scholarly journals Vector Addition Systems Reachability Problem (A Simpler Solution)

10.29007/bnx2 ◽  
2018 ◽  
Author(s):  
Jerome Leroux
Keyword(s):  
General Problem ◽  
Simple Algorithm ◽  
Initial Configuration ◽  
Central Problem ◽  
Reachability Problem ◽  
Final Configuration ◽  
Vector Addition ◽  
Finite Sequences ◽  
Production Relations ◽  
Inductive Invariants

The reachability problem for Vector Addition Systems (VASs) is a central problem of net theory. The general problem is known to be decidable by algorithms based on the classical Kosaraju-Lambert-Mayr-Sacerdote-Tenney decomposition (KLMTS decomposition). Recently from this decomposition, we deduced that a final configuration is not reachable from an initial one if and only if there exists a Presburger inductive invariant that contains the initial configuration but not the final one. Since we can decide if a Preburger formula denotes an inductive invariant, we deduce from this result that there exist checkable certificates of non-reachability in the Presburger arithmetic. In particular, there exists a simple algorithm for deciding the general VAS reachability problem based on two semi-algorithms. A first one that tries to prove the reachability by enumerating finite sequences of actions and a second one that tries to prove the non-reachability by enumerating Presburger formulas. In another recent paper we provided the first proof of the VAS reachability problem that is not based on the KLMST decomposition. The proof is based on the notion of production relations that directly proves the existence of Presburger inductive invariants. In this paper we propose new intermediate results simplifying a bit more this last proof.

The Refinement Algorithm of Blank Outline Based on One-Step Inverse Analysis

2016 ◽  
Vol 725 ◽  
pp. 517-522
Author(s):  
Jia Kai Zhou ◽  
Yi Dong Bao ◽  
Wan Lin Zhou ◽  
Jing Cui ◽  
Hui Ting Wang
Keyword(s):  
Inverse Analysis ◽  
Computation Time ◽  
Initial Configuration ◽  
Final Configuration ◽  
Characteristic Lines ◽  
Rounded Corner ◽  
Part Surface ◽  
One Step ◽  
Blank Shape ◽  
Refinement Algorithm

Blank dimensions and outlines can be obtained in one-step inverse analysis. Applying more accurate mesh will achieve more precise outlines while usually lead to the increase of computation time. To ensure operation efficiency, this paper proposes a new blank outline refinement algorithm based on one-step inverse analysis. Firstly, the initial configuration is obtained from the final configuration by one-step inverse analysis. Secondly, all outline nodes is projected to the nearest element in the final configuration. Thirdly, according to the position of projected nodes in the element, the coordinate of outline nodes in the initial configuration is achieved through mapping. Finally the number of outline nodes is increased in rounded corners, the coordinate of added nodes are calculated through interpolation. At last all outlines corresponding to characteristic lines of part surface are acquired. Using A-pillar as an example, outlines are calculated by the refinement algorithm and commercial software. It proves that under the same mesh quality, outlines obtained by refinement algorithm become more accurate and smooth, especially in rounded corner. The results can contribute to judge the rationality of blank shape and improve the final part forming property. This algorithm refines the accuracy of outlines and ensures the efficiency of one-step inverse analysis.

scholarly journals Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

2021 ◽  
pp. 42-61
Author(s):  
A. R. Balasubramanian ◽  
Javier Esparza ◽  
Mikhail Raskin
Keyword(s):  
Lower Bounds ◽  
Petri Net ◽  
Special Class ◽  
Upper Bounds ◽  
Initial State ◽  
Reachability Problem ◽  
Final State ◽  
Final Configuration ◽  
Finite State ◽  
State Agents

AbstractIn rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number B such that all initial configurations of the protocol with at least B agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper [17], Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in "Image missing" for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to "Image missing" and "Image missing" , respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is "Image missing" -complete for leaderless protocols, "Image missing" -complete for symmetric protocols with a leader, and in "Image missing" for leaderless symmetric protocols, thereby solving all the problems left open in [17].

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.

Motion Planning of a Group of Agents Using the Homotopy Approach

2007 ◽  
Author(s):  
Mayank Lal ◽  
Suhada Jayasuriya ◽  
Swaminathan Sethuraman
Keyword(s):  
Motion Planning ◽  
Mobile Agents ◽  
Single Agent ◽  
Initial Configuration ◽  
Potential Fields ◽  
Polynomial Space ◽  
Time Varying ◽  
Multiple Roots ◽  
Final Configuration ◽  
Discriminant Variety

In this paper motion planning of a group of agents is done to move the group from an initial configuration to a final configuration through obstacles in 2-D. Also we introduce a new homotopy approach which uses potential fields to find paths in polynomial space. We use the homotopy approach for changing the group shape of the mobile agents and at the same time treat the group as a single agent by finding a bounding disc for it to plan the motion of the group through obstacles. A time varying polynomial is constructed, the roots of which represent the current positions of the mobile agents in a frame attached to the bounding disc. The real and imaginary parts of the roots of this polynomial represent the x and y coordinates of the mobile agents in this frame. This polynomial is constructed such that it avoids the discriminant variety or the set of polynomials having multiple roots. This is equivalent to saying that the mobile agents do not collide with each other at all times. The bounding disc is then used to plan the motion of the agents through obstacles such that the group avoids the obstacles at all times.

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