plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this result

Abstract. We show that Csanky’s fast parallel algorithm for computing the characteristic polynomial of a matrix can be formalized in the logical theory LAP, and can be proved correct in LAP from the principle of linear independence. LAP is a natural theory for reasoning about linear algebra introduced in [8]. Further, we show that several principles of matrix algebra, such as linear independence or the Cayley-Hamilton Theorem, can be shown equivalent in the logical theory QLA. Applying the separation between complexity classes AC 0 [2] � DET(GF(2)), we show that these principles are in fact not provable in QLA. In a nutshell, we show that linear independence is “all there is ” to elementary linear algebra (from a proof complexity point of view), and furthermore, linear independence cannot be proved trivially (again, from a proof complexity point of view). Key words: Proof complexity, Csanky’s algorithm, matrix algebra. 1

Representations of boolean functions as polynomials (over rings) have been used to establish lower bounds in complexity theory. Such representations were used to great effect by Smolensky, who established that MOD q / # AC 0 [MOD p] (for distinct primes p, q) by representing AC 0 [MOD p] functions as low-degree multilinear polynomials over fields of characteristic p. Another tool which has yielded insight into small-depth circuit complexity classes is the program-over-monoids model of computation, which has provided characterizations of circuit complexity classes such as AC 0 and NC 1 .

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected. 3 Definitions

this paper describes the complete problems that motivate interest in these classes, discusses some surprising recent discoveries, and points out open problems where progress can reasonably be expected

This is a survey of work on proof complexity and proof search, as motivated by the P versus NP problem. We discuss propositional proof complexity, Cook’s program, proof automatizability, proof search, algorithms for satisfiability, and the state of the art of our (in)ability to separate P and NP.

The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧, ∨, ¬. Suppose L ∈ {0, 1} ∗ is a language. Let Ln = L ∩ {0, 1} n. We say that L is computed by a family of circuits C1, C2,... if on an input x = (x1,..., xn), Cn(x) is 1 when x ∈ Ln and is 0 when x / ∈ Ln. For a circuit C, we define size(C) to be the number of edges in the graph representing C, and depth(C) to be the length of the longest path from an input to the output. We say that L ∈ P /poly if there exists a family C1, C2,... computing L such that size(Cn) = n O(1). It is easy to see that if a Turing machine computes L in time T (n), then there exists a family of circuits C1, C2,... computing L so that size(Cn) ≤ (T (n)) 2. If a Turing machine is given a circuit C and an input x, then it can compute C(x) in time size(C). However, there exist languages that are computable by families of circuits but are not computable by Turing machines. The simplest example is L = Halting problem, Cn(x) = 1 iff L(n) = 1 for |x | = n. But in some sense, this is not an interesting example.