arXiv subjects

Sam Buss

Publications and source records attributed to Sam Buss.

14 recordsLinked to original sources

arXiv2026

Herbrand Game Complexity

The Student-Teacher game is an extension of Herbrand's theorem. In the game, Student and Teacher take turns giving values for existentially and universally quantified variables, and Student is allowed to backtrack to propose other values for existentially quantified variables. The game has become increasingly important for proving lower bounds on provability in theories of bounded arithmetic. In those applications, the game is played in an arithmetical theory; however, this paper studies the Student-Teacher game in the setting of pure first-order logic, so it is more closely related to Herbrand's theorem and the midsequent theorem. When played in pure first-order logic, a formula~$\varphi$ is logically valid if and only if there is a Student-Teacher game with a winning strategy for Student for establishing~$\varphi$. We present a refined version of the Student-Teacher game in arbitrary first-order universal theories and include a proof of the validity of Student-Teacher games from the sequent calculus midsequent theorem in an appendix. The game is presented as a finite tree with vertices and edges labeled by terms, and with a total order on the nodes. The totally ordered tree represents the players' interaction. Our main results show that minimal trees in the Student-Teacher game can be arbitrarily complex. Specifically, for every totally ordered tree~$T$, we construct a valid prenex formula~$\varphi$ such that every Student-Teacher game for~$\varphi$ contains $T$ as a substructure. It follows not only that there is no computable bound on the size of Herbrand disjunctions, which is a well-known fact, but also that there is no bound on their complexity in the sense that it is not possible to restrict the types of totally ordered trees knowing only the length of the formula.

math.LO↗

arXiv2025

Redundancy rules for MaxSAT

The concept of redundancy in SAT leads to more expressive and powerful proof search techniques, e.g., able to express various inprocessing techniques, and originates interesting hierarchies of proof systems [Heule et$.$al'20, Buss-Thapen'19]. Redundancy has also been integrated in MaxSAT [Ihalainen et$.$al'22, Berg et$.$al'23, Bonacina et$.$al'24]. In this paper, we define a structured hierarchy of redundancy proof systems for MaxSAT, with the goal of studying its proof complexity. We obtain MaxSAT variants of proof systems such as SPR, PR, SR, and others, previously defined for SAT. All our rules are polynomially checkable, unlike [Ihalainen et$.$al'22]. Moreover, they are simpler and weaker than [Berg et$.$al'23], and possibly amenable to lower bounds. This work also complements the approach of [Bonacina et$.$al'24]. Their proof systems use different rule sets for soft and hard clauses, while here we propose a system using only hard clauses and blocking variables. This is easier to integrate with current solvers and proof checkers. We discuss the strength of the systems introduced, we show some limitations of them, and we give a short cost-SR proof that any assignment for the weak pigeonhole principle $PHP^{m}_{n}$ falsifies at least $m-n$ clauses. We conclude by discussing the integration of our rules with the MaxSAT resolution proof system, which is a commonly studied proof system for MaxSAT.

cs.LO↗

arXiv2025

A Logspace Constructive Proof of L=SL

We formalize the proof of Reingold's Theorem that SL=L [Rei05] in the theory of bounded arithmetic VL, which corresponds to ``logspace reasoning''. As a consequence, we get that VL=VSL, where VSL is the theory of bounded arithmetic for ``symmetric-logspace reasoning''. This resolves in the affirmative an old open question from Kolokolova [Kol05] (see also Cook-Nguyen [NC10]). Our proof relies on the Rozenman-Vadhan alternative proof of Reingold's Theorem ([RV05]). To formalize this proof in VL, we need to avoid reasoning about eigenvalues and eigenvectors (common in both original proofs of SL=L). We achieve this by using some results from Buss-Kabanets-Kolokolova-Kouck\'y [Bus+20] that allow VL to reason about graph expansion in combinatorial terms.

cs.LO↗

arXiv2024

Extended Resolution Clause Learning via Dual Implication Points

We present a new extended resolution clause learning (ERCL) algorithm, implemented as part of a conflict-driven clause-learning (CDCL) SAT solver, wherein new variables are dynamically introduced as definitions for {\it Dual Implication Points} (DIPs) in the implication graph constructed by the solver at runtime. DIPs are generalizations of unique implication points and can be informally viewed as a pair of dominator nodes, from the decision variable at the highest decision level to the conflict node, in an implication graph. We perform extensive experimental evaluation to establish the efficacy of our ERCL method, implemented as part of the MapleLCM SAT solver and dubbed xMapleLCM, against several leading solvers including the baseline MapleLCM, as well as CDCL solvers such as Kissat 3.1.1, CryptoMiniSat 5.11, and SBVA+CaDiCaL, the winner of SAT Competition 2023. We show that xMapleLCM outperforms these solvers on Tseitin and XORified formulas. We further compare xMapleLCM with GlucoseER, a system that implements extended resolution in a different way, and provide a detailed comparative analysis of their performance.

cs.LO↗

arXiv2024

Regular resolution effectively simulates resolution

Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs in regular resolution while admitting polynomial-size proofs in resolution. Thus, with respect to the usual notion of simulation, regular resolution is separated from resolution. An alternative, and weaker, notion for comparing proof systems is that of an "effective simulation," which allows the translation of the formula along with the proof when moving between proof systems. We prove that regular resolution is equivalent to resolution under effective simulations. As a corollary, we recover in a black-box fashion a recent result on the hardness of automating regular resolution.

