55 research outputs found
Computing downward closures for stacked counter automata
The downward closure of a language of words is the set of all (not
necessarily contiguous) subwords of members of . It is well known that the
downward closure of any language is regular. Although the downward closure
seems to be a promising abstraction, there are only few language classes for
which an automaton for the downward closure is known to be computable.
It is shown here that for stacked counter automata, the downward closure is
computable. Stacked counter automata are finite automata with a storage
mechanism obtained by \emph{adding blind counters} and \emph{building stacks}.
Hence, they generalize pushdown and blind counter automata.
The class of languages accepted by these automata are precisely those in the
hierarchy obtained from the context-free languages by alternating two closure
operators: imposing semilinear constraints and taking the algebraic extension.
The main tool for computing downward closures is the new concept of Parikh
annotations. As a second application of Parikh annotations, it is shown that
the hierarchy above is strict at every level.Comment: 34 pages, 1 figure; submitte
The Complexity of Downward Closure Comparisons
The downward closure of a language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of every language is regular. Moreover, recent results show that downward closures are computable for quite powerful system models.
One advantage of abstracting a language by its downward closure is that then equivalence and inclusion become decidable. In this work, we study the complexity of these two problems. More precisely, we consider the following decision problems: Given languages K and L from classes C and D, respectively, does the downward closure of K include (equal) that of L?
These problems are investigated for finite automata, one-counter automata, context-free grammars, and reversal-bounded counter automata. For each combination, we prove a completeness result either for fixed or for arbitrary alphabets. Moreover, for Petri net languages, we show that both problems are Ackermann-hard and for higher-order pushdown automata of order k, we prove hardness for complements of nondeterministic k-fold exponential time
The Complexity of Knapsack in Graph Groups
Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In LohreyZ16 the authors proved that for each graph group, the knapsack problem can be solved in NP. Here, we determine the exact complexity of the problem for every graph group. While the problem is TC^0-complete for complete graphs, it is LogCFL-complete for each (non-complete) transitive forest. For every remaining graph, the problem is NP-complete
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