Abstract. In this work the computational complexity of two simple string problems on compressed input strings is considered: the querying problem (What is the symbol at a given position in a given input string?) and the embedding problem (Can the first input string embedded into the second input string?). Straightline programs are used for text compression. It is shown that the querying problem becomes P-complete for compressed strings, while the embedding problem becomes hard for the complexity class Θ p 2. 1

Abstract. We study the compressed word problem: a variant of the word problem for finitely generated groups where the input word is given by a context-free grammar that generates exactly one string. We show that finite extensions and free products preserve the complexity of the compressed word problem. Also, the compressed word problem for a graph group can be solved in polynomial time. These results allow us to obtain new upper complexity bounds for the word problem for certain automorphism groups and group extensions. 1

ABSTRACT. Tight connections between leafs languages and strings compressed via straight-line programs (SLPs) are established. It is shown that the compressed membership problem for a language L iscompletefortheleaflanguageclassdefinedby L vialogspacemachines. Amoredifficult variant of the compressed membership problem for L is shown to be complete for the leaf language classdefinedby Lviapolynomialtimemachines. Asacorollary,afixedlinearvisiblypushdownlanguagewithaPSPACE-completecompressedmembershipproblemisobtained. ForXMLlanguages, the compressed membership problem is shown tobe coNP-complete. 1

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.

Abstract. It is shown that the compressed word problem for an HNNextension 〈H,t | t −1 at = ϕ(a)(a ∈ A) 〉 with A finite is polynomial time Turing-reducible to the compressed word problem for the base group H. An analogous result for amalgamated free products is shown as well. 1