Invariant-Free Clausal Temporal Resolution

288 views
244 views

Published on

Invariant-Free Clausal Temporal Resolution.
Jornadas SISTEDES 2012
Universidad de Almería

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
288
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Invariant-Free Clausal Temporal Resolution

  1. 1. Invariant-Free Clausal Temporal Resolution Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogicTheTemporalLogic PLTLClausal J. Gaintzarain, M. Hermo, P. Lucio, M. Navarro, F. OrejasResolutionfor PLTLClausal to appear in Journal of Automated ReasoningNormal Form (Online from December 2th, 2011)Invariant-FreeTemporal PROLE 2012, September 19thResolution Invariant-Free Clausal Temporal Resolution
  2. 2. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  3. 3. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  4. 4. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  5. 5. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  6. 6. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  7. 7. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  8. 8. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  9. 9. Temporal Logic Invariant-Free Clausal Temporal Resolution Significant role in Computer Science.Introductionto Temporal Useful for specification and verification of dynamic systemsLogic RoboticsTheTemporal Agent-Based SystemsLogic PLTLClausal Control SystemsResolution Dynamic Databasesfor PLTLClausal etc.Normal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  10. 10. Temporal Logic Invariant-Free Clausal Temporal Resolution Significant role in Computer Science.Introductionto Temporal Useful for specification and verification of dynamic systemsLogic RoboticsTheTemporal Agent-Based SystemsLogic PLTLClausal Control SystemsResolution Dynamic Databasesfor PLTLClausal etc.Normal FormInvariant- Also important in other fields: Philosophy, Mathematics,FreeTemporal Linguistics, Social Sciences, Systems Biology, etc.Resolution Invariant-Free Clausal Temporal Resolution
  11. 11. Temporal Logic: Example Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  12. 12. Temporal Logic: Specification Invariant-Free Clausal Temporal ResolutionIntroduction 1: Being in error means being neither available nor printingto TemporalLogic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  13. 13. Temporal Logic: Specification Invariant-Free Clausal Temporal ResolutionIntroduction 1: Being in error means being neither available nor printingto TemporalLogic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))TheTemporal 2: A printer will eventually end its job or produce an errorLogic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X))ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  14. 14. Temporal Logic: Specification Invariant-Free Clausal Temporal ResolutionIntroduction 1: Being in error means being neither available nor printingto TemporalLogic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))TheTemporal 2: A printer will eventually end its job or produce an errorLogic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X))ClausalResolutionfor PLTL 3: A non-available printer will not receive a new job until itClausal becomes availableNormal Form ∀X(¬available(X) → ¬new job for(X) U available(X))Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  15. 15. Temporal Logic: Verification Invariant-Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X))Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  16. 16. Temporal Logic: Verification Invariant-Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X))Introductionto Temporal System specificationLogicThe 1: Being in error means being neither available nor printingTemporalLogic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))Clausal 2: . . .Resolutionfor PLTL 3: A non-available printer will not receive a new job until itClausalNormal Form becomes availableInvariant- ∀X(¬available(X) → ¬new job for(X) U available(X))FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  17. 17. Temporal Logic: Verification Invariant-Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X))Introductionto Temporal System specificationLogicThe 1: Being in error means being neither available nor printingTemporalLogic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))Clausal 2: . . .Resolutionfor PLTL 3: A non-available printer will not receive a new job until itClausalNormal Form becomes availableInvariant- ∀X(¬available(X) → ¬new job for(X) U available(X))FreeTemporalResolution Deductive verification methods Tableaux, Sequent calculi, Resolution, etc. Invariant-Free Clausal Temporal Resolution
