# Jan Krajícek's research while affiliated with Math University and other places

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## Publications (87)

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 i...

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 m...

The working conjecture from K'04 that there is a proof complexity generator hard for all proof systems 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 $rng(g)$ intersects all infinite NP sets. W...

We study from the proof complexity perspective the (informal) proof search problem (cf. [17, Sections 1.5 and 21.5]): • Is there an optimal way to search for propositional proofs?
We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexi...

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 bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit...

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 alg...

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 to the value of the function for some $x$ of length $n$. The funct...

We prove a limitation on a variant of the KPT theorem proposed for propositional proof systems by Pich and Santhanam 2020, for all proof systems that prove the disjointness of two NP sets that are hard to distinguish.

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 pr...

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-e...

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 $\mu \in [0,1]$ be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions $y_i(\vec{x})$, and $U_{n...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the eleventh publication in the Lecture Notes in Logic series, collects the proceedings of...

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...

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.

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 sema...

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 AC0[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...

We consider sets $\Gamma(n,s,k)$ 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 $\Gamma(0,s,k)$ (with $s,k$ depending on
the size of the ER-refutation) and, in particul...

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...

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 linear maps on the vector space of...

It is well known (cf. Krajíček and Pudlák [‘Propositional proof systems, the consistency of first order theories and the complexity
of computations’, J. Symbolic Logic 54 (1989) 1063–1079]) that a polynomial time algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1(k...

A method how to construct Boolean-valued models of some fragments of
arithmetic was developed in Krajicek (2011), with the intended applications in
bounded arithmetic and proof complexity. Such a model is formed by a family of
random variables defined on a pseudo-finite sample space. We show that under a
fairly natural condition on the family (call...

For an NP intersect coNP function g of the Nisan-Wigderson type and a string
b outside its range we consider a two player game on a common input a to the
function. One player, a computationally limited Student, tries to find a bit of
g(a) that differs from the corresponding bit of b. He can query a
computationally unlimited Teacher for the witnesse...

We apply classical proof complexity ideas to transfer lengths-of-proofs lower bounds for a propositional proof system P into examples of hard unsatisfiable formulas for a class Alg(P)Alg(P) of SAT algorithms determined by P. The class Alg(P)Alg(P) contains those algorithms M for which P proves in polynomial size tautologies expressing the soundness...

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τ(g) b expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system...

A Ramsey statement denoted \({n \longrightarrow (k)^2_2}\) says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n
k
) and with terms of size \({\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}\) . Let r
k
be...

We shall discuss several situations in which it is possible to extract from a proof, be it a proof in a first-order theory
or a propositional proof, some feasible computational information about the theorem being proved. This includes extracting
feasible algorithms, deterministic or interactive, for witnessing an existential quantifier, a uniform f...

Let L be a first-order language and Φ and Ψ two L -sentences that cannot be satisfied simultaneously in any finite L -structure. Then obviously the following principle Chain L ,Φ,Ψ ( n, m ) holds: For any chain of finite L -structures C 1 , …, C m with the universe [ n ] one of the following conditions must fail:
For each fixed L and parameters n,...

We prove the following results: (i) PV proves NP⊆P/poly iff PV proves
coNP⊆ NP/O(1). (ii) If PV proves NP⊆P/poly then PV proves that
the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S_21 proves
NP⊆P/poly iff S21 proves coNP⊆ NP/O(log n). (iv) If S_21
proves NP⊆P/poly then S21 proves that the Polynomial Hierarchy collapses to
PNP[l...

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories T2 2 and T3 2.

We prove that there is a polynomial time substitution (y1,…,yn):=g(x1,…,xk)(y1,…,yn):=g(x1,…,xk) with k≪nk≪n such that whenever the substitution instance A(g(x1,…,xk))A(g(x1,…,xk)) of a 3DNF formula A(y1,…,yn)A(y1,…,yn) has a short resolution proof it follows that A(y1,…,yn)A(y1,…,yn) is a tautology. The qualification “short” depends on the paramet...

We prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the ha...

