Logical matrix

A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets.

Matrix representation of a relation

If R is a binary relation between the finite indexed sets X and Y (so R ⊆ X×Y), then R can be represented by the logical matrix M whose row and column indices index the elements of X and Y, respectively, such that the entries of M are defined by:

M_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R\\0&(x_{i},y_{j})\not \in R\end{cases}}

In order to designate the row and column numbers of the matrix, the sets X and Y are indexed with positive integers: i ranges from 1 to the cardinality (size) of X and j ranges from 1 to the cardinality of Y. See the entry on indexed sets for more detail.

Example

The binary relation R on the set {1, 2, 3, 4} is defined so that aRb holds if and only if a divides b evenly, with no remainder. For example, 2R4 holds because 2 divides 4 without leaving a remainder, but 3R4 does not hold because when 3 divides 4 there is a remainder of 1. The following set is the set of pairs for which the relation R holds.

{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.

The corresponding representation as a logical matrix is:

{\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}}

which includes a diagonal of ones since each number divides itself.

Other examples

A permutation matrix is a (0,1)-matrix, all of whose columns and rows each have exactly one nonzero element.
- A Costas array is a special case of a permutation matrix
An incidence matrix in combinatorics and finite geometry has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a block design, or edges of a graph (discrete mathematics)
A design matrix in analysis of variance is a (0,1)-matrix with constant row sums.
A logical matrix may represent an adjacency matrix in graph theory: non-symmetric matrices correspond to directed graphs, symmetric matrices to ordinary graphs, and a 1 on the diagonal corresponds to a loop at the corresponding vertex.
The biadjacency matrix of a simple, undirected bipartite graph is a (0,1)-matrix, and any (0,1)-matrix arises in this way.
The prime factors of a list of m square-free, n-smooth numbers can be described as a m×π(n) (0,1)-matrix, where π is the prime-counting function and a_ij is 1 if and only if the jth prime divides the ith number. This representation is useful in the quadratic sieve factoring algorithm.
A bitmap image containing pixels in only two colors can be represented as a (0,1)-matrix in which the 0's represent pixels of one color and the 1's represent pixels of the other color.
A binary matrix can be used to check the game rules in the game of Go ^[1]

Some properties

The matrix representation of the equality relation on a finite set is the identity matrix I, that is, the matrix whose entries on the diagonal are all 1, while the others are all 0. More generally, if relation R satisfies I ⊂ R, then R is a reflexive relation.

If the Boolean domain is viewed as a semiring, where addition corresponds to logical OR and multiplication to logical AND, the matrix representation of the composition of two relations is equal to the matrix product of the matrix representations of these relations. This product can be computed in expected time O(n²).^[2]

Frequently operations on binary matrices are defined in terms of modular arithmetic mod 2—that is, the elements are treated as elements of the Galois field GF(2) = ℤ₂. They arise in a variety of representations and have a number of more restricted special forms. They are applied e.g. in XOR-satisfiability.

The number of distinct m-by-n binary matrices is equal to 2^mn, and is thus finite.

Lattice

Let n and m be given and let U denote the set of all logical m × n matrices. Then U has a partial order given by

m\subset n\quad {\text{when}}\quad \forall i,j\quad m_{ij}=1\implies n_{ij}=1.

In fact, U forms a Boolean algebra with the operations and and or between two matrices applied component-wise. The complement of a logical matrix is obtained by swapping all zeros and ones for their opposite.

Every logical matrix a = ( a _{i j} ) has an transpose a^T = ( a _{j i} ). Suppose a is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, a^T a contains the m × m identity matrix, and the product a a^T contains the n × n identity.

As a mathematical structure, the Boolean algebra U forms a lattice ordered by inclusion; additionally it is a multiplicative lattice due to matrix multiplication.

Every logical matrix in U corresponds to a binary relation. These listed operations on U, and ordering, correspond to a calculus of relations, where the matrix multiplication represents composition of relations.^[3]

Logical vectors

If m or n equals one, then the m × n logical matrix (M_{i j}) is a logical vector. If m = 1 the vector is a row vector, and if n = 1 it is a column vector. In either case the index equaling one is dropped from denotation of the vector.

Suppose $(P_{i}),\quad i=1,2,...m\ \ {\text{and}}\ \ (Q_{j}),\quad j=1,2,...n$ are two logical vectors. The outer product of P and Q results in an m × n rectangular relation:

M_{ij}=P_{i}\land Q_{j}.

A re-ordering of the rows and columns of such a matrix can assemble all the ones into a rectangular part of the matrix. ^[4]

Let h be the vector of all ones. Then if v is an arbitrary logical vector, the relation R = v h^T has constant rows determined by v. In the calculus of relations such an R is called a vector.^[4] A particular instance is the universal relation h h^T.

For a given relation R, a maximal, rectangular relation contained in R is called a concept in R. Relations may be studied by decomposing into concepts, and then noting the induced concept lattice.

Notes

^ Petersen, Kjeld (February 8, 2013). "Binmatrix". Retrieved August 11, 2017.
^ Patrick E. O'Neil, Elizabeth J. O'Neil (1973). "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure" (PDF). Information and Control. 22 (2): 132–138. doi:10.1016/s0019-9958(73)90228-3. — The algorithm relies on addition being idempotent, cf. p.134 (bottom).
^ Irving Copilowish (December 1948) "Matrix development of the calculus of relations", Journal of Symbolic Logic 13(4): 193–203 Jstor link
^ ^a ^b Gunther Schmidt (2013). "6: Relations and Vectors". Relational Mathematics. Cambridge University Press. p. 91. doi:10.1017/CBO9780511778810. ISBN 9780511778810.

References

Hogben, Leslie (2006), Handbook of Linear Algebra (Discrete Mathematics and Its Applications), Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-510-8, section 31.3, Binary Matrices
Kim, Ki Hang (1982), Boolean Matrix Theory and Applications, ISBN 0-8247-1788-0

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Logical matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

[1] Petersen, Kjeld (February 8, 2013). "Binmatrix". Retrieved August 11, 2017.

[2] Patrick E. O'Neil, Elizabeth J. O'Neil (1973). "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure" (PDF). Information and Control. 22 (2): 132–138. doi:10.1016/s0019-9958(73)90228-3. — The algorithm relies on addition being idempotent, cf. p.134 (bottom).

[3] Irving Copilowish (December 1948) "Matrix development of the calculus of relations", Journal of Symbolic Logic 13(4): 193–203 Jstor link

[GS-4] Gunther Schmidt (2013). "6: Relations and Vectors". Relational Mathematics. Cambridge University Press. p. 91. doi:10.1017/CBO9780511778810. ISBN 9780511778810.

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