Declarative languages such as OBJ*, CafeOBJ, and Maude use syntactic annotations to introduce replacement restrictions aimed at improving termination or efficiency of computations. Unfortunately, there is a lack of formal techniques for proving such benefits. We show that context-sensitive rewriting and on-demand rewriting provide a suitable framework to address this problem. We provide methods to analyze termination of on-demand rewriting and apply them to analyze termination of OBJ*, CafeOBJ, and Maude programs. Keywords Program analysis, replacement restrictions, term rewriting, termination 1.

We investigate termination of rewriting computations guided by strategy annotations. We show that proofs of termination can be obtained by proving (innermost) termination of context-sensitive rewriting (CSR). Hence, we investigate how to prove innermost termination of CSR using existing methods for proving termination of CSR.

We propose two conditions of the E-strategy with and without on-demand flags on which an evaluated term is always in head normal form. In rewriting with the E-strategy without (or with) on-demand flags, terms are evaluated according to a list of natural numbers (or integers) given to each function symbol. The first (or second) condition is that if there exists a rule such that a function symbol f occurs in its left-hand side and its i-th argument is not a variable, a list of f must contain i (or i), and if f is also a defined one, a list of f must contain 0 at the end. While there is no restriction w.r.t. the first condition, the second one can only be applied to left-linear constructor TRSs. But, There are cases in which rewriting with the E-strategy with on-demand flags terminates properly while that with the E-strategy without on-demand flags does not. We also propose a method of obtaining normal forms if a way to get head normal forms is given.

Strategy annotations are used in several programming languages as replacement restrictions aimed at improving efficiency and/or reducing the risk of nontermination.

Copyright c ○ 2000 by Keiichirou KUSAKARI Recently, Arts and Giesl introduced the notion of dependency pairs, which gives effective methods for proving termination of term rewriting systems (TRSs). In this thesis, we extend the notion of dependency pairs to AC-TRSs, and introduce new methods for effectively proving AC-termination. Since it is impossible to directly apply the notion of dependency pairs to AC-TRSs, we introduce the head parts in terms and show an analogy between the root positions in infinite reduction sequences by TRSs and the head positions in those by AC-TRSs. Indeed, this analogy is essential for the extension of dependency pairs to AC-TRSs. Based on this analogy, we define AC-dependency pairs. To simplify the task of proving termination and AC-termination, several elimination transformations such as the dummy elimination, the distribution elimination, the general dummy elimination and the improved general dummy elimination, have been proposed. In this thesis, we show that the argument filtering method combined with the AC-dependency pair technique is essential in all the elimination transformations above. We present remarkable simple proofs for the soundness of these elimination transformations based on this observation. Moreover, we propose a new elimination transformation, called the argument filtering transformation, which is not only more powerful than all the other elimination transformations but also especially useful to make clear an essential relationship among them.

We propose two analyses of the evaluation strategy (the E-strategy). Firstly, we analyze a completeness of the E-strategy. For a complete E-strategy each evaluated term is guaranteed to be a normal form. In this paper we define the notion of the completeness for the context-sensitive rewriting, called -completeness, and show a condition of the E-strategy to satisfy the -completeness. Secondarily, we give a strictness analysis for the E-strategy. A function is called strict in a certain argument if the evaluation of that argument does not change the termination behaviour. From the analyses, we can define a method to get a more effective default strategy for the E-strategy.

Term rewriting systems are widely used in computer science as a model of computation to relate syntax and semantics. In order to implement term rewriting system we need to use a strategy since there are many reduction sequences from a term in general. A strategy chooses one from such sequences. It is a function that takes a term to be rewritten and returns a term obtained by rewriting from the input term. There are two wellknown strategies: innermost strategies (or eager evaluation) and outermost strategies (or lazy evaluation). Innermost strategies can be implemented much more efficiently than outermost ones, while outermost strategies often have a better termination behavior than innermost ones. The evaluation strategy (the E-strategy), which is adopted by the family of OBJ algebraic specification languages, is one of the compromises between them. The E-strategy is more flexible than other fixed order of evaluation because each function symbol can have its own local strategy...

We investigate the problem of proving termination of OBJ programs with positive local strategies, i.e., lists of positive (or null) integers that are associated to the symbols of the signature. We establish conditions that fully characterize this problem as the problem of proving innermost termination of context-sensitive rewriting. Finally, we investigate how to prove innermost termination of context-sensitive rewriting using existing methods for proving termination of context-sensitive rewriting. Keywords: Declarative programming, replacement restrictions, term rewriting, termination. 1

journal homepage: www.elsevier.com/locate/tcs On-demand strategy annotations revisited: An improved on-demand evaluation strategy ✩