Fundamental Theorems and Families of Forms for Binary and Multiple-Valued Linearly Independent Logic (original) (raw)

Vectorial Bi-Decompositions of Logic Functions

The bi-decomposition is a very powerful method to find an optimal multiple-level circuit structure that realizes a logic function. So far, the required simplification of the decomposition functions was realized in the strong bi-decomposition by reducing the number of variables. The completeness of this decomposition method was reached by the weak bi-decomposition. The drawback of the weak bi-decomposition is that different path lengths occur because the number of variables is only decreased for one of the two decomposition functions. In this paper, the method of bi-decomposition is extended by the vectorial bi-decomposition which utilized for the first time simpler decomposition functions which depend on all variables as the given function. It will be shown that vectorial bi-decompositions exist for functions for which no strong bi-decomposition is possible. Using the vectorial bi-decomposition together with the strong bi-decomposition, circuit structures can be found which have sho...

Classical and Multivalued Logics: Foundations and Computational Applications (LOGFAC) TIC2001-1577-C03

2003

In this project we are working on classical and multivalued propositional logic as a foundation of computer science. We are looking at the foundations of multivalued logic based on t-norms, and the proof theory of both classical and multivalued logics. Moreover, we are applying this knowledge to automated deduction, hardware verification, and SATbased problem solving. The topics we are studying are: (1) logic and algebraic foundations of multivalued logic underlying fuzzy logics; (2) application of these logics to the study of approximate reasoning in intelligent systems; (3) study of classical deduction systems from the point of view of computational complexity; (4) extension to multivalued logic of the results obtained in classical proof theory; (5) theoretical foundation of automated deduction; and (6) design and implementation of algorithms for classical and multivalued SAT.

On Boolean Functions Encodable as a Single Linear Pseudo-Boolean Constraint

Lecture Notes in Computer Science, 2007

A linear pseudo-Boolean constraint (LPB) is an expression of the form a1 • 1 +. .. + am • m ≥ d, where each i is a literal (it assumes the value 1 or 0 depending on whether a propositional variable xi is true or false) and a1,. .. , am, d are natural numbers. An LPB is a generalisation of a propositional clause, on the other hand it is a restriction of integer linear programming. LPBs can be used to represent Boolean functions more compactly than the well-known conjunctive or disjunctive normal forms. In this paper, we address the question: how much more compactly? We compare the expressiveness of a single LPB to that of related formalisms, and give an algorithm for computing an LPB representation of a given formula if this is possible. Note: This report is the long version of [18] and contains the proofs omitted there for space reasons.

An algorithm for bi-decomposition of logic functions

Proceedings of the 38th …, 2001

We propose a new BDD-based method for decomposition of multi-output incompletely specified logic functions into netlists of two-input logic gates. The algorithm uses the internal don't-cares during the decomposition to produce compact well-balanced netlists with short delay. The resulting netlists are provably nonredundant and facilitate test pattern generation. Experimental results over MCNC benchmarks show that our approach outperforms SIS and other BDD-based decomposition methods in terms of area and delay of the resulting circuits with comparable CPU time.

Vector Logic: A Natural Algebraic Representation of the Fundamental Logical Gates

Journal of Logic and Computation, 2007

Vector logic is a matrix-vector representation of the logical calculus inspired in neural network models. In this algebraic formalism, the truth values map on orthonormal Q-dimensional vectors, the monadic operations are represented by square matrices, and the dyadic operations produce rectangular matrices that act on the Kronecker product of the vector truth values. In this formalism, the theorems and tautologies of classical logic are demonstrated using the rules of matrix algebra. In the present work, we analyse a three-valued vector logic that adds to the 'yes' and 'no' vectors, a third 'uncertain' vector that represents the truth value corresponding to undecidable propositions. Fuzziness is produced both via linear combinations of 'yes' and 'no' vectors, and by the supplementary dimension of the logical vector subspace. We describe the basic matrix operators, and we show that for this three-valued vector logic, the modalities 'possibility' and 'necessity' are simple square matrices instead of infinite recursive processes. Finally, we explore the application of this formalism to represent the complex-valued operator ffiffiffiffiffiffiffiffiffiffiffi NOT p

Compact XOR-Bi-Decomposition for Generalized Lattices of Boolean Functions

2017

Bi-Decomposition is a very powerful approach for the synthesis of multi-level combinational circuits because it utilizes the properties of the given functions to find small circuits, with low power consumption and low delay. Compact bi-decompositions restrict the variables in the support of the decomposition functions as much as possible. Methods to find compact AND-, OR-, or XOR-bi-decompositions for a given completely specified function are well known. Lattices of Boolean Functions significantly increase the possibilities to synthesize a minimal circuit. However, so far only methods to find compact ANDor OR-bi-decompositions for lattices of Boolean functions are known. This gap, i.e., a method to find a compact XOR-bi-decomposition for a lattice of Boolean functions, has been closed by the approach suggested in this paper. A lattice of Boolean functions represents all possible functions which are defined by an incompletely specified function. In the context of vectorial bi-decompo...

Efficient calculation of fixed-polarity polynomial expressions for multiple-valued logic functions

2002

This paper presents a tabular technique for calculation of fixed-polarity polynomial expressions for MV functions. The technique is derived from a generalization of the corresponding methods for Fixed-Polarity Reed-Muller (FPRM) expressions for switching functions. All useful features of tabular techniques for FPRMs, as for example, simplicity of involved operations and high possibilities for parallelization of the calculation procedure, are preserved. The method can be extended to the calculation of coefficients in Kronecker expressions for MV functions.

Two classes of Boolean functions for dependency analysis

Science of Computer Programming, 1998

Many static analyses for declarative programming/database languages use Boolean functions to express dependencies among variables or argument positions. Examples include groundness analysis, arguably the most important analysis for logic programs, finiteness analysis and functional dependency analysis for databases. We identify two classes of Boolean functions that have been used: positive and definite functions, and we systematically investigate these classes and their efficient implementation for dependency analyses. On the theoretical side, we provide syntactic characterizations and study the expressiveness and algebraic properties of the classes. In particular, we show that both are closed under existential quantification. On the practical side, we investigate various representations for the classes based on reduced ordered binary decision diagrams (ROBDDs), disjunctive normal form, conjunctive normal form, Blake canonical form, dual Blake canonical form, and a form specific to definite functions. We compare the resulting implementations of groundness analyzers based on the representations for precision and efficiency.

Satisfiability in multi-valued circuits

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, 2018

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras A there is a polynomial time algorithm that decides if an equation over A has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.