arXiv subjects

Jan Krajicek

Publications and source records attributed to Jan Krajicek.

At least 19 recordsLinked to original sources

arXiv2026

A proof complexity perspective on effectively zero-knowledge proofs

Ilango (FOCS 2025) invented effectively zero-knowledge proofs, a new variant of zero-knowledge. We reformulate it in the language of logic and give simple proofs (under the same assumptions as Ilango (FOCS 2025)) of its existence and of the key property defined in Ilango (FOCS 2025) that it is "indistinguishable from true" (that property is in Ilango (FOCS 2025) a part of the definition of the prover, not its consequence). Using the theory of proof complexity generators we show that the concept can be turned it into a genuinely zero-knowledge proofs, assuming a conjecture from the theory about the existence of a hard generator and allowing the parties to share a common random string.

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arXiv2026

Extended Frege proofs, circuits and rewriting

Inspired by a statement about Extended Frege proof systems by Jain and Jin (FOCS 2022) we prove that: - there is a p-time binary relation $\approx$ between circuits that implies their logical equivalence, - the relation $\approx$ implies that each of the two circuits can be rewritten into the other one by possibly deleting some gates and adding at most seven new gates, - if the equivalence $C \equiv D$ has a size $s$ proof in an Extended Frege or a Circuit Frege proof system then there is a chain of circuits $E_i$ $$ C = E_0 \approx \dots \approx E_t = D $$ with $t \le s^{O(1)}$.

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arXiv2026

On $NP \cap coNP$ proof complexity generators

Motivated by the theory of proof complexity generators we consider the following $Σ^p_2$ search problem $\mbox{DD}_P$ determined by a propositional proof system $P$: given a $P$-proof $π$ of a disjunction $\bigvee_i α_i$, no two $α_i$ having an atom in common, find $i$ such that $α_i \in \mbox{TAUT}$. We formulate a hypothesis (ST) that for some strong proof system $P$ the problem $\mbox{DD}_P$ is not solvable in the student-teacher model with a $p$-time student and a constant number of rounds. The hypothesis follows from the existence of hard one-way permutations. We prove, using a model-theoretic assumption, that (ST) implies $NP \neq coNP$. The assumption concerns the existence of extensions of models of a bounded arithmetic theory and it is open at present if it holds.

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arXiv2026

Failure of the strong feasible disjunction property

A propositional proof system $P$ has the strong feasible disjunction property iff there is a constant $c \geq 1$ such that whenever $P$ admits a size $s$ proof of $\bigvee_i α_i$ with no two $α_i$ sharing an atom then one of $α_i$ has a $P$-proof of size $\le s^c$. We combine the work of Ilango (2025) and Ren et al. (2025) with the gadget proof complexity generator of K. (2007) and rule out the property for strong enough proof systems under the following two hypotheses: - there exists a language in class E that requires exponential size circuits even if they are allowed to query an NP oracle, - there exists a P/poly demi-bit in the sense of Rudich (1997).

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arXiv2023

On the existence of strong proof complexity generators

Cook and Reckhow 1979 pointed out that NP is not closed under complementation iff there is no propositional proof system that admits polynomial size proofs of all tautologies. Theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus at a conjecture from K.2004 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 exist a p-time function $g$ stretching each input by one bit such that its range intersects all infinite NP 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 K.2009 is a good candidate for $g$. We define a new hardness property of generators, the $\bigvee$-hardness, and shows that one specific gadget generator is the $\bigvee$-hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite NP sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite NP sets.

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arXiv2023

Extended Nullstellensatz proof systems

For a finite set $\cal F$ of polynomials over fixed finite prime field of size $p$ containing all polynomials $x^2 - x$ a Nullstellensatz proof of the unsolvability of the system $$ f = 0\ ,\ \mbox{ all } f \in {\cal F} $$ in the field is a linear combination $\sum_{f \in {\cal F}} \ h_f \cdot f$ that equals to $1$ in the ring of polynomails. The measure of complexity of such a proof is its degree: $\max_f deg(h_f f)$. We study the problem to establish degree lower bounds for some {\em extended} NS proof systems: these systems prove the unsolvability of $\cal F$ by proving the unsolvability of a bigger set ${\cal F}\cup {\cal E}$, where set $\cal E$ may use new variables $r$ and contains all polynomials $r^p - r$, and satisfies the following soundness condition: -- - Any $0,1$-assignment $\overline a$ to variables $\overline x$ can be appended by an assignment $\overline b$ to variables $\overline r$ such that for all $g \in {\cal E}$ it holds that $g(\overline a, \overline b) = 0$. We define a notion of pseudo-solutions of $\cal F$ and prove that the existence of pseudo-solutions with suitable parameters implies lower bounds for two extended NS proof systems ENS and UENS defined in Buss et al. (1996/97). Further we give a combinatorial example of $\cal F$ and candidate pseudo-solutions based on the pigeonhole principle.

math.LO↗

arXiv2023

A proof complexity conjecture and the Incompleteness theorem

Given a sound first-order p-time theory $T$ capable of formalizing syntax of first-order logic we define a p-time function $g_T$ 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 $g_T$ 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: - there is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm), - $E \not\subseteq P/poly$, - 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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arXiv2021