cs.LO↗

arXiv2023

On the Consistency of Circuit Lower Bounds for Non-Deterministic Time

We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V$^0_2$ is consistent with the conjecture that NEXP $\not\subseteq$ P/poly, i.e., some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. The same techniques establish the same results with NEXP replaced by the class of problems that are decidable in non-deterministic barely superpolynomial time such as NTIME$(n^{O(\log\log\log n)})$. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds.

cs.CC↗

arXiv2019

Proof complexity of systems of (non-deterministic) decision trees and branching programs

This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs - deterministic and nondeterministic. The systems LDT and LNDT are propositional proof systems in which lines represent deterministic or non-deterministic decision trees. Branching programs are modeled as decision dags. Adding extension to LDT and LNDT gives systems eLDT and eLNDT in which lines represent deterministic and non-deterministic branching programs, respectively. Deterministic and non-deterministic branching programs correspond to log-space (L) and nondeterministic log-space (NL). Thus the systems eLDT and eLNDT are propositional proof systems that reason with (nonuniform) L and NL properties. The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in the systems LDT, LNDT, eLDT, and eLNDT. These systems are also compared with Frege systems, constantdepth Frege systems and extended Frege systems

cs.CC↗

arXiv2019

DRAT and Propagation Redundancy Proofs Without New Variables

We study the complexity of a range of propositional proof systems which allow inference rules of the form: from a set of clauses $\Gamma$ derive the set of clauses $\Gamma \cup \{ C \}$ where, due to some syntactic condition, $\Gamma \cup \{ C \}$ is satisfiable if $\Gamma$ is, but where $\Gamma$ does not necessarily imply $C$. These inference rules include BC, RAT, SPR and PR (respectively short for blocked clauses, resolution asymmetric tautologies, subset propagation redundancy and propagation redundancy), which arose from work in satisfiability (SAT) solving. We introduce a new, more general rule SR (substitution redundancy). If the new clause $C$ is allowed to include new variables then the systems based on these rules are all equivalent to extended resolution. We focus on restricted systems that do not allow new variables. The systems with deletion, where we can delete a clause from our set at any time, are denoted DBC${}^-$, DRAT${}^-$, DSPR${}^-$, DPR${}^-$ and DSR${}^-$. The systems without deletion are BC${}^-$, RAT${}^-$, SPR${}^-$, PR${}^-$ and SR${}^-$. With deletion, we show that DRAT${}^-$, DSPR${}^-$ and DPR${}^-$ are equivalent. By earlier work of Kiesl, Rebola-Pardo and Heule, they are also equivalent to DBC${}^-$. Without deletion, we show that SPR${}^-$ can simulate PR${}^-$ provided only short clauses are inferred by SPR inferences. We also show that many of the well-known "hard" principles have small SPR${}^-$ refutations. These include the pigeonhole principle, bit pigeonhole principle, parity principle, Tseitin tautologies and clique-coloring tautologies. SPR${}^-$ can also handle or-fication and xor-ification, and lifting with an index gadget. Our final result is an exponential size lower bound for RAT${}^-$ refutations, giving exponential separations between RAT${}^-$ and both DRAT${}^-$ and SPR${}^-$.

cs.LO↗

arXiv2018

Strategies for Stable Merge Sorting

We introduce new stable natural merge sort algorithms, called $2$-merge sort and $\alpha$-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shivers' sort, $\alpha$-stack sorts, and our new $2$-merge and $\alpha$-merge sorts. The upper and lower bounds have the forms $c \cdot n \log m$ and $c \cdot n \log n$ for inputs of length~$n$ comprising $m$~monotone runs. For Timsort, we prove a lower bound of $(1.5 - o(1)) n \log n$. For $2$-merge sort, we prove optimal upper and lower bounds of approximately $(1.089 \pm o(1))n \log m$. We prove similar asymptotically matching upper and lower bounds for $\alpha$-merge sort, when $\varphi < \alpha < 2$, where $\varphi$ is the golden ratio. Our bounds are in terms of merge cost; this upper bounds the number of comparisons and accurately models runtime. The merge strategies can be used for any stable merge sort, not just natural merge sorts. The new $2$-merge and $\alpha$-merge sorts have better worst-case merge cost upper bounds and are slightly simpler to implement than the widely-used Timsort; they also perform better in experiments. We report also experimental comparisons with algorithms developed by Munro-Wild and Jug\'e subsequently to the results of the present paper.

cs.DS↗

arXiv2015

Short Proofs of the Kneser-Lov\'asz Coloring Principle

We prove that the propositional translations of the Kneser-Lov\'asz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lov\'asz theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.

math.LO↗

arXiv2014

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.

cs.LO↗

arXiv2012

Unshuffling a Square is NP-Hard

A shuffle of two strings is formed by interleaving the characters into a new string, keeping the characters of each string in order. A string is a square if it is a shuffle of two identical strings. There is a known polynomial time dynamic programming algorithm to determine if a given string z is the shuffle of two given strings x,y; however, it has been an open question whether there is a polynomial time algorithm to determine if a given string z is a square. We resolve this by proving that this problem is NP-complete via a many-one reduction from 3- Partition.

cs.CC↗

arXiv2012

Improved Separations of Regular Resolution from Clause Learning Proof Systems

We prove that the graph tautology formulas of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology formulas can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search. We prove similar results for the guarded, xor-fied pebbling tautologies which Urquhart proved are hard for regular resolution.

cs.LO↗

arXiv2012

An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning

We prove that the graph tautology principles of Alekhnovich, Johannsen, Pitassi and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses and without degenerate resolution inferences. We also prove that these graph tautology principles can be refuted by polynomial size DPLL proofs with clause learning, even when restricted to greedy, unit-propagating DPLL search.

cs.LO↗