Abstract

Topical but classical results concerning the incidence relationship between prime clauses and implicants of a monotone Boolean function are derived by applying a general theory of computational equivalence and replaceability to distributive lattices. A non-standard combinatorial model for the free distributive lattice FDL(n) is described, and a correspondence between monotone Boolean functions and partitions of a standard Cayley diagram for the symmetric group is derived.

Preliminary research on-classifying and characterising the simple paths and circuits that are the blocks of this partition is summarised. It is shown in particular that each path and circuit corresponds to a characteristic configuration of implicants and clauses. The motivation for the research and expected future directions are briefly outlined.

Introduction

Models of Boolean formulae expressed in terms of the incidence relationship between the prime implicants and clauses of a function were first discovered several years ago, but they have recently been independently rediscovered by several authors, and have attracted renewed interest. They have been used in proving lower bounds by Karchmer and Wigderson and subsequently by Razborov. More general investigations aimed at relating the complexity of functions to the model have also been carried out by Newman [20].

This paper demonstrates the close connection between these classical models for monotone Boolean formulae and circuits and a general theory of computational equivalence as it applies to FDL(n): the (finite) distributive lattice freely generated by n elements. It also describes how the incidence relationships between prime implicants and clauses associated with monotone Boolean functions can be viewed as built up from a characteristic class of incidence patterns between relatively small subsets of implicants and clauses.

Abstract

We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mACi, mNCi, mLOGCFL, mBWBP, mL, mNL, mP, mBPP and mNP. We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions.

We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcsényi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity.

We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC1, motivated by Barrington's result [Bar89] that BWBP = NC1. Although we cannot answer this question, we show two preliminary results: every monotone branching program for majority has size Ω(n2) with no width restriction, and no monotone analogue of Barrington's gadget exists.

Introduction

A computation is monotone if it does not use the negation operation. Monotone circuits and formulas have been studied as restricted models of computation with the goal of developing techniques for the general problem of proving lower bounds.

In this paper we seek to unify the theory of monotone complexity along the lines of Babai, Frankl, and Simon who gave a framework for communication complexity theory. We propose a collection of monotone complexity models paralleling the familiar nonmonotone models. This provides a rich classification system for monotone functions including most monotone circuit classes previously considered, as well as monotone space-bounded complexity classes which have previously received little attention.

Abstract

We survey some recent results on read-once Boolean functions. Among them are a characterization theorem, a generalization and a discussion on the randomized Boolean decision tree complexity for read-once functions. A previously unpublished result of Lovás and Newman is also presented.

Introduction

A Boolean formula is a rooted binary tree whose internal nodes are labeled by the Boolean operators ∨ or Λ and in which each leaf is labeled by a Boolean variable or its negation. A Boolean formula computes a Boolean function in a natural way.

A Boolean formula is read-once if every variable appears exactly once. A function is read-once if it has a read-once formula.

Read-once functions have been studied by many authors, since they have the lowest possible formula size (for functions that depend on all their variables). In addition, every NC1 function on n variables is a projection of a read-once function with a polynomial (in n) number of variables.

We present here some recent results in the area. All but one of those results have been published, hence, full proofs will be generally omitted and will be given just for the unpublished result (Theorem 3.4). The results we will discuss cover a characterization theorem, some generalizations and results on the randomized decision tree complexity of read-once functions. There is a recent result on learning of read-once functions, [AHK89], which will not be described.

Definitions and Notations

If g : {0, l}n ↦ {0, 1} has a formula in which no negated variable appears, we say that g is monotone. The size of a Boolean formula is the number of its leaves.

Introduction

In the last decade substantial progress has been made in our understanding of restricted classes of Boolean circuits, in particular those restricted to have constant depth (Furst, Sipser, Saxe, Ajtai, Yao, Haiåstad, Razborov, Smolensky or to be monotone (Razborov, Andreev, Alon and Boppana, Tardos, Karchmer and Wigderson). The question arises, perhaps more urgently than before, as to what approaches could be pursued that might contribute to progress on the unrestricted model.

