Z-reachability Problem for Games on 2-dimensional Vector Addition Systems with States is in P

2013 ◽  
Vol 123 (1) ◽  
pp. 15-42 ◽  
Author(s):  
Jakub Chaloupka
Keyword(s):  
Dimensional Vector ◽  
Reachability Problem ◽  
Vector Addition

scholarly journals The Reachability Problem for Two-Dimensional Vector Addition Systems with States

2021 ◽  
Vol 68 (5) ◽  
pp. 1-43
Author(s):  
Michael Blondin ◽  
Matthias Englert ◽  
Alain Finkel ◽  
Stefan GÖller ◽  
Christoph Haase ◽  
...  
Keyword(s):  
Regular Expression ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Reachability Problem ◽  
Technical Result ◽  
Vector Addition ◽  
Bounded Language

We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

SOME COMPLEXITY RESULTS FOR RINGS OF PETRI NETS

1994 ◽  
Vol 05 (03n04) ◽  
pp. 281-292
Author(s):  
HSU-CHUN YEN ◽  
BOW-YAW WANG ◽  
MING-SHANG YANG
Keyword(s):  
Petri Nets ◽  
State Machines ◽  
Slight Improvement ◽  
Reachability Problem ◽  
Encoding Scheme ◽  
N Cycle ◽  
Vector Addition ◽  
Finite State ◽  
Boundedness Problem ◽  
Compare And Contrast

We define a subclass of Petri nets called m–state n–cycle Petri nets, each of which can be thought of as a ring of n bounded (by m states) Petri nets using n potentially unbounded places as joins. Let Ring(n, l, m) be the class of m–state n–cycle Petri nets in which the largest integer mentioned can be represented in l bits (when the standard binary encoding scheme is used). As it turns out, both the reachability problem and the boundedness problem can be decided in O(n(l+log m)) nondeterministic space. Our results provide a slight improvement over previous results for the so-called cyclic communicating finite state machines. We also compare and contrast our results with that of VASS(n, l, s), which represents the class of n-dimensional s-state vector addition systems with states where the largest integer mentioned can be described in l bits.

Automorphisms of supermaximal subspaces

1985 ◽  
Vol 50 (1) ◽  
pp. 1-9 ◽  
Author(s):  
R. G. Downey ◽  
G. R. Hird
Keyword(s):  
Vector Space ◽  
Recursive Relation ◽  
Effective Procedure ◽  
Dimensional Vector ◽  
Algebraic Systems ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Vector Addition ◽  
State Construction ◽  
Recursive Set

An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

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