We prove lower bounds for a proof system having exponential speed-up over both polynomial calculus and constant-depth Frege systems in DeMorgan language.

A fundamental problem about the strength of non-deterministic computations is the problem whether the complexity class NP{\cal N}{\cal P} is closed under complementation. The set TAUT (w.l.o.g. a subset of {0,1}*) of propositional tautologies (in some fixed, complete language, e.g. DeMorgan language) is coNP{\cal N}{\cal P}-complete. The above prob...

We consider exponentially large finite relational structures (with the universe {0, 1} n ) whose basic relations are computed by polynomial size ( n O (1) ) circuits. We study behaviour of such structures when pulled back by P/poly maps to a bigger or to a smaller universe. In particular, we prove that:
1. If there exists a P/poly map g : {0, 1} n...

The original impetus for writing this expository note was an idea of J. Nešetřil to edit a series of articles (in a series edited at the Institute for Theoretical Informatics at the Charles University in Prague) on the state of Theoretical Computer Science (TCS). Additional excuses were a lecture on the P/NP problem I gave during the celebrations o...

We define the notion of a combinatorics of a first order structure, and a relation of covering between first order structures and propositional proof systems. Namely, a first order structure M
combinatorially satisfies an L-sentence Φ iff Φ holds in all L-structures definable in M. The combinatorics Comb(M) of M is the set of all sentences combinat...

This article is a continuation of our search for tautologies that are hard even for strong propositional proof systems like EF. cf. [14, 15]. The particular tautologies we study, the τ-formulas. are obtained from any /poly map g; they express that a string is outside of the range of g, Maps g considered here are particular pseudorandom generators....

We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardin...

We study diagonalization in the context of implicit proofs of [10]. We prove that at least one of the following three conjectures is true: There is a function f: {0, 1}* → {0, 1} computable in E that has circuit complexity 2Ω(n). NP ≠ co NP. There is no p-optimal propositional proof system. We note that a variant of the statement (either NP ≠ co NP...

We link the Dehn function of finitely presented groups to the length-of-proofs function in propositional proof complexity.

We describe a general method how to construct from a prepositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory and hence, in parti...

We show that the feasible interpolation property is robust for some proof systems but not for others.

For all i ≥ 1, Ti+11(α) is not ∀Σb2(α)-conservative over Ti1(α).

Let $f_0, f_1, \dots, f_k$ be $n$--variable polynomials
over a finite prime field $\fp$. A proof of the ideal
membership $f_0 \in \langle f_1, \dots, f_k \rangle$ in {\em
polynomial calculus} is a sequence of polynomials $h_1, \dots,
h_t$ such that $h_t = f_0$, and such that every $h_i$ is
either an $f_j$, $j \geq 1$, or obtained from $h_1, \dots,...

We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs...

Over a fixed finite field Fp, families of polynomial equations $f_i(x_1, \dots, x_{n_N}) = 0$ for i = 1,…, kN, that are uniformly determined by a parameter N, are considered. The notion of a uniform family is defined in terms of first-order logic. A notion of an abstract Euler characteristic
is used to give sense to a statement that the system has...

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generaliza...

We consider tautologies formed form a pseudo-random number generator, defined in Krajicek [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajicek [11]. Further we give a purely finitary statement, in the form of a hardness conditi...

We investigate the problem how to lift the non - ∀∑b1(α) - conservatively of T22(α) over S22(α) to the expected non - ∀∑bi(α) - conservativity of Ti+12(α) over Si+12(α), for i > 1. We give a non-trivial refinement of the "lifting method" developed in [4,8], and we prove a sufficient condition on a ∀∑b1(f)-consequence of T2(f) to yield the non-conse...

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑i ai
· xi
≥ b (xi
's variables, ai
's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(C...

We investigate the possibility to characterize (multi)functions that are -definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T
2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results:(1) A reformulation of known characterizations of (multi)functions that are and -definable...

We construct a faithful interpretation of Łukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.
We prove a completeness theorem for product logic extended by a unary connective δ of Baaz [1]. We show that Gödel's logic is a sublogic of...

