Equivalence Problems for Circuits over Sets of Natural Numbers

Glaßer, Christian; Herr, Katrin; Reitwießner, Christian; Travers, Stephen; Waldherr, Matthias

doi:10.1007/s00224-008-9144-8

Cited by 6 publications

(16 citation statements)

References 10 publications

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“…Our contribution to questions from previous work. By the equivalence mentioned above, our bounds for EC(∪, ∩, , +, ×) improve the bounds for the problems MC(∪, ∩, , +, ×) [22] and EQ(∪, ∩, , +, ×) [11] as follows. The lower bound is raised from NEXP to L NEXP and the upper bound is slightly reduced from R T (Σ 1 ) to R tt (Σ 1 ).…”

Section: :3mentioning

confidence: 62%

“…We prove that PIT is logspace many-one equivalent to MC(∩, +, ×) studied in [22], MC Z (+, ×), MC Z (∩, +, ×) studied in [32], and EQ(+, ×) studied in [11]. This characterizes the complexity of these problems and shows that the question for improved bounds is equivalent to a well-studied, open problem in algebraic computing complexity.…”

Section: :3mentioning

confidence: 82%

“…Travers [32] and Breunig [6] considered membership problems for circuits over integers (MC Z ) and positive integers (MC N + ), respectively. Glaßer et al [11] investigated equivalence problems for circuits over sets of natural numbers (EQ), i.e., the problem of deciding whether two given circuits compute the same set. Satisfiability problems for circuits over sets of natural numbers, studied by Glaßer et al [13], are a generalization of the membership problems investigated by McKenzie and Wagner [22]: Here the circuits can have unassigned input gates.…”

Section: + ∩mentioning

confidence: 99%

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Emptiness problems for integer circuits

Barth

Beck

Dose

et al. 2020

Theoretical Computer Science

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We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(∪, ∩, , +, ×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(∪, ∩, , +, ×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity: membership problem MC(∩, +, ×) studied by McKenzie and Wagner 2007 integer membership problems MC Z (+, ×) and MC Z (∩, +, ×) studied by Travers 2006 equivalence problem EQ(+, ×) studied by Glaßer et al. 2010

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Section: :3mentioning

confidence: 62%

Section: :3mentioning

confidence: 82%

Section: + ∩mentioning

confidence: 99%

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Emptiness problems for integer circuits

Barth

Beck

Dose

et al. 2020

Theoretical Computer Science

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“…Decision problems for integer expressions have been studied for more than 40 years: Stockmeyer and Meyer [31] showed that for expressions over {+, ∪} compressed membership is NP-complete and equivalence is Π 2 P-complete (universality is, of course, trivial). For recent results on such problems with operations from {+, ∪, ∩, ×, }, see McKenzie and Wagner [23] and Glaßer et al [8].…”

Section: Universality Of Unpdamentioning

confidence: 99%

“…Remark. With circuits instead of formulae (see also [23] and [8]) we would not need doubling. Furthermore, we only use ×N on fixed numbers, so instead we could use any feature for expressing an arithmetic progression with fixed common difference.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Unary Pushdown Automata and Straight-Line Programs

Chistikov

Majumdar

2014

Automata, Languages, and Programming

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Abstract. We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs-one for the prefix, one for the lasso-that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNPhardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is Π2P-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply Π2P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards.

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Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

Glaßer

Jönsson

Martin³

2016

Pursuit of the Universal

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe study interactions between Skolem Arithmetic and certain classes of Circuit Satisfiability and Constraint Satisfaction Problems (CSPs). We revisit results of Glaßer et al. [1] in the context of CSPs and settle the major open question from that paper, finding a certain satisfiability problem on circuits-involving complement, intersection, union and multiplication-to be decidable. This we prove using the decidability of Skolem Arithmetic. Then we solve a second question left open in [1] by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection, union and multiplication. We continue by studying first-order expansions of Skolem Arithmetic without constants, (N; ×), as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.

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Equivalence Problems for Circuits over Sets of Natural Numbers

Cited by 6 publications

References 10 publications

Emptiness problems for integer circuits

Emptiness problems for integer circuits

Unary Pushdown Automata and Straight-Line Programs

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

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