O. Maler and A. Pnueli. Reachability analysis of planar multilinear systems. In C. Courcoubetis, editor, Proceedings of the 4th Computer-Aided Verification. LNCS 697, Springer Verlag, July 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....updatable timed automata [12, 13] and initialized rectangular automata [20, 32] For all these models the decidability depends on existence of a finite bisimulation and holds for systems of any dimensions. Another class of decidability results concerns planar systems. The method was suggested in [27], where decidability was stated for 2 dim PCD (systems with the dynamics given by Piecewise Constant Derivatives) The results were extended to planar multipolynomial systems in [16] and to non deterministic planar polygonal systems (SPDI) in [8] All these results are based on topological ....

....conditions, invariants and graphs of resets are rectangles; 2) for each location , the dynamics has the form x 2 R , where R is a rectangle. Another special case of linear hybrid automata are PCDs, that are described in next section. 2. 3 PCD A piecewise constant derivative system (PCD) [7, 27] is a pair H = P; F ) with P = fP s g s2S a finite family of non overlapping convex polygonal sets in R with non empty interiors, and F = fc s g s2S a family of vectors in R . The dynamics of the PCD is determined by the equation x = c s for x 2 P s . Hence trajectories are broken lines. ....

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O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In CAV'93, LNCS 697, p. 194--209. Springer-Verlag, 1993.

....non dctcrministicfinitc state automata. Piecewise constant derivative systems The notion of simulation used in previous section was the notion of I simulation. We go further and present here systelnS that silnulate analog autolnata using the SI silnulation notion. Actually we pursue the work of [4, 3, 17] about Piecewise Constant Derivative systelnS: Trajectory Direction Figure 3 A PCD system in dimension 2. Definition 4.2 (PCD System[17, 3] A Piecewise Constant Derivative system (PCD) is a pair H = X,g) where X is the state space, g is a (possibly partial) function from X to a finite set ....

....of I simulation. We go further and present here systelnS that silnulate analog autolnata using the SI silnulation notion. Actually we pursue the work of [4, 3, 17] about Piecewise Constant Derivative systelnS: Trajectory Direction Figure 3 A PCD system in dimension 2. Definition 4. 2 (PCD System[17, 3]) A Piecewise Constant Derivative system (PCD) is a pair H = X,g) where X is the state space, g is a (possibly partial) function from X to a finite set of vectors C C X, and for every c G C, g (c) is a finite union of convexpolyhedral sets. The trajectories of the PCD system are given by the ....

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O. Maler and A. Pnuelli. Reachability Analysis of Planar Multi-linear Systems. In Proc. of the 5th Workshop on Computer-Aided Verification. In Leer. Notes in Comp. Sci. vol. 697: 194-209.

....methods, however, do not necessarily result in decision procedures (they are actually not meant to) The other, applicable to two dimensional dynamical systems, relies on the topological properties of the plane, and explicitly focuses on decidability issues. This approach has been proposed in [15]. There, it is shown that the reachability problem for two dimensional systems with piece wise constant derivatives (PCD) is decidable. This result has been extended in [7] for planar piece wise Hamiltonian systems. In [4] it has been shown that the reachability problem for PCD is undecidable for ....

....piece wise constant derivatives (PCD) is decidable. This result has been extended in [7] for planar piece wise Hamiltonian systems. In [4] it has been shown that the reachability problem for PCD is undecidable for dimensions higher than two. piece wise rectangular differential inclusions. As in [15], our procedure is not based on the computation of the reach set but rather on the computation of the limit of individual trajectories. A key idea is the use of one dimensional affine Poincar e maps for which we can easily compute the fixpoints. The decidability result of [15] fundamentally relies ....

[Article contains additional citation context not shown here]

O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In CAV'93. LNCS 697, pages 194--209. Springer Verlag, 1993.

....methods, however, do not necessarily result in decision procedures (they are actually not meant to) The other, applicable to two dimensional dynamical systems, relies on the topological properties of the plane, and explicitly focuses on decidability issues. This approach has been proposed in [16]. There, it is shown that the reachability problem for twodimensional systems with piece wise constant derivatives (PCD) is decidable. This result has been extended in [8] for planar piece wise Hamiltonian systems. In [4] it has been shown that the reachability problem for PCD is undecidable for ....

