On the complexity of containment, equivalence, and reachability for finite and 2-dimensional vector addition systems with states

2006 ◽  
pp. 360-370
Author(s):  
Rodney R. Howell ◽  
Dung T. Huynh ◽  
Louis E. Rosier ◽  
Hsu-Chun Yen
Keyword(s):  
Dimensional Vector ◽  
Vector Addition

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].

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.

scholarly journals Students’ Understanding of Graphical Vector Addition in One and Two Dimensions

2011 ◽  
Vol 3 (2) ◽  
pp. 102-111
Author(s):  
Umporn Wutchana ◽  
Narumon Emarat
Keyword(s):  
High School ◽  
High School Students ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
School Students ◽  
Free Response ◽  
Vector Addition ◽  
Qualitative Understanding ◽  
Graphical Vector

Understanding initial topics in physics (forces, fields, and kinematical quantities) requires a grasp of basic vector concepts. In this study, we intended to explore students’ qualitative understanding of graphical vector addition in one and two dimensions. Using two free-response problems of a diagnostic quiz, one is one-dimensional vector addition and the other is two-dimensional vector addition. Sixty-three grade ten high-school students’ responses were categorized. From the investigation, results represent that from the high-school students who already completed their vector lesson, only 10% of them provided correct answer for vector addition in one-dimension and 32% for the two-dimensional addition.

scholarly journals Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states

1986 ◽  
Vol 46 ◽  
pp. 107-140 ◽  
Author(s):  
Rodney R. Howell ◽  
Louis E. Rosier ◽  
Dung T. Huynh ◽  
Hsu-Chun Yen
Keyword(s):  
Dimensional Vector ◽  
Complexity Bounds ◽  
Vector Addition
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