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

1. Invariant-
Free Clausal
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
J. Gaintzarain, M. Hermo, P. Lucio, M. Navarro, F. Orejas
Resolution
for PLTL
Clausal
to appear in Journal of Automated Reasoning
Normal Form (Online from December 2th, 2011)
Invariant-
Free
Temporal PROLE 2012, September 19th
Resolution
Invariant-Free Clausal Temporal Resolution

2. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

3. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

4. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

5. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

6. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

7. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

8. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

9. Temporal Logic
Invariant-
Free Clausal
Temporal
Resolution
Significant role in Computer Science.
Introduction
to Temporal Useful for specification and verification of dynamic systems
Logic
Robotics
The
Temporal Agent-Based Systems
Logic PLTL
Clausal
Control Systems
Resolution Dynamic Databases
for PLTL
Clausal
etc.
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

10. Temporal Logic
Invariant-
Free Clausal
Temporal
Resolution
Significant role in Computer Science.
Introduction
to Temporal Useful for specification and verification of dynamic systems
Logic
Robotics
The
Temporal Agent-Based Systems
Logic PLTL
Clausal
Control Systems
Resolution Dynamic Databases
for PLTL
Clausal
etc.
Normal Form
Invariant- Also important in other fields: Philosophy, Mathematics,
Free
Temporal
Linguistics, Social Sciences, Systems Biology, etc.
Resolution
Invariant-Free Clausal Temporal Resolution

11. Temporal Logic: Example
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

12. Temporal Logic: Specification
Invariant-
Free Clausal
Temporal
Resolution
Introduction
1: Being in error means being neither available nor printing
to Temporal
Logic
∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

13. Temporal Logic: Specification
Invariant-
Free Clausal
Temporal
Resolution
Introduction
1: Being in error means being neither available nor printing
to Temporal
Logic
∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
The
Temporal
2: A printer will eventually end its job or produce an error
Logic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X))
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

14. Temporal Logic: Specification
Invariant-
Free Clausal
Temporal
Resolution
Introduction
1: Being in error means being neither available nor printing
to Temporal
Logic
∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
The
Temporal
2: A printer will eventually end its job or produce an error
Logic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X))
Clausal
Resolution
for PLTL 3: A non-available printer will not receive a new job until it
Clausal becomes available
Normal Form
∀X(¬available(X) → ¬new job for(X) U available(X))
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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))
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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))
Introduction
to Temporal System specification
Logic
The 1: Being in error means being neither available nor printing
Temporal
Logic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
Clausal 2: . . .
Resolution
for PLTL 3: A non-available printer will not receive a new job until it
Clausal
Normal Form
becomes available
Invariant-
∀X(¬available(X) → ¬new job for(X) U available(X))
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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))
Introduction
to Temporal System specification
Logic
The 1: Being in error means being neither available nor printing
Temporal
Logic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
Clausal 2: . . .
Resolution
for PLTL 3: A non-available printer will not receive a new job until it
Clausal
Normal Form
becomes available
Invariant-
∀X(¬available(X) → ¬new job for(X) U available(X))
Free
Temporal
Resolution Deductive verification methods
Tableaux, Sequent calculi, Resolution, etc.
Invariant-Free Clausal Temporal Resolution

18. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

19. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

20. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The Unbounded versus bounded
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

21. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The Unbounded versus bounded
Temporal
Logic PLTL Discrete versus dense
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

22. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The Unbounded versus bounded
Temporal
Logic PLTL Discrete versus dense
Clausal
Resolution Point-based versus interval-based
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

23. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The Unbounded versus bounded
Temporal
Logic PLTL Discrete versus dense
Clausal
Resolution Point-based versus interval-based
for PLTL
Clausal Only-future versus past-and-future
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

24. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution
Different versions of Temporal Logic:
Introduction
to Temporal
Logic
Linear versus branching
The Unbounded versus bounded
Temporal
Logic PLTL Discrete versus dense
Clausal
Resolution Point-based versus interval-based
for PLTL
Clausal Only-future versus past-and-future
Normal Form
Propositional versus first-order
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

25. The Temporal Logic PLTL
Invariant-
Free Clausal
Temporal
Resolution Different versions of Temporal Logic:
Linear versus branching
Introduction
to Temporal
Logic
Unbounded versus bounded
The Discrete versus dense
Temporal
Logic PLTL Point-based versus interval-based
Clausal
Resolution Only-future versus past-and-future
for PLTL
Clausal
Propositional versus first-order
Normal Form
Invariant- PLTL
Free
Temporal
Resolution
Propositional Linear-time Temporal Logic
Invariant-Free Clausal Temporal Resolution

