Hard Instances of Algorithms and Proof Systems

Chen, Yijia; Flum, Jörg; Müller, Moritz

doi:10.1007/978-3-642-30870-3_13

Cited by 4 publications

(16 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then f ∈ L PV and we claim that there is a nonstandard n ∈ M such that f M {0, 1} n is a mad pseudo proof system in K (F n PV ). We can assume that ∀yFml( f (y)) holds in N and hence in M. So to get the madness property (7) it suffices to get M |= ψ(n) for the formula ψ(x) defined in (8). Now, for g ∈ L PV let B g, f : N → N map m ∈ N to For g ∈ L PV and standard > 0 let ϕ g, f, (x) be the formula…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…(b)⇒(c) and (d)⇒(a) are trivial. To prove (c)⇒(d) we proceed as in [, Proposition 3.2] using a padding function: a polynomial time computable function pad that maps a formula F and a string

y \in {0, 1}^{*}

to a formula

pad (F, y)

of length at least

| F | + | y |

that has the same satisfying assignments as F , and such that there are two polynomial time functions mapping any input of the form

pad (F, y)

to F and y , respectively. Let

double-struckA

be a sat ‐solver and assume (c).…”

Section: Hard Sequencesmentioning

confidence: 99%

“…Note that (H2) in the definition of a polynomial time computable hard sequence is equivalent to the statement that t A (x n ) is not n O(1) , matching the definition in[8].www.mlq-journal.orgC 2018 The Authors. Mathematical Logic Quarterly published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim…”

mentioning

confidence: 99%

See 2 more Smart Citations

A remark on pseudo proof systems and hard instances of the satisfiability problem

Maly

Müller

2018

Mathematical Logic Qtrly

Self Cite

View full text Add to dashboard Cite

We link two concepts from the literature, namely hard sequences for the satisfiability problem sat and so‐called pseudo proof systems proposed for study by Krajíček. Pseudo proof systems are elements of a particular nonstandard model constructed by forcing with random variables. We show that the existence of mad pseudo proof systems is equivalent to the existence of a randomized polynomial time procedure with a highly restrictive use of randomness which produces satisfiable formulas whose satisfying assignments are probably hard to find.

show abstract

Section: Proof Of Theorem 13mentioning

confidence: 99%

y \in {0, 1}^{*}

to a formula

pad (F, y)

of length at least

| F | + | y |

that has the same satisfying assignments as F , and such that there are two polynomial time functions mapping any input of the form

pad (F, y)

to F and y , respectively. Let

double-struckA

be a sat ‐solver and assume (c).…”

Section: Hard Sequencesmentioning

confidence: 99%

See 1 more Smart Citation

A remark on pseudo proof systems and hard instances of the satisfiability problem

Maly

Müller

2018

Mathematical Logic Qtrly

Self Cite

View full text Add to dashboard Cite

show abstract

“…Chen, Flum and Müller [CFM11] generalize this result from optimal acceptors to optimal algorithms. Graph (non)isomorphism and optimality up to permutations of vertices.…”

Section: Introductionmentioning

confidence: 77%

“…While it is easy to see that Messner's proof goes for randomized acceptors as well, it does not apply to computations with advice (where optimal proof systems exist [CK07], but no optimal acceptors are known), heuristic computations (where optimal acceptors do exist [HINS11], but no optimal proof systems are known), or the restricted notion of optimality used in this paper. Recently, it was shown that the existence of optimal acceptors is equivalent for all co -NP-complete languages [CFM11].…”

Section: Introductionmentioning

confidence: 99%

On an optimal randomized acceptor for graph nonisomorphism

Hirsch

Itsykson

2012

Information Processing Letters

View full text Add to dashboard Cite

An acceptor for a language L is an algorithm that accepts every input in L and does not stop on every input not in L. An acceptor Opt for a language L is called optimal if for every acceptor A for this language there exists polynomial p A such that for every x ∈ L, the running time time Opt (x) of Opt on x is bounded by p A (time A (x) + |x|) for every x ∈ L. (Note that the comparison of the running time is done pointwise, i.e., for every word of the language.)The existence of optimal acceptors is an important open question equivalent to the existence of p-optimal proof systems for many important languages [KP89, Sad99, Mes99]. Yet no nontrivial examples of languages in NP ∪ co -NP having optimal acceptors are known.In this short note we construct a randomized acceptor for graph nonisomorphism that is optimal up to permutations of the vertices of the input graphs, i.e., its running time on a pair of graphs (G 1 , G 2 ) is at most polynomially larger than the maximum (in fact, even the median) of the running time of any other acceptor taken over all permuted pairs (π 1 (G 1 ), π 2 (G 2 )). One possible motivation is the (pointwise) optimality in the class of acceptors based on graph invariants where the time needed to compute an invariant does not depend much on the representation of the input pair of nonisomorphic graphs.In fact, our acceptor remains optimal even if the running time is compared to the average-case running time over all permuted pairs. We introduce a natural notion of average-case optimality (not depending on the language of graph nonisomorphism) and show that our acceptor remains average-case optimal for any probability distribution on the inputs that respects permutations of vertices.

show abstract

Average-Case Hardness of Proving Tautologies and Theorems

Monroe¹

2022

Preprint

View full text Add to dashboard Cite

We consolidate two widely believed conjectures about tautologiesno optimal proof system exists, and most require superpolynomial size proofs in any system-into a p-isomorphism-invariant condition satisfied by all paddable coNP-complete languages or none. The condition is: for any Turing machine (TM) M accepting the language, P-uniform input families requiring superpolynomial time by M exist (equivalent to the first conjecture) and appear with positive upper density in an enumeration of input families (implies the second). In that case, no such language is easy on average (in AvgP) for a distribution applying non-negligible weight to the hard families.The hardness of proving tautologies and theorems is likely related. Motivated by the fact that arithmetic sentences encoding "string x is Kolmogorov random" are true but unprovable with positive density in a finitely axiomatized theory T (Calude and Jürgensen), we conjecture that any propositional proof system requires superpolynomial size proofs for a dense set of P-uniform families of tautologies encoding "there is no T proof of size ≤ t showing that string x is Kolmogorov random". This implies the above condition.The conjecture suggests that there is no optimal proof system because undecidable theories help prove tautologies and do so more efficiently as axioms are added, and that constructing hard tautologies seems difficult because it is impossible to construct Kolmogorov random strings. Similar conjectures that computational blind spots are manifestations of noncomputability would resolve other open problems.

show abstract

Hard Instances of Algorithms and Proof Systems

Cited by 4 publications

References 10 publications

A remark on pseudo proof systems and hard instances of the satisfiability problem

A remark on pseudo proof systems and hard instances of the satisfiability problem

On an optimal randomized acceptor for graph nonisomorphism

Average-Case Hardness of Proving Tautologies and Theorems

Contact Info

Product

Resources

About