Soren Riis, Meera Sitharam

The semantics of decision problems are always essentially independent of the

underlying representation. Thus the space of input data (under appropriate

indexing) is closed

under action of the symmetrical group $S_n$ (for a specific data-size)

and the input-output relation is closed under the action of $S_n$.

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Paul Beame

Proof complexity, the study of the lengths of proofs in

propositional logic, is an area of study that is fundamentally connected

both to major open questions of computational complexity theory and

to practical properties of automated theorem provers. In the last

decade, there have been a number of significant advances ...
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Eli Ben-Sasson, Avi Wigderson

The width of a Resolution proof is defined to be the maximal number of

literals in any clause of the proof. In this paper we relate proof width

to proof length (=size), in both general Resolution, and its tree-like

variant. The following consequences of these relations reveal width as ...
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Eli Ben-Sasson, Russell Impagliazzo, Avi Wigderson

We present the best known separation

between tree-like and general resolution, improving

on the recent $\exp(n^\epsilon)$ separation of \cite{BEGJ98}.

This is done by constructing a natural family of contradictions, of

size $n$, that have $O(n)$-size resolution

refutations, but only $\exp (\Omega(n/\log n))$-size tree-like refutations.

This result ...
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Albert Atserias, Nicola Galesi, Ricard Gavaldà

We study the complexity of proving the Pigeon Hole

Principle (PHP) in a monotone variant of the Gentzen Calculus, also

known as Geometric Logic. We show that the standard encoding

of the PHP as a monotone sequent admits quasipolynomial-size proofs

in this system. This result is a consequence of ...
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Eli Ben-Sasson, Nicola Galesi

We study the space complexity of refuting unsatisfiable random $k$-CNFs in

the Resolution proof system. We prove that for any large enough $\Delta$,

with high probability a random $k$-CNF over $n$ variables and $\Delta n$

clauses requires resolution clause space of

$\Omega(n \cdot \Delta^{-\frac{1+\epsilon}{k-2-\epsilon}})$,

for any $0<\epsilon<1/2$. For constant $\Delta$, ...
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Joshua Buresh-Oppenheim, David Mitchell

We prove exponential separations between the sizes of

particular refutations in negative, respectively linear, resolution and

general resolution. Only a superpolynomial separation between negative

and general resolution was previously known. Our examples show that there

is no strong relationship between the size and width of refutations in

negative and ...
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Eli Ben-Sasson, Yonatan Bilu

We present the first example of a natural distribution on instances

of an NP-complete problem, with the following properties.

With high probability a random formula from this

distribution (a) is unsatisfiable,

(b) has a short proof that can be found easily, and (c) does not have a short

(general) resolution ...
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Josh Buresh-Oppenheim, Paul Beame, Ran Raz, Ashish Sabharwal

We prove a quasi-polynomial lower bound on the size of bounded-depth

Frege proofs of the pigeonhole principle $PHP^{m}_n$ where

$m= (1+1/{\polylog n})n$.

This lower bound qualitatively matches the known quasi-polynomial-size

bounded-depth Frege proofs for these principles.

Our technique, which uses a switching lemma argument like other lower bounds

for ...
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Eli Ben-Sasson, Prahladh Harsha

We present a simple proof of the bounded-depth Frege lower bounds of

Pitassi et. al. and Krajicek et. al. for the pigeonhole

principle. Our method uses the interpretation of proofs as two player

games given by Pudlak and Buss. Our lower bound is conceptually

simpler than previous ones, and relies ...
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Paul Beame, Joseph Culberson, David Mitchell, Cristopher Moore

We consider the resolution proof complexity of propositional formulas which encode random instances of graph $k$-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity.

For random graphs with linearly many edges we obtain linear-exponential lower bounds on the length of resolution refutations. For any $\epsilon>0$, ...
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Neil Thapen, Nicola Galesi

We define a collection of Prover-Delayer games that characterize certain subsystems of resolution. This allows us to give some natural criteria which guarantee lower bounds on the resolution width of a formula, and to extend these results to formulas of unbounded initial width.

