Timed Transition Systems Timed Automata Networks of Timed Automata Regions and Region Graph Equivalence Checking Problems Timed Automata and Logics for Real-time Systems Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 1 / 43
Timed Transition Systems Timed Automata Networks of Timed Automata Regions and Region Graph Equivalence Checking Problems Goals of the course Learning outcomes At the end of the course, you will be familiar with the basic theory of timed automata, be able to model and verify real-time systems using UPPAAL, be familiar with some of the behavioural equivalences over time automata and some of the logics for real-time systems and their connections with behavioural equivalences. The course will be evaluated via a small-group project. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 2 / 43
Timed Transition Systems Timed Automata Networks of Timed Automata Regions and Region Graph Equivalence Checking Problems Content of the first part of the course labelled transition systems with time timed automata timed and untimed bisimilarity timed and untimed language equivalence region graph and the reachability problem networks of timed automata model checking of basic properties of timed automata Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 3 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems Need for Introducing Time-related Features Timeouts in protocols: In CCS, say, timeouts can be modelled using nondeterminism. Enough to prove that the protocol is safe. Maybe too abstract for certain questions. (What is the average time to deliver the message?) Many real-life systems depend on timing: Real-time controllers (production lines, computers in cars, railway crossings). Embedded systems (mobile phones, remote controllers, digital watch). ... Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 4 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems Labelled Transition Systems with Time Timed (labelled) transition system (TLTS) a TLTS is a triple ( Proc , Act , { − →| a ∈ Act } ) where Proc is a set of states (or processes), Act = N ∪ R ≥ 0 is a set of actions (consisting of labels and time-elapsing steps), and a for every a ∈ Act , − → ⊆ Proc × Proc is a binary relation on states called the transition relation. We write a → s ′ if a ∈ N and ( s , s ′ ) ∈ a − − → , and s d → s ′ if d ∈ R ≥ 0 and ( s , s ′ ) ∈ d s − − → . Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 5 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems Healthiness conditions on delay transitions Typical requirements on the delay transitions → s ′ and s d d → s ′′ ( d ∈ R ≥ 0 ) then (Determinism) If s − − s ′ = s ′′ . 0 → s ′ if, and only if, s = s ′ . (Zero delay) s − d d ′ → s ′′ then s d + d ′ → s ′ → s ′′ . (Additivity 1) If s − − − → s ′′ and e ≤ d then s → s ′′ for d e → s ′ d − e (Additivity 2) If s − − − some s ′ . Why are these reasonable requirements? Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 6 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems How to Describe Timed Transition Systems? Syntax Semantics − → unknown entity known entity − → Labelled Transition Systems CCS − → Timed Transition Systems ??? Timed Automata [Alur, Dill’90] Finite-state automata equipped with clocks. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 7 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems How to Describe Timed Transition Systems? Syntax Semantics − → unknown entity known entity − → Labelled Transition Systems CCS − → Timed Transition Systems ??? Timed Automata [Alur, Dill’90] Finite-state automata equipped with clocks. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 7 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems How to Describe Timed Transition Systems? Syntax Semantics − → unknown entity known entity − → Labelled Transition Systems CCS − → Timed Transition Systems ??? Timed Automata [Alur, Dill’90] Finite-state automata equipped with clocks. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 7 / 43
Timed Transition Systems Timed Automata Motivation Networks of Timed Automata Definition of TLTS Regions and Region Graph How to Describe Timed Transition Systems Equivalence Checking Problems How to Describe Timed Transition Systems? Syntax Semantics − → unknown entity known entity − → Labelled Transition Systems CCS − → Timed Transition Systems ??? Timed Automata [Alur, Dill’90] Finite-state automata equipped with clocks. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 7 / 43
� � � Timed Transition Systems Timed Automata Clock Constraints and Valuation Networks of Timed Automata Definition of Timed Automata Regions and Region Graph Semantics of Timed Automata Equivalence Checking Problems Example: Light switch press x > 1 . 4 press x ≤ 1 . 4 press x :=0 � Light Bright Off press Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 8 / 43
Timed Transition Systems Timed Automata Clock Constraints and Valuation Networks of Timed Automata Definition of Timed Automata Regions and Region Graph Semantics of Timed Automata Equivalence Checking Problems Definition of TA: Clock Constraints Let C = { x , y , . . . } be a finite set of clocks. Set B ( C ) of clock constraints over C B ( C ) is defined by the following abstract syntax g , g 1 , g 2 ::= x ∼ n | x − y ∼ n | g 1 ∧ g 2 where x , y ∈ C are clocks, n ∈ N and ∼∈ {≤ , <, = , >, ≥} . Example: x ≤ 3 ∧ y > 0 ∧ y − x = 2 Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 9 / 43
Timed Transition Systems Timed Automata Clock Constraints and Valuation Networks of Timed Automata Definition of Timed Automata Regions and Region Graph Semantics of Timed Automata Equivalence Checking Problems Clock Valuation Clock valuation Clock valuation v is a function v : C → R ≥ 0 . Let v be a clock valuation. Then v + d is a clock valuation for any d ∈ R ≥ 0 and it is defined by ( v + d )( x ) = v ( x ) + d for all x ∈ C v [ r ] is a clock valuation for any r ⊆ C and it is defined by � 0 if x ∈ r v [ r ]( x ) v ( x ) otherwise. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 10 / 43
Timed Transition Systems Timed Automata Clock Constraints and Valuation Networks of Timed Automata Definition of Timed Automata Regions and Region Graph Semantics of Timed Automata Equivalence Checking Problems Clock Valuation Clock valuation Clock valuation v is a function v : C → R ≥ 0 . Let v be a clock valuation. Then v + d is a clock valuation for any d ∈ R ≥ 0 and it is defined by ( v + d )( x ) = v ( x ) + d for all x ∈ C v [ r ] is a clock valuation for any r ⊆ C and it is defined by � 0 if x ∈ r v [ r ]( x ) v ( x ) otherwise. Luca Aceto ICE-TCS, School of Computer Science Reykjavik University, Iceland luca@ru.is and luca.aceto@gmail.com Timed Automata and Logics for Real-time Systems 10 / 43
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