PTIME Queries Revisited

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Müller

Yokoyama

2018

The complexity of the parameterized halting problem for nondeterministic Turing machines p-HALT is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-HALT is in para-AC 0 , the parameterized version of the circuit complexity class AC 0 , then AC 0 , or equivalently, (+, ×)-invariant FO, has a logic. Although it is widely believed that p-HALT / ∈ para-AC 0 , we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊆ LINH. Here, LINH denotes the linear time hierarchy.On the other hand, we suggest an approach toward proving NE ⊆ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-DavisPutnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊆ LINH. Interestingly, central to this result is a para-AC 0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.

“…Combined with (17) and (19), for every string s ∈ {0, 1} * with |s| ≥ h(|M h | + |M Q | + e) S(s) |= ϕ I ∧ ϕ I ⇐⇒ s = 1 h(m) for some m ≥ |M h | + |M Q | + e and s ∈ Q,…”

Section: Claimmentioning

confidence: 99%

A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

Müller

Yokoyama

2018

“…By a reformulation of the statement "L ≤ is a P-bounded logic for P" due to Nash et al [14] (see [3] for a proof), we get: …”

Section: Lemmamentioning

confidence: 99%

On p-Optimal Proof Systems and Logics for PTIME

Automata, Languages and Programming

Flum

2010

“…In a different context, namely when trying to show that a certain logic L ≤ for PTIME (introduced in [7]) does not satisfy some effectivity condition, Nash et al introduced implicitly [12] (and this was done explicitly in [1]) the parameterized acceptance problem p-Acc ≤ for nondeterministic Turing machines:…”

Section: P-gödelmentioning

confidence: 99%

On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT

Flum

2010

Computer Science Logic

Abstract. For a reasonable sound and complete proof calculus for first-order logic consider the problem to decide, given a sentence ϕ of first-order logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.