We also use games to give upper ... more >>>

Paul Beame, Nathan Segerlind

We prove that an \omega(log^3 n) lower bound for the three-party number-on-the-forehead (NOF) communication complexity of the set-disjointness function implies an n^\omega(1) size lower bound for tree-like Lovasz-Schrijver systems that refute unsatisfiable CNFs. More generally, we prove that an n^\Omega(1) lower bound for the (k+1)-party NOF communication complexity of set-disjointness ... more >>>

Jakob Nordström

The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of memory cells used if the proof is only allowed to resolve on clauses kept in memory. Both of these measures have previously ... more >>>

Ran Raz, Iddo Tzameret

We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following:

1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial ... more >>>

Eran Ofek, Uriel Feige

Let $\phi$ be a 3CNF formula with n variables and m clauses. A

simple nonconstructive argument shows that when m is

sufficiently large compared to n, most 3CNF formulas are not

satisfiable. It is an open question whether there is an efficient

refutation algorithm that for most such formulas proves ...
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Alex Hertel, Alasdair Urquhart

The importance of {\em width} as a resource in resolution theorem proving

has been emphasized in work of Ben-Sasson and Wigderson. Their results show that lower

bounds on the size of resolution refutations can be proved in a uniform manner by

demonstrating lower bounds on the width ...
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Paul Beame, Russell Impagliazzo, Nathan Segerlind

We consider extensions of the DPLL approach to satisfiability testing that add a version of memoization, in which formulas that the algorithm has previously shown to be unsatisfiable are remembered for later use. Such formula caching algorithms have been suggested for satisfiability and stochastic satisfiability. We formalize these methods by ... more >>>

Jan Krajicek

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. In fact, the lower bound

applies to semantic derivations operating with sets defined by OBDDs.

We do not assume ...
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Nathan Segerlind

We demonstrate a family of propositional formulas in conjunctive normal form

so that a formula of size $N$ requires size $2^{\Omega(\sqrt[7]{N/logN})}$

to refute using the tree-like OBDD refutation system of

Atserias, Kolaitis and Vardi

with respect to all variable orderings.

All known symbolic quantifier elimination algorithms for satisfiability

generate ...
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Nicola Galesi, Massimo Lauria

We introduce an algebraic proof system Pcrk, which combines together {\em Polynomial Calculus} (Pc) and {\em $k$-DNF Resolution} (Resk).

This is a natural generalization to Resk of the well-known {\em Polynomial Calculus with Resolution} (Pcr) system which combines together Pc and Resolution.

We study the complexity of proofs in such ... more >>>

Philipp Hertel

Satisfiability algorithms have become one of the most practical and successful approaches for solving a variety of real-world problems, including hardware verification, experimental design, planning and diagnosis problems. The main reason for the success is due to highly optimized algorithms for SAT based on resolution. The most successful of these ... more >>>

Ran Raz, Iddo Tzameret

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. ... more >>>

Jakob Nordström

We present a greatly simplified proof of the length-space

trade-off result for resolution in Hertel and Pitassi (2007), and

also prove a couple of other theorems in the same vein. We point

out two important ingredients needed for our proofs to work, and

discuss possible conclusions to be drawn regarding ...
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Troy Lee, Adi Shraibman

We show that disjointness requires randomized communication

Omega(\frac{n^{1/2k}}{(k-1)2^{k-1}2^{2^{k-1}}})

in the general k-party number-on-the-forehead model of complexity.

The previous best lower bound was Omega(\frac{log n}{k-1}). By

results of Beame, Pitassi, and Segerlind, this implies

2^{n^{Omega(1)}} lower bounds on the size of tree-like Lovasz-Schrijver

proof systems needed to refute certain unsatisfiable ...
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Kazuo Iwama, Suguru Tamaki

The planar Hajos calculus is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. We prove that the planar Hajos calculus is polynomially bounded iff the HajLos calculus is polynomially bounded.

more >>>Jakob Nordström, Johan Hastad

Most state-of-the-art satisfiability algorithms today are variants of

the DPLL procedure augmented with clause learning. The main bottleneck

for such algorithms, other than the obvious one of time, is the amount

of memory used. In the field of proof complexity, the resources of

time and memory correspond to the length ...
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Eli Ben-Sasson, Jakob Nordström

A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.

