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.     

Mean-payoff games and propositional proofs

We associate a CNF-formula to every instance of the mean-payoff game problem in such a way that if the value of the game is non-negative the formula is satisfiable, and if the value of the game is negative the formula has a polynomial-size refutation in Σ₂-Frege (i.e. DNF-resolution). This reduces mean-payoff games to the weak automatizability of Σ₂-Frege, and to the interpolation problem for Σ_2,2-Frege. Since the interpolation problem for Σ₁-Frege (i.e. resolution) is solvable in polynomial time, our result is close to optimal up to the computational complexity of solving mean-payoff games. The proof of the main result requires building low-depth formulas that compute the bits of the sum of a constant number of integers in binary notation, and low-complexity proofs of the required arithmetic properties.     

A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

We prove a superlinear lower bound on the size of a bounded depth bilinear arithmetical circuit computing cyclic convolution. Our proof uses the strengthening of the Donoho–Stark uncertainty principle [D.L. Donoho, P.B. Stark, Uncertainty principles and signal recovery, SIAM Journal of Applied Mathematics 49 (1989) 906–931] given by Tao [T. Tao, An uncertainty principle for cyclic groups of prime order, Mathematical Research Letters 12 (2005) 121–127], and a combinatorial lemma by Raz and Shpilka [R. Raz, A. Shpilka, Lower bounds for matrix product, in arbitrary circuits with bounded gates, SIAM Journal of Computing 32 (2003) 488–513]. This combination and an observation on ranks of circulant matrices, which we use to give a much shorter proof of the Donoho–Stark principle, may have other applications.     

A lower bound for testing juntas

We show an Ω(m) lower bound on the number of queries required to test whether a Boolean function depends on at most m out of its n variables. This improves a previously known lower bound for testing this property. Our proof is simple and uses only elementary techniques.     

A lower bound method for quantum circuits

Quantum circuits, which are shallow, limited in the number of gates and additional workspace qubits, are popular for quantum computation because they form the simplest possible model similar to the classical model of a network of Boolean gates and capable of performing non-trivial computation. We give a new lower bound technique for such circuits and use it to give another proof that deterministic computation of the parity function cannot be performed by such circuits.     

On the computational complexity of qualitative coalitional games

We study coalitional games in which agents are each assumed to have a goal to be achieved, and where the characteristic property of a coalition is a set of choices, with each choice denoting a set of goals that would be achieved if the choice was made. Such qualitative coalitional games (qcgs) are a natural tool for modelling goal-oriented multiagent systems. After introducing and formally defining qcgs, we systematically formulate fourteen natural decision problems associated with them, and determine the computational complexity of these problems. For example, we formulate a notion of coalitional stability inspired by that of the core from conventional coalitional games, and prove that the problem of showing that the core of a qcg is non-empty is D^p₁-complete. (As an aside, we present what we believe is the first “natural” problem that is proven to be complete for D^p₂.) We conclude by discussing the relationship of our work to other research on coalitional reasoning in multiagent systems, and present some avenues for future research.     

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.     

Complexity of question/answer games

Question/Answer games (Q/A games for short) are a generalization of the Rényi–Ulam game and they are a model for information extraction in parallel. A Q/A game, G=(D,s,(_q1,…,_qk))$G=(D,s,(q_1,\dots ,q_k))$, is played on a directed acyclic graph, D=(V,E)$D=(V,E)$, with a distinguished start vertex s$s$. In the i$i$th round, Paul selects a set, _Qi⊆V$Q_i\subseteq V$, of at most _qi$q_i$ non-terminal vertices. Carole responds by choosing an outgoing edge from each vertex in _Qi$Q_i$. At the end of k$k$ rounds, Paul wins if Carole’s answers define a unique path from the root to one of the terminal vertices in D$D$.     

A simplified way of proving trade-off results for resolution

We present a greatly simplified proof of the length-space trade-off result for resolution in [P. Hertel, T. Pitassi, Exponential time/space speedups for resolution and the PSPACE-completeness of black-white pebbling, in: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS '07), Oct. 2007, pp. 137-149], 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 some possible conclusions. Our key trick is to look at formulas of the type F=G∧H, where G and H are over disjoint sets of variables and have very different length-space properties with respect to resolution.     

On the computational complexity of coalitional resource games

We study Coalitional Resource Games (crgs), a variation of Qualitative Coalitional Games (qcgs) in which each agent is endowed with a set of resources, and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources. We investigate and classify the computational complexity of a number of natural decision problems for crgs, over and above those previously investigated for qcgs in general. For example, we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound (i.e., a limit value for every resource) is co-np-complete. We then investigate the relationship between crgs and qcgs, and in particular the extent to which it is possible to translate between the two models. We first characterise the complexity of determining equivalence between crgs and qcgs. We then show that it is always possible to translate any given crg into a succinct equivalent qcg, and that it is not always possible to translate a qcg into an equivalent crg; we establish some necessary and some sufficient conditions for a translation from qcgs to crgs to be possible, and show that even where an equivalent crg exists, it may have size exponential in the number of goals and agents of its source qcg.     

