On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT

Chen, Yijia; Flum, Jörg

doi:10.1007/978-3-642-15205-4_18

Cited by 5 publications

(5 citation statements)

References 14 publications

(23 reference statements)

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“…Concerning the converse we note that Theorem 7.3 (i), (ii) implies: Adapting a mode of speech from [8], call an almost tally problem (Q, κ) slicewise monotone if (1 n , x) ∈ Q implies (1 m , x) ∈ Q for all x ∈ {0, 1} * and all n, m ∈ N with n < m. One can show that p-HALT is the hardest such problem in para-NP. This is an easy modification of the proof Lemma 3.10 and strengthens [8,Proposition 11]: Corollary 7.5. Every almost tally problem in para-NP that is slicewise monotone is reducible to p-HALT.…”

Section: Provability Of the Mrdp Theoremmentioning

confidence: 96%

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

Chen

Müller

Yokoyama

2018

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

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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.

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Section: Provability Of the Mrdp Theoremmentioning

confidence: 96%

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

Chen

Müller

Yokoyama

2018

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

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“…For the classes P and NP such listings were already considered and put to good use by Sadowski in [16]. This more general notion is also meaningful in the context of logics (for P and NP already remarked in [3]); loosely speaking, if LIST(C, TAUT, C ), then in the order-invariant logic corresponding to C we can axiomatize the classes of structures in C and we can show that there is an algorithm solving its satisfaction relation, which for fixed sentence is of type C . By this general approach we get further new insights (cf.…”

Section: Introductionmentioning

confidence: 93%

“…By methods used in [3] one can show that DTC inv already is a L-bounded logic for L if there is an algorithm deciding |= FO inv , which for fixed first-order ϕ requires logarithmic space. We shall prove this result in the full version of this paper.…”

Section: Hence a Only Needs Space O(log |N|)mentioning

confidence: 99%

Listings and Logics

Chen

Flum

2011

2011 IEEE 26th Annual Symposium on Logic in Computer Science

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There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In [4] we have shown that LFP inv , the "order-invariant least fixed-point logic LFP," captures P (on all finite structures) if and only if there is a listing of the Psubsets of the set TAUT of propositional tautologies. We are able to extend the result to listings of the L-subsets (NL-subsets) of TAUT and the logic DTC inv (TC inv ). As a byproduct we get that LFP inv captures P if DTC inv captures L.Furthermore, we show that the existence of a listing of the L-subsets of TAUT is equivalent to the existence of an almost space optimal algorithm for TAUT. To obtain this result we have to derive a space version of a theorem of Levin on optimal inverters. I . INTRODUCTIONIt is well-known that for standard complexity classes C (as L, NL and P) the existence of a logic capturing C is equivalent to the existence of a listing (or effective enumeration) of the classes of structures closed under isomorphism in C by means of Turing machines of type C that decide them (or equivalently, to such a listing of the classes of graphs closed under isomorphism). Recently [4] we have shown for C = P that such a listing exists if there is a listing of the P-subsets of the set TAUT of propositional tautologies; more explicitly, a listing of the subsets in P of TAUT by means of polynomial time Turing machines deciding them. Even more, it is shown in [4] that the following two statements are equivalent:(i) There is a listing of the P-subsets of TAUT.(ii) The logic LFP inv , the "order-invariant least fixed-point logic LFP," 1 captures P. The starting point for the investigations which led to this paper was the observation (cf. Proposition II.3) that one gets a listing of the P-subsets of TAUT if one assumes that there is a listing of its L-subsets. In particular, then LFP inv captures P. Thus, we asked ourselves whether then we even get that DTC inv , the "order-invariant deterministic transitive closure logic DTC," 1 captures L.At the end we realized that the equivalence of (i) and (ii) lifts to their L-analogues and to their NL-analogues. In 1 It is well-known that LFP captures P on ordered structures and that deterministic transitive closure logic DTC captures L on ordered structures. particular, LFP inv captures P if DTC inv captures L. Note that it is not known whether the existence of a logic capturing P is implied by the existence of a logic capturing L.A more general notion of listing turned out to be helpful. For complexity classes C and C we consider listings of the C-subsets of TAUT by means of Turing machines of type C ; we write LIST(C, TAUT, C ) if such a listing exists. For the classes P and NP such listings were already considered and put to good use by Sadowski in [16]. This more general notion is also meaningful in the context of logics (for P and NP already remarked in [3]); loosely speaking, if LIST(C, TAUT, C ), then in the order-invariant logic corresponding to C we can axiomatize the classe...

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“…Interestingly, each problem identified by Chen and Flum [5] as having the same complexity as coBHP under fixed parameter tractable reductions, such as the set of arithmetic statements φ with no proof of fewer than t steps, is also the resource-bounded version of a non-c.e. language.…”

Section: Speedup For Conp-complete Languagesmentioning

confidence: 99%

Speedup for natural problems and noncomputability

Monroe

2011

Theoretical Computer Science

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A resource-bounded version of the statement "no algorithm recognizes all non-halting Turing machines" is equivalent to an infinitely often (i.o.) superpolynomial speedup for the time required to accept any (paddable) coNPcomplete language and also equivalent to a superpolynomial speedup in proof length in propositional proof systems for tautologies, each of which implies P = NP. This suggests a correspondence between the properties "has no algorithm at all" and "has no best algorithm" which seems relevant to open problems in computational and proof complexity.

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On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT

Cited by 5 publications

References 14 publications

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

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

Listings and Logics

Speedup for natural problems and noncomputability

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