topics in timed automata
play

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software - PowerPoint PPT Presentation

Jul 15, 2023 •414 likes •1.38k views

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group 1/25 Reachability: Does something bad happen? The gate is still open when the train is 2 minutes away from the crossing This problem is

  1. Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group 1/25

  2. Reachability: Does something bad happen? “The gate is still open when the train is 2 minutes away from the crossing” This problem is PSPACE-complete A theory of timed automata R. Alur and D.L. Dill, TCS’94 2/25

  3. Tools ◮ UPPAAL: Uppsaala university (Sweden) , Aalborg university (Denmark) ◮ KRONOS: Verimag (France) ◮ RED National Taiwan University (Taiwan) ◮ Rabbit Brandenburg TU Cottbus (Germany) 3/25

  4. Tools ◮ UPPAAL: Uppsaala university (Sweden) , Aalborg university (Denmark) ◮ KRONOS: Verimag (France) ◮ RED National Taiwan University (Taiwan) ◮ Rabbit Brandenburg TU Cottbus (Germany) and still research on for efficient algorithms . . . 3/25

  5. Lecture 6: Reachability 4/25

  6. Timed Automata s 2 ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) { y } s 0 s 1 s 3 ( x > 6 ) ( y < 1 ) { y } Run: finite sequence of transitions s 0 s 1 s 3 0 . 4 0 . 5 x 0 0 . 4 0 . 9 y 0 0 0 . 5 ◮ accepting if ends in green state 5/25

  7. Reachability problem Given a TA, does it have an accepting run s 2 ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) { y } s 0 s 1 s 3 ( x > 6 ) ( y < 1 ) { y } Theorem [AD94] This problem is PSPACE-complete first solution based on Regions 6/25

  8. Key idea: Maintain sets of valuations reachable along a path y y y y x x x x ( x ≤ 5 ) ( y ≥ 7 ) q 0 q 1 q 2 q 3 { x } 7/25

  9. Key idea: Maintain sets of valuations reachable along a path x = y ≥ 0 x = y ≥ 0 y − x ≥ 7 y − x ≥ 7 y y y y x x x x ( x ≤ 5 ) ( y ≥ 7 ) q 0 q 1 q 2 q 3 { x } Easy to describe convex sets 7/25

  10. Zones and zone graph ◮ Zone: set of valuations defined by conjunctions of constraints: x ∼ c x − y ∼ c e.g. ( x − y ≥ 1 ) ∧ ( y < 2 ) ◮ Representation: by DBM [Dil89] Sound and complete [DT98] Zone graph preserves state reachability 8/25

  11. Problem of non-termination y x ( y = 1 ) { y } { x , y } q 0 q 1 9/25

  12. Abstractions Zone graph q 0 , Z 0 × × q 1 , Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  13. Abstractions q 0 , Z 0 Zone graph q 0 , Z 0 × × q 1 , Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  14. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph q 0 , Z 0 × × q 1 , Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  15. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × q 1 , Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  16. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × q 1 , Z 1 W 1 q 1 , × Z 1 q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  17. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × a ( W 1 ) q 1 , Z 1 W 1 q 1 , × Z 1 q 2 , q 3 , Z 2 Z 3 . . . . . . potentially infinite... 10/25

  18. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × a ( W 1 ) q 1 , Z 1 W 1 q 1 , × Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . . . q 2 , q 3 , W 2 W 3 Z 3 Z 2 potentially infinite... 10/25

  19. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × a ( W 1 ) q 1 , Z 1 W 1 q 1 , × Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . a ( W 3 ) . . q 2 , q 3 , W 2 W 3 Z 3 Z 2 potentially infinite... a ( W 2 ) 10/25

  20. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × a ( W 1 ) q 1 , Z 1 W 1 q 1 , × Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . a ( W 3 ) q 2 , q 3 , . . W 2 W 3 Z 2 Z 3 potentially infinite... a ( W 2 ) Find a such that number of abstracted sets is finite 10/25