  18. 18. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  19. 19. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  20. 20. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingThe Unbounded versus boundedTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  21. 21. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingThe Unbounded versus boundedTemporalLogic PLTL Discrete versus denseClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  22. 22. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingThe Unbounded versus boundedTemporalLogic PLTL Discrete versus denseClausalResolution Point-based versus interval-basedfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  23. 23. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingThe Unbounded versus boundedTemporalLogic PLTL Discrete versus denseClausalResolution Point-based versus interval-basedfor PLTLClausal Only-future versus past-and-futureNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  24. 24. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic:Introductionto TemporalLogic Linear versus branchingThe Unbounded versus boundedTemporalLogic PLTL Discrete versus denseClausalResolution Point-based versus interval-basedfor PLTLClausal Only-future versus past-and-futureNormal Form Propositional versus first-orderInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  25. 25. The Temporal Logic PLTL Invariant-Free Clausal Temporal Resolution Different versions of Temporal Logic: Linear versus branchingIntroductionto TemporalLogic Unbounded versus boundedThe Discrete versus denseTemporalLogic PLTL Point-based versus interval-basedClausalResolution Only-future versus past-and-futurefor PLTLClausal Propositional versus first-orderNormal FormInvariant- PLTLFreeTemporalResolution Propositional Linear-time Temporal Logic Invariant-Free Clausal Temporal Resolution
  26. 26. PLTL: minimal language Invariant-Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”)Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  27. 27. PLTL: minimal language Invariant-Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”)Introductionto Temporal pLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  28. 28. PLTL: minimal language Invariant-Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”)Introductionto Temporal pLogicTheTemporalLogic PLTLClausal ◦pResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  29. 29. PLTL: minimal language Invariant-Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”)Introductionto Temporal pLogicTheTemporalLogic PLTLClausal ◦pResolutionfor PLTLClausalNormal FormInvariant- qU pFreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  30. 30. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . .Introductionto Temporal -VM : SM → 2Prop where Prop is the set of all the possibleLogic atomic propositions.TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  31. 31. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . .Introductionto Temporal -VM : SM → 2Prop where Prop is the set of all the possibleLogic atomic propositions.TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  32. 32. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . .Introductionto Temporal -VM : SM → 2Prop where Prop is the set of all the possibleLogic atomic propositions.TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-Free M, sj |= ϕ denotes that the formula ϕ is true in the stateTemporalResolution sj of M. Invariant-Free Clausal Temporal Resolution
  33. 33. PLTL: Model Theory Invariant-Free Clausal The connective ◦ (“next”) Temporal Resolution M, sj |= ◦ϕ iff M, sj+1 |= ϕIntroductionto TemporalLogicTheTemporal M, sj |= ◦pLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  34. 34. PLTL: Model Theory Invariant-Free Clausal The connective U (“until”) Temporal Resolution M, sj |= ϕ U ψ iff M, sk |= ψ for some k ≥ j and M, si |= ϕ for every i ∈ {j, . . . , k − 1}Introductionto TemporalLogicTheTemporalLogic PLTL M, sj |= p U qClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  35. 35. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution ModelIntroductionto Temporal M |= ψ iff M, s0 |= ψLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  36. 36. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution ModelIntroductionto Temporal M |= ψ iff M, s0 |= ψLogicTheTemporal Logical consequenceLogic PLTLClausal Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM :Resolutionfor PLTL if M, sj |= Φ then M, sj |= ψClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  37. 37. PLTL: Model Theory Invariant-Free Clausal Temporal Resolution ModelIntroductionto Temporal M |= ψ iff M, s0 |= ψLogicTheTemporal Logical consequenceLogic PLTLClausal Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM :Resolutionfor PLTL if M, sj |= Φ then M, sj |= ψClausalNormal Form SatisfiabilityInvariant-FreeTemporal ψ is satisfiable iff there exists a model of ψResolution Invariant-Free Clausal Temporal Resolution
