More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding

Fröhlich, Andreas; Kovásznai, Gergely; Biere, Armin

doi:10.1007/978-3-642-38536-0_33

Cited by 10 publications

(5 citation statements)

References 18 publications

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“…Completeness results for various bit-vector logics and encodings. This is the table presented by Fröhlich et al[15] extended by the result proved in this paper.…”

supporting

confidence: 79%

On the complexity of the quantified bit-vector arithmetic with binary encoding

Jonas

Strejček

2018

Information Processing Letters

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We study the precise computational complexity of deciding satisfiability of first-order quantified formulas over the theory of fixed-size bit-vectors with binary-encoded bit-widths and constants. This problem is known to be in EXPSPACE and to be NEXPTIME-hard. We show that this problem is complete for the complexity class AEXP(poly) -the class of problems decidable by an alternating Turing machine using exponential time, but only a polynomial number of alternations between existential and universal states.

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“…Completeness results for various bit-vector logics and encodings. This is the table presented by Fröhlich et al[15] extended by the result proved in this paper.…”

supporting

confidence: 79%

On the complexity of the quantified bit-vector arithmetic with binary encoding

Jonas

Strejček

2018

Information Processing Letters

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“…• It is interesting that restricting shifts to be used only with c = 1 causes the complexity to drop to PSpace-completeness, as being proved for QF_BV 1 in [17].…”

Section: Fragmentsmentioning

confidence: 96%

“…After proving the computation complexity of certain logics, the natural question arises: Are there any practically reasonable fragments which have lower complexity? [17,18] investigates how the set of bit-vector operators used in formulas affects the computational complexity. We defined three fragments:…”

Section: Fragmentsmentioning

confidence: 99%

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How Hard is Bit-Precise Reasoning?

Kovásznai¹

2018

Proceedings of the 10th International Conference on Applied Informatics

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Formal verification of hardware and software is important in the case of critical systems, as shown by several accidents due to such problems as overflow, rounding error, etc. Bit-precise reasoning over bit-vectors is a fundamental tool for attacking such verification problems. But how hard is reasoning over bit-vector logics? It is essential to answer this question before inventing solving approaches for bit-vector logics. The talk gives an overview on the complexity of bit-vector logics. Complexity depends on what bit-vector operators are used and whether quantifiers are permitted. For instance, the quantifier-free bit-vector logic (QF_BV) with all the common operators is NEXPTIME-complete, which seems extremely high considering the fact that QF_BV is commonly used in hardware verification. We will investigate how that complexity can be reduced to NPcompleteness or PSPACE-completeness by restricting the set of operators. Quantifiers are often used in software verification problems such as termination analysis and invariant synthesis. The bit-vector logic with quantifiers and uninterpreted functions (UFBV) is 2-NEXPTIME-complete. Interestingly, as it was proved recently, that logic without uninterpreted functions (BV) is AEXP(poly)-complete.

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“…Despite a wide literature studying the complexity of single theories or of families of theories (e.g. [21,20,19,17,10,7,15,14,11,8,13,5]) and some more general work on complexity of T -solving [3,21,20], we are not aware of any previous work explicitly addressing NP-hardness of T -solving for a generic theory T .…”

Section: Introductionmentioning

confidence: 99%

Colors Make Theories Hard

Sebastiani

2016

Automated Reasoning

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The satisfiability problem for conjunctions of quantifier-free literals in first-order theories T of interest-"T-solving" for short-has been deeply investigated for more than three decades from both theoretical and practical perspectives, and it is currently a core issue of state-of-the-art SMT solving. Given some theory T of interest, a key theoretical problem is to establish the computational (in)tractability of T-solving, or to identify intractable fragments of T. In this paper we investigate this problem from a general perspective, and we present a simple and general criterion for establishing the NP-hardness of Tsolving, which is based on the novel concept of "colorer" for a theory T. As a proof of concept, we show the effectiveness and simplicity of this novel criterion by easily producing very simple proofs of the NP-hardness for many theories of interest for SMT, or of some of their fragments.

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More on the Complexity of Quantifier-Free Fixed-Size Bit-Vector Logics with Binary Encoding

Cited by 10 publications

References 18 publications

On the complexity of the quantified bit-vector arithmetic with binary encoding

On the complexity of the quantified bit-vector arithmetic with binary encoding

How Hard is Bit-Precise Reasoning?

Colors Make Theories Hard

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