26. PLTL: minimal language
Invariant-
Free Clausal Atomic propositions: p, q, r, . . .
Temporal
Resolution Classical connectives: ¬, ∧ (“not”, “and”)
Temporal connectives: ◦, U (“next”, “until”)
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

27. PLTL: minimal language
Invariant-
Free Clausal Atomic propositions: p, q, r, . . .
Temporal
Resolution Classical connectives: ¬, ∧ (“not”, “and”)
Temporal connectives: ◦, U (“next”, “until”)
Introduction
to Temporal
p
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

28. PLTL: minimal language
Invariant-
Free Clausal Atomic propositions: p, q, r, . . .
Temporal
Resolution Classical connectives: ¬, ∧ (“not”, “and”)
Temporal connectives: ◦, U (“next”, “until”)
Introduction
to Temporal
p
Logic
The
Temporal
Logic PLTL
Clausal ◦p
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

29. PLTL: minimal language
Invariant-
Free Clausal Atomic propositions: p, q, r, . . .
Temporal
Resolution Classical connectives: ¬, ∧ (“not”, “and”)
Temporal connectives: ◦, U (“next”, “until”)
Introduction
to Temporal
p
Logic
The
Temporal
Logic PLTL
Clausal ◦p
Resolution
for PLTL
Clausal
Normal Form
Invariant- qU p
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

30. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution PLTL-structure: M = (SM , VM )
-SM : denumerable sequence of states s0 , s1 , s2 , . . .
Introduction
to Temporal -VM : SM → 2Prop where Prop is the set of all the possible
Logic
atomic propositions.
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

31. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution PLTL-structure: M = (SM , VM )
-SM : denumerable sequence of states s0 , s1 , s2 , . . .
Introduction
to Temporal -VM : SM → 2Prop where Prop is the set of all the possible
Logic
atomic propositions.
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

32. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution PLTL-structure: M = (SM , VM )
-SM : denumerable sequence of states s0 , s1 , s2 , . . .
Introduction
to Temporal -VM : SM → 2Prop where Prop is the set of all the possible
Logic
atomic propositions.
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free M, sj |= ϕ denotes that the formula ϕ is true in the state
Temporal
Resolution sj of M.
Invariant-Free Clausal Temporal Resolution

33. PLTL: Model Theory
Invariant-
Free Clausal The connective ◦ (“next”)
Temporal
Resolution
M, sj |= ◦ϕ iff M, sj+1 |= ϕ
Introduction
to Temporal
Logic
The
Temporal M, sj |= ◦p
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
M, sj |= p U q
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

35. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution
Model
Introduction
to Temporal
M |= ψ iff M, s0 |= ψ
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

36. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution
Model
Introduction
to Temporal
M |= ψ iff M, s0 |= ψ
Logic
The
Temporal Logical consequence
Logic PLTL
Clausal
Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM :
Resolution
for PLTL
if M, sj |= Φ then M, sj |= ψ
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

37. PLTL: Model Theory
Invariant-
Free Clausal
Temporal
Resolution
Model
Introduction
to Temporal
M |= ψ iff M, s0 |= ψ
Logic
The
Temporal Logical consequence
Logic PLTL
Clausal
Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM :
Resolution
for PLTL
if M, sj |= Φ then M, sj |= ψ
Clausal
Normal Form
Satisfiability
Invariant-
Free
Temporal
ψ is satisfiable iff there exists a model of ψ
Resolution
Invariant-Free Clausal Temporal Resolution

38. PLTL: Defined Connectives
Invariant-
Free Clausal
Temporal The connective (“eventually” or “some time”)
Resolution
ϕ ≡ TU ϕ
Introduction
to Temporal M, sj |= p
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

39. PLTL: Defined Connectives
Invariant-
Free Clausal
Temporal The connective (“eventually” or “some time”)
Resolution
ϕ ≡ TU ϕ
Introduction
to Temporal M, sj |= p
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL The connective (“always”)
Clausal
Normal Form
ϕ ≡ ¬ ¬ϕ
Invariant-
Free M, sj |= p
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