A large lower bound on the query complexity of a simple boolean function

Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron (1998) and inspired by Rubinfeld and Sudan (1996), deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input satisfies the property P or is far from having the property. For a family of boolean functions f=(fn) the associated property is the set of 1-inputs of f. Here, the known lower bounds on the query complexity of properties identified by boolean functions representable by (very) restricted branching programs of small size is improved up to Ω(n1/2), where n is the input length.     

On the complexity of constrained Nash equilibria in graphical games

A widely accepted rational behavior for non-cooperative players is based on the notion of Nash equilibrium. Although the existence of a Nash equilibrium is guaranteed in the mixed framework (i.e., when players select their actions in a randomized manner) in many real-world applications the existence of “any” equilibrium is not enough. Rather, it is often desirable to single out equilibria satisfying some additional requirements (in order, for instance, to guarantee a minimum payoff to certain players), which we call constrained Nash equilibria.In this paper, a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games, where a player p may directly be interested in some of the other players only, called the neighbors of p. This setting is very useful for modeling large population games, where typically each player does not directly depend on all the players, and representing her utility function extensively is either inconvenient or infeasible.Based on this framework, the complexity of deciding the existence and of computing constrained equilibria is then investigated, in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players’ interactions. The analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies, i.e., when players are forced to deterministically choose the action to perform. In particular, for this latter case, restrictions on players’ interactions and on constraints are identified, that make the computation of Nash equilibria an easy problem, for which polynomial and highly-parallelizable algorithms are presented.     

A lower bound for computing Oja depth

Let S={s₁,…,sn} be a set of points in the plane. The Oja depth of a query point θ with respect to S is the sum of the areas of all triangles (θ,si,sj). This depth may be computed in O(nlogn) time in the RAM model of computation. We show that a matching lower bound holds in the algebraic decision tree model. This bound also applies to the computation of the Oja gradient, the Oja sign test, and to the problem of computing the sum of pairwise distances among points on a line.     

A polynomial space construction of tree-like models for logics with local chains of modal connectives

The LA-logics (“logics with Local Agreement”) are polymodal logics defined semantically such that at any world of a model, the sets of successors for the different accessibility relations can be linearly ordered and the accessibility relations are equivalence relations. In a previous work, we have shown that every LA-logic defined with a finite set of modal indices has an NP-complete satisfiability problem. In this paper, we introduce a class of LA-logics with a countably infinite set of modal indices and we show that the satisfiability problem is PSPACE-complete for every logic of such a class. The upper bound is shown by exhibiting a tree structure of the models. This allows us to establish a surprising correspondence between the modal depth of formulae and the number of occurrences of distinct modal connectives. More importantly, as a consequence, we can show the PSPACE-completeness of Gargov's logic DALLA and Nakamura's logic LGM restricted to modal indices that are rational numbers, for which the computational complexity characterization has been open until now. These logics are known to belong to the class of information logics and fuzzy modal logics, respectively.     

A metric for positional games

We define an extended real-valued metric, ρ, for positional games and prove that this class of games is a topological semigroup. We then show that two games are finitely separated iff they are path-connected and iff two closely related Conway games are equivalent. If two games are at a finite distance then this distance is bounded by the maximum difference of any two atoms found in the games. We may improve on this estimate when two games have the same form, as given by a form match. Finally, we show that if ρ(G,H)=∞ then for all X we have G+X H+X, a step towards proving cancellation for positional games.     

Top-down lower bounds for depth-three circuits

We present a top-down lower bound method for depth-three , , ¬-circuits which is simpler than the previous methods and in some cases gives better lower bounds. In particular, we prove that depth-three , , ¬-circuits that compute parity (or majority) require size at least , respectively). This is the first simple proof of a strong lower bound by a top-down argument for non-monotone circuits.     

Quantum lower bounds for the Goldreich-Levin problem

At the heart of the Goldreich-Levin theorem is the problem of determining an n-bit string a by making queries to two oracles, referred to as IP (inner product) and EQ (equivalence). The IP oracle, on input x, returns a bit that is biased towards a⋅x (the modulo two inner product of a with x) in the following sense. For a random x, the probability that IP(x)=a⋅x is at least . The EQ oracle, on input x, returns a bit specifying whether or not x=a. It has been shown that a quantum algorithm can solve this problem with O(1/?) IP and EQ queries, whereas any classical algorithm requires Ω(n/?²) such queries. Also, the quantum algorithm requires only O(n/?) auxiliary one- and two-qubit gates in addition to its queries. We show that the above quantum algorithm is optimal in terms of both EQ and IP queries. Specifically, Ω(1/?) EQ queries are necessary, and Ω(1/?) IP queries are necessary if the number of EQ queries is .