arXiv subjects

Leszek Aleksander Kołodziejczyk

Publications and source records attributed to Leszek Aleksander Kołodziejczyk.

13 recordsLinked to original sources

arXiv2026

Tightness and solidity in fragments of Peano Arithmetic

It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as \emph{tightness} and to carry out a more systematic study of tightness and its stronger variants that he called neatness and solidity. Enayat proved that not only $\mathsf{PA}$, but also $\mathsf{ZF}$ and $\mathsf{Z}_2$ are solid. On the other hand, it was shown in later work by a number of authors that many natural proper fragments of those theories are not even tight. Enayat asked whether there is a proper solid subtheory of the theories listed above. We answer that question in the case of $\mathsf{PA}$ by proving that for every $n$, there exist both a solid theory and a tight but not neat theory strictly between $\mathsf{I}Σ_n$ and $\mathsf{PA}$. Moreover, the solid subtheories of $\mathsf{PA}$ can be required to be unable to interpret $\mathsf{PA}$. We also provide simple examples of proper solid subtheories of $\mathsf{ZF}$ and $\mathsf{Z}_2$, as well as further separations between properties related to tightness, including an example of a sequential theory that is neat but not semantically tight in the sense of Freire and Hamkins.

math.LO↗

arXiv2026

The cohesive and stable Ramsey theorems and proof size over a weak base theory

We show that over the weak base theory $\mathrm{RCA}_0^*$, cohesive Ramsey's theorem for pairs $\mathrm{CRT}^2_2$ implies exponential closure of the definable cut $\mathrm{I}^0_1$, which is the intersection of all $Σ^0_1$-definable cuts. Consequences include non-elementary proof speedup of $\mathrm{RCA}_0^* + \mathrm{CRT}^2_2$ over $\mathrm{RCA}_0^*$ for $Π_1$ sentences and the unprovability of $\mathrm{CRT}^2_2$ in $\mathrm{RCA}_0^* + \mathrm{CAC}$. On the other hand, we show that $\mathrm{RCA}_0^* + \mathrm{SRT}^2_2$, where $\mathrm{SRT}^2_2$ is stable Ramsey's theorem for pairs, is polynomially simulated by $\mathrm{RCA}_0^*$ with respect to proofs of $\forall Π^0_3$ sentences. Nevertheless, $\mathrm{SRT}^2_2$ also implies a nontrivial property of $\mathrm{I}^0_1$, specifically closure under functions of quasipolynomial growth rate.

math.LO↗

arXiv2024

The strength of the dominance rule

It has become standard that, when a SAT solver decides that a CNF $Γ$ is unsatisfiable, it produces a certificate of unsatisfiability in the form of a refutation of $Γ$ in some proof system. The system typically used is DRAT, which is equivalent to extended resolution (ER) -- for example, until this year DRAT refutations were required in the annual SAT competition. Recently [Bogaerts et al.~2023] introduced a new proof system, associated with the tool VeriPB, which is at least as strong as DRAT and is further able to handle certain symmetry-breaking techniques. We show that this system simulates the proof system $G_1$, which allows limited reasoning with QBFs and forms the first level above ER in a natural hierarchy of proof systems. This hierarchy is not known to be strict, but nevertheless this is evidence that the system of [Bogaerts et al. 2023] is plausibly strictly stronger than ER and DRAT. In the other direction, we show that symmetry-breaking for a single symmetry can be handled inside ER.

cs.LO↗

arXiv2022

An isomorphism theorem for models of Weak König's Lemma without primitive recursion

We prove that if $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are countable models of the theory $\mathrm{WKL}^*_0$ such that $\mathrm{I}Σ_1(A)$ fails for some $A \in \mathcal{X} \cap \mathcal{Y}$, then $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are isomorphic. As a consequence, the analytic hierarchy collapses to $Δ^1_1$ provably in $\mathrm{WKL}^*_0 + \neg\mathrm{I}Σ^0_1$, and $\mathrm{WKL}$ is the strongest $Π^1_2$ statement that is $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg\mathrm{I}Σ^0_1$. Applying our results to the $Δ^0_n$-definable sets in models of $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ that also satisfy an appropriate relativization of Weak König's Lemma, we prove that for each $n \ge 1$, the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ is c.e. In contrast, we prove that the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n$ is $Π_2$-complete. This answers a question of Towsner. We also show that $\mathrm{RCA}_0 + \mathrm{RT}^2_2$ is $Π^1_1$-conservative over $\mathrm{B}Σ^0_2$ if and only if it is conservative over $\mathrm{B}Σ^0_2$ with respect to $\forall Π^0_5$ sentences.

