Verifying proofs in constant depth

ACM Trans. Comput. Theory

Limaye

Mahajan

et al. 2016

Self Cite

A proof system for a language L is a function f such that Range(f ) is exactly L. In this paper, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC 0 ), or of O(log log n) depth but with O(1) alternations (poly log AC 0 ). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error-correction on a theorem-proof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular language have poly log AC 0 proof systems, and from a previous result (Beyersdorff et al, MFCS 2011, where NC 0 proof systems were first introduced), this is tight. Our technique also shows that Maj has poly log AC 0 proof system. We explore the question of whether Taut has NC 0 proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that both Undirected Reachability and Directed UnReachability have NC 0 proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC 0 ; they have SAC 0 or coSAC 0 proof systems. arXiv:1307.4897v1 [cs.CC] 18 Jul 2013 2 PreliminariesUnless otherwise stated, we consider only bounded fanin circuits over ∨, ∧, ¬.Definition 1 ([11]). A circuit family {C n } n>0 is a proof system for a language L if there is a function m : N −→ N such that for each n where L =n = ∅,

Section: Proof Systems For Regular Languagesmentioning

confidence: 96%

“…Using Lemma 1 and the known lower bound for Maj from [11], we can show that the following languages have no NC 0 proof systems:…”

Section: Pushing the Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Small Depth Proof Systems

ACM Trans. Comput. Theory

Limaye

Mahajan

et al. 2016

Self Cite

“…Namely, with d(u, v) = |u| + |v| − |u ∧ v|, where u ∧ v is the largest common prefix of u and v, the latter property is that d(τ (u), τ (v)) ≤ k × d(u, v), a strong form of uniform continuity. Continuity thus appears as a natural invariant when characterizing transductions-the forward behaviors of τ , that is, its images, are less relevant, as any NP problem is the image of Σ * under an AC 0 function [4].…”

Section: Introductionmentioning

confidence: 99%

A Circuit Complexity Approach to Transductions

Cadilhac

Lecture Notes in Computer Science

Ludwig

et al. 2015

Self Cite

We investigate the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, we first propose a framework of independent interest to express functions of variable output length using circuits, and argue for its pertinence. We then provide a general characterization of the set of transductions realizable by such circuits, relying on a notion of continuity. We deduce that it is decidable whether a transduction is definable in AC 0 and, assuming a well-established conjecture, the same for ACC 0 .

Small Depth Proof Systems

Mathematical Foundations of Computer Science 2013

Limaye

Mahajan

et al. 2013

Self Cite

Abstract. A proof system for a language L is a function f such that Range(f ) is exactly L. In this paper, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by bounded fanin circuits of constant depth (NC 0 ), or of O(log log n) depth but with O(1) alternations (poly log AC 0 ). Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error-correction on a theorem-proof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular language have poly log AC 0 proof systems, and from a previous result (Beyersdorff et al, MFCS 2011, where NC 0 proof systems were first introduced), this is tight. Our technique also shows that Maj has poly log AC 0 proof system. We explore the question of whether Taut has NC 0 proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that both Undirected Reachability and Directed UnReachability have NC 0 proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of AC 0 ; they have SAC 0 or coSAC 0 proof systems.