  38. 38. PLTL: Defined Connectives Invariant-Free Clausal Temporal The connective (“eventually” or “some time”) Resolution ϕ ≡ TU ϕIntroductionto Temporal M, sj |= pLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  39. 39. PLTL: Defined Connectives Invariant-Free Clausal Temporal The connective (“eventually” or “some time”) Resolution ϕ ≡ TU ϕIntroductionto Temporal M, sj |= pLogicTheTemporalLogic PLTLClausalResolutionfor PLTL The connective (“always”)ClausalNormal Form ϕ ≡ ¬ ¬ϕInvariant-Free M, sj |= pTemporalResolution Invariant-Free Clausal Temporal Resolution
  40. 40. PLTL: Defined Connectives Invariant-Free Clausal Temporal Resolution The connective R (“release”) ϕ R ψ ≡ ¬(¬ϕ U ¬ψ)Introductionto TemporalLogic M, sj |= q R pTheTemporal EitherLogic PLTLClausalResolutionfor PLTLClausalNormal Form orInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  41. 41. PLTL: Eventualities and Invariants Invariant-Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives:Introductionto Temporal ϕ U ψ, ψLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  42. 42. PLTL: Eventualities and Invariants Invariant-Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives:Introductionto Temporal ϕ U ψ, ψLogicTheTemporal InvariantsLogic PLTL They assert that a formula is always true from some momentClausalResolution onwardsfor PLTL They are often expressed in an intricate way by means of setsClausalNormal Form of formulas:Invariant- ψFreeTemporal {ψ, (ψ → ◦ψ)} ψ is a logical consequenceResolution {ψ, (ψ → ◦ϕ), (ϕ → ψ)} ψ is a logical consequence Invariant-Free Clausal Temporal Resolution
  43. 43. PLTL: Eventualities and Invariants Invariant-Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives:Introductionto Temporal ϕ U ψ, ψLogicTheTemporal InvariantsLogic PLTL They assert that a formula is always true from some momentClausalResolution onwardsfor PLTL They are often expressed in an intricate way by means of setsClausalNormal Form of formulas:Invariant- ψFreeTemporal {ψ, (ψ → ◦ψ)} ψ is a logical consequenceResolution {ψ, (ψ → ◦ϕ), (ϕ → ψ)} ψ is a logical consequence Usually, their syntactic detection is not trivial: “hidden” invariants Invariant-Free Clausal Temporal Resolution
  44. 44. PLTL: Decidability Invariant-Free Clausal Temporal ResolutionIntroduction PLTL is decidableto TemporalLogic PSPACE-completeTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  45. 45. PLTL: Decidability Invariant-Free Clausal Temporal ResolutionIntroduction PLTL is decidableto TemporalLogic PSPACE-completeTheTemporalLogic PLTL Key issue in every deduction method for PLTLClausalResolution Given a set of formulas Φ and an eventuality ψ, how tofor PLTLClausal detect whether or not Φ contains a “hidden” invariant thatNormal Form prevents the satisfaction of ψ?Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  46. 46. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  47. 47. Clausal Resolution for PLTL Invariant-Free Clausal Temporal Resolution Fisher’s Clausal Temporal Resolution for PLTL:Introduction Clauses are in the so-called Separated Normal Form.to TemporalLogic Requires invariant generation for solving eventualities.The Invariant generation is carried out by means of anTemporalLogic PLTL algorithm based on graph search.ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  48. 48. Clausal Resolution for PLTL Invariant-Free Clausal Temporal Resolution Fisher’s Clausal Temporal Resolution for PLTL:Introduction Clauses are in the so-called Separated Normal Form.to TemporalLogic Requires invariant generation for solving eventualities.The Invariant generation is carried out by means of anTemporalLogic PLTL algorithm based on graph search.ClausalResolutionfor PLTL Our Clausal Temporal Resolution for PLTL:Clausal Different clausal normal form.Normal FormInvariant- New rule for solving eventualities ( U )FreeTemporal that does not require invariant generation.Resolution Invariant-Free Clausal Temporal Resolution
  49. 49. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  50. 50. Clausal Normal Form Invariant-Free Clausal Temporal Propositional literals P ::= p | ¬p ResolutionIntroduction Temporal literals T ::= P1 U P2 | P1 R P2 | P | Pto TemporalLogicThe Literals L ::= ◦i P | ◦i T for i ∈ I NTemporalLogic PLTLClausal Now-clauses N ::= ⊥ | L ∨ NResolutionfor PLTLClausal Clauses C ::= N | NNormal FormInvariant- Always-clausesFreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  51. 51. Transformation into Clausal Normal Form Invariant-Free Clausal Temporal Resolution PLTL-formula ϕ → Translation → CNF(ϕ) Conjunction of clausesIntroductionto TemporalLogic Set of clausesTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  52. 52. Transformation into Clausal Normal Form Invariant-Free Clausal Temporal Resolution PLTL-formula ϕ → Translation → CNF(ϕ) Conjunction of clausesIntroductionto TemporalLogic Set of clausesTheTemporalLogic PLTL a U ¬r,ClausalResolution (¬a ∨ p),for PLTL ((p ∧ q) U ¬r) ∧ ¬◦(p ∨ q) → (¬a ∨ q),ClausalNormal Form ◦¬p,Invariant- ◦¬qFreeTemporal New propositional variables.Resolution Satisfiability is preserved. Invariant-Free Clausal Temporal Resolution
