Timed Automata II CS 5270 Lecture 5 17.02.05 Lecture5

Goals Timed Transition Systems: UPPAAL Review Definition Timed behaviors. Parallel composition Reachability UPPAAL 17.02.05 Lecture5

Timed Transition Systems Transition Systems + Clock Variables. Clock variables. Used to record the passage of (real) time. Act like Timers. Can be read. Transitions constrained (guarded) by current values of clock variables. Can be reset to 0 during a transition. Continue to evolve (in real time) until reset. 17.02.05 Lecture5

Using Clock Variables Hot; y y ≤ 2 On-ac; x OK Off-ac Off-ac x  5 Spec. : Turn off ac if the temperature is OK or any time after 5 units of time has elapsed since turning it on. Turn on ac within 2 time units after receiving Hot signal. 17.02.05 Lecture5

Using Clock Variables Hot; y y ≤ 2 On-ac; x OK Off-ac Off-ac x  5 Three components: Action on-ac Reset x Guard y ≤ 2 17.02.05 Lecture5

Using Clock Variables Hot; y y ≤ 2 On-ac; x OK Off-ac Off-ac x ¸ 5 Three components: Action on-ac Reset x In general, more than one clock can be reset. Guard y ≤ 2 In general, more than one constraint can be used.

Timed Transitions a ; X g a, an action X, a set of clock variables; the clock variables set to 0. g, a guard; a predicate based on the values of the clock variables.

State Invariants A clock constraint is associated with each state: state invariant The system can stay in the state only as long as the state’s invariant is not violated. For time points which violate the invariant one expects an output transition to be enabled. Otherwise a time deadlock. The progress of time is blocked (in the model!). 17.02.05 Lecture5

State Invariants x ≤ 2 b a ; x 17.02.05 Lecture5

State Invariants SAME AS b a ; x x ≤ 2 b a ; x x > 2 17.02.05 Lecture5

State Invariants s1 s2 s0 b a ; x x ≤ 2 x > 3 At (s1, x = 2.4) the behavior is undefined! 17.02.05 Lecture5

State Invariants g g1 g2 g3 At all “times” g OR g1 OR g2 OR g3 is satisfied. If more than one output transition is enabled, the choice is made non-deterministically. 17.02.05 Lecture5

Timed Transition Systems: Definition Clock Constraints: X, a finite set of clocks (clock variables). x  X  (X), the set of clock constraints over X. (X) = x ≤ c | x ≥ c | x < c | x >c | 1  2 | c  Q Q , the set of non-negative rational numbers. 2X --- The set of subsets of X X = {x, y} 2X = { , {x}, {y}, {x, y} } 17.02.05 Lecture5

Timed Transition Systems TTS = (S, sin, Act, X, I, ) S is a finite set of states. sin, initial state Act, a finite set of actions X, a finite set of clock variables. I : S (X), assigns a clock invariant to each state.   S  Act  2X  (X)  S , the transition relation. 17.02.05 Lecture5

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2  y  3 (s0, add, {x}, True, s1) is a transition in TTS.

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 X = {x0, x1,..xn} True = x0 ≥ 0 X = {x, y} True = x  0 (s0, add, {x}, True, s1) is a transition in TTS.

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2  (s1, data, {y}, True, s2) is a transition in TTS. (s2, ack.add, ;, x > 2, s3) is a transition in TTS. (s3, ack.data, ;, y ≤ 5, s0) is a transition in TTS

Behaviors s3 c b a s0 s1 s2 d s4 s0 a s1 b s2 c s3 s0 a s1 b s2 d s4 Are both paths (runs). 17.02.05 Lecture5

Behaviors s3 c x < 1 b a ; x s0 s1 s2 x = 1 x ≥ 2 d s4 s0 a s1 b s2 c s3 is NOT a run! s0 a s1 b s2 d s4 is a run. Computing behaviors is difficult. 17.02.05 Lecture5

Behaviors TTS = (S, sin, Act, X, I, ) We associate a “normal” transition system with TTS while taking time into account: TSTTS = (S, sin, Act  R, ) R, non-negative reals   S  Act  R  S TSTTS is, almost always, an infinite transition system! 17.02.05 Lecture5

