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..., model learning, timed systems 1 Introduction Research during the last decades have developed powerful techniques for using models of reactive systems in specification, automated verification (e.g., =-=[CGP99]-=-), test case generation (e.g., [FJJV97,SEG00]), implementation (e.g., [HLN+90]), and validation of reactive systems in telecommunication, embedded control, and related application areas. Typically, su...

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...ed at certain moments in time, and that the learner is able to control and record precisely the timing of the occurrence of each action. We consider systems that can be described by a timed automaton =-=[AD94]-=-, i.e., a finite automaton equipped with clocks that constrain the possible absolute times of occurrences of actions. There are some properties of timed automata that make the design of learning algor...

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... that the time difference between the first b and the subsequent a is greater or equal to 3. 2 4 The L∗ algorithm for learning DFAs In this section, we shortly review the L∗ algorithm, due to Angluin =-=[Ang87]-=- for learning a regular (untimed) language, L(A) ⊆ Γ∗, accepted by a minimal deterministic finite automaton (DFA) A = (Γ, Q, q0, δ, Qf ). In this algorithm a so-called Learner , who initially knows no...

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...m contains inconsistent constraints, or requires some clock to be negative, then the clock constraint is unsatisfiable. The canonical form for an unsatisfiable clock constraint is defined to be false =-=[Dil89]-=-. Clock constraints satisfy an important closure property Proposition 1 For a clock constraint ϕ and two clock valuations γ, γ′, if γ |= ϕ and γ′ |= ϕ, then min(γ, γ′) |= ϕ and max(γ, γ′) |= ϕ, where ...

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...f learning algorithms difficult: the set of clocks is not known a priori, and they cannot in general be determinized [AD94]. We therefore restrict consideration to a class of event-recording automata =-=[AFH99]-=-. These are timed automata that, for every action a, use a clock that records the time of the last occurrence of a. Event-recording automata can be determinized, and are sufficiently expressive to mod...

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...ing all conjuncts involving xa, and adding the conjunct xa ≤ 0. • We define the time elapsing of ϕ, denoted by ϕ↑ , as Can(ϕ′), where ϕ′ is obtained from Can(ϕ) by removing all upper bounds on clocks =-=[DT98]-=-. It is a standard result that these operations mirror the corresponding operations on clock valuations, in the sense that • γ′ |= ϕ[xa 7→ 0] iff γ′ = γ[xa 7→ 0] for some γ with γ |= ϕ, and • γ′ |=ϕ↑ ...

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...o create a good test suite and test oracle for testing subsequent versions. This has been demonstrated, e.g., by Hungar et al. [HHNS02,HNS03]). • Black-box techniques, such as adaptive model checking =-=[GPY02]-=-, have been developed to check correctness properties, even when source code or formal models are not available. • Tools that analyze the source code statically depend heavily on the implementation la...

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...rty of region equivalence is that it is preserved by reset and time elapsing operations. If γ ∼K γ′ then γ[xa 7→ 0] ∼K γ′[xa 7→ 0], and for each t ∈ R≥0 there is a t′ ∈ R≥0 such that γ + t ∼K γ′ + t′ =-=[Yov96]-=-. If we combine this fact with the fact that the reset and time elapsing operations on constraints mirror the same operations on clock valuations, we infer that the relation ≈K on clock constraints is...

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...obtained by adapting the L∗ algorithm. A central idea in the L∗ algorithm is to let each state be identified by the words that reach it from the initial state (such words are called access strings in =-=[BDG97]-=-). States are congruent if, according to the queries submitted so far, the same continuations of their access strings are accepted. This idea is naturally based on the properties of Nerode’s right con...

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... is answered either by yes or by a counterexample on which the hypothesis and the correct language disagree. Techniques for learning finite automata have been successfully used for regression testing =-=[HHNS02]-=- and model checking [GPY02] of finite-state systems for which no model or source code is available. In this paper, we extend the techniques for automata learning developed by Angluin and others to the...

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...utomata in a systematic way, and it is hard to generalize the algorithm to timed automata with more than one clock. Previous work on learning of infinite-state systems include the work by Berg et al. =-=[BJR08]-=-, who consider Mealy machines extended with state variables that can assume values from a potentially unbounded domain. These values can be passed as parameters in input and output messages, and can b...

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...to being clock guards: one reason is that this is sufficient for many applications, another is allowing them to be general clock constraints would make analysis algorithms more complicated (see e.g., =-=[BY03]-=-). (3) (3)TODO: The last sentence is new/BJ. A guarded word is a sequence wg = (a1, g1)(a2, g2) . . . (an, gn) of symbols ai ∈ Σ that are paired with clock guards. For a clocked word wc = (a1, γ1)(a2,...

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...7)TODO: new sentence known fact, that despite theoretically lower worst-case complexity, algorithms based on region graphs perform less efficiently than algorithms based on zone graphs. Together with =-=[GJP06]-=-, this paper describes initial efforts on learning algorithms for timed systems. Future work has to show the benefit of these algorithms in practical applications as well as to examine the most suitab...

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...ossible at the occurrence of future actions. Our approach to overcoming the problem that DERAs have no canonical form is to define a subclass of EDERAs which do have a canonical form, and which 1 See =-=[MP04]-=- for a study on right-congruences on timed languages. 19 furthermore can be understood as a DFA over Σ × GΣ where GΣ is the set of clock guards. We can then use Angluin’s algorithm to learn this DFA, ...

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...esults of queries. Its theoretical worst-case complexity is significantly higher than that of TL∗s. The only other work on learning of timed systems we are aware is by Verwer, de Weerdt and Witteveen =-=[VdWW06]-=-, who present an algorithm for learning of timed automata with one clock which is reset at every transition. Their setting differs from ours in that the learner does not choose the words to be 4 used ...