In this paper we resolve the question by answering ... more >>>

Alex Hertel, Alasdair Urquhart

We discovered a serious error in one of our previous submissions to ECCC and wish to make sure that this mistake is publicly known.

The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (Exists,k)-pebble game, ... more >>>

Eli Ben-Sasson, Jakob Nordström

For current state-of-the-art satisfiability algorithms based on the

DPLL procedure and clause learning, the two main bottlenecks are the

amounts of time and memory used. Understanding time and memory

consumption, and how they are related to one another, is therefore a

question of considerable practical importance. In the field of ...
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Eli Ben-Sasson, Jakob Nordström

The k-DNF resolution proof systems are a family of systems indexed by

the integer k, where the kth member is restricted to operating with

formulas in disjunctive normal form with all terms of bounded arity k

(k-DNF formulas). This family was introduced in [Krajicek 2001] as an

extension of the ...
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Paul Beame, Trinh Huynh

We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most $k$ (known as ${\rm Th}(k)$ proofs). Such systems include: Lovasz-Schrijver ... more >>>

Olga Tveretina, Carsten Sinz, Hans Zantema

Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential

size and that limited OBDD derivations cannot simulate resolution polynomially. Here we show that any arbitrary OBDD Apply refutation of the pigeonhole formula has an exponential

size: we prove that the size of one ...
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Jakob Nordström, Alexander Razborov

In the context of proving lower bounds on proof space in $k$-DNF

resolution, [Ben-Sasson and Nordström 2009] introduced the concept of

minimally unsatisfiable sets of $k$-DNF formulas and proved that a

minimally unsatisfiable $k$-DNF set with $m$ formulas can have at most

$O((mk)^{k+1})$ variables. They also gave an example of ...
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Jakob Nordström

The last decade has seen a revival of interest in pebble games in the

context of proof complexity. Pebbling has proven to be a useful tool

for studying resolution-based proof systems when comparing the

strength of different subsystems, showing bounds on proof space, and

establishing size-space trade-offs. The typical approach ...
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Jan Krajicek

Let $g$ be a map defined as the Nisan-Wigderson generator

but based on an $NP \cap coNP$-function $f$. Any string $b$ outside the range of

$g$ determines a propositional tautology $\tau(g)_b$ expressing this

fact. Razborov \cite{Raz03} has conjectured that if $f$ is hard on average for

P/poly then these ...
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Olaf Beyersdorff, Nicola Galesi, Massimo Lauria

Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles.

We broadly investigate Parameterized Resolution obtaining the following ...
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Shir Ben-Israel, Eli Ben-Sasson, David Karger

This paper shows that the use of ``local symmetry breaking'' can dramatically reduce the length of propositional refutations. For each of the three propositional proof systems known as (i) treelike resolution, (ii) resolution, and (iii) k-DNF resolution, we describe families of unsatisfiable formulas in conjunctive normal form (CNF) that are ... more >>>

Olaf Beyersdorff, Nicola Galesi, Massimo Lauria

In this note we show that the asymmetric Prover-Delayer game developed by Beyersdorff, Galesi, and Lauria (ECCC TR10-059) for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form $2^{\Omega(n\log n)}$ for the pigeonhole ... more >>>

Iddo Tzameret

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege--yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analogue of Frege proofs, different from that given in Buss ... more >>>

Eli Ben-Sasson, Jakob Nordström

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There ... more >>>

Olaf Beyersdorff, Nicola Galesi, Massimo Lauria, Alexander Razborov

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that framework the parameterized version of any proof system is not fpt-bounded for some technical reasons, but we remark that this question becomes much more interesting if we restrict ourselves to those parameterized contradictions ... more >>>

