A Dependency Pair Framework for A ∨ C-Termination

1993

This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in [Nip9l, Nip93]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. Our result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a well-founded monotone algebra" described in [Zan93]. The resulting technique is as follows: Choose a higher-order algebra with operations for each function symbol in the HRS, equipped with some well-founded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule and for each valuation of the free variables in that rule the value of the left hand side is greater than the value of the right hand side, then the HRS is terminating. At the end of the paper two applications of this technique are given, which show that this technique is natural and can easily be applied. A nice characterisation of termination is given in [Zan93]. The function symbols of a TRS 1Z have to be interpreted as strictly monotonic operations in some well-founded algebra. This interpretation is extended to closed terms as a usual algebraic homomorphism. Now the associated rewrite relation is terminating if every left hand side is greater (under the chosen interpretation) than the belonging right hand side, for each possible interpretation of the variables in that rule. The strength of this characterisation is that one can concentrate on the "intuitive reason" for termination. This intuition can be translated in suited operations on well-founded orderings, thus using semantical arguments. The real termination proof consists of testing a simple condition on the rules only instead of on all possible rewrite steps or all possible redexes. This semantical approach is more convenient than a syntactical technique. The aim of this paper is to generalise this semantical characterisation of termination for TRSs to one for HRSs. We use an extension of the definition of an HRS in [Nip93] because we do not need the restrictions of the formalism (for instance, that the rules should be of base type and the left hand sides should be patterns). Our result is also applicable to the HRSs of the definition in [Nip9l, Nip93]. The main result is that such a generalisation is possible. The interpretation of terms can be extended to the interpretation of higher-order terms. The orderings and the notion of strictness can also be generalised. The techniques to achieve this are similar to those used in [Gan8O, dV87]. Moreover, the result that termination proofs can be given with a well-founded monotone algebra in [Zan93] carries over to HRSs with simple conditions on the well-founded ordering. With this technique some natural HRSs are proved to be terminating (see Section 7.

2013

Well-foundedness is related to an important property of rewriting systems, namely termination. A well-known technique to prove well-foundedness on term orderings is Kruskal's theorem, which implies that a monotonic term ordering over a finite signature satisfying the subterm property is well-founded. However, it does not seem to work for a number of terminating term rewriting systems. In this paper, it is shown that a term ordering possessing subterm property and decomposability ( in place of monotonicity) does yield a simpler proof of well-foundedness for terminating term rewriting systems than the techniques depending on Kruskal's theorem.

Carolina Digital Repository (University of North Carolina at Chapel Hill), 1985

In this paper we describe a new class of orderings-associative path orderings-for proving termination of associative-commutative term rewriting systems. These orderings are based on the concept of simplification orderings and extend the well-known recursive path orderings to Econgruence classes, where E is an equational theory consisting of associativity and commutativity axioms. Associative path orderings are applicable to term rewriting systems for which a precedence ordering on the set of operator symbols can be defined that satisfies a certain condition, the associative path condition. The precedence ordering can often be derived from the structure of the reduction rules. We include termination proofs for various term rewriting systems (for rings, boolean algebra, etc .) and, in addition, point out ways to handle situations where the associative path condition is too restrictive .