  21. Abstractions a ( Z 0 ) q 0 , Z 0 Zone graph × q 0 , × Z 0 × × a ( W 1 ) q 1 , Z 1 W 1 q 1 , × Z 1 × q 2 , q 3 , Z 2 Z 3 . . . . a ( W 3 ) q 2 , q 3 , . . W 2 W 3 Z 2 Z 3 potentially infinite... a ( W 2 ) Coarser the abstraction, smaller the abstracted graph 10/25

  22. Condition 1 : Abstractions should have finite range Condition 2 : Abstractions should be sound ⇒ a ( W ) can contain only valuations simulated by W a ( W ) g 5 g 4 g 2 g 1 g 3 R v 5 R 4 R 2 R 1 R 3 q , W v ′ g 1 g 2 g 3 g 4 g 5 R 1 R 2 R 3 R 5 R 4 11/25

  23. Condition 1 : Abstractions should have finite range Condition 2 : Abstractions should be sound ⇒ a ( W ) can contain only valuations simulated by W a ( W ) g 5 g 4 g 2 g 1 g 3 R v 5 R 4 R 2 R 1 R 3 q , W v ′ g 1 g 2 g 3 g 4 g 5 R 1 R 2 R 3 R 5 R 4 Question: Why not add all the valuations simulated by W ? 11/25

  24. Bounds and abstractions Theorem [LS00] Coarsest simulation relation is EXPTIME-hard s 2 ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) { y } s 0 s 1 s 3 ( x > 6 ) ( y < 1 ) { y } 12/25

  25. Bounds and abstractions Theorem [LS00] Coarsest simulation relation is EXPTIME-hard ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) ( x > 6 ) ( y < 1 ) 12/25

  26. Bounds and abstractions Theorem [LS00] Coarsest simulation relation is EXPTIME-hard ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) ( x > 6 ) ( y < 1 ) M-bounds [AD94] M ( x ) = 6 , M ( y ) = 3 v � M v ′ 12/25

  27. Bounds and abstractions Theorem [LS00] Coarsest simulation relation is EXPTIME-hard ( y ≤ 3 ) ( x < 4 ) ( x < 1 ) ( x > 6 ) ( y < 1 ) M-bounds [AD94] LU-bounds [BBLP04] L ( x ) = 6 , L ( y ) = −∞ M ( x ) = 6 , M ( y ) = 3 U ( x ) = 4 , U ( y ) = 3 v � M v ′ v � LU v ′ 12/25

  28. Abstractions in literature [BBLP04, Bou04] ( � LU ) a � LU ( � M ) Closure M 13/25

  29. Abstractions in literature [BBLP04, Bou04] ( � LU ) a � LU ( � M ) Closure M Non-convex 13/25

  30. Abstractions in literature [BBLP04, Bou04] Extra + ( � LU ) a � LU LU Extra + ( � M ) Extra LU Closure M M Non-convex Extra M Convex Only convex abstractions used in implementations ! 13/25

  31. Timed automata Zone graph Problem of non-termination Use finite abstractions Zones Bounds as parameters are efficient Restriction to convex abstractions Non-convex abstr. are coarser 14/25

  32. Timed automata Zone graph Problem of non-termination Use finite abstractions Zones Bounds as parameters are efficient Restriction to convex abstractions Non-convex abstr. are coarser Question: Can we benefit from both together? 14/25

  33. In this lecture... Efficient use of the non-convex Closure approximation Using non-convex approximations for efficient analysis of timed automata F. Herbreteau, D. Kini, B. Srivathsan, I. Walukiewicz. FSTTCS’11 15/25

  34. Observation 1 : We can use abstractions without storing them 16/25

  35. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 a ( W 1 ) q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 Standard algorithm: covering tree q 3 Z 3 17/25

  36. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 a ( W 1 ) q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 q 3 Pick simulation based a Z 3 17/25

  37. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 a ( W 1 ) q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 q 3 Pick simulation based a Z 3 17/25

  38. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 a ( W 1 ) q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 q 3 Pick simulation based a Z 3 17/25

  39. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 � a ( W 1 ) q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 q 3 Pick simulation based a Z 3 17/25

  40. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 � a ( W 1 ) � q 1 q 3 = q 1 ∧ W 1 , a ( W 5 ) Z 1 W 5 q 5 a ( W 3 ) ⊆ a ( W 1 )? , Z 5 a ( W 2 ) a ( W 4 ) W 2 q 2 q 4 Z 4 , Z 2 , W 4 a ( W 3 ) , W 3 q 3 Pick simulation based a Z 3 17/25