Information in propositional proofs and algorithmic proof search

We study from the proof complexity perspective the (informal) proof search problem: Is there an optimal way to search for propositional proofs? We note that for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists. To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system $P$ we attach {\bf information-efficiency function} $i_P(τ)$ assigning to a tautology a natural number, and we show that: - $i_P(τ)$ characterizes time any $P$-proof search algorithm has to use on $τ$ and that for a fixed $P$ there is such an information-optimal algorithm, - a proof system is information-efficiency optimal iff it is p-optimal, - for non-automatizable systems $P$ there are formulas $τ$ with short proofs but having large information measure $i_P(τ)$. We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P(τ)$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search.

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arXiv2020

Small circuits and dual weak PHP in the universal theory of p-time algorithms

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending $n$ bits to $m \geq n^2$ bits violates the dual weak pigeonhole principle: every string $y$ of length $m$ equals the value of the function for some $x$ of length $n$. The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size $n^d$.

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arXiv2020

Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$.

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arXiv2019

On monotone circuits with local oracles and clique lower bounds

We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs $y_i = y_i(\vec{x})$ that can perform unstructured computations on the input string $\vec{x}$. Let $μ\in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n,k}, V_{n,k} \subseteq \{0,1\}^m$ be the set of $k$-cliques and the set of complete $(k-1)$-partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-$2$ monotone circuits with local oracles, we show that the size of the smallest circuits separating $U_{n,3}$ (triangles) and $V_{n,3}$ (complete bipartite graphs) undergoes two phase transitions according to $μ$. 2. For $5 \leq k(n) \leq n^{1/4}$, arbitrary depth, and $μ\leq 1/50$, we prove that the monotone circuit size complexity of separating the sets $U_{n,k}$ and $V_{n,k}$ is $n^{Θ(\sqrt{k})}$, under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of $k$-clique obtained by Alon and Boppana (1987).

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arXiv2019

The Cook-Reckhow definition

The Cook-Reckhow 1979 paper defined the area of research we now call Proof Complexity. There were earlier papers which contributed to the subject as we understand it today, the most significant being Tseitin's 1968 paper, but none of them introduced general notions that would allow to make an explicit and universal link between lengths-of-proofs problems and computational complexity theory. In this note we shall highlight three particular definitions from the paper: of proof systems, p-simulations and the pigeonhole principle formula, and discuss their role in defining the field. We will also mention some related developments and open problems.

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arXiv2018

Randomized feasible interpolation and monotone circuits with a local oracle

We generalize the feasible interpolation theorem for semantic derivations from K.(1997) by allowing randomized protocols (protocols in the sense of K.(1997). We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for the monotone Karchmer-Wigderson multi-function $KW^m[U,V]$ making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of $U, V$, $U$ closed upwards, yields a small randomized protocol for and hence a small monotone CLO separating the sets. To establish a lower bound for monotone CLOs separating two NP sets, one closed upwards, is an open problem. This research is motivated by the open problem to establish a lower bound for proof system $R(LIN)$ operating with clauses formed by linear Boolean functions over the 2-element field. The new randomized feasible interpolation applies to this proof system and also to the semantic versions of the cutting planes proof system CP (as randomized protocols efficiently simulate protocols with small real communication complexity of K.(1998)), to small width resolution over CP, R(CP), and to random resolution RR of Buss, Kolodziejczyk and Thapen (2014).

math.LO↗

arXiv2017

Unprovability of circuit upper bounds in Cook's theory PV

We establish unconditionally that for every integer $k \geq 1$ there is a language $L \in \mbox{P}$ such that it is consistent with Cook's theory PV that $L \notin Size(n^k)$. Our argument is non-constructive and does not provide an explicit description of this language.

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arXiv2017

A feasible interpolation for random resolution

Random resolution, defined by Buss, Kolodziejczyk and Thapen (JSL, 2014), is a sound propositional proof system that extends the resolution proof system by the possibility to augment any set of initial clauses by a set of randomly chosen clauses (modulo a technical condition). We show how to apply the general feasible interpolation theorem for semantic derivations of Krajicek (JSL, 1997) to random resolution. As a consequence we get a lower bound for random resolution refutations of the clique-coloring formulas.

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arXiv2015

Consistency of circuit evaluation, extended resolution and total NP search problems

We consider sets $Γ(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF shortly refutable in Extended R, ER, can be easily reduced to an instance of $Γ(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that $Γ(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V^1_1$. We use the ideas of implicit proofs to define from $Γ(0,s,k)$ a non-relativized NP search problem $iΓ$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V^1_2$. The reductions are definable in $S^1_2$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

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arXiv2015

Expansions of pseudofinite structures and circuit and proof complexity

I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a suitable expansion would imply that, assuming a one-way permutation exists, the computational class NP is not closed under complementation.

math.LO↗

arXiv2014

A reduction of proof complexity to computational complexity for $AC^0[p]$ Frege systems

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime $p$ (the so called $AC^0[p]$ Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over the finite filed with $p$ elements.

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