Vector logic

Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators.

Overview

Classic binary logic is represented by a small set of mathematical functions depending on one (monadic or two (dyadic) variables. In the binary set, the value 1 corresponds to true and the value 0 to false. For instance, a two-dimensional vector logic requires a correspondence between the truth-values true (t) and false (f), and two q-dimensional normalized column vectors composed by real numbers s and n, hence:

$t=s$ and $f=n$

(where $q\geq 2$ is an arbitrary natural number, and “normalized” means that the [| length] of the vector is 1). This correspondence generates a space of vector truth-values: V₂={s,n}. The basic logical operations defined using this set of vectors lead to matrix operators.

The operations of vector logic are based on the scalar product between q-dimensional column vectors: $u^{T}v=<u,v>$: the orthonormality between vectors s and n implies that $<u,v>=1$ if $u=v$, and $<u,v>=0$ if $u\neq v$.

Monadic operators

The monadic operators result from the application $Mon:V_{2}\to V_{2}$, and the associated matrices have q rows and q columns. The two basic monadic operators for this two-dimensional vector logic are the identity and the negation:

• Identity: A logical identity ID(p)is represented by matrix $I=ss^{T}+nn^{T}$. This matrix operates as follows: Ip=p, p∈V₂; due to the orthogonality of s respect to n, we have $Is=ss^{T}s+nn^{T}s=s<s,s>+n<n,n>$ and conversely $In=n$.

• Negation: A logical negation ¬p is represented by matrix $N=ns^{T}+sn^{T}$ Consequently, Ns=n and Nn=s. The involutory behavior of the logical negation, namely that ¬(¬p) equals p, corresponds with the fact that N²=I. Is important to note that this vector logic identity matrix is not generally an identity matrix in the sense of matrix algebra.

Dyadic operators

The 16 two-valued dyadic operators correspond to functions of the type $Dyad:V_{2}\otimes V_{2}\to V_{2}$; the dyadic matrices have q rows and q² columns. The matrices that execute these dyadic operations are based on the properties of the Kronecker product.

Two properties of this product are essential for the formalism of vector logic:

1. The mixed-product property If A, B, C and D are matrices of such size that one can form the matrix products AC and BD, then

   $(A\otimes B)(C\otimes D)=AC\otimes BD$

2. Distributive transpose The operation of transposition is distributive over the Kronecker product:

   $(A\otimes B)^{T}=A^{T}\otimes B^{T}.$

Using these properties, expressions for dyadic logic functions can be obtained:

• Conjunction. The conjunction (p∧q) is executed by a matrix that acts on two vector truth-values: $C(u\otimes v)$ .This matrix reproduces the features of the classical conjunction truth-table in its formulation:

$C=s(s\otimes s^{T})+n(s\otimes n^{T})+n(n\otimes s^{T})+n(n\otimes n^{T})$

and verifies

$C(s\otimes s)=s$, and

$C(s\otimes n)=C(n\otimes s)=C(n\otimes n)=n$.

• Disjunction. The disjunction is executed by the matrix

$D=s(s\otimes s^{T})+s(s\otimes n^{T})+s(n\otimes s^{T})+n(n\otimes n^{T})$, resulting in

$D(s\otimes s)=D(s\otimes n)=D(n\otimes s)=s$ and

$D(n\otimes n)=n$.

• Implication. The implication corresponds in classical logic to the expression p→q≡¬p∨q. The vector logic version of this equivalence leads to a matrix that represents this implication in vector logic: $L=D(N\otimes I)$. The explicit expression for this implication is:

$L=s(s\otimes s^{T})+n(s\otimes n^{T})+s(n\otimes s^{T})+n(n\otimes n^{T})$,

and the properties of classical implication are satisfied:

$L(s\otimes s)=L(n\otimes s)=L(n\otimes n)=s$ and

$L(s\otimes n)=n$.

• Equivalence and Exclusive or.