  41. Using non-convex abstractions a ( Z 0 ) q 0 , Z 0 a ( Z 1 ) q 1 q 3 = q 1 ∧ , a ( Z 5 ) Z 1 q 5 a ( Z 3 ) ⊆ a ( Z 1 )? , Z 5 a ( Z 2 ) a ( Z 4 ) q 2 q 4 Z 4 , Z 2 , a ( Z 3 ) , q 3 Pick simulation based a Z 3 17/25

  42. Using non-convex abstractions q 0 , Z 0 q 1 q 3 = q 1 ∧ , Z 1 q 5 a ( Z 3 ) ⊆ a ( Z 1 )? , Z 5 q 2 q 4 Z 4 , Z 2 , , Need to store only concrete semantics q 3 Z 3 17/25

  43. Using non-convex abstractions q 0 , Z 0 q 1 q 3 = q 1 ∧ , Z 1 q 5 Z 3 ⊆ a ( Z 1 )? , Z 5 q 2 q 4 Z 4 , Z 2 , Use Z ⊆ a ( Z ′ ) for termination , q 3 Z 3 17/25

  44. Observation 1 : We can use abstractions without storing them Observation 2 : We can do the inclusion test efficiently 18/25

  45. Coming next... The inclusion test Z ⊆ Closure M ( Z ′ ) 19/25

  46. What is Closure M ? y M ( y ) x 0 M ( x ) 20/25

  47. What is Closure M ? y M ( y ) Z x 0 M ( x ) 20/25

  48. What is Closure M ? y M ( y ) Z x 0 M ( x ) Closure M ( Z ) : set of regions that Z intersects 20/25

  49. Z ⊆ Closure M ( Z ′ ) ? y Z Z ′ M ( y ) x 0 M ( x ) 21/25

  50. Z ⊆ Closure M ( Z ′ ) ? y Closure M ( Z ′ ) Z Z ′ M ( y ) x 0 M ( x ) 21/25

  51. Z ⊆ Closure M ( Z ′ ) ? y Closure M ( Z ′ ) Z Z ′ M ( y ) x 0 M ( x ) 21/25

Recommend

OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A

OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A

OUTLINE Model Checking in a Nutshell Timed automata and TCTL Timed Automata, TCTL A UPPAAL Tutorial &amp; Verification Problems Data stuctures &amp; central algorithms UPPAAL input languages 1 2 Timed Automata: Syntax

408 views • 11 slides
Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group 1/29 Theorem (Lecture 2) Deterministic timed automata are closed under complement 2/29 Theorem (Lecture 2) Deterministic timed automata are closed under

654 views • 50 slides
Formalized Timed Automata Simon Wimmer Fakultt fr Informatik, Technische Universitt

Formalized Timed Automata Simon Wimmer Fakultt fr Informatik, Technische Universitt

Formalized Timed Automata Simon Wimmer Fakultt fr Informatik, Technische Universitt Mnchen ITP Talk on August 24, 2016 Timed Automata Timed Automata (TA) Finite Automata with clocks Clock guards on transitions and clock

735 views • 35 slides
Seminar: Automata Theory Timed Automata Jennifer Nist 11 th February 2016 Chair of Software

Seminar: Automata Theory Timed Automata Jennifer Nist 11 th February 2016 Chair of Software

Seminar: Automata Theory Timed Automata Jennifer Nist 11 th February 2016 Chair of Software Engineering Albert-Ludwigs Universit at Freiburg 11 th February 2016 Jennifer Nist Timed Automata 1 / 28 Outline 1 Timed Automata 2 Timed Language

676 views • 30 slides
Timed Automata and Logics for Real-time Systems Luca Aceto ICE-TCS, School of Computer Science

Timed Automata and Logics for Real-time Systems Luca Aceto ICE-TCS, School of Computer Science

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,

1.12k views • 65 slides
Decomposition of Weighted Timed Automata Vitaly Perevo shchi kov she (joint work with

Decomposition of Weighted Timed Automata Vitaly Perevo shchi kov she (joint work with

