Transfinite Rewriting Semantics for Term Rewriting Systems (original) (raw)

Natural Rewriting for General Term Rewriting Systems

2004

We address the problem of an efficient rewriting strategy for general term rewriting systems. Several strategies have been proposed over the last two decades for rewriting, the most efficient of all being the natural rewriting strategy . All the strategies so far, including natural rewriting, assume that the given term rewriting system is a left-linear constructor system. Although these restrictions are reasonable for some functional programming languages, they limit the expressive power of equational languages, and they preclude certain applications of rewriting to equational theorem proving and to languages combining equational and logic programming. In this paper, we propose a conservative generalization of natural rewriting that does not require the rules to be left-linear and constructor-based. We also establish the soundness and completeness of this generalization.

Semantics of non-terminating rewrite systems using minimal coverings

1995

We propose a new semantics for rewrite systems ba.~ed on interpreting rewrite rules as in equatioIlB between terms in an ordered algebra. In part.icular, we show thai the algebra. of normal forms in a terminating system is a uniqnely minimal covering of the term algebra. In the non-terminating ca..~e, the existence of this minimal covering is established in the comple tion of an ordered algebra formed by rewrit.ing sequences. We thus generalize the properties of normal forms far: non-terminating systelil~ to this minimal covering. ThesE' include the exi~tence of normal forms for arbitrary rewrite ~ystems, and their uniqueness for conBue-nt ~ystems, in which Ca<le the algebra of normal forms i~ isomorphic to the canonical quotient. algebra associated with the rule~ when seen as eqnations. This extend!> the benefits of alge braic semantics to systems with non-determinist.ic and non-t.erminating computations. V•le first study properties of abstract. order~, and then instantiat.e the~e to term rewriting sy~tems.

Decidable call-by-need computations in term rewriting

Information and Computation, 2005

The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.

Completeness of context-sensitive rewriting

Information Processing Letters, 2015

Restrictions of rewriting may turn normal forms of some terms unreachable, leading to incomplete computations. Context-sensitive rewriting (csr) is the restriction of rewriting that only permits reductions on arguments selected by a replacement map µ, which associates a subset µ(f) of argument indices to each function symbol f. Hendrix and Meseguer defined an algebraic semantics for Term Rewriting Systems (TRSs) executing csr that can be used to reason about programs written in programming languages like CafeOBJ and Maude, where such replacement restrictions can be specified in programs. Semantic completeness of csr was also defined. In this paper we show that canonical replacement maps, which play a prominent role in simulating rewriting computations with csr, are necessary for completeness in important classes of TRSs.

A General Natural Rewriting Strategy

We define an efficient rewriting strategy for general term rewriting systems. Several strategies have been proposed over the last two decades for rewriting, the most efficient of all being the natural rewriting strategy of Escobar. All the strategies so far, including natural rewriting, assume that the given term rewriting system is left-linear and constructor-based. Although these restrictions are reasonable for some functional programming languages, they limit the expressive power of equational programming languages, and they preclude certain applications of rewriting to equational theorem proving and to languages combining equational and logic programming. In [5], we proposed a conservative generalization of the natural rewriting strategy that does not require the previous assumptions for the rules and we established its soundness and completeness.

A compact fixpoint semantics for term rewriting systems

Theoretical Computer Science, 2010

This work is motivated by the fact that a "compact" semantics for term rewriting systems, which is essential for the development of effective semantics-based program manipulation tools (e.g. automatic program analyzers and debuggers), does not exist. The big-step rewriting semantics that is most commonly considered in functional programming is the set of values/normal forms that the program is able to compute for any input expression. Such a big-step semantics is unnecessarily oversized, as it contains many "semantically useless" elements that can be retrieved from a smaller set of terms. Therefore, in this article, we present a compressed, goal-independent collecting fixpoint semantics that contains the smallest set of terms that are sufficient to describe, by semantic closure, all possible rewritings. We prove soundness and completeness under ascertained conditions. The compactness of the semantics makes it suitable for applications. Actually, our semantics can be finite whereas the big-step semantics is generally not, and even when both semantics are infinite, the fixpoint computation of our semantics produces fewer elements at each step. To support this claim we report several experiments performed with a prototypical implementation.

On constructor rewrite systems and the lambda-calculus

Automata, Languages and Programming, 2009

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by-value betareduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in an term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.

On Ground-Confluence of Term Rewriting Systems

Information and Computation, 1990

It is undecidable whether or not a finite Noetherian term rewriting system is ground-confluent. This undecidability result holds even when systems involving only unary function symbols and one constant are being considered, or when left-linear or right-linear ...

A PVS Theory for Term Rewriting Systems

Electronic Notes in Theoretical Computer Science, 2009

A theory, called trs, for Term Rewriting Systems in the theorem Prover PVS is described. This theory is built on the PVS libraries for finite sequences and sets and a previously developed PVS theory named ars for Abstract Reduction Systems which was built on the PVS libraries for sets. Theories for dealing with the structure of terms, for replacements and substitutions jointly with ars allow for adequate specifications of notions of term rewriting such as critical pairs and formalization of elaborated criteria from the theory of Term Rewriting Systems such as the Knuth-Bendix Critical Pair Theorem. On the other hand, ars specifies definitions and notions such as reduction, confluence and normal forms as well as non basic concepts such as Noetherianity.