In vector logic the equivalence p≡q is represented by the following matrix:

$E=s(s\otimes s^{T})+n(s\otimes n^{T})+n(n\otimes s^{T})+s(n\otimes n^{T})$ with

$E(s\otimes s)=E(n\otimes n)=s$ and

$E(s\otimes n)=E(n\otimes s)=n$.

The Exclusive or is the negation of the equivalence, ¬(p≡q); it corresponds with the matrix $X=NE$ given by

$X=n(s\otimes s^{T})+s(s\otimes n^{T})+s(n\otimes s^{T})+n(n\otimes n^{T})$,

with $X(s\otimes s)=X(n\otimes n)=n$ and

$X(s\otimes n)=X(n\otimes s)=s$.

De Morgan Law

In the two-valued logic, the conjunction and the disjunction operations satisfy the De Morgan Law: p∧q≡¬(¬p∨¬q), and its dual: p∨q≡¬(¬p∧¬q)). For the two-dimensional vector logic this Law is also verified:

$C(u\otimes v)=ND(Nu\otimes Nv)$, where u and v are two logic vectors.

The Kronecker product implies the following factorization:

$C(u\otimes v)=ND(N\otimes N)(u\otimes v)$.

Then it can be proved that in the two–dimensional vector logic the De Morgan Law is a law involving operators, and not only a law concerning operations :

$C=ND(N\otimes N)$

The matrices S=NC and P=ND, correspond to the Sheffer (NAND) and the Peirce (NOR) operations, respectively.

Law of Contraposition

In the classical propositional calculus, the [[Contraposition (traditional logic) | Law of Contraposition] p→q≡¬q→¬p is proved because the equivalence holds for all the possible combinations of truth-values of p and q . Instead, in vector logic, the law of contraposition emerges from a chain of equalities within the rules of matrix algebra and Kronecker products, as shown in what follows:

$L(u\otimes v)=D(N\otimes I)(u\otimes v)=D(Nu\otimes v)=D(Nu\otimes NNv)=D(NNv\otimes Nu)=D(N\otimes I)(Nv\otimes Nu)=L(Nv\otimes Nu)$

This result is based in the fact that D, the disjunction matrix, represents a commutative operation.

Many-valued two-dimensional logic

Many-valued logic was developed by many researchers, particularly by Jan Łukasiewicz and allows extending logical operations to truth-values that include uncertainties . In modern times, these extended logical systems have been adapted to the development of fuzzy logics and its applications (G. J. Klir, and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice-Hall, New Jersey, 1995). The case of two-valued vector logic, uncertainties in the truth values can be introduced using vectors with s and n weighted by probabilities. Let , with be this kind of “probabilistic” vectors. Here, the many-valued character of the logic is introduced a posteriori via the uncertainties introduced in the inputs .

Scalar projections of vector outputs

The outputs of this many-valued logic can be projected on scalar functions and generate a particular class of probabilistic logic with similarities with the many-valued logic of Reichenbach (N. Rescher. Many-Valued Logic. McGraw-Hill, New York, 1969; R. Blanché, Introduction à la Logique Contemporaine, Armand Colin, Paris, 1968; G. J. Klir, and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice-Hall, New Jersey, 1995). Given two vectors and and a dyadic logical matrix G, a scalar probabilistic logic is provided by the projection over vector s :

.

Here are the main results of these projections:

The associated negations are:

It is interesting that if the scalar values belong to the set {0, ½, 1} these many value scalar logic is for many of the operators almost identical to the 3-valued logic of Łukasiewitz (N. Rescher. Many-Valued Logic. McGraw-Hill, New York, 1969).

It has been proved that when the monadic or dyadic operators act over probabilistic vectors belonging to this set, the output is also an element of this set (E. Mizraji. The operators of vector logic. Mathematical Logic Quarterly, 42, 27–39, 1996).