In this note we first argue that if P ≠ NP then any circuit-theoretic proof of this would have to be preceded by analogous results for the more constrained arithmetic model. This is because, as we shall observe, there are proven implications showing that if, for example, the Hamiltonian cycle problem (HC) requires exponential circuit size, then so does the analogous problem on arithmetic circuits. Since the set of valid algebraic identities in the latter model form a proper subset of those in the former, a lower bound proof for it should be strictly easier.

In spite of the above relationship the algebraic model is often regarded as an alternative, rather than a restriction of the Boolean model. One reason for this is that specific computations are usually understandable in one of these models, and not in both. In particular, the main power of the algebraic model derives from the possibility of cancellations, and it is usually difficult to express explicitly how these help in computing combinatorial problems.

Introduction

In recent years several methods have been developed for obtaining superpolynomial lower bounds on the monotone formula and circuit size of explicitly given Boolean functions. Among these are the method of approximations, the combinatorial analysis of a communication problem related to monotone depth and the use of matrices with very particular rank properties. Now it can be said almost surely that each of these methods would need considerable strengthening to yield nontrivial lower bounds for the size of circuits or formulae over a complete basis. So, it seems interesting to try to understand from the formal point of view what kind of machinery we lack.

The first step in that direction was undertaken by the author in. In that paper two possible formalizations of the method of approximations were considered. The restrictive version forbids the method to use extra variables. This version was proven to be practically useless for circuits over a complete basis. If extra variables are allowed (the second formalization) then the method becomes universal, i.e. for any Boolean function f there exists an approximating model giving a lower bound for the circuit size of f which is tight up to a polynomial. Then the burden of proving lower bounds for the circuit size shifts to estimating from below the minimal number of covering sets in a particular instance of “MINIMUM COVER”. One application of an analogous model appears in where the first nonlinear lower bound was proven for the complexity of MAJORITY with respect to switching-and-rectifiers networks.

Abstract

Let f be an arbitrary Boolean function depending on n variables and let A be a network computing them, i.e., A has n inputs and one output and for an arbitrary Boolean vector a of length n outputs f(a). Assume we have to compute simultaneously the values f(a1), …,f(ar) of f on r arbitrary Boolean vectors a1, …,ar. Then we can do it by r copies of A. But in most cases it can be done more efficiently (with a smaller complexity) by one network with nr inputs and r outputs (as already shown in Uhlig (1974)). In this paper we present a new and simple proof of this fact based on a new construction method. Furthermore, we show that the depth of our network is “almost” minimal.

Introduction

Let us consider (combinatorial) networks. Precise definitions are given in [Lu58, Lu65, Sa76, We87]. We assume that a complete set G of gates is given, i.e., every Boolean function can be computed (realized) by a network consisting of gates of G. For example, the set consisting of 2-input AND, 2-input OR and the NOT function is complete. A cost C(Gi) (a positive number) is associated with each of the gates Gi ∈ G. The complexity C(A) of a network A is the sum of the costs of its gates. The complexity C(f) of a Boolean function f is defined by C(f) = min C(A) where A ranges over all networks computing f.

By Bn we denote the set of Boolean functions {0, l}n → {0, 1}.

This paper is a survey of some results by the author in the study of the subgroup structure of the finite simple groups of Lie type. Throughout the paper G is a simple algebraic group; if G is denned over a finite field Fq then σ is some Steinberg endomorphism of G. We shall omit index G in notations like NG(X), CG(X).

Subgroups of simple groups of exceptional type.

The first result of the paper is a reduction theorem for the maximal subgroups of finite exceptional groups similar to the well-known result of M. Aschbacher for finite classical groups. In the case of classical groups Theorem 1 doesn't give any new information, but it may be useful in the study of simple groups of exceptional type. Another and more explicit version of a reduction theorem was obtained recently by M. W. Liebeck and G. M. Seitz.

