Context-sensitive Innermost Reachability is Decidable for Linear Right-shallow Term Rewriting Systems (original) (raw)
Related papers
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.
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.
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.
Reachability Analysis of Innermost Rewriting - extended version
2017
We consider the problem of inferring a grammar describing the output of a functional program given a grammar describing its input. Solutions to this problem are helpful for detecting bugs or proving safety properties of functional programs, and several rewriting tools exist for solving this problem. However, known grammar inference techniques are not able to take evaluation strategies of the program into account. This yields very imprecise results when the evaluation strategy matters. In this work, we adapt the Tree Automata Completion algorithm to approximate accurately the set of terms reachable by rewriting under the innermost strategy. We formally prove that the proposed technique is sound and precise w.r.t. innermost rewriting. We show that those results can be extended to the leftmost and rightmost innermost case. The algorithms for the general innermost case have been implemented in the Timbuk reachability tool. Experiments show that it noticeably improves the accuracy of sta...
Decidable call-by-need computations in term rewriting
Information and Computation, 2005
The theorem of Huet and Lévy stating that for orthogonal rewrite systems (i) every reducible term contains a needed redex and (ii) repeated contraction of needed redexes results in a normal form if the term under consideration has a normal form, forms the basis of all results on optimal normalizing strategies for orthogonal rewrite systems. However, needed redexes are not computable in general. In the paper we show how the use of approximations and elementary tree automata techniques allows one to obtain decidable conditions in a simple and elegant way. Surprisingly, by avoiding complicated concepts like index and sequentiality we are able to cover much larger classes of rewrite systems. We also study modularity aspects of the classes in our hierarchy. It turns out that none of the classes is preserved under signature extension. By imposing various conditions we recover the preservation under signature extension. By imposing some more conditions we are able to strengthen the signature extension results to modularity for disjoint and constructor-sharing combinations.
Head-Needed Strategy of Higher-Order Rewrite Systems and Its Decidable Classes
IPSJ Online Transactions, 2009
The present paper discusses a head-needed strategy and its decidable classes of higher-order rewrite systems (HRSs), which is an extension of the headneeded strategy of term rewriting systems (TRSs). We discuss strong sequential and NV-sequential classes having the following three properties, which are mandatory for practical use: (1) the strategy reducing a head-needed redex is head normalizing (2) whether a redex is head-needed is decidable, and (3) whether an HRS belongs to the class is decidable. The main difficulty in realizing (1) is caused by the β-reductions induced from the higher-order reductions. Since β-reduction changes the structure of higher-order terms, the definition of descendants for HRSs becomes complicated. In order to overcome this difficulty, we introduce a function, PV, to follow occurrences moved by βreductions. We present a concrete definition of descendants for HRSs by using PV and then show property (1) for orthogonal systems. We also show properties (2) and (3) using tree automata techniques, a ground tree transducer (GTT), and recognizability of redexes.
Decidability of Innermost Termination for Semi-Constructor Term Rewriting Systems
Yi and Sakai [7] showed that the termination problem is decidable for a daae of semiconstructor term rewriting systems, which is a superclepsilonssupercl\epsilon\ ssuperclepsilons of the class of right ground term rewriting systems. The decidability w&8 8hown8hown8hown by the fact that every non-terminating TRS in the cl&ae has aloop. In this paper we modify the prinftyfpr\infty fprinftyf of [7] to show that innermost termination is decidable for the class of semi-constnictor TRSs.
On tree automata that certify termination of left-linear term rewriting systems
Information and Computation, 2007
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.
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 ...