J. Mller, J. Lichtenberg, H. R. Andersen, H. Hulgaard. Dierence Decision Diagrams. In Proceeding of the 13th Annual Conference of the European Association for Computer Science Logic (CSL'99), volume 1683 of LNCS, pages 111-125. Springer, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....atom or the negation of an atom. In Boolean logic, two logically equivalent Boolean formulas are represented by the same BDD. A (reduced ordered) binary decision diagram is a canonical representation of a formula. implies modi cations of BDD handling functions. In a similar way, M ller et al. [18] introduce Di erence Decision Diagrams which allow to eciently represent and manipulate a Boolean logic over inequalities of the form x y c. Several BDD based decision procedures have also been given for quanti ed rst order logic (without equality) 10, 21] In the context of formal veri ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111-125. Springer-Verlag, 1999.

....All inconsistent paths can be removed, resulting in an OBDD with only consistent paths. However, due to sharing subterms, an OBDD can have exponentially many paths, so still there is a computational bottleneck. A typical example of this approach are the DDDs (di#erence decision diagrams) of [17], where all atoms are of the form x y c, for variables x and y and a constant c (known as separation predicates [22] or di#erence logic) In both the DNF and the BDD method, the boolean structure is flattened out immediately, and the arithmetical part is dealt with in a second step. In the ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Di#erence decision diagrams. In Computer Science Logic, Denmark, September 1999.

....All inconsistent paths can be removed, resulting in an OBDD with only consistent paths. However, due to sharing subterms, an OBDD can have exponentially many paths, so still there is a computational bottleneck. A typical example of this approach are the DDDs (di#erence decision diagrams) of [17], where all atoms are of the form x y c, for variables x and y and a constant c (known as separation predicates [22] or di#erence logic) In both the DNF and the BDD method, the boolean structure is flattened out immediately, and the arithmetical part is dealt with in a second step. In the ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Di#erence decision diagrams. In Computer Science Logic, Denmark, September 1999.

....are other symbolic data structures that do not feature this problem. The possibly most notable ones are BDD like datastructures [Bry86] where tests correspond to traversals of a directed acyclic graph. The interested reader is referred to the literature on Di#erence Decision Diagrams (DDDs) MLAH99] Clock Di#erence Diagrams (CDDs) LWYP99] Region Encoding Diagrams (REDs) Wan00] and Clock Restriction Diagrams (CRDs) Wan01] 4.3 Forward State Space Exploration Though the state space of dense real time interpretations is typically infinite, it can be be quotiented into a finite number ....

Jesper Mller, Jakob Lichtenberg, Henrik R. Andersen, and Henrik Hulgaard. Di#erence Decision Diagrams. In Computer Science Logic, The IT University of Copenhagen, Denmark, September 1999. 83

....y. The reachability algorithm in Tempo is parameterized over data structures for representing and manipulating clock constraints. Thus, it is a suitable platform for comparing di erent representations. Currently, Tempo includes a polyhedra library [Wil93] and a di erence decision diagram library [MLAH99] for representing clock constraints. The rest of the paper is organized as follows. Event clock automata and their properties are presented in Section 2. Section 3 presents the veri cation tool Tempo, and contains the description of the reachability algorithm and some examples that we have veri ....

....value 1 for de ned clocks, and value 0 for unde ned clocks it can be speci ed that time elapses only for de ned clocks. In the DDD representation, elapse of time is implemented by performing the operation of existential quanti cation of a special variable, which denotes the actual time (see [MLAH99] for more details) However, this operation lets time elapse for all clocks of the system, which occurs also for the unde ned clocks. Therefore, after performing the existential quanti cation, we must set the value of the unde ned clocks to ( 1) the value that represents . The operation ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Computer Science Logic, The IT University of Copenhagen, Denmark, September 1999.

.... Project 26270 VHS (Veri cation of Hybrid systems) and the French Israeli collaboration project 970maefut5 (Hybrid Models of Industrial Plants) Consequently, the search for new representation schemes for timed polyhedra is a very active domain of research [ABK 97] BMPY97] S99] MLAH99b] BLP 99] W00] In this paper, we propose a new representation scheme for non convex timed polyhedra based on a reformulation and extension of our previous work in [BMP99] where we proposed a canonical representation for non convex orthogonal polyhedra. As in [BMP99] the representation is ....

