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an improved on-the-fly tableau construction for a real-time temporal logic Marc Geilen 12 July 2003 /e

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Overview Introduction RT Temporal Logic Model-Checking Ingredients of the tableau procedure Example Conclusions

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/e Temporal Logic Model Checking Timed Automaton A : ' A S System S Logical Property ' Product Automaton A S £ A : ' L (A S ) \L (A : ' )= S satisfies ' L (A S £ A ' )= iff

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/e Real-time temporal logic Linear Temporal Logic, extended with quantitative dense time (e.g., pos real numbers) Timed state sequence is a sequence of states (valuations of propositions) and intervals p | : ' | ' 1 _ ' 2 | ' 1 U 6 d ' 2 or in positive normal form: p | : p | ' 1 _ ' 2 | ' 1 ^ ' 2 | ' 1 U 6 d ' 2 | ' 1 V 6 d ' 2 ( {p,q}, [0,1) ) ( {p},[1,4] ) ( {q}, (4, 7) ) …

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/e Goal To have an efficient algorithm for translating real-time temporal logic formulas into timed automata to enable temporal logic model- checking for timed systems.

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/e Previous work Alur, Feder and Henzinger, 96 Tableau construction for dense time Metric Interval Temporal Logic. Linear Temporal Logic with ' 1 U I ' 2 Establishing the connection between MITL and Timed Automata, not meant for implementation

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/e Previous work Geilen, Dams, 00 Attempt at an on-the-fly tableau construction For fragment of MITL: ' 1 U I ' 2 where I=[0,d] Relied on restriction to timed state sequences with special type of intervals [a, b)

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/e This work… introduces an on-the-fly tableau construction for a fragment of MITL ( ' 1 U I ' 2 where I=[0,d] ) without the restriction on intervals

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/e Untimed OTF tableau revisited Label states with formulas Separation of constraints on current state and remainder of the state sequence Normal form: °'°'

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/e Untimed OTF tableau revisited p U q = q _ p ^° p U q p p U q q ° p p U q q ° p p U q q °

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/e Real-time tableaux Timed automata Intervals and locations

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/e Real-time tableaux

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/e Real-time tableaux

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/e Real-time tableaux

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/e Timers Timers measure/constrain distance between transitions p,x>0 p U 6 d q q x:=d [t 1,... p,x > 0 p U 6 d q q x:=d (t 1,...

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/e p,x>0 p U 6 x q p U 6 d q p U 6 x q q x:=d [t 1,... p,x > 0 p U 6 x+ " q p U 6 d q p U 6 " q q x:=d (t 1,...

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/e Release Formulas

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/e Parts of the tableau automaton Locations: sets of formulas Propositional and timer constraints in locations are derived from the formulas Timers: for every bounded Until or Release formula (counting down) Edges: determined by a normal form procedure from singular to open and from open to singular intervals.

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/e Disjunctive Temporal Normal forms Extended logic x 0, x>0 TS. ' (e.g., {x:=5}. ' ') ' 1 U 6 x ' 2, ' 1 U 6 x + " ' 2 ' 1 V 6 x ' 2, ' 1 V < x ' 2 ° '

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/e Disjunctive Temporal Normal forms Extended logic and interpretation with timers ( ¾,I) |= À ' º : Timers --> IR ( ¾,I) |= À x>0if º (x)>0 ( ¾,I) |= À TS. ' if ( ¾,I) |= TS. À '

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/e Disjunctive Temporal Normal Forms Normal form °Ã°Ã

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/e Normal form rewrite rules Depend on interval type (s/o) ´ : equivalent for first singular interval {0} ´ : equivalent in initial open interval (0,… s o

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/e Equivalences / rewrite rules Some examples: ' 1 U 6 d ' 2 ´ {x:=5}.( ' 1 U 6 x ' 2 ) ' 1 U 6 d ' 2 ´ {x:=5}.( ' 1 U 6 x+ " ' 2 ) ' 1 U 6 d ' 2 ^ ' 1 U 6 x ' 2 ´ ' 1 U 6 x ' 2 (if x 6 d) ' 1 U 6 x ' 2 ´ ' 2 _ ( x > 0 ^ ' 1 ^ ° ' 1 U 6 x ' 2 ) (`the Next operator refers to the next interval) s o

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/e Example

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/e Example

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/e Example

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/e Dealing with interval types timed automata cannot directly enforce interval types But alternation of singular and open intervals can be enforced by a well-known trick

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/e Outline of the algorithm Get initial locations from NF of initial formula As long as there are locations that have not been expanded –Expand one of these locations from its Next formulas according to its type s/o

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/e Application of the algorithm § 6 5 p = true U 6 5 p

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/e Application of the algorithm true U 6 5 p ´ [x:=5]. true U 6 x p s

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/e Application of the algorithm [x:=5]. true U 6 x p ´ [x:=5].p _ [x:=5].x>0 ^° true U 6 x p s

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/e Application of the algorithm

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/e Application of the algorithm true U 6 x p ´ p _ x>0 ^° true U 6 x p o

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/e Application of the algorithm true U 6 x p ´ p _ x>0 ^° true U 6 x p s

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/e Application of the algorithm true ´ true ^ ° true

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/e ¤ § 6 5 p = false V (true U 6 5 p)

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/e Conclusions and Future work OTF tableau construction algorithm Lifts the constraints imposed in an earlier paper Optimizations possible Weakly monotonic time for interleaving semantics Simple extensions of the logic Implementation

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/e Thanks!

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