Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over an extended signature. They are not sound w.r.t. reduction for every CTRS, while they are complete w.r.t. reduction. Here, soundness w.r.t. reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch's unraveling for a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled TRS is left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies that of Ohlebusch's unraveling. Finally, we show that soundness of Ohlebusch's unraveling is the weakest in soundness of the other unravelings and a transformation, proposed by Serbanuta and Rosu, for (normal) deterministic CTRSs, i.e., soundness of them respectively implies that of Ohlebusch's unraveling.Comment: 49 pages, 1 table, publication in Special Issue: Selected papers of the "22nd International Conference on Rewriting Techniques and Applications (RTA'11)

Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped \lambda-calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of M\"uller and Dougherty for unconditional rewriting. Two cases are considered, whether \beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. 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 part of rewrite rules

Polynomial Path Orders: A Maximal Model

This paper is concerned with the automated complexity analysis of term rewrite systems (TRSs for short) and the ramification of these in implicit computational complexity theory (ICC for short). We introduce a novel path order with multiset status, the polynomial path order POP*. Essentially relying on the principle of predicative recursion as proposed by Bellantoni and Cook, its distinct feature is the tight control of resources on compatible TRSs: The (innermost) runtime complexity of compatible TRSs is polynomially bounded. We have implemented the technique, as underpinned by our experimental evidence our approach to the automated runtime complexity analysis is not only feasible, but compared to existing methods incredibly fast. As an application in the context of ICC we provide an order-theoretic characterisation of the polytime computable functions. To be precise, the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*

Rewrite, rewrite, rewrite, rewrite, rewrite, …

We study properties of rewrite systems that are not necessarily terminating, but allow instead for transfinite 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 infinitary theories. We also consider sufficient completeness of hierarchical systems

Termination proofs by multiset path orderings imply primitive recursive derivation lengths

AbstractIt is shown that a termination proof for a term-rewriting system using multiset path orderings (i.e. recursive path orderings with multiset status only) yields a primitive recursive bound on the length of derivations, measured in the size of the starting term, confirming a conjecture of Plaisted (1978). This result holds for a great variety of path orderings, including path of subterms ordering, recursive decomposition ordering, and the path ordering of Kapur (1985) if lexicographic status is not incorporated. The result is essentially optimal as such derivation lengths can be found in each level of the Grzegorczyk hierarchy, even for string-rewriting systems

Classes of Terminating Logic Programs

Termination of logic programs depends critically on the selection rule, i.e. the rule that determines which atom is selected in each resolution step. In this article, we classify programs (and queries) according to the selection rules for which they terminate. This is a survey and unified view on different approaches in the literature. For each class, we present a sufficient, for most classes even necessary, criterion for determining that a program is in that class. We study six classes: a program strongly terminates if it terminates for all selection rules; a program input terminates if it terminates for selection rules which only select atoms that are sufficiently instantiated in their input positions, so that these arguments do not get instantiated any further by the unification; a program local delay terminates if it terminates for local selection rules which only select atoms that are bounded w.r.t. an appropriate level mapping; a program left-terminates if it terminates for the usual left-to-right selection rule; a program exists-terminates if there exists a selection rule for which it terminates; finally, a program has bounded nondeterminism if it only has finitely many refutations. We propose a semantics-preserving transformation from programs with bounded nondeterminism into strongly terminating programs. Moreover, by unifying different formalisms and making appropriate assumptions, we are able to establish a formal hierarchy between the different classes.Comment: 50 pages. The following mistake was corrected: In figure 5, the first clause for insert was insert([],X,[X]

On Proving Soundness of the Computationally Equivalent Transformation for Normal Conditional Term Rewriting Systems by Using Unravelings

In this paper, we show that the SR transformation, a computationally equivalent transformation proposed by Serbanuta and Rosu, is sound for weakly left-linear normal conditional term rewriting systems (CTRS). Here, soundness for a CTRS means that reduction of the transformed unconditional term rewriting system (TRS) creates no undesired reduction for the CTRS. We first show that every reduction sequence of the transformed TRS starting with a term corresponding to the one considered on the CTRS is simulated by the reduction of the TRS obtained by the simultaneous unraveling. Then, we use the fact that the unraveling is sound for weakly left-linear normal CTRSs

Termination of rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page