Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 De-randomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 De-randomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

We show that neither Resolution nor tree-like Resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is xed-parameter tractable by randomized algorithms with one-sided error.

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the Dense Linear Order Principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the size-width relationship is tight.

We consider tautologies formed from a pseudo-random number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-logicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any text-book system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...

A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomial-mapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f: C → Cm, such that, the image of f is not contained in the image of any polynomial-mapping Γ: Cm−1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness2), of degree up to 2mo(1) , implies super-polynomial lower bounds for computing the permanent over C. More generally, we say that a polynomial-mapping f: Fn → Fm is (s, r)-elusive, if for every polynomial-mapping Γ: Fs → Fm of degree r, Image(f) � ⊂ Image(Γ).

Abstract. We show that constant-depth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constant-depth Frege with counting axioms systems. 1

Abstract. Goldreich (ECCC 2000) proposed a candidate one-way function construction which is parameterized by the choice of a small predicate (over d = O(1) variables) and of a bipartite expanding graph of right-degree d. The function is computed by labeling the n vertices on the left with the bits of the input, labeling each of the n vertices on the right with the value of the predicate applied to the neighbors, and outputting the n-bit string of labels of the vertices on the right. Inverting Goldreich’s one-way function is equivalent to finding solutions to a certain constraint satisfaction problem (which easily reduces to SAT) having a “planted solution, ” and so the use of SAT solvers constitutes a natural class of attacks. We perform an experimental analysis using MiniSat, which is one of the best publicly available algorithms for SAT. Our experiment shows that the running time required to invert the function grows exponentially with the length of the input, and that such an attack becomes infeasible

We answer this question by showing a polynomial encoding of Graph Ordering Principle on m variables which requires PC and PCR refutations of degree Ω ( √ m). Trade-offs optimality follows from our result and from the short refutations of Graph Ordering Principle in [Bonet and Galesi 1999; 2001]. We then introduce the algebraic proof system PCRk which combines together Polynomial Calculus (PC) and k-DNF Resolution (RESk). We show a size hierarchy theorem for PCRk: PCRk is exponentially separated from PCRk+1. This follows from the previous degree lower bound and from techniques developed for RESk. Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in PCRk.

Explicit constructions of selectors and related combinatorial structures, with applications

We suggest a candidate one-way function using combinatorial constructs such as expander graphs. These graphs are used to determine a sequence of small overlapping subsets of input bits, to which a hard-wired random predicate is applied. Thus, the function is extremely easy to evaluate: all that is needed is to take multiple projections of the input bits, and to use these as entries to a look-up table. It is feasible for the adversary to scan the look-up table, but we believe it would be infeasible to find an input that fits a given sequence of values obtained for these overlapping projections. The conjectured difficulty of inverting the suggested function does not seem to follow from any well-known assumption. Instead, we propose the study of the complexity of inverting this function as an interesting open problem, with the hope that further research will provide evidence to our belief that the inversion task is intractable. Supported by MINERVA Foundation, Germany. 0 1 In...