Boolean Semiring Research Articles

Diameters of commuting graphs of matrices over semirings

We calculate the diameters of commuting graphs of matrices over the binary Boolean semiring, the tropical semiring and an arbitrary nonentire commutative semiring. We also find the lower bound for the diameter of the commuting graph of the semigroup of matrices over an arbitrary commutative entire antinegative semiring.

Representation theory of finite semigroups over semirings

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory.

Bisimulation Minimisation of Weighted Automata on Unranked Trees

Several models of automata are available that operate unranked trees. Two well-known examples are the stepwise unranked tree automaton (suta) and the parallel unranked tree automaton (puta). By adding a weight, taken from some semiring, to every transition we generalise these two qualitative automata models to quantitative models, thereby obtaining weighted stepwise unranked tree automata (wsuta) and weighted parallel unranked tree automata (wputa); the qualitative automata models are reobtained by choosing the BOOLEAN semiring. The weighted versions have applications in natural language processing, XML-based data management and quantitative information retrieval. We address the minimisation problem of wsuta and wputa by using (forward and backward) bisimulations and we prove the following results: (1) for every wsuta an equivalent forward (resp. backward) bisimulation minimal wsuta can be computed in time O(mn) where n is the number of states and m is the number of transitions of the given wsuta; (2) the same result is proved for wputa instead of wsuta; (3) if the semiring is additive cancellative or the BOOLEAN semiring, then the bound can be improved to O(mlog n) for both wsuta and wputa; (4) for every deterministic puta we can compute a minimal equivalent deterministic puta in time O(mlog n); (5) the automata models wsuta, wputa, and weighted unranked tree automaton have the same computational power.

Linear stationary control systems over the Boolean semiring and their graphs of modules

We consider linear stationary dynamical systems over the Boolean semiring. We analyze the properties of complete observability, identifiability, attainability, and controllability of a system. We define the notion of the “graph of modules” of totally controllable totally attainable Boolean linear stationary systems by analogy with spaces of modules in the case of systems over fields. The above-mentioned graph is described in the simplest case of one-dimensional inputs and outputs. We prove the weak connectedness of this oriented graph.

The Complexity of Tensor Circuit Evaluation

We continue the study of tensor calculus over semirings in terms of complexity theory initiated by Damm et al. (2003). First, we look at tensor circuits, a natural generalization of tensor formulas; we show that the problem to determine whether the circuit output over a certain semiring is non-zero is complete for NE = NTime(2 O(n)) over the Boolean semiring, for $$\oplus$$ E over the field $$\mathbb{F}_{2}$$ , and for analogous classes over other semirings. Moreover, common-sense restrictions such as imposing bounds on circuit and/or tensor depth, are shown to elegantly capture the classes P, NTime $$(2^{O(log^{k} n)})$$ , NTimeSpace( $$2^{O(log^{k} n)}$$ , n O(1)), for k ? 1, PSPACE, and their counting counterparts. The proofs of these results use a model of algebraic Turing machines over a semiring together with a predicate-based approach on counting, which is similar to that of Toda (1991). This allows characterizations of the classes $$\oplus$$ P, NP, co-NP, co-DP, C=P, SPP, USP, and UP, and their exponential time counterparts, in a single framework. Finally, we show that a number of natural problems concerning tensor formulas and circuits, such as asking whether the output of a formula/circuit is a diagonal matrix, or the identity matrix, or a permutation matrix, capture the classes $$\prod^{p}_{2}$$ for formulas and $$\prod^{e}_{2}$$ for circuits over the Boolean semiring; other semirings are also discussed.

Efficient transitive closure of sparse matrices over closed semirings

This paper surveys several alternative data structures and algorithms for multiplying sparse upper-triangular matrices over closed semirings, and evaluates their efficiency in computing transitive closures of matrices over the Boolean semiring. Two new variants are introduced that outperform previously known methods on a collection of large data-sets drawn from linguistic applications.

Representations and complete semiring morphisms

We investigate the connection between representations and complete semiring morphisms. In some semirings, e.g., for power series over the Boolean semiring, both concepts coincide. But there exist also semirings, e.g., the semiring of power series over the tropical semiring, where the concept of a complete semiring morphism is more general than that of a representation.