40. PLTL: Defined Connectives
Invariant-
Free Clausal
Temporal
Resolution
The connective R (“release”)
ϕ R ψ ≡ ¬(¬ϕ U ¬ψ)
Introduction
to Temporal
Logic M, sj |= q R p
The
Temporal Either
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form or
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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:
Introduction
to Temporal ϕ U ψ, ψ
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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:
Introduction
to Temporal ϕ U ψ, ψ
Logic
The
Temporal
Invariants
Logic PLTL
They assert that a formula is always true from some moment
Clausal
Resolution onwards
for PLTL
They are often expressed in an intricate way by means of sets
Clausal
Normal Form of formulas:
Invariant- ψ
Free
Temporal {ψ, (ψ → ◦ψ)} ψ is a logical consequence
Resolution
{ψ, (ψ → ◦ϕ), (ϕ → ψ)} ψ is a logical consequence
Invariant-Free Clausal Temporal Resolution

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:
Introduction
to Temporal ϕ U ψ, ψ
Logic
The
Temporal
Invariants
Logic PLTL
They assert that a formula is always true from some moment
Clausal
Resolution onwards
for PLTL
They are often expressed in an intricate way by means of sets
Clausal
Normal Form of formulas:
Invariant- ψ
Free
Temporal {ψ, (ψ → ◦ψ)} ψ is a logical consequence
Resolution
{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. PLTL: Decidability
Invariant-
Free Clausal
Temporal
Resolution
Introduction PLTL is decidable
to Temporal
Logic PSPACE-complete
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

45. PLTL: Decidability
Invariant-
Free Clausal
Temporal
Resolution
Introduction PLTL is decidable
to Temporal
Logic PSPACE-complete
The
Temporal
Logic PLTL
Key issue in every deduction method for PLTL
Clausal
Resolution Given a set of formulas Φ and an eventuality ψ, how to
for PLTL
Clausal
detect whether or not Φ contains a “hidden” invariant that
Normal Form prevents the satisfaction of ψ?
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

46. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic Requires invariant generation for solving eventualities.
The Invariant generation is carried out by means of an
Temporal
Logic PLTL algorithm based on graph search.
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic Requires invariant generation for solving eventualities.
The Invariant generation is carried out by means of an
Temporal
Logic PLTL algorithm based on graph search.
Clausal
Resolution
for PLTL Our Clausal Temporal Resolution for PLTL:
Clausal Different clausal normal form.
Normal Form
Invariant-
New rule for solving eventualities ( U )
Free
Temporal
that does not require invariant generation.
Resolution
Invariant-Free Clausal Temporal Resolution

49. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

50. Clausal Normal Form
Invariant-
Free Clausal
Temporal Propositional literals P ::= p | ¬p
Resolution
Introduction
Temporal literals T ::= P1 U P2 | P1 R P2 | P | P
to Temporal
Logic
The
Literals L ::= ◦i P | ◦i T for i ∈ I
N
Temporal
Logic PLTL
Clausal Now-clauses N ::= ⊥ | L ∨ N
Resolution
for PLTL
Clausal Clauses C ::= N | N
Normal Form
Invariant- Always-clauses
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

51. Transformation into Clausal Normal Form
Invariant-
Free Clausal
Temporal
Resolution
PLTL-formula ϕ → Translation → CNF(ϕ)
Conjunction of clauses
Introduction
to Temporal
Logic Set of clauses
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

52. Transformation into Clausal Normal Form
Invariant-
Free Clausal
Temporal
Resolution
PLTL-formula ϕ → Translation → CNF(ϕ)
Conjunction of clauses
Introduction
to Temporal
Logic Set of clauses
The
Temporal
Logic PLTL a U ¬r,
Clausal
Resolution
(¬a ∨ p),
for PLTL
((p ∧ q) U ¬r) ∧ ¬◦(p ∨ q) → (¬a ∨ q),
Clausal
Normal Form ◦¬p,
Invariant- ◦¬q
Free
Temporal New propositional variables.
Resolution
Satisfiability is preserved.
Invariant-Free Clausal Temporal Resolution

53. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
1 Introduction to Temporal Logic
Logic
2 The Temporal Logic PLTL
The
Temporal
Logic PLTL
3 Clausal Resolution for PLTL
Clausal 4 Clausal Normal Form
Resolution
for PLTL 5 Invariant-Free Temporal Resolution
Clausal
Normal Form 6 Ongoing and Future Work
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

54. Resolution Procedure
Invariant-
Free Clausal
Temporal Derivation
Resolution
A derivation D for a set of clauses Γ is a sequence
Introduction
to Temporal
Logic
Γ0 → Γ1 → . . . → Γi → . . .
The
Temporal where
Logic PLTL
Clausal
Γ0 = Γ
Resolution and
for PLTL
Clausal
Γi is obtained from Γi−1 by applying some of the rules
Normal Form for every i ≥ 1
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