Sebastian Müller, Iddo Tzameret

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>>

Paul Beame, Chris Beck, Russell Impagliazzo

We give the first time-space tradeoff lower bounds for Resolution proofs that apply to superlinear space. In particular, we show that there are formulas of size $N$ that have Resolution refutations of space and size each roughly $N^{\log_2 N}$ (and like all formulas have Resolution refutations of space $N$) for ... more >>>

Pavel Hrubes, Iddo Tzameret

We study arithmetic proof systems $\mathbb{P}_c(\mathbb{F})$ and $\mathbb{P}_f(\mathbb{F})$ operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field $\mathbb{F}$. We establish a series of structural theorems about these proof systems, the main one stating that $\mathbb{P}_c(\mathbb{F})$ proofs can be balanced: if a polynomial identity of ... more >>>

Joerg Flum, Moritz Müller

We introduce a (new) notion of parameterized proof system. For parameterized versions of standard proof systems such as Extended Frege and Substitution Frege we compare their complexity with respect to parameterized simulations.

more >>>Massimo Lauria

Ramsey Theorem is a cornerstone of combinatorics and logic. In its

simplest formulation it says that there is a function $r$ such that

any simple graph with $r(k,s)$ vertices contains either a clique of

size $k$ or an independent set of size $s$. We study the complexity

of proving upper ...
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Yuval Filmus, Massimo Lauria, Jakob Nordström, Noga Ron-Zewi, Neil Thapen

During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems ... more >>>

Massimo Lauria, Pavel Pudlak, Vojtech Rodl, Neil Thapen

We say that a graph with $n$ vertices is $c$-Ramsey if it does not contain either a clique or an independent set of size $c \log n$. We define a CNF formula which expresses this property for a graph $G$. We show a superpolynomial lower bound on the length of ... more >>>

Siu Man Chan

The two-player pebble game of Dymond–Tompa is identified as a barrier for existing techniques to save space or to speed up parallel algorithms for evaluation problems.

Many combinatorial lower bounds to study L versus NL and NC versus P under different restricted settings scale in the same way as the ... more >>>

Iddo Tzameret

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficient *deterministic* refutation algorithm for random 3SAT with ... more >>>

Moritz Müller, Stefan Szeider

So-called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs. The ordered pathwidth of a proof is shown to be roughly the same as its formula space. Length-space lower bounds for R(k)-refutations ... more >>>

Fu Li, Iddo Tzameret

Motivated by the fundamental lower bounds questions in proof complexity, we investigate the complexity of generating identities of matrix rings, and related problems. Specifically, for a field $\mathbb{F}$ let $A$ be a non-commutative (associative) $\mathbb{F}$-algebra (e.g., the algebra Mat$_d(\mathbb{F})\;$ of $d\times d$ matrices over $\mathbb{F}$). We say that a non-commutative ... more >>>

Joshua Grochow, Toniann Pitassi

We introduce a new and very natural algebraic proof system, which has tight connections to (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits ($VNP \neq VP$). As a ... more >>>

Yuval Filmus, Massimo Lauria, Mladen Mikša, Jakob Nordström, Marc Vinyals

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of CNF formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools ... more >>>

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF ... more >>>

Mladen Mikša, Jakob Nordström

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03] established that if the clause-variable incidence graph of a CNF formula F is a good ... more >>>

Fu Li, Iddo Tzameret, Zhengyu Wang

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional-calculus (i.e., Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the ... more >>>

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

The groundbreaking paper `Short proofs are narrow - resolution made simple' by Ben-Sasson and Wigderson (J. ACM 2001) introduces what is today arguably the main technique to obtain resolution lower bounds: to show a lower bound for the width of proofs. Another important measure for resolution is space, and in ... more >>>

Toniann Pitassi, Iddo Tzameret

We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to old problems, and new directions of research. In particular, we focus on tight ... more >>>

