Confluence of Shallow Right-Linear Rewrite Systems (original) (raw)
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Confluence of Right Ground Term Rewriting Systems is Decidable
Foundations of Software Science and Computational …, 2005
Term rewriting systems provide a versatile model of computation. An important property which allows to abstract from potential nondeterminism of parallel execution of the modelled program is confluence. In this paper we prove that confluence of a fairly large class of systems, namely right ground term rewriting systems, is decidable. We introduce a labelling of variables with colours and constrain substitutions according to these colours. We show how right ground rewriting systems can be reduced to simple systems with coloured variables. Such systems can be analysed using reduction-automata techniques which leads to an interesting decision procedure for confluence.
IEICE Transactions on Information and Systems, 2010
In this paper, we show that the termination and the innermost termination properties are decidable for the class of term rewriting systems (TRSs for short) all of whose dependency pairs are right-linear and right-shallow. We also show that the innermost termination is decidable for the class of TRSs all of whose dependency pairs are shallow. The key observation common to these two classes is as follows: for every TRS in the class, we can construct, by using the dependency-pairs information, a finite set of terms such that if the TRS is non-terminating then there is a looping sequence beginning with a term in the finite set. This fact is obtained by modifying the analysis of argument propagation in shallow dependency pairs proposed by Wang and Sakai in 2006. However we gained a great benefit that the resulted procedures do not require any decision procedure of reachability problem used in Wang's procedure for shallow case, because known decidable classes of reachability problem are not larger than classes discussing in this paper.
New Undecidability Results for Properties of Term Rewrite Systems
Electronic Notes in Theoretical Computer Science, 2012
This paper is on several basic properties of term rewrite systems: reachability, joinability, uniqueness of normal forms, unique normalization, confluence, and existence of normal forms, for subclasses of rewrite systems defined by syntactic restrictions on variables. All these properties are known to be undecidable for the general class and decidable for ground (variable-free) systems. Recently, there has been impressive progress on efficient algorithms or decidability results for many of these properties. The aim of this paper is to present new results and organize existing ones to clarify further the boundary between decidability and undecidability for these properties. Another goal is to spur research towards a complete classification of these properties for subclasses defined by syntactic restrictions on variables. The proofs of the presented results may be intrinsically interesting as well due to their economy, which is partly based on improved reductions between some of the properties.
IPSJ Online Transactions, 2009
The reachability problem for given an initial term, a goal term, and a term rewriting system (TRS) is to decide whether the initial one is reachable to the goal one by the TRS or not. A term is shallow if each variable in the term occurs at depth 0 or 1. Innermost reduction is a strategy that rewrites innermost redexes, and context-sensitive reduction is a strategy in which rewritable positions are indicated by specifying arguments of function symbols. In this paper, we show that the reachability problem under context-sensitive innermost reduction is decidable for linear right-shallow TRSs. Our approach is based on the tree automata technique that is commonly used for analysis of reachability and its related properties. We show a procedure to construct tree automata accepting the sets of terms reachable from a given term by context-sensitive innermost reduction of a given linear right-shallow TRS.
Decidability of Reachability for Right-shallow Context-sensitive Term Rewriting Systems
IPSJ Online Transactions, 2011
The reachability problem for an initial term, a goal term, and a rewrite system is to decide whether the initial term is reachable to goal one by the rewrite system or not. The innermost reachability problem is to decide whether the initial term is reachable to goal one by innermost reductions of the rewrite system or not. A context-sensitive term rewriting system (CS-TRS) is a pair of a term rewriting system and a mapping that specifies arguments of function symbols and determines rewritable positions of terms. In this paper, we show that both reachability for right-linear right-shallow CS-TRSs and innermost reachability for shallow CS-TRSs are decidable. We prove these claims by presenting algorithms to construct a tree automaton accepting the set of terms reachable from a given term by (innermost) reductions of a given CS-TRS.
2016
The ground term reachability problem consists in determining whether a given variable-free term t can be transformed into a given variable-free term t' by the application of rules from a term rewriting system R. The joinability problem, on the other hand, consists in determining whether there exists a variable-free term t'' which is reachable both from t and from t'. Both problems have proven to be of fundamental importance for several subfields of computer science. Nevertheless, these problems are undecidable even when restricted to linear term rewriting systems. In this work, we approach reachability and joinability in linear term rewriting systems from the perspective of parameterized complexity theory, and show that these problems are fixed parameter tractable with respect to the depth of derivations. More precisely, we consider a notion of parallel rewriting, in which an unbounded number of rules can be applied simultaneously to a term as long as these rules do ...
Uniqueness of Normal Forms for Shallow Term Rewrite Systems
ACM Transactions on Computational Logic
Uniqueness of normal forms (UN =) is an important property of term rewrite systems. UN = is decidable for ground (i.e., variable-free) systems and undecidable in general. Recently it was shown to be decidable for linear, shallow systems. We generalize this previous result and show that this property is decidable for shallow rewrite systems, in contrast to confluence, reachability and other properties, which are all undecidable for flat systems. Our result is also optimal in some sense, since we prove that the UN = property is undecidable for two classes of linear rewrite systems: left-flat systems in which right-hand sides are of depth at most two and right-flat systems in which left-hand sides are of depth at most two.
Deciding Fundamental Properties of Right-(Ground or Variable) Rewrite Systems by Rewrite Closure
Lecture Notes in Computer Science, 2004
Right-(ground or variable) rewrite systems (RGV systems for short) are term rewrite systems where all right hand sides of rules are restricted to be either ground or a variable. We define a minimal rewrite extension R of the rewrite relation induced by a RGV system R. This extension admits a rewrite closure presentation, which can be effectively constructed from R. The rewrite closure is used to obtain decidability of the reachability, joinability, termination, and confluence properties of the RGV system R. We also show that the word problem and the unification problem are decidable for confluent RGV systems. We analyze the time complexity of the obtained procedures; for shallow RGV systems, termination and confluence are exponential, which is the best possible result since all these problems are EXPTIME-hard for shallow RGV systems.