threshold circuits of small majority depth we investigate the complexity of computations with constant depth threshold circuits such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold when restricted to polynomial size these circuits compute exactly the functions in the class tc sp0 these circuits are usually studied by measuring their efficiency in terms of their total depth using this point of view the best division and iterated multiplication circuits have depth three and four respectively in this thesis we propose a different approach since threshold gates are much more powerful than and or gates we allow the explicit use of and or gates and consider the main measure of complexity to be the majority depth of the circuit i e the maximum number of threshold gates on any path in the circuit using this approach we obtain division and iterated multiplication circuits of total depth four and five but of majority depth two and three the technique used is called chinese remaindering we present this technique as a general tool for computing functions with integer values and use it to obtain depth four threshold circuits of majority depth two for other arithmetic problems such as the logarithm and power series approximation we also consider the iterated multiplication problem for integers modulo q and for finite fields the notion of majority depth naturally leads to a hierarchy of subclasses of tc sp0 we investigate this hierarchy and show that it is closely related to the usual depth hierarchy

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

some consequences of cryptographical conjectures for s 1 2 and ef we show that there is a pair of disjoint np sets whose disjointness is provable in s 2 1 and which cannot be separated by a set in p poly if the cryptosystem rsa is secure further we show that factoring and the discrete logarithm are implicitly definable in any extension of s 2 1 admitting an np definition of primes about which it can prove that no number satisfying the definition is composite

classes of representable disjoint np pairs for a propositional proof system p we introduce the complexity class dnpp p of all disjoint np pairs for which the disjointness of the pair is efficiently provable in the proof system p we exhibit structural properties of proof systems which make canonical np pairs associated with these proof systems hard or complete for dnpp p moreover we demonstrate that non equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for dnpp p and the reductions between the canonical pairs exist

nondeterministic functions and the existence of optimal proof systems we provide new characterizations of two previously studied questions on nondeterministic function classes q1 do nondeterministic functions admit efficient deterministic refinements q2 do nondeterministic function classes contain complete functions we show that q1 for the class npmv t is equivalent to the question whether the standard proof system for sat is p optimal and to the assumption that every optimal proof system is p optimal assuming only the existence of a p optimal proof system for sat we show that every set with an optimal proof system has a p optimal proof system under the latter assumption we also obtain a positive answer to q2 for the class npmv t an alternative view on nondeterministic functions is provided by disjoint sets and tuples we pursue this approach for disjoint np pairs and its generalizations to tuples of sets from np and conp with disjointness conditions of varying strength in this way we obtain new characterizations of q2 for the class npsv question q1 for npsv is equivalent to the question of whether every disjoint np pair is easy to separate in addition we characterize this problem by the question of whether every propositional proof system has the effective interpolation property again these interpolation properties are intimately connected to disjoint np pairs and we show how different interpolation properties can be modeled by np pairs associated with the underlying proof system

on noncommutative quantum field theories and dimensionless insertions in this paper we discuss the precise definition of the tree level action for the noncommutative scalar and gauge field theories on four dimensional canonically deformed euclidean space as tools to achieve this goal we employ power counting and normalization conditions

translation invariant models for non commutative gauge fields motivated by the recent construction of a translation invariant renormalizable non commutative model for a scalar field 1 we introduce models for non commutative u 1 gauge fields along the same lines more precisely we include some extra terms into the action with the aim of getting rid of the uv ir mixing

