Controlled Term Rewriting

Abstract

Abstract. Motivated by the problem of verification of imperative tree transformation programs, we study the combination, called controlled term rewriting systems (CntTRS), of term rewriting rules with constraints selecting the possible rewrite positions. These constraints are specified, for each rewrite rule, by a selection automaton which defines a set of positions in a term based on tree automata computations. We show that reachability is PSPACE-complete for so-called monotonic CntTRS, such that the size of every left-hand-side of every rewrite rule is larger or equal to the size of the corresponding right-hand-side, and also for the class of context-free non-collapsing CntTRS, which transform Context-Free (CF) tree language into CF tree languages. When allowing size-reducing rules, reachability becomes undecidable, even for flat CntTRS (both sides of rewrite rules are of depth at most one) when restricting to words (i.e. function symbols have arity at most one), and for ground CntTRS (rewrite rules have no variables). We also consider a restricted version of the control such that a position is selected if the sequence of symbols on the path from that position to the root of the tree belongs to a given regular language. This restriction enables decision results in the above cases.

Transforming Prefix-constrained or Controlled Rewrite Systems

“…The language recognized by A is L(A) = {t ∈ T (Σ) | sruns(A, t) = ∅}. Selection Automaton [10]. A selection automaton A is a 5-tuple (Σ, Q, Q f , S, ∆) where (Σ, Q, Q f , ∆) is a tree automaton denoted ta(A) and S is a set of states of Q called selection states.…”

“…Controlled Term Rewrite System (cntTRS) [10]. A controlled rewrite system R is a finite set of controlled rewrite rules of the form A : l → r, composed of a selection automaton A and a rewrite rule l → r. A term t rewrites into t in one step by a cntTRS R, denoted t → R t , if there exist a controlled rewrite rule A : l → r in R, a position p ∈ sel(A, t), and a substitution σ s.t.…”

“…Theorem 1 of [10] shows that starting from a context-sensitive tree grammar G and a monotonic cntTRS R, one can construct a context-sensitive tree grammar G * that generates the closure of L(G) by R. Roughly speaking, it amounts to consider the set ∆ of production rules of G as a rewrite system, and to transform the cntTRS ∆ ∪ R into an ordinary TRS R , which will be considered as the production rules of G * . This transformation of a cntTRS into a TRS has some similarities with ours, if we consider that λ corresponds to top, and f in to j k .…”

“…Prefix-constrained rewriting [11] is an extension of context-sensitive rewriting, where rewritable positions are defined by a finite string automaton that indicates the allowed prefixes. Controlled rewriting [10] is an extension of prefix-constrained rewriting, where rewritable positions are defined by a finite tree automaton that considers the entire term (i.e. not only prefixes).…”

“…Section 4 presents a transformation of any cntTRS into an ordinary TRS. The relationship with [10] is discussed in Section 4.3.…”