....planar piece wise Hamiltonian systems. In [4] it has been shown that the reachability problem for PCD is undecidable for dimensions higher than two. In this paper we develop an algorithm for solving the reachability problem of two dimensional piece wise rectangular di#erential inclusions. As in [16], our procedure is not based on the computation of the reach set but rather on the computation of the limit of individual trajectories. A key idea is the use of onedimensional a#ne Poincar maps for which we can easily compute the fixpoints. The decidability result of [16] fundamentally relies on ....

[Article contains additional citation context not shown here]

O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In CAV'93. LNCS 697, 194--209. Springer Verlag, 1993.

....non deterministic finite state automata. 4.2 Piecewise constant derivative systems The notion of simulation used in previous section was the notion of I simulation. We go further and present here systems that simulate analog automata using the SI simulation notion. Actually we pursue the work of [4, 3, 17] about Piecewise Constant Derivative systems: 23 Trajectory Direction Figure 3: A PCD system in dimension 2. Definition 4.2 (PCD System[17, 3] A Piecewise Constant Derivative system (PCD) is a pair H = X; g) where X is the state space, g is a (possibly partial) function from X to a finite ....

....of I simulation. We go further and present here systems that simulate analog automata using the SI simulation notion. Actually we pursue the work of [4, 3, 17] about Piecewise Constant Derivative systems: 23 Trajectory Direction Figure 3: A PCD system in dimension 2. Definition 4. 2 (PCD System[17, 3]) A Piecewise Constant Derivative system (PCD) is a pair H = X; g) where X is the state space, g is a (possibly partial) function from X to a finite set of vectors C ae X, and for every c 2 C, g Gamma1 (c) is a finite union of convex polyhedral sets. The trajectories of the PCD system are ....

[Article contains additional citation context not shown here]

O. Maler and A. Pnuelli. Reachability Analysis of Planar Multi-linear Systems. In Proc. of the 5th Workshop on Computer-Aided Verification. In Lect. Notes in Comp. Sci. vol. 697: 194-209.

....this problem is in general very difficult to solve. In worst case it requires (in principle) an explicit solution of the ODE:s why there is little hope of generalizing the solution strategy. This is true except for some rudimentary problem classes for which semi decidable algorithms do exist [Maler and Pnueli, 1993, Alur et al. 1993] Practically interesting approaches are typically based on various smart application of numerical simulation [Pettit and Wellstead, 1993] Of significant importance is to handle the previously mentioned problem associated with fatal causal conflicts. Next we will only ....

O. Maler and A. Pnueli. Reachability Analysis of Planar Multi-linear Systems. In C. Courcoubetis, editor, Proc. 5th int. conf. on Computer Aided Verification, pages 194--209. Springer Verlag, 1993.

....1993; Brockett, 1993; Back, Guckenheimer and Myers, 1993) This paper and several others define the dynamics of systems in terms of a polyhedral partition of state space and a simple linear flow defined within each polytope. The resulting system dynamics is locally linear, yet globally nonlinear. Maler and Pnueli (1993) examined a particular class of hybrid systems known as two dimensional multi linear systems. They found that many reachability question (point, region, and face) can be decided in a finite amount of time. Their proof exploits the fact that no flow lines in two dimensions can intersect. Asarin, ....

....with being true (or false) are not connected. In fact, the number of regions grows exponentially with . Variable assignment testing Before building clauses, we need to explain how we will test the assignment of a particular variable. One of the limitations of the 2D multi linear system that Maler and Pnueli (1993) took advantage of was the inability of two flows to converge and become a single flow. Figure 1) Since the apparatus will combine the outputs of several filters to produce a single flow, another dimension is added to the variable assignment. This extra dimension also serves as a temporary ....

Maler, O., & A. Pnueli, (1993). Reachability Analysis of Planar Multi-linear Systems. In C. Courcoubetis, Ed. Proceedings of the 5th Workshop on Computer-Aided Verification. Lecture Notes in Computer Science. 697. 194-209. Springer-Verlag.

