentences are called propositions. Definition 2.1. An atom is a proposition that can not be broken down into more simple propositions. In the propositional logic five logical connectives are used: ¬, ∧, ∨, →, and ↔ with their usual meaning and truth tables. Next, the propositional language can be defined. Definition 2.2. (Syntax of propositional logic) If σ is a countable set of atoms, we define the language of the σ-propositional formulas as the set of elements that are generated by the following rules: i. Every atom of σ is a σ-formula. ii. If φ is a σ-formula, then (¬φ) is also a σ-formula. iii. If φ 1 ,. .. , φ n are σ-formulas, then (φ 1 ∨. .. ∨ φ n) and (φ 1 ∧. .. ∧ φ n) are also σ-formulas. iv. If φ and ψ are σ-formulas, then (φ → ψ) and (φ ↔ ψ) are also σ-formulas. Remark 2.3. Often parentheses are suppressed to keep formulas readable if this results in no ambiguity. Definition 2.4. (Semantics of propositional logic) If σ is a countable set of atoms, a σ-interpretation is a mapping I: σ −→ {T, F }, where T denotes the truth value TRUE and F denotes the truth value FALSE.

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Termination of rewriting

Cited by 602 publications

References 53 publications

Hierarchical termination

Hierarchical termination

Theorem Proving

Automated Theorem Proving

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