Behaviors TTS = (S, sin, Act, X, I, ) TSTTS = (S, sin, Act  R, ) S = S  V V --- Valuations A valuation says what the current values of each clock variable is. v : X R 17.02.05 Lecture5

Behaviors s3 c x < 1 b a ; x s0 s1 s2 x = 1 x ≥ 2 d s4 (s1, 0) (s2, 1.8) (s4, ) are t-states. (s3, 5) is a t-state but not reachable. 17.02.05 Lecture5

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s1, (2, 5) ) is a state; (s1, V) V(x) = 2 V(y) = 5 (s2, V’) is a state. V’(x) = 15 V’(y) = 0 (s2, (15, 0) ) 17.02.05 Lecture5

Behaviors TTS = (S, sin, Act, X, I, !) TSTTS = (S, sin, Act  R, ) R, non-negative reals   S  Act  R  S S = S  V sin = (sin, VZERO) VZERO (x) = 0 for every x in X. 17.02.05 Lecture5

Behaviors There will be two types of transitions. Time pass move: (s, v)  (s, v’) t units of time pass starting from V. V’ (x) = V(x) + t for every x. Instantaneous transition. (s, v)  (s’, v’) Some transition in the timed automaton is taken at v; the guard is satisfied; v’ is the same as v except for resets. t a 17.02.05 Lecture5

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 1 Time- passing move?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 1 Time- passing move?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s1, (0, 5)) (s1, (0, 5) ) Time- passing move?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 2 Time- passing move?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s1, (0, 3)) a transition in TS?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s3, (0, 3)) a transition in TS?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s1, (0, 4)) a transition in TS?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s1, (0, 0)) a transition in TS?

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 Is this possible? data (s2, (0, 0) )

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2

Example add ; x (s0, (0, 0)) s0 s1 .8 1 ………….. data ; y (s0, (.8, .8)) ack.data ack.add s2 s3 x > 2 TSTTS will have (uncountably) infinite number of states and transitions.

Timed Behaviors TTS = (S, S0, Act, X, I, !) TSTTS = (S, sin, Act  R, ) The timed behavior of TTS is defined as the behavior of TSTTS Runs computations 17.02.05 Lecture5

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0)) ..…

Timed Behaviors Two consecutive time-passing moves can be amalgamated into one time passing move. (s1, (2,5)) 2 (s1, (4, 7)) 1.3 (s1, (5.3, 8.3)) can also be recorded as : (s1, (2,5)) 3.3 (s1, (5.3, 8.3)) (s, V) 1 (s, V+1) 2 (s, (V+1)+2) can also be recorded as: (s, V) 1+2 (s, V+(1+2)) 17.02.05 Lecture5

Timed Computations TTS = (S, S0, Act, X, I, !) TSTTS = (S, S0, Act [ R, )) (s0, V0) 0 (s0, V0’) a0 (s1, V1) 1 (s1, V1’) a1 (s2, V2) …… induces the timed computation (a0, 0) (a1, 0+ 1) …… (an, 0+1+..n) Transition systems --- Computations Timed Transition Systems --- Timed Computations. 17.02.05 Lecture5

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0)) (add, ? ) (data, ?) ….

Example add ; x s0 s1 data ; y y ≤ 5 ack.data ack.add s2 s3 x > 2 (s0, (0, 0)) 1.6 (s0, (1.6, 1.6)) add (s1, (0, 1.6)) 2 (s1, (2, 3.6) data (s2, (2, 0)) (add, 1.6 ) (data, 3.6) ….

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (add, 1) (d, 10) (ack.a , 3) A timed computation? 17.02.05 Lecture5

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (add, 1) (d, 10) (ack.a , 3) A tmed computation? No! Time must increase monotonically. 17.02.05 Lecture5

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (a, 1) (ack.a , 3) (d, 10) A timed computation? 17.02.05 Lecture5

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (a, 1) (ack.a , 3) (d, 10) A timed computation? No! The order of execution must respect the specification!