....within this region. We prove the decidability of point to point, edge to edge and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree. Our results are a generalization of those of [10], where the subcase of our problem with the vector fields within the regions being constant (the so called multi linear model) was considered. We are able to reuse also a part of the proof from [10] to show that every infinite trajectory of the system either intersects only with a finite number of ....

....can be described by polynomials of arbitrary degree. Our results are a generalization of those of [10] where the subcase of our problem with the vector fields within the regions being constant (the so called multi linear model) was considered. We are able to reuse also a part of the proof from [10] to show that every infinite trajectory of the system either intersects only with a finite number of the region boundaries, or starting from some point will repeatingly intersect certain fixed sequence of region boundaries (in fact this result holds even for much wider classes of systems, its ....

[Article contains additional citation context not shown here]

O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In Proceedings of the 5th International Conference of Computer Aided Verification, volume 697 of Lecture Notes in Computer Science, pages 194--209, 1993.

....Paris 12 June 6, 1994 [summary by Fr ed eric Chyzak] Abstract E. Asarin deals with simple differential systems: dynamical systems with piecewiseconstant derivative in short, PCD systems. Recently, it has been proved that the reachability problem is decidable in a 2 dimensional space [2]. Asarin and Maler proved in [1] that every Turing machine can be simulated by a 3 dimensional PCD system. Thus, the reachability problem is proved to be undecidable in more than two dimensions. Connections with automata theory and first order logic are also given. This summary is based on ....

....Face 1 0 0 1 1 2 0 1 0 1 0 1 1 2 0 1 2 1 POP1 PUSH1(1) PUSH1(0) 0 1 S1 S2 0 1 1 0 1 1 Input Face Figure 4. push and pop with two stacks The reachability problem for the class of polyhedral PCD systems was proved to be decidable in the case of two dimensions by Maler and Pnueli in [2]. The last result about automata can be easily generalised to prove that it becomes undecidable in higher dimensions. Theorem 3. Any pushdown automaton with 2 stacks can be simulated with a 4 dimensional PCD system. The idea of the proof is that adding a third dimension to the pipes makes it ....

Maler (O.) and Pnueli (A.). -- Reachability analysis of planar multi-linear systems. In Courcoubetis (C.) (editor), Computer Aided Verification: Proceedings. Lecture Notes in Computer Science, vol. 697, pp. 194--209. -- Springer Verlag, 1993. Proceedings of the 5th International conference, CAV'93, Elounda, Greece. 136

.... dynamics (Theorem 9) a class of hybrid automata that includes the water level controller W 1 shown above (Theorem 11) and several product classes of finitary hybrid automata (Theorem 13) While also geometric in its intuition, the hybrid automaton model differs from the related approach of [18]. First, their dynamical systems are deterministic and our euclidean automata are nondeterministic, both as far as discrete and continuous progress is concerned. Second, we consider the product of euclidean automata with boolean automata, which results in multiple copies of euclidean state spaces. ....

O. Maler, A. Pnueli. Reachability analysis of planar multi-linear systems. Computer-aided Verification, Springer LNCS 697, pp. 194--209, 1993.

....non deterministic finite state automata. 4.2 Piecewise constant derivative systems The notion of simulation used in previous section was the notion of I simulation. We go further and present here systems that simulate analog automata using the SI simulation notion. Actually we pursue the work of [3, 5, 17] about Piecewise Constant Derivative systems. Note that similar systems have also been studied in [27, 26] Trajectory Direction Figure 3: A PCD system in dimension 2. Definition 4.2 (PCD System[5, 17] A Piecewise Constant Derivative system (PCD) is a pair H = X; g) where X is the state space, g ....

....that simulate analog automata using the SI simulation notion. Actually we pursue the work of [3, 5, 17] about Piecewise Constant Derivative systems. Note that similar systems have also been studied in [27, 26] Trajectory Direction Figure 3: A PCD system in dimension 2. Definition 4. 2 (PCD System[5, 17]) A Piecewise Constant Derivative system (PCD) is a pair H = X; g) where X is the state space, g is a (possibly partial) function from X to a finite set of vectors C ae X, and for every c 2 C, g Gamma1 (c) is a finite union of convex polyhedral sets. The trajectories of the PCD system are ....

