Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Abstract. We consider proof complexity in light of the unusual binary encoding of certain combinatorial principles. We contrast this proof complexity with the normal unary encoding in several refutation systems, based on Resolution and Sherali–Adams. We first consider [Formula: see text], which is an extension of Resolution working on [Formula: see text]-DNFs (Disjunctive Normal Form formulas). We prove an exponential lower bound of [Formula: see text] for the size of refutations of the binary version of the [Formula: see text]-Clique Principle in [Formula: see text], where [Formula: see text] and [Formula: see text] is a doubly exponential function. Our result improves that of Lauria et al., who proved a similar lower bound for [Formula: see text], i.e., Resolution. For the [Formula: see text]-Clique and other principles we study, we show how lower bounds in Resolution for the unary version follow from lower bounds in [Formula: see text] for the binary version, so we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the (weak) Pigeonhole Principle [Formula: see text]. We prove that for any [Formula: see text], [Formula: see text] requires refutations of size [Formula: see text] in [Formula: see text] for [Formula: see text]. Our lower bound cannot be improved substantially with the same method since for [Formula: see text] we can prove there are [Formula: see text] size refutations of [Formula: see text] in [Formula: see text]. This is a consequence of the same upper bound for the unary weak Pigeonhole Principle of Buss and Pitassi. We contrast unary versus binary encoding in the Sherali–Adams (SA) refutation system where we prove lower bounds for both rank and size. For the unary encoding of the Pigeonhole Principle and the Ordering Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the (weak) Pigeonhole Principle [Formula: see text] requires exponentially sized (in [Formula: see text]) SA refutations, whereas the binary encoding of the Ordering Principle admits logarithmic rank, polynomially sized SA refutations. We continue by considering a natural refutation system we call “SA+Squares,” which is intermediate between SA and Lasserre (Sum-of-Squares). This has been studied under the name static-[Formula: see text] by Grigoriev et al. In this system, the unary encoding of the Linear Ordering Principle [Formula: see text] requires [Formula: see text] rank while the unary encoding of the Pigeonhole Principle becomes constant rank. Since Potechin has shown that the rank of [Formula: see text] in Lasserre is [Formula: see text], we uncover an almost quadratic separation between SA+Squares and Lasserre in terms of rank. Grigoriev et al. noted that the unary Pigeonhole Principle has rank 2 in SA+Squares and therefore polynomial size. Since we show the same applies to the binary [Formula: see text], we deduce an exponential separation for size between SA and SA+Squares.

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(k)} lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [Massimo Lauria et al., 2017] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [Nathan Segerlind et al., 2004], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in Pi_2-form and with no finite model.

We propose a defi nition of Cellular Automaton in which links between cells can change during the computation. This is done locally by each cell, which can reach the neighbours of its neighbours in a single computational step. We suggest that Dynamic Neighbourhood Cellular Automata can serve as a theoretical model for studying Algorithmic and Computational Complexity issues in the are of Ubiquitous Computing. We illustrate this approach by giving an optimal logarithmic time solution of the Firing Squad Synchronisation problem in our model, which is an exponential speed-up over classical Cellular Automata. 1.

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional CNF formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter (we call such a formula a parameterized contradiction). One could separate the parameterized complexity classes FPT and W(M. Cesati, 2006) by showing that there is no fpt-bounded parameterized proof system, i.e., that there is no proof system that admits proofs of size f(k)nO(1)where f is a computable function and n represents the size of the propositional formula. By way of a first step, we introduce the system of parameterized tree-like resolution, and show that this system is not fpt-bounded. Indeed we give a general result on the size of shortest tree-like resolution proofs of parameterized contradictions that uniformly encode first-order principles over a universe of size n. We establish a dichotomy theorem that splits the exponential case of Riis's complexity-gap Theorem into two sub-cases, one that admits proofs of size f(k)nO(1)and one that does not. We also discuss how the set of parameterized contradictions may be embedded into the set of (ordinary) contradictions by the addition of new axioms. When embedded into general (DAG-like) resolution, we demonstrate that the pigeonhole principle has a proof of size 2kn2. This contrasts with the case of tree-like resolution where the embedded pigeonhole principle falls into the "non-FPT" category of our dichotomy.

We prove a dichotomy theorem for the rank of the uniformly generated(i.e. expressible in First-Order (FO) Logic) propositional tautologiesin both the Lovász-Schrijver (LS) and Sherali-Adams (SA) proofsystems. More precisely, we first show that the propositional translationsof FO formulae that are universally true, i.e. hold in all finiteand infinite models, have LS proofs whose rank is constant, independentlyfrom the size of the (finite) universe. In contrast to that, we provethat the propositional formulae that hold in all finite models butfail in some infinite structure require proofs whose SA rank grows poly-logarithmically with the size of the universe.

We prove the following two results: (1) There is a resolution proof of the Weak Pigeon-Hole Principle, WPHP/sub n//sup m/of size 2/sup O([n log n/log m]+log m)/ for any number of pigeons m and any number of holes n. (2) Any resolution proof of WPHP/sub n//sup m/ of width (1/16 - /spl epsi/) n/sup 2/ has to be of size 2/sup /spl Omega/(n)/, independently from m.. These results give not only a resolution size-width tradeoff for the Weak Pigeon-Hole Principle, but also almost optimal such trade-off for resolution in general. The upper bound (1) may be of independent interest, as it has been known for the two extreme values of m, m = n + 1 and in = 2/sup /spl radic/(n log n)/, only.

We prove that any optimal tree resolution proof of PHP/sub n//sup m/ is of size 2/sup /spl theta/(n log n)/, independently from m, even if it is infinity. So far, only a 2/sup /spl Omega/(n)/ lower bound has been known in the general case. We also show that any, not necessarily optimal, regular tree resolution proof PHP/sub n//sup m/ is bounded by 2/sup O(n log m)/. To the best of our knowledge, this is the first time the worst case proof complexity has been considered. Finally, we discuss possible connections of our result to Riis' (1999) complexity gap theorem for tree resolution.

We prove exponential lower bounds on the resolution proofs of some tautologies, based on rectangular grid graphs. More specifically, we show a 2/sup /spl Omega/(n)/ lower bound for any resolution proof of the mutilated chessboard problem on a 2n/spl times/2n chessboard as well as for the Tseitin tautology (G. Tseitin, 1968) based on the n/spl times/n rectangular grid graph. The former result answers a 35 year old conjecture by J. McCarthy (1964).