We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semi-linear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.

We prove three new undecidability results for computational mechanisms over finite trees: There is a linear, ultra-shallow, noetherian and strongly confluent rewrite system R such that the 9 8 -fragment of the first-order theory of one-step-rewriting by R is undecidable; the emptiness problem for tree automata with equality tests between cousins is undecidable; and the 9 8 - fragment of the first-order theory of set constraints with the union operator is undecidable. The common feature of these three computational mechanisms is that they allow us to describe the set of first-order terms that represent grids. We extend our representation of grids by terms to a representation of linear two-dimensional patterns by linear terms, which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results. 1 Introduction The grid structure provides convenient means for encoding computation sequences of Turing machines. A classical encoding...

: The goal of this paper is both to give a E-unification procedure that always terminates, and to decide unifiability. For this, we assume that the equational theory is specified by a confluent and constructor-based rewrite system, and that four additional restrictions are satisfied. We give a procedure that represents the (possibly infinite) set of solutions thanks to a tree tuple synchronized grammar, and that can decide unifiability thanks to an emptiness test. Moreover we show that if only three of the four additional restrictions are satisfied then unifiability is undecidable. 1 Introduction First order E-unification [29] is a tool that plays an important role in automated deduction, in particular in functional logic programming and for solving symbolic constraints (see [4] for an extensive survey of the area). It consists in finding instances to variables that make two terms equal modulo an equational theory given by a set of equalities, i.e. it amounts to solve an equation (ca...

Abstract. We prove that the properties of reachability, joinability and confluence are undecidable for flat TRSs. Here, a TRS is flat if the heights of the left and right-hand sides of each rewrite rule are at most one. Key words: Term rewriting system, Decision problem, Confluence, Flat. 1

This paper describes several classes of term rewriting systems (TRS’s) where narrowing has a finite search space and is still (strongly) complete as a mechanism for solving reachability goals. These classes do not assume confluence of the TRS. We also ascertain purely syntactic criteria that suffice to ensure the termination of narrowing and include several subclasses of popular TRS’s such as right-linear TRS’s, almost orthogonal TRS’s, topmost TRS’s, and left-flat TRS’s. Our results improve and/or generalize previous criteria in the literature regarding narrowing termination.

We give a set of inference rules for E-unification, similar to the inference rules for Syntactic Mutation. If the E is finitely saturated by paramodulation, then we can block certain terms from further inferences. Therefore,