K. Cer¯ans. Algorithmic Problems in Analysis of Real-time System Specifications. PhD thesis, University of Latvia, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[PV94] In [BES93, KPSY93, BER94, MV94, BR95, ACH97] it was shown that, under various strong side conditions, reachability is decidable for timed automata with one stopwatch, but the general problem of one stopwatch automata was left open. As far as undecidability results are concerned, in [Cer92] it was shown that reachability is undecidable for timed automata with three stopwatches, as well as for timed automata with one memory cell (a variable of constant slope 0) and assignments between variables. It was also known that reachability is undecidable for timed automata with six memory ....

....W (or some multiple thereof) units of time. The wrapping edges ensure that variables take the same values at the beginning and end of a round, unless they are explicitly reassigned by a nonwrapping edge. This is the content of the wrapping lemma. A similar wrapping technique can be found in [Cer92] In figures of simple automata, we use the following conventions. First, all variables whose slopes are not listed are clocks, i.e. they have slope 1. Second, wrapping conditions are left implicit; in particular, we omit invariants from every figure after those regarding the three basic lemmas, ....

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K. Cer¯ans. Algorithmic Problems in Analysis of Real-time System Specifications. PhD thesis, University of Latvia, 1992.

....allow arbitrary rational numbers as the initial delay bounds is straightforward 37 . As a corollary of the theorem we get al..so the decidability of M for TMS. The main constructions used in the deciding algorithm are those of timed region graph [AD90] and symbolic checking of timed bisimulation [Cer92a, Cer92b], for exposition see also [CGL94, Go94] The full version will discuss this issue in more detail. 35 n is an injective mapping of M into S, introduced for the sake of further convenience. 36 This requirement is to be applied to the defining equations of all variables which the initial ....

K. Cer¯ans, Algorithmic Problems in Analysis of Real Time System Specifications, Dr.sc.comp. theses, University of Latvia, 1992.

....of timed systems done in [1, 11] concentrated on minimization of the region graph. For testing timed systems and many other purposes, minimization of the region graph results in a structure that is too course, and the more fundamental operation of minimization of transition systems is required. In [10, 9, 2], bisimulations between timed automata are studied, but minimization up to bisimulation is not dealt with. The paper is organized as follows. In Section 2, we present some examples that motivate the MTA model. BTDA s and their operational semantics are defined in Section 3. In Section 4, we prove ....

....three steps: 1. Say that a clock x is free in a region ff if ff contains two different states s; s 0 with s = x s 0 . Take, for each clock x and region ff for which x is free, two arbitrary states s and s 0 and decide whether they are bisimilar. This can be done using the result of Cer ans [10, 9] who proved that bisimulation is decidable for timed automata (this result carries over to our setting) According to Lemma 6.7, the outcome is independent of the choice of s and s 0 . If s and s 0 are not bisimilar then we declare that clock x is relevant for ff, otherwise we declare that x ....

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K. Cer¯ans. Algorithmic Problems in Analysis of Real Time System Specifications. Dr.sc.comp. thesis, University of Latvia, R¯iga, 1992.

....= 0 and, for all j with v(x i j ) 6= 1, ae 0 (fract(v(x i j ) fract(v 0 (x i j ) We extend ae 0 to a uniform mapping ae with the required property by defining ae(1) 1 and, for t 2 R, ae(t) Delta = btc ae 0 (fract(t) Theta Lemma 4.6. ae(v) e) ae(v(e) 2 Cer ans [ Cer92b, Cer92a] does not require uniform mappings to be continuous as he should have since his proof of Lemma 3.7 in [ Cer92b] which coincides with Lemma 11.7 in [ Cer92a] uses the property that a uniform mapping has an inverse that is also uniform. Lemma 4.7. Whenever ae is a uniform mapping then for every ....

....by defining ae(1) 1 and, for t 2 R, ae(t) Delta = btc ae 0 (fract(t) Theta Lemma 4.6. ae(v) e) ae(v(e) 2 Cer ans [ Cer92b, Cer92a] does not require uniform mappings to be continuous as he should have since his proof of Lemma 3. 7 in [ Cer92b] which coincides with Lemma 11.7 in [ Cer92a] uses the property that a uniform mapping has an inverse that is also uniform. Lemma 4.7. Whenever ae is a uniform mapping then for every d 2 R 0 the mapping ae d , defined by ae d (t) Delta = ae(t Gamma d) Gamma ae( Gammad) for every t 2 R 1 , is also uniform. Lemma 4.8. ae d (v ....

K. Cer¯ans. Algorithmic Problems in Analysis of Real Time System Specifications. Dr.sc.comp. thesis, University of Latvia, R¯iga, 1992.

....and a two slope variable with slopes 0 and 1 is a stopwatch. In [ACHH93] it is shown that reachability is undecidable for timed automata with two skewed clocks. In [KPSY93] it is shown that reachability is undecidable for timed automata with two threeslope variables and restriction (2) In [Cer92] it is shown that reachability is undecidable for timed automata with three stopwatches and restriction (2) In [ACH93, BES93, KPSY93, BER94] it is shown that, under various strong side conditions, reachability is decidable for timed automata with one stopwatch, but the general problem is left ....

....4 Delta act (v) a) and (2) for every vertex v, and every variable a, inv (v) a) 0; 4k a ] where k a = maxw2V max act(w) a) is the largest slope allowed to a in any vertex. In figures we leave these wrapping conditions implicit. We use the following wrapping technique found originally in [Cer92] Wrapping lemma. Let k 1 ; k 2 2 Q 0 , and consider the simple wrapping automaton fragment of Figure 6. Suppose that when edge e 1 is traversed into v 1 , c = fl and d = ffi, where 0 fl 2k 1 and 0 ffi 2k 2 . Then the next time e 4 is traversed out of v 1 , c has value fl and d has value ....

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K. Cer¯ans. Algorithmic Problems in Analysis of Real-time System Specifications. PhD thesis, Univ. of Latvia, 1992.

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