Focusing and polarization in linear,intuitionistic, and classical logics

A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.     

Exponential lower bounds for the pigeonhole principle

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 (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 polynomial-size, logn-depth Frege proofs for the PHP. The main lemma in our proof can be viewed as a general Håstad-style Switching Lemma for restrictions that are partial matchings. Our lower bounds for the pigeonhole principle improve on previous superpolynomial lower bounds.     

Cut-elimination and Redundancy-elimination by Resolution

A new cut-elimination method for Gentzen’s LK is defined. First cut-elimination is generalized to the problem of redundancy-elimination. Then the elimination of redundancy in LK-proofs is performed by a resolution method in the following way. A set of clauses C is assigned to an LK-proof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LK-proof ψ′ with atomic cuts;ψ′ can be constructed from the resolution proof and ψ by a projection method. In the final step the atomic cuts are eliminated and a cut-free proof is obtained. The complexity of the method is analyzed and it is shown that a non-elementary speed-up over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudo-cuts) can be transformed into formal ones by the method of redundancy-elimination; moreover, the method can even be used to transform incorrect proofs into correct ones.     

Property Testing Lower Bounds via Communication Complexity

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of ??(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.     

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

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 form 2 ^N(1) for the number of formulas in a constant-depth Frege proof of the modular counting principleCount _q ^N from instances of the counting principleCount _m ^M .We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations.Further we shwo that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connectiveMOD _p follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987) and Smolensky (1987) (in fact, these two proof systems are basically equivalent).     

Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Razborov (1996) recently proved that polynomial calculus proofs of the pigeonhole principle must have degree at least ⌈n/2⌉ + 1 over any field. We present a simplified proof of the same result.?Furthermore, we show a matching upper bound on polynomial calculus proofs of the pigeonhole principle for any field of suficiently large characteristic, and an ⌈n/2⌉ + 1 lower bound for any subset sum problem over the field of reals.?We show that these degree lower bounds also translate into lower bounds on the number of monomials in any polynomial calculus proof, and hence on the running time of most implementations of the Gr?bner basis algorithm. Received: October 14, 1997.     

A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover-Delayer games

In this note we show that the asymmetric Prover-Delayer game developed in Beyersdorff et al. (2010) [2] 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 ²Ω(nlogn) for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki (1999) [7] and Dantchev and Riis (2001) [5].     

Short proofs of the pigeonhole formulas based on the connection method

Quadratic proofs of pigeonhole formulas are presented using connection method proof techniques. The interest of this result derives from the fact that for this class of formulas exponential lower bounds are known for the length of resolution refutations.     

Relations between diagonalization,proof systems,and complexity gaps

In this paper we study diagonal processes over time bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. These diagonalization methods also show that the Gap Theorem for resource bounded computations can hold only for those complexity classes which differ from the corresponding provable complexity classes. Furthermore, we show that there exist recursive time bounds T(n) such that the class of languages for which we can formally prove the existence of Turing machines which accept them in time T(n) differs from the class of languages accepted by Turing machines for which we can prove formally that they run in time T(n). We also investigate the corresponding problems for tape bound computations and discuss the difference time and tapebounded computations.     

Parameterized Proof Complexity

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 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[SAT] by showing that there is no fpt-bounded parameterized proof system for parameterized contradictions, i.e., that there is no proof system that admits proofs of size f(k)n ^O(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 subcases, one that admits proofs of size f(k)n ^O(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 2 ^k n ². This contrasts with the case of tree-like resolution where the embedded pigeonhole principle falls into the “non-FPT” category of our dichotomy.     

Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates

Consider circuits consisting of a threshold gate at the top, Mod _m gates at the next level (for a fixed m ), and polylog fan-in AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for restricted versions of this model, in which each Mod _m -of-AND subcircuit is a symmetric function of the inputs to that subcircuit. We show that if q is a prime not dividing m , the Mod _q function requires exponential-size circuits of this type. This generalizes recent results and techniques of Cai et al. [CGT] (which held only for q=2 ) and Goldmann (which held only for depth two threshold over Mod _m circuits). As a further generalization of the [CGT] result, the symmetry condition on the Mod _m subcircuits can be relaxed somewhat, still resulting in an exponential lower bound. The basis of the proof is to reduce the problem to estimating an exponential sum, which generalizes the notion of ``correlation" studied by [CGT]. This identifies the type of exponential sum that will be instrumental in proving the general case. Along the way we substantially simplify previous proofs. Received June 1997, and in final form October 1998.     