Decomposition of Weighted Timed Automata Vitaly Perevo shchi kov she (joint work with Manfred Droste) Leipzig University, QuantLA FFM 2015 Timed automata 1 Nondeterministic finite automata A with clocks a , x 2, y 0 Edges: a

374 views • 36 slides
Modelling and Verification Timed Automata: A Formalism for Real-time Systems Labelled transition

Modelling and Verification Timed Automata: A Formalism for Real-time Systems Labelled transition

Timed Transition Systems Timed Automata Networks of Timed Automata Equivalence Checking Problems Regions and Region Graph Modelling and Verification Timed Automata: A Formalism for Real-time Systems Labelled transition systems with time

1.21k views • 62 slides
Abstractions for timed automata B. Srivathsan Chennai Mathematical Institute Joint work with F.

Abstractions for timed automata B. Srivathsan Chennai Mathematical Institute Joint work with F.

Abstractions for timed automata B. Srivathsan Chennai Mathematical Institute Joint work with F. Herbreteau, I. Walukiewicz (LaBRI, Bordeaux) 1 / 35 Reachability for timed automata Observation 1 Observation 2 2 / 35 Timed Automata q 2 ( y

1.71k views • 131 slides
The Timestamp of Timed Automata Amnon Rosenmann Graz University of Technology

The Timestamp of Timed Automata Amnon Rosenmann Graz University of Technology

The Timestamp of Timed Automata Amnon Rosenmann Graz University of Technology rosenmann@math.tugraz.at Amnon Rosenmann (TU Graz) The Timestamp of Timed Automata 1 / 30 Introduction Timed automata (TA) are finite automata extended with clocks

1.02k views • 30 slides
Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I.

Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I.

Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I. Walukiewicz Universit e de Bordeaux, LaBRI - CNRS August 2010 Efficient Emptiness Check for Timed B uchi Automata - 1/24 Timed B uchi Automata

653 views • 54 slides
A Nivat Theorem for Weighted Timed Automata and Weighted Relative Distance Logic Manfred Droste

A Nivat Theorem for Weighted Timed Automata and Weighted Relative Distance Logic Manfred Droste

A Nivat Theorem for Weighted Timed Automata and Weighted Relative Distance Logic Manfred Droste and Vitaly Perevoshchikov Leipzig University ICALP, Track B 9 th of July 2014 Timed automata 1 Nondeterministic finite automata A with clocks a ,

947 views • 32 slides
Lazy abstractions for timed automata F. Herbreteau 1 , B. Srivathsan 2 , I. Walukiewicz 1 LaBRI,

Lazy abstractions for timed automata F. Herbreteau 1 , B. Srivathsan 2 , I. Walukiewicz 1 LaBRI,

Lazy abstractions for timed automata F. Herbreteau 1 , B. Srivathsan 2 , I. Walukiewicz 1 LaBRI, Universit e Bordeaux 1 Software modeling and verification group, RWTH-Aachen MOVES Seminar, March 2013 Lazy abstractions for timed automata -

609 views • 60 slides
Coinduction in concurrent timed systems Jan Komenda Institute of Mathematics, Czech Academy of

Coinduction in concurrent timed systems Jan Komenda Institute of Mathematics, Czech Academy of

Mealy and weighted automata as coalgebras Functional stream calculus (max,+)-automata and timed automata Synchronous composition Produc Coinduction in concurrent timed systems Jan Komenda Institute of Mathematics, Czech Academy of Sciences,

573 views • 43 slides
Are Timed Automata Bad for a Specification Language? Language Inclusion Checking for Timed

Are Timed Automata Bad for a Specification Language? Language Inclusion Checking for Timed

Are Timed Automata Bad for a Specification Language? Language Inclusion Checking for Timed Automata Contributors Ting Wang, Zhejiang University Jun Sun, SUTD Yang Liu, NTU Xinyu Wang, Zhejiang University Shanping Li, Zhejiang University

706 views • 41 slides
Decidability of Timed Communicating Automata L. Clemente, University of Warsaw Praha, July 2018

Decidability of Timed Communicating Automata L. Clemente, University of Warsaw Praha, July 2018

