Results 11  20
of
22
Feasible proofs of matrix properties with Csanky’s algorithm
 in: Computer Science Logic (CSL’05), 2005
"... 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 introduc ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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 CayleyHamilton 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
Lower Bounds for (MOD p  MOD m) Circuits
 Proc. 39th IEEE FOCS
, 1998
"... Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circui ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Modular gates are known to be immune for the random restriction techniques of Ajtai (1983), Furst, Saxe, Sipser (1984), Yao (1985) and Hastad (1986). We demonstrate here a random clustering technique which overcomes this diculty and is capable to prove generalizations of several known modular circuit lower bounds of Barrington, Straubing, Therien (1990), Krause and Pudlak (1994), and others, characterizing symmetric functions computable by small (MOD p ; AND t ; MODm ) circuits. Applying a degreedecreasing technique together with random restriction methods for the AND gates at the bottom level, we also prove a hard special case of the Constant Degree Hypothesis of Barrington, Straubing, Therien (1990), and other related lower bounds for certain (MOD p ; MODm ; AND) circuits. Most of the previous lower bounds on circuits with modular gates used special denitions of the modular gates (i.e., the gate outputs one if the sum of its inputs is divisible by m, or is not divisible by m), and were not valid for more general MODm gates. Our methods are applicable, and our lower bounds are valid, for the most general modular gates as well. 1
A DegreeDecreasing Lemma for (MOD q  MOD p) Circuits
, 2001
"... 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 complexit ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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
Polynomial Programs and the RazborovSmolensky Method
 Electronic Colloquium on Computational Complexity
, 2001
"... 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] fun ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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 lowdegree multilinear polynomials over fields of characteristic p. Another tool which has yielded insight into smalldepth circuit complexity classes is the programovermonoids model of computation, which has provided characterizations of circuit complexity classes such as AC 0 and NC 1 .
unknown title
"... 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. ..."
Abstract
 Add to MetaCart
(Show Context)
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.
U.C. Berkeley — CS278: Computational Complexity Handout N25 Professor Luca Trevisan 12/1/2004 Notes for Lecture 25
"... In this lecture we prove a lower bound on the size of a constant depth circuit which computes the XOR of n bits. Before we talk about bounds on the size of a circuit, let us first clarify what we mean by circuit depth and circuit size. The depth of a circuit is defined as the length of the longest p ..."
Abstract
 Add to MetaCart
(Show Context)
In this lecture we prove a lower bound on the size of a constant depth circuit which computes the XOR of n bits. Before we talk about bounds on the size of a circuit, let us first clarify what we mean by circuit depth and circuit size. The depth of a circuit is defined as the length of the longest path from the input to output. The size of a circuit is the number of AND and OR gates in the circuit. Note that, for our purpose, we assume all the gates have unlimited fanin and fanout. We define AC 0 to be the class of decision problems solvable by circuits of polynomial size and constant depth. We want to prove the result that PARITY is not in AC 0. There are two known techniques to prove this result. In this class, we will talk about a proof which uses polynomials; in the next class we will look at a different proof which uses random restrictions. 1 Circuit Upper Bounds for PARITY Before we go into our proof, let us first look at a circuit of constant depth d that computes
On the Relation Between BDDs and FDDs (Extended Abstract)
, 1994
"... Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art datastructure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branc ..."
Abstract
 Add to MetaCart
(Show Context)
Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art datastructure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXORbased logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. BDDs (FDDs) are directed acyclic graphs, whe...
SIGACT News Complexity Theory Column 19
"... 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 ..."
Abstract
 Add to MetaCart
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
Towards NP−P via Proof Complexity and Search
, 2009
"... 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. ..."
Abstract
 Add to MetaCart
(Show Context)
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.
SIGACT News Complexity Theory Column 19
, 1997
"... 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 ..."
Abstract
 Add to MetaCart
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