Dieter Hofbauer

Termination proofs by multiset path orderings imply primitive recursive derivation lengths

It 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 [Pla78]. This result holds for a great variety of path orderings including path of subterms ordering, recursive decomposition ordering, and the path ordering of Kapur, Narendran, Sivakumar 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.

Termination proofs and the length of derivations

The derivation height of a term t, relative to a set R of rewrite rules, dh R (t), is the length of a longest derivation from t. We investigate in which way certain termination proof methods impose bounds on dh R . In particular we show that, if termination of R can be proved by polynomial interpretation then dh R is bounded from above by a doubly exponential function, whereas termination proofs by Knuth-Bendix ordering are possible even for systems where dh R cannot be bounded by any primitive recursive functions. For both methods, conditions are given which guarantee a singly exponential upper bound on dh R . Moreover, all upper bounds are tight.

Match-Bounded String Rewriting Systems

Abstract.We introduce a new class of automated proof methods for the termination of rewriting systems on strings. The basis of all these methods is to show that rewriting preserves regular languages. To this end, letters are annotated with natural numbers, called match heights. If the minimal height of all positions in a redex is h then every position in the reduct will get height h+1. In a match-bounded system, match heights are globally bounded. Using recent results on deleting systems, we prove that rewriting by a match-bounded system preserves regular languages. Hence it is decidable whether a given rewriting system has a given match bound. We also provide a criterion for the absence of a match-bound. It is still open whether match-boundedness is decidable. Match-boundedness for all strings can be used as an automated criterion for termination, for match-bounded systems are terminating. This criterion can be strengthened by requiring match-boundedness only for a restricted set of strings, namely the set of right hand sides of forward closures.

Termination of string rewriting with matrix interpretations

A rewriting system can be shown terminating by an order-preserving mapping into a well-founded domain. We present an instance of this scheme for string rewriting where the domain is a set of square matrices of natural numbers, equipped with a well-founded ordering that is not total. The coefficients of the matrices can be found via a transformation to a boolean satisfiability problem. The matrix method also supports relative termination, thus it fits with the dependency pair method as well. Our implementation is able to automatically solve hard termination problems.

Termination Proofs by Context-Dependent Interpretations

Proving termination of a rewrite system by an interpretation over the natural numbers directly implies an upper bound on the derivational complexity of the system. In this way, however, the derivation height of terms is often heavily overestimated. Here we present a generalization of termination proofs by interpretations that can avoid this drawback of the traditional approach. A number of simple examples illustrate how to achieve tight or even optimal bounds on the derivation height. The method is general enough to capture cases where simplification orderings fail.

Termination Proofs for String Rewriting Systems via Inverse Match-Bounds

Annotating a letter by a number, one can record information about its history during a rewrite derivation. In each rewrite step, numbers in the reduct are updated depending on the redex numbering. A string rewriting system is called match-bounded if there is a global upper bound to these numbers. Match-boundedness is known to be a strong sufficient criterion for both termination and preservation of regular languages. We show that the string rewriting systems whose inverse (left and right hand sides exchanged) is match-bounded, also have exceptional properties, but slightly different ones. Inverse match-bounded systems need not terminate; they effectively preserve context-free languages; their sets of normalizable strings and their sets of immortal strings are effectively regular. These languages can be used to decide the normalization, the uniform normalization, the termination and the uniform termination problem for inverse match-bounded systems. We also prove that the termination problem is decidable in linear time, and that a certain strong reachability problem is decidable, thereby solving two open problems of McNaughton’s. Like match-bounds, inverse match-bounds entail linear derivational complexity on the set of terminating strings.

On state-alternating context-free grammars

State-alternating context-free grammars are introduced, and the language classes obtained from them are compared to the classes of the Chomsky hierarchy as well as to some well-known complexity classes. In particular, state-alternating context-free grammars are compared to alternating context-free grammars (Theoret. Comput. Sci. 67 (1989) 75-85) and to alternating pushdown automata. Further, various derivation strategies are considered, and their influence on the expressive power of (state-) alternating context-free grammars is investigated.

On tree automata that certify termination of left-linear term rewriting systems

We present a new method for proving termination of term rewriting systems automatically. It is a generalization of the match bound method for string rewriting. To prove that a term rewriting system terminates on a given regular language of terms, we first construct an enriched system over a new signature that simulates the original derivations. The enriched system is an infinite system over an infinite signature, but it is locally terminating: every restriction of the enriched system to a finite signature is terminating. We then construct iteratively a finite tree automaton that accepts the enriched given regular language and is closed under rewriting modulo the enriched system. If this procedure stops, then the enriched system is compact: every enriched derivation involves only a finite signature. Therefore, the original system terminates. We present three methods to construct the enrichment: top heights, roof heights, and match heights. Top and roof heights work for left-linear systems, while match heights give a powerful method for linear systems. For linear systems, the method is strengthened further by a forward closure construction. Using these methods, we give examples for automated termination proofs that cannot be obtained by standard methods.

Termination of {aa ⅾ bc, bb ⅾ ac, cc ⅾ ab}

0020-0190/

Linearizing Term Rewriting Systems Using Test Sets

– see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2005.12.011 participating programs nor by their authors, and to our knowledge it remained open until the solution given below was found. We find the problem attractive due to its seeming simplicity. Firstly, the system R is highly symmetric as a renaming of its own reverse. Secondly, it is small: 3 rules with 12 letters. Note however that (uniform) termination is undecidable even for string rewriting systems with three rules [6]. Finally, R is length-preserving. However, also for length-preserving string rewriting systems the termination problem remains undecidable [2].