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (a, 1) (d, 1) (ack.add, 10) A time computation? 17.02.05 Lecture5

Example add ; x data ; y y ≤ 5 ack.data ack.add x > 2 (a, 1) (d, 1) (ack.add, 10) A timed computation? Yes! Transitions are assumed to take zero time. Must rule out an infinite number of occurrences in finite time. 17.02.05 Lecture5

Zeno Computations a b x < 1 … (b, ½) ….(b, 3/4) ….(b, 15/16) …. (b, 31/32)…. We must define computations of timed transition systems carefully! 17.02.05 Lecture5

Parallel Composition TTS = TTS_1 || TTS_2 || …… || TTS_n Same principle as before: Do common actions together Take union of clock variables. Take conjunction of the guards! 17.02.05 Lecture5

Reachability of Control States For a finite TS it is trivial to decide whether s is reachable in TS. For finite TTS, whether s is reachable in TTS is not easy to decide because TSTTS is an infinite object! But this can be done and this verification process can be automated. More involved (liveness) properties can also be verified effectively but not always efficiently. 17.02.05 Lecture5

An introduction to UPPAAL 17.02.05 CS5272 - 18 Feb 2005Lecture5

What is UPPAAL? A toolbox for modeling, simulation and verification of real-time systems Developed jointly by Uppsala University and Aalborg University Often used for real-time controllers, communication protocols, etc. Consists of 3 main parts a system description a simulator a verifier 17.02.05 Lecture5

System Description 17.02.05 Lecture5

System Description A system in UPPAAL: a set of concurrent processes A process: a timed-automaton with: Clocks and data variables data variables: int, bool, array of int/bool Scopes: global or local (to a process) Both clocks and data variables! Guards (on transitions) and invariants (on locations) are conditions on data variables and clocks 17.02.05 Lecture5

System Description Synchronization actions are described using communication channels Two processes communicate with each other via global data variables or synchronization on communication channels 17.02.05 Lecture5

System Description A system description in UPPAAL is composed of Global declarations Process templates Process assignments A system definition 17.02.05 Lecture5

System Description Overview of UPPAAL 17.02.05 Lecture5

Global declarations All the clocks, variables, constants and channels declared here are visible to all the processes in the system 17.02.05 Lecture5

Global declarations clock x, y; const a 1; int a; int[1,10] b := 2; two clocks x and y const a 1; a constant a with value 1 int a; an integer variable a in the default range from -32768 to 32767 int[1,10] b := 2; an integer variable b in the range 1,…,10 that is initialized to 2 int a[2][3] := {{0,1,2},{3,4,5}}; a two dimensional integer array a default range and its initialization bool a[3], b[5]; two bit arrays a and b, with 3 and 5 elements respectively. 17.02.05 Lecture5

Global Declarations chan d; urgent chan d; broadcast chan e; A communication channel d Actions performed by two processes when they synchronize with each other on channel d are denoted as d! (sending) and d? (receiving) urgent chan d; When two processes are able to synchronize on channel d, they have to do it immediately. No guard is allowed on transitions labeled with synchronization actions (eg. d!,d?) on urgent channels broadcast chan e; Allows 1-to-many synchronization A transition with label e! emits a broadcast on the channel e. Any enabled transition with label e? will synchronise with the emitting process. No guard is allowed on these transitions 17.02.05 Lecture5

Process Templates Define a common control structures for the processes that are alike Each process is an instantiation of a template Each template can have symbolic variables and constants as parameters Parameters are syntactically similar to declarations, but with no initializations Eg. clock x, y; int i1, i2; chan a, b local clocks and variables Declared in the same way as global clocks and variables 17.02.05 Lecture5

Declare symbolic constants and variables here! Name of the template Process Templates Draw the structure of the template here! 17.02.05 Lecture5

Process Templates All clocks, variables, constants declared here are visible to process template P only 17.02.05 Lecture5

Process Templates 17.02.05 An example of process template Lecture5

Process Templates Locations 3 types: normal (with/without invariant), committed(C), urgent (U) When the system enters a committed location, it must leave the location immediately: Just a way of breaking up the effect of a transition into a sequence of steps. urgent location: Only 0 units of time can be spent in this location. 17.02.05 Lecture5

example 17.02.05 Lecture5

Process Templates A guard Timing expression: x ~ e or x – y ~ e A conjunction of timing and data expressions Written as a list of expressions, separated by commas x < y, x <= 5, (a[0] + 1) != (a[1]*5) Timing expression: x ~ e or x – y ~ e x, y : clocks; e: an integer expression ~  { <=, >=, ==, <, > } Data expression: involves constants and variables same as in C/C++/Java 17.02.05 Lecture5