J. Mller, J. Lichtenberg, H.R. Andersen, and H. Hulgaard, Dierence Decision Diagrams, Proc. CSL'99, 1999.

....(e.g. Omega Library [21] and Redlog [8] or less expressive theories such as quanti ed Boolean formulae (e.g. SATO [25] and BuDDy [16] This paper presents a library called DDDLIB for manipulating quanti ed di erence inequalities. Formulae are represented in a graph data structure called DDDs [18], and the library implements a number of classical algorithms, such as Bryant s Apply algorithm [6] for combining formulae with Boolean operators, Fourier Motzkin s algorithm [9] for eliminating quanti ers, and Bellman Ford s shortest path algorithm [5] for determining satis ability. A preliminary ....

.... Or : ddd ddd ddd val Imp : ddd ddd ddd val Exists : var ddd ddd val Replace : ddd var var ddd The following functions determine functional properties of a formula: val Tautology : ddd bool val Equivalent : ddd ddd bool 3 Implementation DDDLIB is based on DDDs [18] which are directed acyclic graphs with two terminal vertices, 0 and 1, and a set of non terminal vertices. Each non terminal x2 1 2 3 1 2 3 (a) 0 1 x1 x0 3 (b) Fig. 1. The formula = 1 x1 x0 3) x2 x0 2) x2 x1 0) as (a) an (x1 ; x2) plot for x0 = 0, and (b) a ....

[Article contains additional citation context not shown here]

J. Mller, J. Lichtenberg, H.R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proc. Computer Science Logic, LNCS 1683, pages 111125, 1999.

.... [4] and timed guarded commands [12] Logics for specifying requirements (e.g. timed computation tree logic [1] and metric interval temporal logic [3] Data structures and algorithms for representing and manipulating logics (e.g. di erence bound matrices [9] di erence decision diagrams [22], clock di erence diagrams [15] and quanti er elimination [13] Techniques for formal analysis (e.g. partial order techniques [10, 19] and compositional and on the y techniques [16, 5] Tools for automatic veri cation (e.g. Kronos [29] Uppaal [18] RED [27] and Tempo [25] The core ....

....from z i to z i . The local time model results in an over approximation of the least xpoint, but in some cases it is still possible to use this over approximation for verifying a temporal property. I have implemented the two techniques using a data structure called di erence decision diagrams [22], and evaluated the e ciency of the techniques on some real time case studies. Experiments show that backwards reachability performs better than forward reachability with respect to both time and space consumptions. In conjunction with the local time model, it is furthermore possible to verify ....

[Article contains additional citation context not shown here]

J. Mller, J. Lichtenberg, H.R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proc. 13th International Conference on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111125, Madrid, Spain, September 1999.

....is also easier than satis ability of slightly more general theories, such as the rst order theories of reals with addition and order, which is NEXP hard [15] and integers with addition and order, which is 2 NEXP hard [28] 6 4. 2 Di erence Decision Diagrams Di erence decision diagrams (ddds) [23] are a data structure for representing and deciding satis ability of expressions. Similar to how a bdd [7] represents the meaning of a Boolean formula implicitly, a ddd represents the meaning [ of an expression using a decision diagram. A ddd is a directed acyclic graph with two terminals, ....

....high and low branches, respectively. Figure 1 shows an example of a ddd. x y 1 2 3 1 2 (a) 0 1 y x 0 y z 2 x z 3 x z 1 (b) Fig. 1. The expression = 1 x z 3) y z 2) y x 0) as (a) an (x; y) plot for z = 0, and (b) a di erence decision diagram. As shown in [23], ddds can be ordered and path reduced, yielding a semicanonical form, which makes it possible to check for validity and satis ability in constant time (as for bdds) The ddd data structure is not canonical, however, so equivalence checking must be performed as a validity check. The operations for ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proceedings 13th International Workshop on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111125, Madrid, Spain, September 1999.