  53. 53. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto Temporal 1 Introduction to Temporal LogicLogic 2 The Temporal Logic PLTLTheTemporalLogic PLTL 3 Clausal Resolution for PLTLClausal 4 Clausal Normal FormResolutionfor PLTL 5 Invariant-Free Temporal ResolutionClausalNormal Form 6 Ongoing and Future WorkInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  54. 54. Resolution Procedure Invariant-Free Clausal Temporal Derivation Resolution A derivation D for a set of clauses Γ is a sequenceIntroductionto TemporalLogic Γ0 → Γ1 → . . . → Γi → . . .TheTemporal whereLogic PLTLClausal Γ0 = ΓResolution andfor PLTLClausal Γi is obtained from Γi−1 by applying some of the rulesNormal Form for every i ≥ 1Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  55. 55. Resolution Procedure Invariant-Free Clausal Temporal Derivation Resolution A derivation D for a set of clauses Γ is a sequenceIntroductionto TemporalLogic Γ0 → Γ1 → . . . → Γi → . . .TheTemporal whereLogic PLTLClausal Γ0 = ΓResolution andfor PLTLClausal Γi is obtained from Γi−1 by applying some of the rulesNormal Form for every i ≥ 1Invariant-FreeTemporalResolution Refutation If D contains the empty clause, then D is a refutation for Γ. Invariant-Free Clausal Temporal Resolution
  56. 56. Our Rules Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogic Clasical-like RulesThe Resolution ruleTemporal Subsumption ruleLogic PLTLClausal Temporal RulesResolutionfor PLTL Temporal decomposition rulesClausal The unnext rule.Normal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  57. 57. Resolution Rule Invariant-Free Clausal Temporal b (L ∨ N) b (L ∨ N ) Resolution (Res) where b, b ∈ {0, 1} b×b (N ∨ N )Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  58. 58. Resolution Rule Invariant-Free Clausal Temporal b (L ∨ N) b (L ∨ N ) Resolution (Res) where b, b ∈ {0, 1} b×b (N ∨ N )Introductionto TemporalLogicThe Complement of a literal:TemporalLogic PLTLClausal p = ¬p ¬p = pResolutionfor PLTLClausal ◦L = ◦LNormal FormInvariant-FreeTemporal P1 U P2 = P1 R P2 P1 R P2 = P1 U P2Resolution P= P P= P Invariant-Free Clausal Temporal Resolution
  59. 59. Subsumption Rule Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogic (Sbm) { b N, bN } −→ { bN } if N ⊆ NTheTemporalLogic PLTLClausal Required for completeness unlike in classical propositionalResolutionfor PLTL logic.ClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  60. 60. Temporal Decomposition Rules Invariant-Free Clausal Temporal Resolution The usual inductive decomposition rule for the connective UIntroductionto TemporalLogicThe pU q ∨ N −→Inductive def. (q ∨ (p ∧ ◦(p U q))) ∨ N ≡TemporalLogic PLTL Original clauseClausal −→Distribution ((q ∨ p) ∧ (q ∨ ◦(p U q))) ∨ N ≡Resolutionfor PLTLClausalNormal Form −→Distribution (q ∨ p ∨ N)∧(q ∨ ◦(p U q) ∨ N)Invariant- Two new clausesFreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  61. 61. Temporal Decomposition Rules Invariant-Free Clausal Temporal Usual inductive definition of U Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  62. 62. Temporal Decomposition Rules Invariant-Free Clausal Temporal Usual inductive definition of U Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  63. 63. Temporal Decomposition Rules Invariant-Free Clausal Usual inductive definition of U Temporal Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  64. 64. Temporal Decomposition Rules Invariant-Free Clausal Usual inductive definition of U Temporal Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution New context-based rule for the connective U ∆ ∪ {ϕ U ψ} −→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆) U ψ) )} Invariant-Free Clausal Temporal Resolution
  65. 65. Temporal Decomposition Rules Invariant-Free Clausal Temporal Resolution New context-based rule for the connective U ∆ ∪ {p U q ∨ N} −→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆) U q)) ∨ N}Introductionto TemporalLogic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧TheTemporalLogic PLTL CNF( (a → (p ∧ ¬∆)))ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  66. 66. Temporal Decomposition Rules Invariant-Free Clausal Temporal Resolution New context-based rule for the connective U ∆ ∪ {p U q ∨ N} −→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆) U q)) ∨ N}Introductionto TemporalLogic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧TheTemporalLogic PLTL CNF( (a → (p ∧ ¬∆)))ClausalResolutionfor PLTLClausal p ∧ ¬∆ is not a propositional literal:Normal Form New propositional variable for replacing p ∧ ¬∆Invariant-FreeTemporal New clauses to define the meaning of the new variableResolution Always-clauses in ∆ are excluded from ¬∆ Invariant-Free Clausal Temporal Resolution