Regaining cut admissibility in deduction modulo using abstract completion

Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore.We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system.Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth–Bendix completion in a non-trivial way, using the framework of abstract canonical systems.These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.     

Upper and lower bounds for some depth-3 circuit classes

We investigate the complexity of depth-3 threshold circuits with majority gates at the output, possibly negated AND gates at level two, and MOD _m gates at level one. We show that the fan-in of the AND gates can be reduced toO(logn) in the case wherem is unbounded, and to a constant in the case wherem is constant. We then use these upper bounds to derive exponential lower bounds for this class of circuits. In the unboundedm case, this yields a new proof of a lower bound of Grolmusz; in the constantm case, our result sharpens his lower bound. In addition, we prove an exponential lower bound if OR gates are also permitted on level two andm is a constant prime power.Dedicated to the memory of Roman Smolensky     

Lower bounds for maximal and convex layers problems

In this paper we derive tight lower bounds for the maximal and convex layers problems in the plane. Our lower bound proofs for the maxima problem and convex hull problem are simpler than those previously known. We also obtain an Ω(nlog n) lower bound for the maximal depth problem, and the convex depth problem, when the points are given in sorted order of their x-coordinates.     

An exponential lower bound on the size of algebraic decision trees for Max

We prove an exponential lower bound on the size of any fixed degree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n— 1 lower bound of [Rabin (1972)] on the depth of algebraic decision trees for this problem. The proof in fact gives an exponential lower bound on the size of the polyhedral decision problem MAX= for testing whether the j-th number is the maximum among a list of n real numbers. Previously, except for linear decision trees, no nontrivial lower bounds on the size of algebraic decision trees for any familiar problems are known. We also establish an interesting connection between our lower bound and the maximum number of minimal cutsets for any rank-d hypergraph on n vertices. Received: 15 April 1996     

Lower Bounds for Randomized Exclusive Write PRAMs

In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and we present randomized lower bounds which match the best deterministic lower bounds known. (For the CREW PRAM model, the lower bound is asymptotically optimal.) These are the first nontrivial randomized lower bounds known for the compaction problem on these models. We show that our lower bounds also apply to the problem of approximate compaction. Next we examine the problem of computing boolean functions on the CREW PRAM model, and we present a randomized lower bound which improves on the previous best randomized lower bound for many boolean functions, including the OR function. (The previous lower bounds for these functions were asymptotically optimal, but we improve the constant multiplicative factor.) We also give an alternate proof for the randomized lower bound on PARITY, which was already optimal to within a constant additive factor. Lastly, we give a randomized lower bound for integer merging on an EREW PRAM which matches the best deterministic lower bound known. In all our proofs, we use the Random Adversary method, which has previously only been used for proving lower bounds on models with Concurrent Write capabilities. Thus this paper also serves to illustrate the power and generality of this method for proving parallel randomized lower bounds. Received October 2, 1995, and in final form January 29, 1997.     

Throughput bounds for closed queueing networks

Analytical lower and upper bounds for the throughput of closed queueing networks with single and delay (infinite) servers are studied in this paper. The numerical evaluation of these bounds requires a small number of significant operations which is independent of the population N. This is in contrast to the exact computation of the throughput which requires at least O(N) operations as N tends to infinity. The bounds are given by simple closed-form analytical expressions and may be more suitable for various performance studies than the algorithmical form of the exact solution.In this paper, the previously known balanced-job bounds are generalized to networks containing delay servers (terminals) and a hierarchy of bounds is obtained for single and multiple class networks. For the single class network, further new bounds are derived: lower and upper bounds that require the evaluation of one square root and an upper bound that requires a constant number of exponentiations. This upper bound does not employ the balancing of server loadings and is especially useful for asymptotic analysis in the case of a large number of customers N.     

Algebraic proofs over noncommutative formulas

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 analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitin?s formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991).The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.