We show that there is a pair of disjoint NP-sets, whose disjointness is provable inS12and which cannot be separated by a set in P/poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension ofS12admitting an NP-definition of primes about which it can prove that no numb...

We show that there is a pair of disjoint 𝒩𝒫 -sets, whose disjointness is provable in S1 2 and which cannot be separated by a set in 𝒫 /poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S1 2 admitting an 𝒩𝒫 -definition of primes abo...

We introduce the notions of a real game (a generalisation of the Karchmer-Wigderson game [M. Karchmer and A. Wigderson, SIAM J. Discrete Math. 3, 255-265 (1990; Zbl 0695.94021)] and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monoto...

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k . We give a new proof of the interpolation theorem ba...

This paper is based on my lecture [26]. It examines the problem of proving non-trivial lower bounds for the length of proofs in propositional logic from the perspective of methods available rather than surveying known partial results (i.e., lower bounds for weaker proof systems). We discuss neither motivations for proving lower bounds for propositi...

We prove a lower bound of the formN
(1) on the degree of polynomials in a Nullstellensatz refutation of theCount
q
polynomials over
m
, whereq is a prime not dividingm. In addition, we give an explicit construction of a degreeN
(1)
design for theCount
q
principle over
m
. As a corollary, using Beameet al. (1994) we obtain a lower bound of the...

A fundamental open problem of mathematical logic and simultaneously the main problem of computational complexity theory is the following one.

We prove lower bounds of the form exp(nε d), εd > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d. This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.

We propose a framework for proving lower bounds to the size of EF-proofs (equivalently, to the number of proof-steps in F-proofs) in terms of boolean valuations. The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the pape...

The weak form of the Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q<sub>i</sub>(x¯)=0, does not have a solution in the algebraic closure iff 1 is in the ideal generated by the polynomials Q<sub>i</sub>(x¯). We shall prove a lower bound on the degrees of polynomials P<sub>i</sub>(x¯) such that Σ <sub>i</sub> P<su...

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and ∧,∨ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set of depth d sequents of total size O(n
3+d
) which are refutable in LK by depth d + 1 proof of size exp(O(log2
n)) but such that every depth d refutation must have the si...

We prove lower bounds of the form exp (n " d ) ; " d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known. Introductio...

The so-called weak form of Hilbert’s Nullstellensatz says that a system of algebraic equations over a field, Q i (x ¯)=0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials Q i (x ¯). We shall prove a lower bound on the degrees of polynomials P i (x ¯) such that ∑ i P i (x ¯)Q i (x ¯)=1....

We characterize functions and predicates Σ i+1 b -definable in S 2 i . In particular, predicates Σ i+1 b -definable in S 2 i are precisely those in bounded query class P Σip [O(log n)] (which equals to Log Space Σip by [B-H,W]). This implies that S 2 i ¬= T 2 i unless P Σip [O(log n)] = Δ i+1 p . Further we construct oracle A such that for all i ≥...

In this paper we prove an exponential lower bound on the size of bounded-depth Frege proofs for the pigeonhole principle (PHP). We also obtain an O(loglogn)-depth lower bound for any polynomial-sized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact complexity of the PHP, as S. Buss has constructed polyn...

No counter-example interpretation for bounded arithmetic is employed to derive recent witnessing theorem for S
2i+1, functions □
i+1P−computable with counterexamples are shown to include all □
i+2P−functions, and two separation results for fragments of S
2(α) are proved.

(1)Ti2=Si+12 implies ∑pi+1⊆Δpi+1⧸poly.(2)S2(α) and are not finitely axiomatizable.The main tool is a Herbrand-type witnessing theorem for ∃∀∃Пbi-formulas provable in Ti2 where the witnessing functions are □pi+1.

(1) Vi2⊢A(a) iff for some term t:Si2⊢ “2i(a) exists→ A(a)”, a bounded first-order formula, i≥1. (2) Vi2 (resp. V2) is not Πb1-conservative over Si2 (resp. over S2). (3) Any model of V2 not satisfying Exp satisfies the collection scheme BΣ01. (4) V13 is not Πb1-conservative over S2.