  67. 67. The unnext rule Invariant-Free Clausal Temporal Resolution (unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ}Introductionto Temporal ∪ { N | N ∈ Γ}Logic where b ∈ {0, 1}TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  68. 68. The unnext rule Invariant-Free Clausal Temporal Resolution (unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ}Introductionto Temporal ∪ { N | N ∈ Γ}Logic where b ∈ {0, 1}TheTemporalLogic PLTL ExampleClausalResolutionfor PLTLClausal {p ∨ ◦q, (◦◦x ∨ ◦w), ◦t, (◦r ∨ s)} −→Normal FormInvariant-Free { ◦x ∨ w, t, (◦◦x ∨ ◦w), (◦r ∨ s)}TemporalResolution Invariant-Free Clausal Temporal Resolution
  69. 69. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p}Free Clausal Temporal ResolutionIntroductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  70. 70. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal ResolutionIntroductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  71. 71. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (¬a ∨ p), (¬a ∨ ¬p)}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  72. 72. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  73. 73. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p),to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  74. 74. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  75. 75. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p),Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  76. 76. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  77. 77. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p),for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  78. 78. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  79. 79. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a}Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  80. 80. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  81. 81. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)Invariant-FreeTemporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p}Resolution Invariant-Free Clausal Temporal Resolution
  82. 82. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)Invariant-FreeTemporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res)Resolution Invariant-Free Clausal Temporal Resolution
  83. 83. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}TheTemporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)Invariant-FreeTemporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res)Resolution Γ7 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p, ◦(a U ¬p)} Invariant-Free Clausal Temporal Resolution
  84. 84. Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)}Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res)to TemporalLogic (¬a ∨ p), (¬a ∨ ¬p)}The Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)TemporalLogic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}ClausalResolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)for PLTL (¬a ∨ ¬p), ¬a}ClausalNormal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)Invariant-FreeTemporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res)Resolution Γ7 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p, (Sbm) ◦(a U ¬p)} Invariant-Free Clausal Temporal Resolution
  85. 85. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)}Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  86. 86. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  87. 87. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p}TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  88. 88. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  89. 89. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  90. 90. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  91. 91. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}Normal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  92. 92. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}Normal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  93. 93. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}Normal FormInvariant- Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ }TemporalResolution Invariant-Free Clausal Temporal Resolution
