06111 Executive Summary -- Complexity of Boolean Functions

Theory of Computing Systems / Mathematical Systems Theory, 2007

Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, what Boolean functions can be defined depends on these gate functions: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes have been known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover, we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case, and that the problem is either in P or coNP-complete.

2008

We describe Haskell implementations of interesting combinatorial generation algorithms with focus on boolean functions and logic circuit representations. First, a complete exact combinational logic circuit synthesizer is described as a combination of catamorphisms and anamorphisms. Using pairing and unpairing functions on natural number representations of truth tables, we derive an encoding for Binary Decision Diagrams (BDDs) with the unique property that its boolean evaluation faithfully mimics its structural conversion to a a natural number through recursive application of a matching pairing function. We then use this result to derive ranking and unranking functions for BDDs and reduced BDDs. Finally, a generalization of the encoding techniques to Multi-Terminal BDDs is provided. The paper is organized as a self-contained literate Haskell program, available at http://logic.csci.unt.edu/tarau/ research/2008/fBDD.zip. Keywords exact combinational logic synthesis, binary decision diagrams, encodings of boolean functions, pairing/unpairing functions, ranking/unranking functions for BDDs and MTBDDs, declarative combinatorics in Haskell

Theoretical Computer Science, 1995

We consider the size of the representation of Boolean functions by several classes of binary decision diagrams (BDDs) (also called branching programs), namely the classes of arbitrary BDDs of real time BDD (RBDD) (i.e. BDDs where each computation path is limited to the number of variables), of free BDDs (FBDDs) (also called read-once-only branching programs), of ordered BDDs (OBDDS) i.e. FBDDs where variables are tested in the same order along all paths), and binary decision trees (BDTs).

Journal of Logic and Computation, 2007

is an informal network of European scientists working on computability theory, including its foundations, technical development, and applications. Among the aims of the network is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory or relativity. Computations may be very general, depending upon the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, and computational learning. Applications are everywhere, especially, in algebra, analysis and geometry, or data types and programming.

Information Processing Letters, 2005

In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {¬, ∧, ∨}.

IAS/Park City Mathematics Series, 2004

2011

This report documents the program and the outcomes of Dagstuhl Seminar 11121 “Computational Complexity of Discrete Problems”. The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in chronological order. The last section contains the abstracts of the talks. Seminar 20.–25. March, 2011 – www.dagstuhl.de/11121 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, F.2 Analysis of Algorithms and Problem Complexity

Journal of Computer and System Sciences, 2001

Theoretical Computer Science, 1995

An infinite sequence F = { f"}z= 1 of one-output Boolean functions with the following two properties is constructed: (1) fn can be computed by a Boolean circuit with O(n) gates. (2) For any positive, nondecreasing, and unbounded function h : N + R, each Boolean circuit having an m/h(m) separator requires a nonlinear number Q(nh(n)) of gates to computef, (e.g., each planar Boolean circuit requires Q(n') gates to computef,). Thus, one can say that f" has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have separators in O(m/log, m).

Information Processing Letters, 1990

We use Karchmer and Wigderson's recent characterization of circuit depth in terms of communication complexity to design shallow Boolean circuits for the counting functions. We show that the MOD, counting function on n arguments can be computed by Boolean networks which contain negations and binary OR-and AND-gates in depth c logrn, where c A 2.881. This is an improvement over the obvious depth upper bound of 3 logan. We can also design circuits for the MOD, and MOD,, functions having depth 3.475 logan and 4.930 logan, respectively.