Journal of Symbolic Computation, 1995

Conditional and Typed Rewriting Systems, 1991

It is well known that the disjoint union of terminating term rewriting systems does not yield a terminating system in general. We show that this undesirable phenomenon vanishes if one implements term rewriting by graph reduction: given two terminating term rewrite systems 7~o and 7~1, the graph reduction system implementing 7~0-F T~1 is terminating. In fact, we prove the stronger result that the graph reduction system for the union ~o U T~ 1 is terminating provided that the left-hand sides of ~i have no common function symbols with the right-hand sides of 7~1-i (i = O, 1). The implementation is complete in the sense that it computes a normal form for each term over the signature of R0 u 7~1. 1 I n t r o d u c t i o n The operational semantics of algebraic specification languages are usually based on term rewriting (see, e.g., [BCV 85], [BHK 89], [EM 85], [FG3M 85], [GH 86]). In this context, confluence and termination are particularly relevant properties of term rewriting systems. Hence, when specifications are structured as combinations of smaller subspecifications, the question arises whether these properties are preserved by a given combination mechanism. Recently, research in this direction has been started by,considering the disjoint union T£0 +7~1 of two term rewriting systems 7"£o and 7~1: the rule set of 7£o ~-7~1 is the union of the rules of T~ and T£1 where the function symbols occurring in T~ and 7~1 are disjoint (or are made disjoint by renaming). Toyama [Toy 87a] proves that 7~o + 7~1 is confluent *Work supported by ESPRIT project #390, PROSPECTRA, and by ESPRIT Basic Research Working Group #3264, COMPASS.

1995

We study the combination of the following already known ideas for showing confluence of unconditional or conditional term rewriting systems into practically more useful confluence criteria for conditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet's introduction and Toyama's generalization of parallel closedness for non-terminating unconditional systems, the use of shallow confluence for proving confluence of terminating and non-terminating conditional systems, the idea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibility of the conditions of conditional critical pairs, and the idea that (when termination is given) only prime superpositions have to be considered and certain normalization restrictions can be applied for the substitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving already known methods, we present the following new ideas and results: We strengthen the criterion for overlay joinable terminating systems, and, by using the expressiveness of our syntactic separation into constructor and non-constructor symbols, we are able to present criteria for level confluence that are not criteria for shallow confluence actually and also able to weaken the severe requirement of normality (stiffened with left-linearity) in the criteria for shallow confluence of terminating and non-terminating conditional systems to the easily satisfied requirement of quasi-normality. Finally, the whole paper also gives a practically useful overview of the syntactic means for showing confluence of conditional term rewriting systems. This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt) 8 When termination is assumed, there are approaches to prove confluence automatically, cf. Becker (1993) and Becker (1994). 9 Cf. our theorems 13.3, 13.4, and 15.3. 10 Cf. Walther (1994). Note that we can even keep the notation style similar to this function specification style, cf. Wirth & Lunde (1994). Let E be a finite set of equations and X a finite subset of V. A substitution σ ∈ S UB(V, T) is called a unifier for E if Eσ ⊆ id. Such a unifier is called most general on X if for each unifier µ for E there is some τ ∈ S UB(V, T) such that X ↿(στ) = X ↿µ. If E has a unifier, then it also has a most general unifier 13 on X, denoted by mgu(E, X). 13 For this most general unifier σ we could, as usual, even require σσ = σ but not V (σ[V (E)]) ⊆ V (E). * ←−v. −→ is called terminating below u if there is no s : N → dom(−→) such that u = s 0 ∧ ∀i ∈ N. s i −→s i+1 .

Logic for Programming, Artificial Intelligence, and …, 2010

Sufficient completeness has been throughly studied for equational specifications, where function symbols are classified into constructors and defined symbols. But what should sufficient completeness mean for a rewrite theory R = (Σ, E, R) with equations E and non-equational rules R describing concurrent transitions in a system? This work argues that a rewrite theory naturally has two notions of constructor: the usual one for its equations E, and a different one for its rules R. The sufficient completeness of constructors for the rules R turns out to be intimately related with deadlock freedom, i.e., R has no deadlocks outside the constructors for R. The relation between these two notions is studied in the setting of unconditional order-sorted rewrite theories. Sufficient conditions are given allowing the automatic checking of sufficient completeness, deadlock freedom, and other related properties, by propositional tree automata modulo equational axioms such as associativity, commutativity, and identity. They are used to extend the Maude Sufficient Completeness Checker from the checking of equational theories to that of both equational and rewrite theories. Finally, the usefulness of the proposed notion of constructors in proving inductive theorems about the reachability rewrite relation →R associated to a rewrite theory R (and also about the joinability relation ↓R) is both characterized and illustrated with an example.