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

primitive ideal spaces characters and kirillov theory for discrete nilpotent groups abstract this paper is concerned with several topics in the harmonic analysis of discrete nilpotent groups we study two topologies on the primitive ideal space and the space of characters the separation of points in the former with the hull kernel topology a weak frobenius reciprocity property and a bochner theorem for traces on complete nilpotent groups

on the automatizability of resolution and related propositional proof systems a propositional proof system is automatizable if there is an algorithm that given a tautology produces a proof in time polynomial in the size of its smallest proof this notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound this new notion is called weak automatizability among other characterizations we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time after studying the robustness of the definition we prove the equivalence between i resolution is weakly automatizable ii res k is weakly automatizable and iii res k has feasible interpolation when k 1 in order to prove this result we show that res 2 has polynomial size proofs of the reflection principle of resolution which is a version of consistency we also show that res k for every k 1 proves its consistency in polynomial size while resolution does not in fact we show that resolution proofs of its own consistency require almost exponential size this gives a better lower bound for the monotone interpolation of res 2 and a separation from resolution as a byproduct our techniques also give us a way to obtain a large class of examples that have small resolution refutations but require relatively large width this answers a question of alekhnovich and razborov related to whether resolution is automatizable in quasipolynomial time

frege systems for quantified boolean logic we define and investigate frege systems for quantified boolean formulas qbf for these new proof systems we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class c to the qbf frege system operating with lines from c such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work this leads to strong lower bounds for restricted versions of qbf frege in particular an exponential lower bound for qbf frege systems operating with ac0 p circuits in contrast any non trivial lower bound for propositional ac0 p frege constitutes a major open problem improving these lower bounds to unrestricted qbf frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity in particular proving a lower bound for qbf frege systems operating with arbitrary p poly circuits is equivalent to either showing a lower bound for p poly or for propositional extended frege which operates with p poly circuits we also compare our new qbf frege systems to standard sequent calculi for qbf and establish a correspondence to intuitionistic bounded arithmetic

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

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 s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 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

on the renormalizability of noncommutative u 1 gauge theory an algebraic approach we investigate the quantum effects of the nonlocal gauge invariant operator in the noncommutative u 1 action and its consequences to the infrared sector of the theory nonlocal operators of such kind were proposed to solve the infrared problem of the noncommutative gauge theories evading the questions on the explicit breaking of the lorentz invariance more recently a first step in the localization of this operator was accomplished by means of the introduction of an extra tensorial matter field and the first loop analysis was carried out blaschke et al 2009 eur phys j c 62 433 43 we will complete this localization avoiding the introduction of new degrees of freedom beyond those of the original action by using only brst doublets this will allow us to conduct a complete brst algebraic study of the renormalizability of the theory following zwanziger s method of localization of nonlocal operators in qft

characterizing propositional proofs as noncommutative formulas does every boolean tautology have a short propositional calculus proof here a propositional calculus i e frege proof is any proof starting from a set of axioms and deriving new boolean formul

pseudorandom generators hard for k dnf resolution and polynomial calculus resolution a pseudorandom generator gn f0 1g n f0 1g m is hard for a propositional proof system p if roughly speaking p cannot eciently

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

on the complexity of resolution with bounded conjunctions we analyze size and space complexity of res k a family of propositional proof systems introduced by krajicek in fund math 170 1 3 2001 123 which extend resolution by allowing disjunctions of conjunctions of up to k 1 literals we show that the treelike res k proof systems form a strict hierarchy with respect to proof size and also with respect to space moreover resolution while simulating treelike res k is almost exponentially separated from treelike res k to study space complexity for general res k we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for resolution and to extend them to res k

polynomial size frege and resolution proofs of st connectivity and hex tautologies a grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices the st connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere we prove that the propositional tautologies which encode the st connectivity principle have polynomial size frege proofs and polynomial size tc0 frege proofs for bounded width grid graphs the st connectivity tautologies have polynomial size resolution proofs a key part of the proof is to show that the group with two generators both of order two has word problem in alternating logtime alogtime and even in tc0 conversely we show that constant depth frege proofs of the st connectivity tautologies require near exponential size the proof uses a reduction from the pigeonhole principle via tautologies that express a directed single source principle sink which is related to papadimitriou s search classes ppad and ppads or psk the st connectivity principle is related to urquhart s propositional hex tautologies and we establish the same upper and lower bounds on proof complexity for the hex tautologies in addition the hex tautology is shown to be equivalent to the sink tautologies and to the one to one onto pigeonhole principle