History

The approach has been inspired in neural network models based on the use of high-dimensional matrices and vectors . Vector logic is a direct translation into a matrix-vector formalism of the classical Boolean polynomials . This kind of formalism has been applied to develop a fuzzy logic in terms of complex numbers . Other matrix and vector approaches to logical calculus have been developed in the framework of quantum physics, computer Science and optics . Early attempts to use liner algebra to represent logic operations can be referred to Peirce and Copilowich . The Indian biophysicist G.N. Ramachandran developed a formalism using algebraic matrices and vectors to represent many operations of classical Indian Logic .

Boolean polynomials

George Boole established the development of logical operations as polynomials . For the case of monadic operators ( identity, ¬ or negation, tautolog1y and contradiction), the Boolean polynomial looks as follows:

$f(x)=f(1)x+f(0)(1-x)$

The four different monadic operations result from the different binary values for the coefficients. Identity operation requires f(1) = 1 and f(0) = 0, and negation occurs if f(1) = 0 and f(0) = 1. For the 16 dyadic operators, the Boolean polynomials are of the form:

$f(x,y)=f(1,1)xy+f(1,0)x(1-y)+f(0,1)(1-x)y+f(0,0)(1-x)(1-y)$

The dyadic operations can be translated to this polynomial format when the coefficients f take the values indicated in the respective truth tables. For instance: the NAND operation requires that f(1,1) = 0 and f(1,0) = f(0,1) = f(0,0) = 1. These Boolean polynomials can be immediately extended to any number of variables, producing a large potential variety of logical operators.

Extensions

1. Vector logic can be extended to many dimensions, and large dimensional vector-spaces allows to create many truth values and the corresponding logical matrices (E. Mizraji, Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation 18, 97–121, 2008)

1. Logical modalities can be fully represented in this context, with recursive process inspired in neural models proposed by McCulloch and Pitts (W. S. McCulloch, and W. Pitts. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5, 115–133, 1943; E. Mizraji. Modalities in vector logic. Notre Dame Journal of Formal Logic, 35, 272–283, 1994; E. Mizraji, Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation 18, 97–121, 2008)

1. Some cognitive problems about logical computations can be analyzed using this formalism, in particular recursive decisions (E. Mizraji, and J. Lin. The dynamics of logical decisions. Physica D, 168–169, 386–396, 2002; P. beim Graben, and R. Potthast. (2009). Inverse problems in dynamic cognitive modeling. Chaos, 19, 015103, 2009; P. beim Graben, D. Pinotsis, D. Saddyand R Potthast (2008a). Language processing with dynamic fields. Cogn. Neurodyn., 2, 79–88, 2008; P. beim Graben, S. Gerth, and S. Vasishth. Towards dynamical system models of language-related brain potentials. Cogn. Neurodyn., 2, 229–255, 2008; A. Binazzi. Cognizione logica e modelli mentali, Studi sulla formazione, 1-2012, pag. 69-84 © Firenze University Press-http://www.fupress.net/index.php/sf/article/view/11649).

1. The computation via reversible operations as the Fredkin gate (E. Fredkin, and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21, 219–253, 1982; R. P. Feynman. Lectures on Computation, Perseus, Cambridge, 1999) can be implemented in vector logic. This implementations provides explicit expressions for matrix operators that produce the input format and the output filtering necessary for obtaining computations (E. Mizraji. The operators of vector logic. Mathematical Logic Quarterly, 42, 27–39, 1996; E. Mizraji, Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation 18, 97–121, 2008 ).

1. Elementary Celular Automata (ECA) can be analyzing using the operator structure of this vector logic; this analysis lead to an spectral decomposition of the laws governing ECA (E. Mizraji, The parts and the whole: inquiring how the interaction of simple subsystems generates complexity. International Journal of General Systems, 35 , p. 395-415, 2006;.C. Arruti, E, Mizraji, Hidden potentialities. International Journal of General Systems, 35 , 461-469, 2006).