math.LO↗

arXiv2021

Approximate counting and NP search problems

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory $\mathrm{APC}_2$ of [Jeřábek 2009]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory $T^2_2$, this shows that $\mathrm{APC}_2$ does not prove every $\forall Σ^b_1$ sentence which is provable in bounded arithmetic. This answers the question posed in [Buss, Kołodziejczyk, Thapen 2014] and represents some progress in the programme of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the "fixing lemma" from [Pudlák, Thapen 2017], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a $\textrm{P}^{\textrm{NP}}$ oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.

math.LO↗

arXiv2021

Weaker cousins of Ramsey's theorem over a weak base theory

The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the Cohesive Ramsey Theorem for pairs $\mathsf{CRT}^2_2$. We study these principles over the base theory $\mathsf{RCA}^*_0$, which is weaker than the usual base theory $\mathsf{RCA}_0$ considered in reverse mathematics in that it allows only $Δ^0_1$-induction as opposed to $Σ^0_1$-induction. In $\mathsf{RCA}^*_0$, it may happen that an unbounded subset of $\mathbb{N}$ is not in bijective correspondence with $\mathbb{N}$. Accordingly, Ramsey-theoretic principles split into at least two variants, "normal" and "long", depending on the sense in which the set witnessing the principle is required to be infinite. We prove that the normal versions of our principles, like that of Ramsey's theorem for pairs and two colours, are equivalent to their relativizations to proper $Σ^0_1$-definable cuts. Because of this, they are all $Π^0_3$- but not $Π^1_1$-conservative over $\mathsf{RCA}^*_0$, and, in any model of $\mathsf{RCA}^*_0 + \neg \mathsf{RCA}_0$, if they are true then they are computably true relative to some set. The long versions exhibit one of two behaviours: they either imply $\mathsf{RCA}_0$ over $\mathsf{RCA}^*_0$ or are $Π^0_3$-conservative over $\mathsf{RCA}^*_0$. The conservation results are obtained using a variant of the so-called grouping principle. We also show that the cohesion principle $\mathsf{COH}$, a strengthening of $\mathsf{CRT}^2_2$, is never computably true in a model of $\mathsf{RCA}^*_0$ and, as a consequence, does not follow from $\mathsf{RT}^2_2$ over $\mathsf{RCA}^*_0$.

math.LO↗

arXiv2021

Ramsey's theorem for pairs, collection, and proof size

We prove that any proof of a $\forall Σ^0_2$ sentence in the theory $\mathrm{WKL}_0 + \mathrm{RT}^2_2$ can be translated into a proof in $\mathrm{RCA}_0$ at the cost of a polynomial increase in size. In fact, the proof in $\mathrm{RCA}_0$ can be found by a polynomial-time algorithm. On the other hand, $\mathrm{RT}^2_2$ has non-elementary speedup over the weaker base theory $\mathrm{RCA}^*_0$ for proofs of $Σ_1$ sentences. We also show that for $n \ge 0$, proofs of $Π_{n+2}$ sentences in $\mathrm{B}Σ_{n+1}+\exp$ can be translated into proofs in $\mathrm{I}Σ_{n} + \exp$ at polynomial cost. Moreover, the $Π_{n+2}$-conservativity of $\mathrm{B}Σ_{n+1} + \exp$ over $\mathrm{I}Σ_{n} + \exp$ can be proved in $\mathrm{PV}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n \ge 1$, this answers a question of Clote, Hájek, and Paris.

math.LO↗

arXiv2021

How strong is Ramsey's theorem if infinity can be weak?