[Article contains additional citation context not shown here]

Oded Maler and Amir Pnueli. Reachability Analysis of Planar Multi-linear Systems. Lecture Notes in Computer Science, 697:194--209, 1995.

....based control synthesis that projects target regions in state space. Hence, it is important to characterize the computational properties of these reachability algorithms and if they are intractable, identify classes of hybrid systems for which these algorithms are practical. Maler and Pnueli [11] examined a class of hybrid systems known as twodimensional multi linear systems. They found that many reachability question (point, region, and face) can be decided in a finite amount of time. Their proof exploits the fact that no flow lines in two dimensions can intersect. Asarin, Maler, and ....

....(or false) are not connected. In fact, the number of regions grows exponentially with i. 3.2 Variable assignment testing Before building clauses, we need to explain how we will test the assignment of a particular variable. One of the limitations of the 2D multi linear system that Maler and Pnueli [11] took advantage of was the inability of two flows to converge and become a single flow (Figure 1) Since the apparatus will combine the outputs of several filters to produce a single flow, another dimension is added to the variable assignment. This extra dimension also serves as a temporary ....

Maler, O., Pnueli, A.: Reachability Analysis of Planar Multi-linear Systems. In C. Courcoubetis (ed.), Proceedings of the 5th Workshop on Computer-Aided Verification. Lecture Notes in Computer Science 697. (1993) 194--209. Springer-Verlag.

....In section 6 we demonstrate the computational power of PCD systems and show that three dimensions are sufficient for simulating two stack machines, and hence the reachability problem for 3 dimensional systems is undecidable. The paper is a combination and elaboration of results presented in [14] and [2] q 1 i ( Gammaa 1 ; Gammab 1 ) a 2 ; Gammab 1 ) a 2 ; b 2 ) Gammaa 1 ; b 2 ) 2 x y Figure 2: Pacman x and the ghost y viewed as a planar PCD system. 2 Preliminaries Throughout this paper, we deal with the d dimensional Euclidean space 1 X = IR d . Points (vectors) in X ....

O. Maler and A. Pnueli, Reachability analysis of planar multilinear systems, In C. Courcoubetis, editor, Proc. of the 5th Workshop on Computer-Aided Verification, Lect. Notes in Comp. Sci. 697, pages 194--209. Springer-Verlag, 1993.

....even without explicit discrete jumps (i.e. the trajectories are continuous but not smooth) can simulate in a reasonable sense every effective transition system and hence their reachability problem is in general undecidable. This fact should be compared with the decision procedure introduced in [10] for such systems with 2 dimensions. The rest of the paper is organized as follows: In section 2 we give the necessary preliminary definitions. In section 3 we show how PCD systems can realize deterministic finite and push down automata. For non deterministic automata we show in section 4 that ....

.... system H, the reachability problem for H, denoted by Reach(H; x; x 0 ) is the following: Given x; x 0 2 X, are there 2 M (H; x) and t 0 such that (t) x 0 For PCD systems with 2 variables (described by linear inequalities and slope vectors) this problem has been proven decidable in [10]. 5 3 Realization of Finite and Push down Automata First we show how every finite state deterministic automaton can be realized by a 3 dimensional PCD system (a deterministic finite automaton without input is a rather trivial object and the construction is presented here just because it ....

O. Maler and A. Pnueli, Reachability analysis of planar multi-linear systems, In C. Courcoubetis, editor, Proc. of the 5th Workshop on Computer-Aided Verification, Elounda, Greece, volume 697 of Lect. Notes in Comp. Sci., pages 194--209. SpringerVerlag, 1993.

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O. Maler and A. Pnueli. Reachability analysis of planar multilinear systems. In C. Courcoubetis, editor, Proceedings of the 4th Computer-Aided Verification. LNCS 697, Springer Verlag, July 1993.

No context found.

O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In C. Courcoubetis, editor, Proc. of the 5th Workshop onComputerided Verification, number 697 in LNCS, pages 194--209. Springer-Verlag, 1993.

No context found.

O. Maler and A. Pnueli. Reachability analysis of planar multi-linear systems. In C. Courcoubetis, editor, Proceedings of the 4th Computerided Verification. LNCS 697, Springer Verlag, July 1993.

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