Theorem 1Let G be defined over the finite field Fq, Gσ ≅ G(Fq) and. Let G0 ≤ G1 ≤ Aut G0and let M be a subgroup in G1. Then one of the following statements is valid:

1. (a) for some proper connected nontrivial σ-invariant subgroup H ≤ G;

2. (b) M is an almost simple group, i.e. S ≤ M ≤ Aut S for some simple group S;

3. (c) M ≤ NG(J) for some Jordan subgroup J in G;

4. (d) G is of type E8, charFq = p > 5 and M ≤ NGl(X), where X ≅ Alt5 × Alt6.

Introduction

McLaughlin's sporadic simple group McL was originally constructed (see) as a permutation group on 275 letters. It is a simple group of order 898128000 = 27.36.53.7.11. It is now known to be the pointwise stabilizer of a 2-dimensional sublattice in the Leech lattice. Its maximal subgroups were found by Finkelstein (see). The modular character tables for the relevant primes p = 2, 7 and 11 were found by Thackray. The 5-modular character tables were found by Hiss, Lux and Parker, up to a few ambiguities (see). These ambiguities together with others in the values of the 5- modular characters 560 and 3038 of the automorphism group of McL, denoted by McL.2, were resolved by Suleiman (see). The main purpose of this paper is to complete the 3-modular character table of McL and to find the 3-modular character table of McL.2.

The 3-modular character table of McL

In this section we are going to complete what has been done by R. Parker on the 3-modular characters of McL. To do so we have to work out again most of the 3-modular characters using the techniques of the ‘Meat-Axe’ which is the main tool in our work. We then use the method of ‘condensation’ (see) to complete the 3-modular character table of McL.

The central characters modulo 3 give the block distribution of the ordinary irreducible characters. There are three blocks of defect zero. These blocks are B1 = {5103}, B2 = {8019a} and B3 = {8019b}. Hence, 5103, 8019a and 8019b are three 3-modular irreducible characters in McL.

Introduction.

The first theorem given here asserts that a geometric hyperplane H of a near hexagon, which intersects each quad at a star must be a generalized hexagon. The second theorem tells us that if a finite near hexagon with parameters possesses such a geometric hyperplane, then that near hexagon Γ must be the dual of a rank 3 polar space Δ. Moreover, there is a bijection H ↔ quads of Γ, which induces an embedding of the hexagon H into the polar space A which is an epimorphism on points. Conceivably, there is a possibility that generalized hexagons might be represented as geometric hyperplanes of some of the “other” dual polar spaces, such as Ω(n, ℝ) (with signature (n – 3,3)), Sp(6, κ), Ω−(8, κ), U(6, κ) or U(7, κ). But the final theorem shows that if Γ is finite, such possibilities cannot happen; that in fact Γ (and Δ) are type Ω(7, q) (or Sp(6, q) if q is even) and H is the hexagon of type G2(q) associated with the standard embedding of G2(q) (either as the stabilizer of an appropriate hyperplane in the 8-dimensional spin module for Ω(7, q) or as the stabilizer of a trilinear form in its natural 7-dimensional module – or the factor of this 7-space module by a 1-dimensional radical when q is even).

The author thanks Professor J. Tan for a valuable discussion, Queen Mary College, U. of London, and the Mathematisches Institute, Albert-Ludwigs Universität Freiburg for their kind hospitality during the writing of this work, and the Alexander von Humboldt Stiftung whose support made the research possible.

INTRODUCTION

A conjugacy class D of 3-transpositions in the group G is a class of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. Such groups were introduced and studied by Bernd Fischer who classified all finite 3-transposition groups with no nontrivial normal, solvable subgroups. His work was of great importance in the classification of finite simple groups.

The basic example of a class of 3-transpositions is the class of transpositions in any symmetric group. This was the only class which Fischer originally considered, but Roger Carter pointed out that examples could be found in several of the classical groups as well. The transvections of symplectic groups over GF(2) form a class of 3-transpositions, so additionally any subgroup of the symplectic group generated by a class of transvections is also a 3-transposition group. The symmetric groups arise in this way as do the orthogonal groups over GF(2). Symplectic transvections over GF(2) are special cases of unitary transvections over GF(4), and this unitary class is still a class of 3-transpositions. The final classical examples are given by the reflection classes of orthogonal groups over GF(3).