We express dominance constraints in the once-only nesting fragment of stratified context unification, which therefore is NP-complete.

Abstract. Monadic second-order unification is second-order unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete, where we use the technique of compressing solutions using singleton context-free grammars. We prove that monadic second-order matching is also NP-complete.

Second-Order Unification is undecidable even for very specialized fragments. The signature plays a crucial role in these fragments. If all symbols are monadic, then the problem is NP-complete, whereas it is enough to have just one binary constant to lose decidability. In this work we reduce Second-Order Unification to Second-Order Unification with a signature that contains just one binary function symbol and constants. The reduction is based on partially currying the equations by using the binary function symbol for explicit application @. Our work simplifies the study of Second-Order Unification and some of its variants, like Context Unification and Bounded Second-Order Unification.

Abstract. The word-problem for a finite set of equational axioms between ground terms is the question whether for terms s, t the equation s = t is a consequence. We consider this problem under grammar based compression of terms, in particular compression with singleton tree grammars (STGs) and with directed acyclic graphs (DAGs) as a special case. We show that given a DAG-compressed ground and reduced term rewriting system T, the T-normal form of an STG-compressed term s can be computed in polynomial time, and hence the T-word problem can be solved in polynomial time. This implies that the word problem of STG-compressed terms w.r.t. a set of DAG-compressed ground equations can be decided in polynomial time. If the ground term rewriting system (gTRS) T is STG-compressed, we show NP-hardness of T-normal-form computation. For compressed, reduced gTRSs we show a PSPACE upper bound on the complexity of the normal form computation of STGcompressed terms. Also special cases are considered and a prototypical implementation is presented.

Bounded Second-Order Unification is a decidable variant of undecidable Second-Order Unification. Stratified Context Unification is a decidable restriction of Context Unification, whose decidability is a long-standing open problem. This paper is a join of two separate previous, preliminary papers on NP-completeness of Bounded Second-Order Unification and Stratified Context Unification. It clarifies some omissions in these papers, joins the algorithmic parts that construct a minimal solution, and gives a clear account of a method of using singleton tree grammars for compression that may have potential usage for other algorithmic questions in related areas.