Process Templates Invariants Updates Same as guards, with no lower bounds on clocks Updates A list of assignment expressions Each assignment is of the form: x := e or a := e’ In x: = e, almost always e is 0. x: a clock a: a variable e: an integer expression e’: a data expression Evaluated sequentially a := 1, b := 2*a sets a to 1, b to 2. 17.02.05 Lecture5

Process Assignments Declare instances of process templates Each assignment has the form P := T(A); P : a process name T : a process template name A : an argument list Examples: P := Q(); Process P is an instance of template Q that has no parameter S:=R(x,1); Process S is an instance of template R that has two parameters - a clock and a constant 17.02.05 Lecture5

Process Assignments Instantiate your processes here! 17.02.05 Lecture5

System Definition Declare the set of processes in the system Eg. system P, Q; A system consists of two processes P and Q Each process in the system must be either a process that appears on the left-hand-side of a process assignment, or a template with no parameter 17.02.05 Lecture5

Simulator 17.02.05 Lecture5

Simulator A validation tool that Interface: divided into 4 sub-windows allows users to interact and observe the behavior of the system during execution visualise executions (i.e. symbolic traces) generated by the verifier Interface: divided into 4 sub-windows Simulation control Variable window Process window Message Sequence Chart 17.02.05 Lecture5

Message sequence chart simulator control variable window process window Message sequence chart simulator control variable window 17.02.05 Lecture5

Simulator Control Used to control the simulation and to select the (symbolic) state or transition to be visualized Upper part: for step-by-step simulation Displays all enabled transitions Enables users to select an enabled transition to be fired Lower part: displays the generated trace A trace is an alternating sequence of location vectors and transitions. 17.02.05 Lecture5

Simulation Control use the mouse to select the transition to be fired press “Reset” resets the simulated system to its initial state press “Next” causes the system to fire the selected transition 17.02.05 Lecture5

Simulation Control the simulation always progresses from this highlighted element press “Prev” to highlight the element immediately preceding the current selection in the trace. press “Replay” to replay the trace starting from the currently selected element. press “Open” to open a file selector window for loading a trace from file. 17.02.05 Lecture5 control the speed of simulation

Variable window Displays the values of data and clocks in the current state or transition selected in the trace 17.02.05 Lecture5

Process window Displays the process instances of the system The current location of each automaton is marked with a red token The transitions currently selected in the simulation control panel are highlighted 17.02.05 Lecture5

the selected transition 17.02.05 Lecture5

Message Sequence Chart Displays an MSC view of the generated trace a vertical lines for each process a horizontal line for each synchronisation point. synchronization point 17.02.05 Lecture5

Verifier 17.02.05 Lecture5

Verifier Used to check invariant and liveness properties of the system Requirement Specification Language A[ ] p For all paths, p always holds E<> p There exists a path where p will eventually hold E[ ] p There exists a path where p always holds A<> p For all paths, p will eventually hold p --> q Whenever p holds, q will eventually hold p : an expression that has no side-effect or state formula (conjunction of state locations) 17.02.05 Lecture5

Verifier Examples A[] not deadlock E<> p1.cs and p2.cs true if the system is not deadlocked. E<> p1.cs and p2.cs true if the system can reach a state where both process p1 and p2 are in their locations cs. A[ ] p1.cs imply not p2.cs true if whenever process p1 is in its location cs, the process is not in its location cs. 17.02.05 Lecture5

Press to verify the selected property The selected property to be verified Insert a new property Remove the selected property Show the comments in overview window the system is deadlock free verification results are shown here! 17.02.05 Lecture5

Downloading UPPAAL http://www.uppaal.com Version 3.4.7 You must use the Windows platform version! References Demos provided in the software Papers and tutorials can be downloaded from the website (under Documentation section) 17.02.05 Lecture5