Ulla Koppenhagen and Ernst W. Mayr. The complexity of the equivalence problem for commutative semigroups. Technical Report TUM-I9603, Institut fur Informatik, Technische Universitat Munchen, January 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....up to now, no tool implementing these algorithms has been provided. There exist, however, some other algorithms for the verification of subclasses of Petri nets such as free choice Petri nets [6] conflict free Petri nets c fl1997 Published by Elsevier Science B. V. 9] reversible Petri nets [3,4,11,14] and marked graphs [5,6] See [7] for a recent panorama on the decidability questions in Petri nets) We add to this list cyclic Petri nets. A Petri net is said to be cyclic if and only if its initial marking is a home state marking. Cyclic Petri nets are used in the design and control of ....

....Petri net reachability sets can be given in a form of finite unions of linear combinations of nonnegative vectors whose sizes are exponential in the size of the Petri nets. Our proof is organized as follows. First, we note that what makes the reachability sets for reversible Petri nets semi linear [2,3,11] is the fact that, for a reversible Petri net (N; M 0 ) we have RS(N;M 0 ) RS( N ; M 0 ) where N is the Petri net inverse of N ( However, for cyclic Petri nets, we only have RS(N;M 0 ) RS( N ; M 0 ) Hopefully, we proved, in section 3, that for every Petri net (N; M 0 ) such that ....

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Koppenhagen E., Mayr E. "The complexity of the equivalence problem for commutative semigroups", Technical Report TUM-I9603, 1996.

....u, and t Delta v 1 Delta v j s. Let OE be any lexicographic term ordering satisfying s t s s s s y s z ; for all s 2 V Gamma (Y [ Z) s V Gamma (Y [ Z) s y 2 Y and s z 2 Z. Let m s be the minimal element wrt OE of the congruence class [s] P t of s in P t . As shown in [34], the binomial s Gamma m s is an element of the reduced Grobner basis of I(P t ) the ideal generated by the congruences in P t considered as binomials. It is also shown that, because of the particular term ordering, there is a u 2 [u] P such that u = v Delta y Delta w 2 (Y [ Z) ....

....ss 2 s 3 ; ts 2 Gamma s ; ts 1 Gamma ss 3 ; s Gamma ss 3 g : The binomial s Gamma ss 3 provides the solution w = s 3 and u = s 1 s 3 , resp. which can be verified by the following derivation in P: u = s 1 s 2 s 3 s 2 s 3 s 1 s 3 = u (P) In further constructions in [34], the algorithm for the generalized subword problem is then used to obtain explicit semilinear representations of congruence classes in finitely presented commutative semigroups, and, using these, to solve the equivalence problem, i.e. to decide, given two commutative semigroup presentations P ....

Ulla Koppenhagen and Ernst W. Mayr. The complexity of the equivalence problem for commutative semigroups. Technical Report TUM-I9603, Institut fur Informatik, Technische Universitat Munchen, January 1996.

....has a number of applications to finitely presented commutative semigroups. Besides those mentioned in Section 3. 3, we are able to derive exponential space complete decision procedures for the subword, finite enumeration, finite containment, and equivalence problems for commutative semigroups (see [KM96]) Furthermore, as shown in Section 4, we obtain an algorithm for transforming any given basis into the reduced Grobner basis for binomial ideals in general, which also requires at most space 2 for some constant d 0 independent of the the size n of the problem instance. Since, in the worst ....

U. Koppenhagen and E.W. Mayr. The Complexity of the Equivalence Problem for Commutative Semigroups. Technical Report TUM-I9603, Institut fur Informatik, Technische Universitat Munchen, January 1996.

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