Decidability of Timed Communicating Automata L. Clemente, University of Warsaw Praha, July 2018 Summary 1. The model: Timed communicating automata (TCA). 2. The problem: control-state reachability. 3. Solution technique: quantifier

263 views • 24 slides
News on Safety Properties for Timed Petri Nets Patrick Totzke Edinburgh 2018-09-24 1 / 15 2 /

News on Safety Properties for Timed Petri Nets Patrick Totzke Edinburgh 2018-09-24 1 / 15 2 /

News on Safety Properties for Timed Petri Nets Patrick Totzke Edinburgh 2018-09-24 1 / 15 2 / 15 Networks of Timed Automata - a (finite-state) control program ( S ) - K indistinguishable d -clock Timed Automata ( C ) 2 / 15 Networks of Timed

1.58k views • 121 slides
Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification

Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification

Real-time Model Checking Priced timed automata Nicolas M ARKEY Lav. Sp ecification &amp; V erification CNRS &amp; ENS Cachan France March 3, 2010 Time is not always sufficient Timed automata are (rather) well understood

1.58k views • 129 slides
On Interleaving in Timed Automata Oded Maler, Marius Bozga, Ramzi Ben Salah VERIMAG 24th August

On Interleaving in Timed Automata Oded Maler, Marius Bozga, Ramzi Ben Salah VERIMAG 24th August

On Interleaving in Timed Automata Oded Maler, Marius Bozga, Ramzi Ben Salah VERIMAG 24th August 2006 Introduction Exploring the state space of Timed Automata is important (circuits timing analysis, scheduling, etc). However, its a very

657 views • 37 slides
Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group

Topics in Timed Automata B. Srivathsan RWTH-Aachen Software modeling and Verification group 1/34 System Specification L ( A ) L ( B ) L ( A ) L ( B ) empty? Is 2/34 System Specification L ( A ) L ( B ) L ( A ) L ( B )

960 views • 60 slides
01/07/2010 Two Parallel Stories Timed and Hybrid Automata Symbolic model checking: Timed and

01/07/2010 Two Parallel Stories Timed and Hybrid Automata Symbolic model checking: Timed and

01/07/2010 Two Parallel Stories Timed and Hybrid Automata Symbolic model checking: Timed and hybrid systems: -abstract data type region algebra yp g g specific region algebra p g g -termination analysis (e.g. clock regions, polyhedra)

807 views • 49 slides
Timed pushdown automata and branching vector addition systems S awomir Lasota University of

Timed pushdown automata and branching vector addition systems S awomir Lasota University of

Timed pushdown automata and branching vector addition systems S awomir Lasota University of Warsaw joint work with Lorenzo Clemente, Filip Mazowiecki and Ranko Lazic AVERTS 2016, Chennai 1 Definable sets offer a right setting for timed

2.11k views • 181 slides
A Concurrency-Preserving Translation from Time Petri Nets to Networks of Timed Automata Sandie

A Concurrency-Preserving Translation from Time Petri Nets to Networks of Timed Automata Sandie

A Concurrency-Preserving Translation from Time Petri Nets to Networks of Timed Automata Sandie Balaguer, Thomas Chatain, Stefan Haar LSV ENS Cachan, INRIA, CNRS France ACTS January 28, 2011 Introduction 1 Motivation Timed and

684 views • 50 slides
Timed automata with diagonal constraints B. Srivathsan Chennai Mathematical Institute, India In

Timed automata with diagonal constraints B. Srivathsan Chennai Mathematical Institute, India In

Timed automata with diagonal constraints B. Srivathsan Chennai Mathematical Institute, India In this lecture, we will consider timed automata that can additionally take guards of the form x y 5, z x &gt; 20, etc. Such guards are

346 views • 5 slides
Scheduling Optim al &amp; Real Tim e using CORA CORA CORA Overview Timed Automata &amp;

Scheduling Optim al &amp; Real Tim e using CORA CORA CORA Overview Timed Automata &amp;

Model Checking Technology Scheduling Optim al &amp; Real Tim e using CORA CORA CORA Overview Timed Automata &amp; Scheduling Informationsteknologi Priced Timed Automata and Optimal Scheduling Optimal Infinite Scheduling

1.04k views • 76 slides

More recommend