Subtree replacement systems

Theory and computer applications of subtree replacement system

Transfinite reductions in orthogonal term rewriting systems

Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, which we allow to be infinite, are unique, in contrast to ω-normal forms. Strongly converging fair reductions result in normal forms. In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for Böhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by ⊥) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of non-unifiable OTRSs

Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Combining constructive and equational geometric constraint solving techniques

In the past few years, there has been a strong trend towards developing parametric, computer aided design systems based on geometric constraint solving. An efective way to capture the design intent in these systems is to define relationships between geometric and technological variables. In general, geometric constraint solving including functional relationships requires a general approach and appropiate techniques toachieve the expected functional capabilities. This work reports on a hybrid method which combines two geometric constraint solving techniques: Constructive and equational. The hybrid solver has the capability of managing functional relationships between dimension variables and variables representing conditions external to the geometric problem. The hybrid solver is described as a rewriting system and is shown to be correct.Postprint (published version

Confluence of Layered Rewrite Systems

We investigate a new, Turing-complete class of layered systems, whose linearized lefthand sides of rules can only be overlapped at the root position. Layered systems define a natural notion of rank for terms: the maximal number of redexes along a path from the root to a leaf. Overlappings are allowed in finite or infinite trees. Rules may be non-terminating, non-left-linear, or non-right- linear. Using a novel unification technique, cyclic unification, we show that rank non-increasing layered systems are confluent provided their cyclic critical pairs have cyclic-joinable decreasing diagrams

Data structure implementation and correctness

Recently, implementation of data structures and correctness proofs of data structure implementations have become important problems in the construction of data abstraction languages, data base systems, and software engineering. The research reported here is primarily concerned with the definition and implementation of data structures, and how to prove an implementation correct;This thesis develops a general technique to implement the source data structure (d1) in terms of the target data structures (d2). That is, given data structures d1 and d2, an implementation of d1 by d2 is defined separately on the syntactical and semantical levels of data structure elements. This work makes a sharp distinction between the specification of a data structure and its implementation. Specification of data structures is considered to be abstract, that is, implementation independent of any specific programming language;This thesis deals primarily with developing criteria for providing provably correct implementation of data structures. The correctness of data structure implementation is developed on two levels: syntactical and semantical. Syntactically, correct implementations deal with algebraic equations (conditional and unconditional) that specify a data structure, while the semantically correct implementations define correctness on the basis of the semantic algebra in the data structure specifications. All the implementations are specified by tree transducers, reducing the problem of implementation to a problem of translation;The issue of tree transducers is addressed on the syntactic and semantic levels. Two key syntactical properties of tree transducers have been investigated. They are, the consistency and semiconsistency of tree transducers with respect to the algebraic equations defining the source and target data structures. These properties have been used for the syntactical correctness proof of the implementation of data structures. One of the key results of this thesis is the development of syntactically honest tree transducers, which is based on a lattice. It has been proved that syntactically honest tree transducers form a base for syntactically correct implementation of data structures