The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group G and takes as input group elements g1, . . . , gn, g ∈ G and asks whether there are x1, . . . , xn ≥ 0 withWe study the knapsack problem for wreath products G ≀ H of groups G and H.Our main result is a characterization of those wreath products G ≀ H for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors G and H. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem.Moreover, we apply our main result to H3(Z), the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1, q) for q ≥ 1. First, we show that the knapsack problem is undecidable for G ≀ H3(Z) for any G ̸ = 1. This implies that for G ̸ = 1 and for infinite and virtually nilpotent groups H, the knapsack problem for G ≀ H is decidable if and only if H is virtually abelian and solvability of systems of exponent equations is decidable for G. Second, we show that the knapsack problem is decidable for G ≀ BS(1, q) if and only if solvability of systems of exponent equations is decidable for G.
ACM Subject ClassificationTheory of computation → Problems, reductions and completeness; Theory of computation → Theory and algorithms for application domains Keywords and phrases knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups Digital Object Identifier 10.4230/LIPIcs...
Automatic structures are infinite structures that are finitely represented by synchronized finite-state automata. This paper concerns specifically automatic structures over finite words and trees (ranked/unranked). We investigate the "directed version" of Ramsey quantifiers, which express the existence of an infinite directed clique. This subsumes the standard "undirected version" of Ramsey quantifiers. Interesting connections between Ramsey quantifiers and two problems in verification are firstly observed: (1) reachability with Büchi and generalized Büchi conditions in regular model checking can be seen as Ramsey quantification over transitive automatic graphs (i.e., whose edge relations are transitive), (2) checking monadic decomposability (a.k.a. recognizability) of automatic relations can be viewed as Ramsey quantification over co-transitive automatic graphs (i.e., the complements of whose edge relations are transitive). We provide a comprehensive complexity landscape of Ramsey quantifiers in these three cases (general, transitive, cotransitive), all between NL and EXP. In turn, this yields a wealth of new results with precise complexity, e.g., verification of subtree/flat prefix rewriting, as well as monadic decomposability over tree-automatic relations. We also obtain substantially simpler proofs, e.g., for NL complexity for monadic decomposability over word-automatic relations (given by DFAs).
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