  94. 94. Example Invariant-Free Clausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)Introductionto TemporalLogic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)TheTemporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}ClausalResolutionfor PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}Normal FormInvariant- Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ }TemporalResolution Invariant-Free Clausal Temporal Resolution
  95. 95. Systematic resolution: Decision procedure Invariant-Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable.Introductionto TemporalLogicTheTemporalLogic PLTLClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  96. 96. Systematic resolution: Decision procedure Invariant-Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable.Introductionto TemporalLogicThe Refutational completeness: If Γ is unsatisfiable thenTemporalLogic PLTL there exists a systematic refutation for Γ.ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  97. 97. Systematic resolution: Decision procedure Invariant-Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable.Introductionto TemporalLogicThe Refutational completeness: If Γ is unsatisfiable thenTemporalLogic PLTL there exists a systematic refutation for Γ.ClausalResolutionfor PLTL Completeness: If Γ is satisfiable then there exists aClausal systematic cyclic derivation for Γ that yields aNormal Form model for Γ.Invariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  98. 98. Systematic resolution: Decision procedure Invariant-Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable.Introductionto TemporalLogicThe Refutational completeness: If Γ is unsatisfiable thenTemporalLogic PLTL there exists a systematic refutation for Γ.ClausalResolutionfor PLTL Completeness: If Γ is satisfiable then there exists aClausal systematic cyclic derivation for Γ that yields aNormal Form model for Γ.Invariant-FreeTemporalResolution Resolution-based decision procedure for PLTL Invariant-Free Clausal Temporal Resolution
  99. 99. Systematic Resolution Invariant-Free Clausal Temporal unnext: only when no other rule can be applied. Resolution New rule for U : only to one selected eventuality betweenIntroductionto Temporal two consecutive applications of unnext.LogicTheTemporal New rule for U : applied just after unnext.Logic PLTLClausalResolution The usual rule is applied to the other eventualities.for PLTLClausalNormal Form The selection process of eventualities must be fair.Invariant-FreeTemporal The new eventualities generated by the new rule for UResolution have priority for being selected. Invariant-Free Clausal Temporal Resolution
  100. 100. Systematic resolution: Termination Invariant-Free Clausal Temporal Resolution Eventualities and definitions generated from p U q pU qIntroduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 )))to TemporalLogic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 )))The ... Finite sequence?TemporalLogic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 )))ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  101. 101. Systematic resolution: Termination Invariant-Free Clausal Temporal Resolution Eventualities and definitions generated from p U q pU qIntroduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 )))to TemporalLogic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 )))The ... Finite sequence?TemporalLogic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 )))ClausalResolutionfor PLTL Always-clauses: not in the negation of the context.Clausal The new variables a1 , a2 , . . . only appear inNormal Form always-clauses.Invariant-Free The number of possible contexts is always finite.TemporalResolution Repetition of contexts produces a refutation. Invariant-Free Clausal Temporal Resolution
  102. 102. Outline of the presentation Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogic 1 Introduction to Temporal LogicTheTemporal 2 The Temporal Logic PLTLLogic PLTLClausal 3 Invariant-Free Clausal Temporal ResolutionResolutionfor PLTL 4 Ongoing and Future WorkClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution
  103. 103. Ongoing and Future Work Invariant-Free Clausal Temporal Resolution Implementation (from preliminary prototypes to ...) Tableau system:Introductionto Temporal http://www.sc.ehu.es/jiwlucap/TTM.htmlLogic Resolution method:The http://www.sc.ehu.es/jiwlucap/TRS.htmlTemporalLogic PLTL TeDiLog: Resolution-based Declarative Temporal LogicClausalResolution Programming Language (to appear)for PLTLClausal Application to CTL (Full Computation Tree Logic)Normal Form Decidable fragments of First-Order Linear-timeInvariant-Free Temporal Logic (FLTL)TemporalResolution etc. Invariant-Free Clausal Temporal Resolution
  104. 104. Invariant-Free Clausal Temporal ResolutionIntroductionto TemporalLogicTheTemporalLogic PLTL Thank you!ClausalResolutionfor PLTLClausalNormal FormInvariant-FreeTemporalResolution Invariant-Free Clausal Temporal Resolution

×