Lecture Notes in Computer Science, 1981

i. Introduction 2. Combinatory Logic Rewriting Systems and preliminary definitions. 3. The Non-Ascending Property as a sufficient condition for strong termination 4. Comparing the Non-Ascending Property, Recursive Path Orderings and Simplification Orderings.

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 ...

2010

Sufficient completeness has been throughly studied for equational specifications, where function symbols are classified into constructors and defined symbols. But what should sufficient completeness mean for a rewrite theory R = (Σ, E, R) with equations E and non-equational rules R describing concurrent transitions in a system? This work argues that a rewrite theory naturally has two notions of constructor: the usual one for its equations E, and a different one for its rules R. The sufficient completeness of constructors for the rules R turns out to be intimately related with deadlock freedom, i.e., R has no deadlocks outside the constructors for R. The relation between these two notions is studied in the setting of unconditional order-sorted rewrite theories with (i) a frozenness map restricting rewriting with R, and (ii) a context-sensitive map restricting rewriting with the equations E, as it is possible for specifications in the Maude language. Sufficient conditions are given allowing the automatic checking of sufficient completeness, and other related properties, by equational tree automata modulo equational axioms such as associativity, commutativity, and identity. They are used to extend the Maude Sufficient Completeness Checker from the checking of equational theories to that of both equational and rewrite theories. Finally, the usefulness of the proposed notion of constructors in proving inductive theorems about the reachability rewrite relation → R associated to a rewrite theory R (and also about the joinability relation ↓ R ) is both characterized and illustrated with examples.

Foundations of Software Science and Computation Structures, 2004

We show how to generate well-founded and stable term orderings based on polynomial interpretations over the real numbers. Monotonicity (another usual requirement in termination proofs) can, then, be gradually introduced in the interpretations to deal with different applications. For instance, the dependency pairs method for proving termination of rewriting, and some restrictions of the rewrite relation which are not monotonic in all arguments of the function symbols, can benefit from this approach. The latter is the case for context-sensitive rewriting (CSR), a simple restriction of rewriting which has been proved useful for describing the semantics of several programming languages (e.g., Maude) and analyzing the properties of the corresponding programs. We show how to automatically generate polynomial interpretations over the real numbers for proving termination of CSR.

Mathematical Foundations of …, 1985

A new partial ordering scheme for proving uniform termination of term rewriting systems is presented. The basic idea is that two terms are compared by comparing the paths through them. It is shown that the ordering is a well-founded simplification ordering and also a ...

2004

We carry out a detailed analysis of Thatte's transformation of term rewriting systems. We refute an earlier claim that this transformation preserves confluence for weakly persistent systems. We prove the preservation of weak normalization, and of confluence in weakly normalizing systems and in nonoverlapping systems with linear subtemplates. We conclude by proving that weak persistence is an undecidable property of term rewriting systems.

Logical Methods in Computer Science, 2013

Terminal coalgebras for a functor serve as semantic domains for state-based systems of various types. For example, behaviors of CCS processes, streams, infinite trees, formal languages and non-well-founded sets form terminal coalgebras. We present a uniform account of the semantics of recursive definitions in terminal coalgebras by combining two ideas: (1) abstract GSOS rules ℓ specify additional algebraic operations on a terminal coalgebra; (2) terminal coalgebras are also initial completely iterative algebras (cias). We also show that an abstract GSOS rule leads to new extended cia structures on the terminal coalgebra. Then we formalize recursive function definitions involving given operations specified by ℓ as recursive program schemes for ℓ, and we prove that unique solutions exist in the extended cias. From our results it follows that the solutions of recursive (function) definitions in terminal coalgebras may be used in subsequent recursive definitions which still have unique solutions. We call this principle modularity. We illustrate our results by the five concrete terminal coalgebras mentioned above, e. g., a finite stream circuit defines a unique stream function.

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.