We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{RCA}^*_0$. Using the Chong-Mourad coding lemma, we show that in a model of $\mathrm{RCA}^*_0 + \neg \mathrm{I}Σ^0_1$, $\mathrm{RT}^n_k$ is equivalent to its own relativization to any proper $Σ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give an axiomatization of the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of PA whose $Π_3$ fragment is $\mathrm{B}Σ_1 + \exp$ and whose $Π_{\ell+3}$ fragment for $\ell \ge 1$ lies between $\mathrm{I}Σ_\ell \Rightarrow \mathrm{B}Σ_{\ell+1}$ and $\mathrm{B}Σ_{\ell+1}$. We also consider the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^2_k$. We show that they form a subtheory of $\mathrm{I}Σ_2$ whose $Π_3$ fragment is $\mathrm{B}Σ_1 + \exp$ and whose $Π_4$ fragment is strictly weaker than $\mathrm{B}Σ_2$ but not contained in $\mathrm{I}Σ_1$. Additionally, we consider a principle $Δ^0_2$-$\mathrm{RT}^2_2$, defined like $\mathrm{RT}^2_2$ but with both the $2$-colourings and the solutions allowed to be $Δ^0_2$-sets. We show that the behaviour of $Δ^0_2$-$\mathrm{RT}^2_2$ over $\mathrm{RCA}_0 + \mathrm{B}Σ^0_2$ is similar to that of $\mathrm{RT}^2_2$ over $\mathrm{RCA}^*_0$, and that $\mathrm{RCA}_0 + \mathrm{B}Σ^0_2 + Δ^0_2$-$\mathrm{RT}^2_2$ is $Π_4$- but not $Π_5$-conservative over $\mathrm{B}Σ_2$. However, the statement we use to witness lack of $Π_5$-conservativity is not provable in $\mathrm{RCA}_0 +\mathrm{RT}^2_2$.

math.LO↗

arXiv2018

Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs

We study Ramsey's theorem for pairs and two colours in the context of the theory of $α$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $ω^{300n}$-large set admits an $ω^n$-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama [Adv. Math. 330 (2018), 1034--1070] stating that Ramsey's theorem for pairs and two colours is $\forallΣ^0_2$-conservative over the axiomatic theory $\mathsf{RCA}_0$ (recursive comprehension).

math.CO↗

arXiv2017

New bounds on the strength of some restrictions of Hindman's Theorem

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies $\mathsf{ACA}_0$. An emerging {\em leitmotiv} is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.

math.LO↗

arXiv2015

How unprovable is Rabin's decidability theorem?

We study the strength of set-theoretic axioms needed to prove Rabin's theorem on the decidability of the MSO theory of the infinite binary tree. We first show that the complementation theorem for tree automata, which forms the technical core of typical proofs of Rabin's theorem, is equivalent over the moderately strong second-order arithmetic theory $\mathsf{ACA}_0$ to a determinacy principle implied by the positional determinacy of all parity games and implying the determinacy of all Gale-Stewart games given by boolean combinations of ${\bf Σ^0_2}$ sets. It follows that complementation for tree automata is provable from $Π^1_3$- but not $Δ^1_3$-comprehension. We then use results due to MedSalem-Tanaka, Möllerfeld and Heinatsch-Möllerfeld to prove that over $Π^1_2$-comprehension, the complementation theorem for tree automata, decidability of the MSO theory of the infinite binary tree, positional determinacy of parity games and determinacy of $\mathrm{Bool}({\bf Σ^0_2})$ Gale-Stewart games are all equivalent. Moreover, these statements are equivalent to the $Π^1_3$-reflection principle for $Π^1_2$-comprehension. It follows in particular that Rabin's decidability theorem is not provable in $Δ^1_3$-comprehension.

math.LO↗

arXiv2014

End-extensions of models of weak arithmetic from complexity-theoretic containments

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of $Π_1(\mathbb{N}) + \neg Ω_1$ has a proper end-extension to a model of $Π_1(\mathbb{N})$, and so $Π_1(\mathbb{N}) + \neg Ω_1 \vdash \mathrm{B}Σ_1$. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, $Π_1(\mathbb{N}) + \neg \mathrm{Exp} \vdash \mathrm{B}Σ_1$. Both assumptions can be modified to versions which make it possible to replace $Π_1(\mathbb{N})$ by $\mathrm{I}Δ_0$ as the base theory. We also show that any proof that $\mathrm{I}Δ_0 + \neg \exp$ does not prove a given finite fragment of $\mathrm{B}Σ_1$ has to be "non-relativizing", in the sense that it will not work in the presence of an arbitrary oracle.

math.LO↗

arXiv2014

Categorical characterizations of the natural numbers require primitive recursion

Simpson and the second author asked whether there exists a characterization of the natural numbers by a second-order sentence which is provably categorical in the theory RCA$^*_0$. We answer in the negative, showing that for any characterization of the natural numbers which is provably true in WKL$^*_0$, the categoricity theorem implies $Σ^0_1$ induction. On the other hand, we show that RCA$^*_0$ does make it possible to characterize the natural numbers categorically by means of a set of second-order sentences. We also show that a certain $Π^1_2$-conservative extension of RCA$^*_0$ admits a provably categorical single-sentence characterization of the naturals, but each such characterization has to be inconsistent with WKL$^*_0$+superexp.

math.LO↗