Information in Propositional Proofs and Algorithmic Proof Search

Krajíček, Jan

doi:10.1017/jsl.2021.75

Cited by 5 publications

(7 citation statements)

References 23 publications

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“…Note the first statement is by [5,Thm.2.4] equivalent to the non-existence of a time-optimal propositional proof search algorithm.…”

Section: Down To Propositional Logicmentioning

confidence: 99%

A proof complexity conjecture and the Incompleteness theorem

Krajíček¹

2023

Preprint

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Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function gT that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of gT intersects all infinite NP sets (i.e. whether it is a proof complexity generator hard for all proof systems).A propositional version of the construction shows that at least one of the following three statements is true:1. there is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm),2. E ⊆ P/poly, 3. there exists function h that stretches all inputs by one bit, is computable in sub-exponential time and its range Rng(h) intersects all infinite NP sets.

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“…Note the first statement is by [5,Thm.2.4] equivalent to the non-existence of a time-optimal propositional proof search algorithm.…”

Section: Down To Propositional Logicmentioning

confidence: 99%

A proof complexity conjecture and the Incompleteness theorem

Krajíček¹

2023

Preprint

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“…While Kt complexity is tightly related to optimal search algorithms (see [Kra21] for a recent application), K t is particularly useful in settings where maintaining a polynomial bound on the running time t is desired (see, e.g., [Hir18]). Antunes and Fortnow [AF09] introduced techniques that can be used to establish (conditional) coding theorems for K t and Kt.…”

Section: Kt(x) = Minmentioning

confidence: 99%

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

Lu¹,

Oliveira²,

Zimand³

2022

Preprint

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The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤ log(1/δ) + O(1). In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that if x can be efficiently sampled with probability δ then rKt(x) = O(log(1/δ)) + O(log n), where rKt denotes the randomized analogue of Levin's Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a O(log(1/δ)) bound instead of the informationtheoretic optimal log(1/δ).Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting. Our main contributions can be summarised as follows.

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“…Our second example follows [24, Remark 6.1] and concerns time-bounded Kolmogorov complexity. Recall that the complexity measure is the minimal size of a program that prints w in time at most (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The point is that a proof complexity generator with stretch produces strings w of complexity smaller than . For example, if g stretches n bits to bits and runs in p-time (which is also polynomial in n ) then for all size m strings and , In fact, as discussed in [24, Section 6.1], for a fixed polynomial time sufficient for the computation of g one can consider the universal Turing machine underlying the definition of as a generator itself 4 . Then for any pps P simulating EF, if some -formulas have short P -proofs (e.g., by proving tautologies expressing the lower bound ), so do some -formulas.…”

Section: Introductionmentioning

confidence: 99%

On the Existence of Strong Proof Complexity Generators

KRAJÍČEK

2023

Bull. symb. log

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Cook and Reckhow [5] pointed out that N P = coN P iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:• There exists a p-time function g stretching each input by one bit such that its range rng(g) intersects all infinite N P sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the -hardness, and show that one specific gadget generator is the -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite N P sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite N P sets.

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Information in Propositional Proofs and Algorithmic Proof Search

Cited by 5 publications

References 23 publications

A proof complexity conjecture and the Incompleteness theorem

A proof complexity conjecture and the Incompleteness theorem

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

On the Existence of Strong Proof Complexity Generators

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