If we consider the real numbers extended by minus infinity with the operations maximum and addition, we obtain the max-algebra or the max-plus semiring. The analog of linear algebra for these operations extended to matrices and vectors has been widely studied. We outline some facts on semirings and max-plus linear algebra, in particular, the solution of maxplus linear systems. As an application...

In this paper, we first study the conversion of weighted two-way automata to one-way automata. We show that this conversion preserves the unambiguity but does not preserve the determinism. Yet, we prove that the conversion of an unambiguous weighted one-way automaton into a two-way automaton leads to a deterministic two-way automaton. As a consequence, we prove that unambiguous weighted two-way...

In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To purpose prove that can be characterized as the concatenation of finite tree (WFTA) and specific class homomorphism. We show series recognized by is closed under inverses homomorphisms, but not homomorphisms. logical automata, which uses MSO logic trees. Finally investigat...

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over commutative semiring S. In particular, we obtain complete characterization congruence-simpleness for such Steinberg algebras, extending well-known characterizations when S is field or ring. also provide criterion AS(GE) graph groupoid GE associated to an arbitrary E be congruence-simple. Motivated result Cla...

We introduce a weighted logic with discounting and we establish Büchi’s and Elgot’s theorem for weighted automata over finite words and arbitrary commutative semirings. Then we investigate Büchi and Muller automata with discounting over the max-plus and the min-plus semiring. We show their expressive equivalence with weighted MSO-sentences with discounting. In this case our logic has a purely s...

Adapting Lindström’s well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of “universal” quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of...

We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations X = f (X) is equal to the least fixed-point of a linear system obtained by “linearizing” the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first use...

The algorithms such as RSA, ElGamal and ECC work on integers. Commutative operations integer multiplication leave these vulnerable to attack by eavesdroppers. For this reason, experts develop the concept of non-commutative algebra in public key cryptosystem adding properties groups, semirings, semiring division, matrices matrix decomposition. However, generating process some cryptosystems is qu...

Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid, subsuming a wide range of systems with quantitative aspects. In this paper we extend this theory towards other behavioural equivalences, by considering semirings of weights. Taking advantage of this extra structure, we consider a general notion of weak weighted bisimulation, which ...