We characterize the bounded first order consequences of theory in U
21 terms of a limited use of exponentiation, we construct a simulation of U
21 by the quantified propositional calculus, and we prove that U
21 is not conservative over IΔ0 and that it is stronger than a conservative Δ
11,b
-extension of S
2. As corollaries we obtain that U
21 is n...

We consider the problem about the length of proofs of the sentences Cons(!!) saying that there is no proof of contradiction in S whose length is::; n. We show the relation of this problem to some problems about propositional proof systems. §O. Introduction. For a finitely axiomatized theory S let Cons(~) denote the statement that there is no proof...

For any countable nonstandard modelM of a sufficiently strong fragment of arithmeticT, and any nonstandard numbersa, c
\leqq 2ac \leqq 2^{a^c }
.

In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbert-style formal systems such as in (17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of...

## Citations

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

... We now give an application of the conditional unprovability result of [15]. Consider theory T PV whose language has a k-ary function symbol f M attached to every p-time clocked machine M with k inputs, all k ≥ 1. Symbol f M is naturally interpreted on N by the function M computes. ...

... Lower bounds for protocols called protocols with equality in this paper would directly lead to lower bounds for R(LIN). A different approach which could work for this proof system is randomized feasible interpolation due to Krajíček [Kra18], studied also by Krajíček and Oliveira [KO18]. ...

... Lower bounds for protocols called protocols with equality in this paper would directly lead to lower bounds for R(LIN). A different approach which could work for this proof system is randomized feasible interpolation due to Krajíček [Kra18], studied also by Krajíček and Oliveira [KO18]. ...

... Using similar methods, a recent line of works [24,6,7,8] achieved unconditional consistency results for fixed-polynomial lower bounds, even for P instead of NP (based on [36]). For example, the main result in [7] implies that S 2 2 + ¬α c ϕ and S 1 2 + ¬α c ψ are consistent for certain formulas ϕ(x) and ψ(x) that define problems in P NP and NP, respectively. ...

... However, if we extend the De Morgan language with counting connectives such as unbounded fan-in mod p (AC 0 [p]-Frege) or threshold gates (TC 0 -Frege), then we step again into the dark- ness: proving super-polynomial lower bounds for these systems is a long-standing open problem on what can be characterized as the "frontiers" of proof complexity. Recent works by Krajícek (2017), Garlik & Ko lodziejczyk (2018) and Krajícek & Oliveira (2018) had suggested possible approaches to attack dag-like Res(lin F 2 ) lower bounds (though this problem remains open to date). ...

... Recall that FP Σ P i [O(g(n)),wit] denotes the class of total search problems computable by a polynomial function that makes O(g(n)) queries to a witnessing Σ P i oracle, meaning that for any positive answer, the oracle also has to produce a witness to the outermost existential quantifier. For any [31]). ...

Reference: Induction rules in bounded arithmetic

... Например, принцип Дирихле, как и другие комбинаторные принципы, использующие мощностные соображения, не может иметь сложного доказательства в системах Фреге [Bus87], поэтому любые комбинаторные принципы, требующие суперполиномиальных доказательств, должны быть очень сложными [BBP95]. В некоторых случаях вопрос о существовании нижних оценок удалось свести к оценке числа раундов интерактивной игры, но получить при этом нетривиальные нижние оценки не удалось [PB95,Pud00,Kra15]. ...

Reference: The complexity of propositional proofs

... Krajíček [23] introduced the notion of concisely-represented proofs of exponential length, there called implicit proofs, here we refer to them as circuit-generated proofs (Definition 1). The search for a contradiction in such a proof is known as a consistency search problem, and such problems have been studied in work of Krajíček [24] and Skelley and Thapen [38]. A recent paper of Beckmann and Buss [3], also within the tradition of bounded arithmetic, proves certain results that appear to strengthen the present ones, by reducing the problems of interest to consistency search problems in less powerful systems. ...

... Essentially all problems have already been stated before, sometimes in different forms. The reader interested in problems should consult monographs [6, 20], survey articles [11, 36] and other lists of problems [11, 21]. ...

Reference: Twelve Problems in Proof Complexity