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Quantitative Verification

Quantitative Verification. Arindam Chakrabarti * Krishnendu Chatterjee * Thomas A. Henzinger * Orna Kupferman ** Rupak Majumdar *** * UC Berkeley ** Hebrew University *** UC Los Angeles. Outline. What is the proposal ? What benefits do we get out of it ?

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Quantitative Verification

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  1. Quantitative Verification Arindam Chakrabarti* Krishnendu Chatterjee* Thomas A. Henzinger* Orna Kupferman** Rupak Majumdar*** *UC Berkeley **Hebrew University ***UC Los Angeles

  2. Outline • What is the proposal ? • What benefits do we get out of it ? • Nailing down some details… • Some interesting results. • Summary 4th OSQ Retreat, Santa Cruz, CA

  3. Formal Verification: Traditional approach • Model: Labelled transition structure. • Property: Classification of finite and/or infinite sequences of states into good and bad sets. • Model-checking: Verification that all sequences of states generated by model are in good set. 4th OSQ Retreat, Santa Cruz, CA

  4. {a} {a,b} {a} {c} {b,c} Traditional approach: Models 4th OSQ Retreat, Santa Cruz, CA

  5. {a} {a,b} {a} {c} {b,c} Traditional approach: Models Each proposition maps each state to TRUE or FALSE. 4th OSQ Retreat, Santa Cruz, CA

  6. {a} {a,b} {a} {c} {b,c} Traditional approach: Models Proposition: a Each proposition maps each state to TRUE or FALSE. 4th OSQ Retreat, Santa Cruz, CA

  7. {a} {a,b} {a} {c} {b,c} Traditional approach: Models Proposition: b Each proposition maps each state to a boolean. 4th OSQ Retreat, Santa Cruz, CA

  8. 3,2,4 8,4,9 1,3,4 0,2,5 34,23,1 Extension 1: Quantitative Propositions, Models Propositions: <a,b,c> Each proposition maps each state to an integer. 4th OSQ Retreat, Santa Cruz, CA

  9. {a} {a,b} {a} {c} {b,c} Traditional approach: Properties A(a U c) 4th OSQ Retreat, Santa Cruz, CA

  10. {a} {a,b} {a} {c} {b,c} Traditional approach: Properties A(a U c) A property maps each path to TRUE or FALSE. 4th OSQ Retreat, Santa Cruz, CA

  11. 3,2,4 8,4,9 1,3,4 0,2,5 34,23,1 Extension 2: Quantitative Properties max(sum(a)) while (sum(b) < 100) 4th OSQ Retreat, Santa Cruz, CA

  12. 3,2,4 8,4,9 1,3,4 0,2,5 112 34,23,1 Extension 2: Quantitative Properties max(sum(a)) while (sum(b) < 100) 4th OSQ Retreat, Santa Cruz, CA

  13. 3,2,4 8,4,9 1,3,4 0,2,5 115 34,23,1 Extension 2: Quantitative Properties max(sum(a)) while (sum(b) < 100) 4th OSQ Retreat, Santa Cruz, CA

  14. 3,2,4 8,4,9 1,3,4 0,2,5 188 34,23,1 Extension 2: Quantitative Properties max(sum(a)) while (sum(b) < 100) A property maps each path to an integer. 4th OSQ Retreat, Santa Cruz, CA

  15. {a} {a,b} {a} {c} {b,c} Traditional approach: Model-checking problem A(a U c) Check if any path in model violates the property (is mapped to FALSE). 4th OSQ Retreat, Santa Cruz, CA

  16. 3,2,4 8,4,9 1,3,4 0,2,5 188 34,23,1 Extension 3: Quantitative Model-checking problem max(sum(a)) while (sum(b) < 100) Find the maximum (or minimum) value of the property on any path in the model. 4th OSQ Retreat, Santa Cruz, CA

  17. Outline • What is the proposal ? • What benefits do we get out of it ? • Nailing down some details… • Some interesting results. • Summary 4th OSQ Retreat, Santa Cruz, CA

  18. stop slow fast fast? stop? slow? slow? fast? fast? 0 1 2 stop? slow? stop? Motor driver in a robot 4th OSQ Retreat, Santa Cruz, CA

  19. receive send receive? receive? receive? 0 1 2 3 send? send? send? Sensornet node with buffer of size 3 4th OSQ Retreat, Santa Cruz, CA

  20. Outline • What is the proposal ? • What benefits do we get out of it ? • Nailing down some details… • Some interesting results. • Summary 4th OSQ Retreat, Santa Cruz, CA

  21. Specifying properties using quantitative automata • Property: maps each sequence of states to an integer. • Quantitative automaton: States, input symbols, counters, guarded instructions on transitions, nondeterminism. • Value of a run is given by limsup of values of a designated counter R0. 4th OSQ Retreat, Santa Cruz, CA

  22. R1 := R1 + a R2 := R2 - b if R1 = R2 then R0 := c R1 := R1 + a R2 := R2 + b if R1 = R2 then R0 := c A Quantitative Automaton Maps each infinite sequence  = hai,bi,cii… to limsup ci such that  ai =  (-1)i¢ bi 4th OSQ Retreat, Santa Cruz, CA

  23. Outline • What is the proposal ? • What benefits do we get out of it ? • Nailing down some details… • Some interesting results. • Summary 4th OSQ Retreat, Santa Cruz, CA

  24. Some interesting results • Infinite det- and nondet- hierarchies. • Power of non-determinism. • Undecidability of model-checking. • Absence of finite-memory determinacy. • Parametric-bounds, decidability, complexity. • Parameter-finding cannot be automated. • Quantitative -calculus, correlations. 4th OSQ Retreat, Santa Cruz, CA

  25. Some interesting results • Infinite det- and nondet- hierarchies. • Power of non-determinism. • Undecidability of model-checking. • Absence of finite-memory determinacy. • Parametric-bounds, decidability, complexity. • Parameter-finding cannot be automated. • Quantitative -calculus, correlations. 4th OSQ Retreat, Santa Cruz, CA

  26. Examples • Response time • Fair maximum • Resoure lifetime 4th OSQ Retreat, Santa Cruz, CA

  27. Summary • Quantitative extension to boolean verification framework. • Motivation for doing so. • Extended definitions for propositions, properties, and the model-checking problem. • Some results (+ problems, solutions), examples. 4th OSQ Retreat, Santa Cruz, CA

  28. Thanks for listening ! Questions, Comments, Suggestions ? 4th OSQ Retreat, Santa Cruz, CA

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