....expression is recast into disjunctive normal form. Di erence bound matrices are used in many real time veri cation tools, such as Kronos [54] and Uppaal [39] but they su er from a number of problems as discussed in Sect. 1.1. 4. 4 Di erence Decision Diagrams Di erence decision diagrams (ddds) [42] are another candidate for a data structure for representing di erence constraint expressions. Similar to how a bdd [11] represents the meaning of a Boolean formula implicitly, a ddd represents the meaning [ of a di erence constraint expression using a decision diagram in which the vertices ....

....as (a) an (x; y) plot for z = 0, and (b) a di erence decision diagram. expression given by: v) high(v) low(v) where (v) is the di erence constraint of v, and high(v) and low(v) are the high and low branches, respectively. Figure 2 shows an example of a ddd. As shown in [42], ddds can be ordered and reduced, yielding a semicanonical form, which makes it possible to check for validity and satis ability in constant time (as for bdds) The ddd data structure is not canonical, however, so equivalence checking must be performed as a validity check. The operations for ....

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proceedings 13th International Conference on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111125, Madrid, Spain, September 1999. 30

....of the form x y d (x and y are clock variables and d is a constant) If we have a compact representation of such formulae and can decide valid implications (for performing the check in the line marked with ( we can use the algorithm in Fig. 1(a) immediately. Di erence decision diagrams [38] are a candidate for such a data structure which furthermore allows reuse of sub formula among the discrete states. Initial experiments with this approach implemented in Uppaal [6] show a signi cant improvement in memory consumption, even though the discrete states are still enumerated ....

....expression is recast into disjunctive normal form. Di erence bound matrices are used in many real time veri cation tools, such as Kronos [50] and Uppaal [36] but they su er from a number of problems as discussed in Sect. 1.1. 4. 4 Di erence Decision Diagrams Di erence decision diagrams (ddds) [38] are another candidate for a data structure for representing di erence constraint expressions. Similar to how a bdd [13] represents the meaning of a Boolean formula implicitly, a ddd represents the meaning [ of a di erence constraint expression using a decision diagram in which the vertices ....

[Article contains additional citation context not shown here]

J. Mller, J. Lichtenberg, H.R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proc. 13th International Conference on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111125, Madrid, Spain, September 1999.

....Omega Library [10] and Redlog [7] or less expressive theories such as quanti ed Boolean formulae (e.g. SATO [22] and BuDDy [15] This paper presents a library called dddlib for manipulating quanti ed di erence inequalities. Expressions are represented in a graph data structure called ddds [18], and the library implements a number of classical algorithms, such as Bryant s Apply algorithm [5] for combining expressions with Boolean operators, Fourier Motzkin s method [8] for eliminating quanti ers, and Bellman Ford s shortest path algorithm [4] for determining satis ability. The rest of ....

.... ddd ddd ddd val Imp : ddd ddd ddd val Exists : var ddd ddd val Replace : ddd var var ddd The following functions determine functional properties of an expression: val Tautology : ddd bool val Equivalent : ddd ddd bool 3 Implementation dddlib is based on (ddds) [18] which are directed acyclic graphs with two terminal vertices, 0 and 1, and a set of non terminal vertices. Each non terminal vertex u = h; l denotes the expression = l ) The test expression is a di erence constraint of the form x i x j d or x i x j d. A ....

[Article contains additional citation context not shown here]

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proc. 13th International Workshop on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111125, September 1999.

No context found.

J. Mller, J. Lichtenberg, H. R. Andersen, H. Hulgaard. Dierence Decision Diagrams. In Proceeding of the 13th Annual Conference of the European Association for Computer Science Logic (CSL'99), volume 1683 of LNCS, pages 111-125. Springer, 1999.

No context found.

J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Di#erence decision diagrams. In Computer Science Logic, The IT University of Copenhagen, Denmark, September 1999.

No context found.

J. Mller, J. Lichtenberg, H. R. Andersen, H. Hulgaard, Dierence Decision Diagrams. In Proc. Computer Science Logic, 1999.

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J. Mller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Dierence decision diagrams. In Proceedings 13th International Conference on Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111-125, Madrid, Spain, September 1999.

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