Milan Bašić

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.

Perfect state transfer in integral circulant graphs

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. We give the simple condition for characterizing integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. Using that, we complete the proof of results stated by Saxena, Severini and Shparlinski. Moreover, it is shown that in the class of unitary Cayley graphs there are only two of them allowing perfect state transfer.

On the clique number of integral circulant graphs

Abstract The concept of gcd-graphs is introduced by Klotz and Sander; they arise as a generalization of unitary Cayley graphs. The gcd-graph X n ( d 1 , … , d k ) has vertices 0 , 1 , … , n − 1 , and two vertices x and y are adjacent iff gcd ( x − y , n ) ∈ D = { d 1 , d 2 , … , d k } . These graphs are exactly the same as circulant graphs with integral eigenvalues characterized by So. In this work we deal with the clique number of integral circulant graphs and investigate the conjecture proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, The Electronic Journal of Combinatorics 14 (2007) #R45] that the clique number divides the number of vertices in the graph X n ( D ) . We completely solve the problem of finding the clique number for integral circulant graphs with exactly one and two divisors. For k ⩾ 3 , we construct a family of counterexamples and disprove the conjecture in this case.

Some classes of integral circulant graphs either allowing or not allowing perfect state transfer

Abstract The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. Motivated by the aforementioned work, Basic, Petkovic and Stevanovic give the simple condition for the characterization of integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. They stated that the integral circulant graphs with minimal vertices allowing perfect state transfer, other than unitary Cayley graphs, are ICG 8 ( { 1 , 2 } ) and ICG 8 ( { 1 , 4 } ) . Moreover, it is also conjectured that two classes of integral circulant graphs ICG n ( { 1 , n / 4 } ) and ICG n ( { 1 , n / 2 } ) allow PST where n ∈ 8 N . These conjectures are confirmed in this work. Moreover, it is shown that there are no integral circulant graphs allowing perfect state transfer in the class of graphs where the number of vertices is a square-free integer.

Nondeterministic automata: Equivalence, bisimulations, and uniform relations

In this paper we study the equivalence of nondeterministic automata pairing the concept of a bisimulation with the recently introduced concept of a uniform relation. In this symbiosis, uniform relations serve as equivalence relations which relate states of two possibly different nondeterministic automata, and bisimulations ensure compatibility with the transitions, initial and terminal states of these automata. We define six types of bisimulations, but due to the duality we discuss three of them: forward, backward-forward, and weak forward bisimulations. For each of these three types of bisimulations we provide a procedure which decides whether there is a bisimulation of this type between two automata, and when it exists, the same procedure computes the greatest one. We also show that there is a uniform forward bisimulation between two automata if and only if the factor automata with respect to the greatest forward bisimulation equivalences on these automata are isomorphic. We prove a similar theorem for weak forward bisimulations, using the concept of a weak forward isomorphism instead of an isomorphism. We also give examples that explain the relationships between the considered types of bisimulations.

Which weighted circulant networks have perfect state transfer

The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini and Shparlinski and characterizing integral graphs among weighted circulant graphs. Finally, classes of weighted circulant graphs supporting perfect state transfer are found. These classes completely cover the class of circulant graphs having perfect state transfer in the unweighted case. In fact, we show that there exists an weighted integral circulant graph with n vertices having perfect state transfer if and only if n is even. Moreover we prove the nonexistence of perfect state transfer for several other classes of weighted integral circulant graphs of even order.

Ranks of fuzzy matrices. Applications in state reduction of fuzzy automata

Abstract In this paper we consider different types of ranks of fuzzy matrices over residuated lattices. We investigate relations between ranks and prove that row rank, column rank and Schein rank of idempotent fuzzy matrices are equal. In particular, ranks and corresponding decompositions of fuzzy matrices representing fuzzy quasi-orders are studied in detail. We show that fuzzy matrix decomposition by ranks can be used in the state reduction of fuzzy automata. Moreover, we prove that using rank decomposition of fuzzy matrices improves results of any state reduction method based on merging indistinguishable states of fuzzy automata.