Dmitry Sokolov

In 1990 Karchmer and Widgerson considered the following communication problem $Bit$: Alice and Bob know a function $f: \{0, 1\}^n \to \{0, 1\}$, Alice receives a point $x \in f^{-1}(1)$, Bob receives $y \in f^{-1}(0)$, and their goal is to find a position $i$ such that $x_i \neq y_i$. Karchmer ... more >>>

Christoph Berkholz, Jakob Nordström

We show that there are CNF formulas which can be refuted in resolution

in both small space and small width, but for which any small-width

proof must have space exceeding by far the linear worst-case upper

bound. This significantly strengthens the space-width trade-offs in

[Ben-Sasson '09]}, and provides one more ...
more >>>

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

We define a cutting planes system CP+$\forall$red for quantified Boolean formulas (QBF) and analyse the proof-theoretic strength of this new calculus. While in the propositional case, Cutting Planes is of intermediate strength between resolution and Frege, our findings here show that the situation in QBF is slightly more complex: while ... more >>>

Olaf Beyersdorff, Luke Hinde, Ján Pich

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction.

The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we ... more >>>

Olaf Beyersdorff, Joshua Blinkhorn, Luke Hinde

As a natural extension of the SAT problem, different proof systems for quantified Boolean formulas (QBF) have been proposed. Many of these extend a propositional system to handle universal quantifiers.

By formalising the construction of the QBF proof system from a propositional proof system, by the addition of the ... more >>>

Moritz Müller, Ján Pich

We ask for feasibly constructive proofs of known circuit lower bounds for explicit functions on bit strings of length $n$. In 1995 Razborov showed that many can be proved in Cook’s theory $PV_1$, a bounded arithmetic formalizing polynomial time reasoning. He formalized circuit lower bound statements for small $n$ of ... more >>>

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, Robert Robere

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by

branching on an inequality and its "integer negation.'' That is, we can (nondeterministically ...
more >>>

Olaf Beyersdorff, Judith Clymo, Stefan Dantchev, Barnaby Martin

We give an analogue of the Riis Complexity Gap Theorem for Quanti fied Boolean Formulas (QBFs). Every fi rst-order sentence $\phi$ without finite models gives rise to a sequence of QBFs whose minimal refutations in tree-like Q-Resolution are either of polynomial size (if $\phi$ has no models) or at least ... more >>>

Olaf Beyersdorff, Judith Clymo

In their influential paper `Short proofs are narrow -- resolution made simple', Ben-Sasson and Wigderson introduced a crucial tool for proving lower bounds on the lengths of proofs in the resolution calculus. Over a decade later their technique for showing lower bounds on the size of proofs, by examining the ... more >>>

Sam Buss, Dmitry Itsykson, Alexander Knop, Dmitry Sokolov

Atserias, Kolaitis, and Vardi [AKV04] showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD($\land$, weakening), simulates CP* (Cutting Planes with unary coefficients). We show that OBDD($\land$, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring ... more >>>

Fedor Part, Iddo Tzameret

Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin$_R$), operates with disjunction of linear equations over a ring $R$. On the one hand, the system captures a natural ``minimal'' extension of resolution in which efficient counting can be achieved; ... more >>>

Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan

Strategy extraction is of paramount importance for quantified Boolean formulas (QBF), both in solving and proof complexity. It extracts (counter)models for a QBF from a run of the solver resp. the proof of the QBF, thereby allowing to certify the solver's answer resp. establish soundness of the system. So far ... more >>>

Leroy Chew

Quantified Boolean Formulas (QBFs) extend propositional formulas with Boolean quantifiers. Working with QBF differs from propositional logic in its quantifier handling, but as propositional satisfiability (SAT) is a subproblem of QBF, all SAT hardness in solving and proof complexity transfers to QBF. This makes it difficult to analyse efforts dealing ... more >>>

Iddo Tzameret, Stephen Cook

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over $GF(2)$ in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the ... more >>>