55. Resolution Procedure
Invariant-
Free Clausal
Temporal Derivation
Resolution
A derivation D for a set of clauses Γ is a sequence
Introduction
to Temporal
Logic
Γ0 → Γ1 → . . . → Γi → . . .
The
Temporal where
Logic PLTL
Clausal
Γ0 = Γ
Resolution and
for PLTL
Clausal
Γi is obtained from Γi−1 by applying some of the rules
Normal Form for every i ≥ 1
Invariant-
Free
Temporal
Resolution Refutation
If D contains the empty clause, then D is a refutation for Γ.
Invariant-Free Clausal Temporal Resolution

56. Our Rules
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
Clasical-like Rules
The
Resolution rule
Temporal Subsumption rule
Logic PLTL
Clausal
Temporal Rules
Resolution
for PLTL
Temporal decomposition rules
Clausal
The unnext rule.
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

57. Resolution Rule
Invariant-
Free Clausal
Temporal b (L ∨ N) b (L ∨ N )
Resolution (Res) where b, b ∈ {0, 1}
b×b (N ∨ N )
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

58. Resolution Rule
Invariant-
Free Clausal
Temporal b (L ∨ N) b (L ∨ N )
Resolution (Res) where b, b ∈ {0, 1}
b×b (N ∨ N )
Introduction
to Temporal
Logic
The
Complement of a literal:
Temporal
Logic PLTL
Clausal p = ¬p ¬p = p
Resolution
for PLTL
Clausal ◦L = ◦L
Normal Form
Invariant-
Free
Temporal
P1 U P2 = P1 R P2 P1 R P2 = P1 U P2
Resolution
P= P P= P
Invariant-Free Clausal Temporal Resolution

59. Subsumption Rule
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
(Sbm) { b N, bN } −→ { bN } if N ⊆ N
The
Temporal
Logic PLTL
Clausal Required for completeness unlike in classical propositional
Resolution
for PLTL logic.
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

60. Temporal Decomposition Rules
Invariant-
Free Clausal
Temporal
Resolution
The usual inductive decomposition rule for the connective U
Introduction
to Temporal
Logic
The pU q ∨ N −→Inductive def. (q ∨ (p ∧ ◦(p U q))) ∨ N ≡
Temporal
Logic PLTL Original clause
Clausal −→Distribution ((q ∨ p) ∧ (q ∨ ◦(p U q))) ∨ N ≡
Resolution
for PLTL
Clausal
Normal Form
−→Distribution (q ∨ p ∨ N)∧(q ∨ ◦(p U q) ∨ N)
Invariant- Two new clauses
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

61. Temporal Decomposition Rules
Invariant-
Free Clausal
Temporal
Usual inductive definition of U
Resolution
{ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

62. Temporal Decomposition Rules
Invariant-
Free Clausal
Temporal
Usual inductive definition of U
Resolution
{ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

63. Temporal Decomposition Rules
Invariant-
Free Clausal Usual inductive definition of U
Temporal
Resolution
{ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

64. Temporal Decomposition Rules
Invariant-
Free Clausal Usual inductive definition of U
Temporal
Resolution
{ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
New context-based rule for the connective U
∆ ∪ {ϕ U ψ} −→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆) U ψ) )}
Invariant-Free Clausal Temporal Resolution

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}
Introduction
to Temporal
Logic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧
The
Temporal
Logic PLTL
CNF( (a → (p ∧ ¬∆)))
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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}
Introduction
to Temporal
Logic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧
The
Temporal
Logic PLTL
CNF( (a → (p ∧ ¬∆)))
Clausal
Resolution
for PLTL
Clausal
p ∧ ¬∆ is not a propositional literal:
Normal Form
New propositional variable for replacing p ∧ ¬∆
Invariant-
Free
Temporal
New clauses to define the meaning of the new variable
Resolution
Always-clauses in ∆ are excluded from ¬∆
Invariant-Free Clausal Temporal Resolution

67. The unnext rule
Invariant-
Free Clausal
Temporal
Resolution
(unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ}
Introduction
to Temporal
∪ { N | N ∈ Γ}
Logic where b ∈ {0, 1}
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

68. The unnext rule
Invariant-
Free Clausal
Temporal
Resolution
(unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ}
Introduction
to Temporal
∪ { N | N ∈ Γ}
Logic where b ∈ {0, 1}
The
Temporal
Logic PLTL
Example
Clausal
Resolution
for PLTL
Clausal {p ∨ ◦q, (◦◦x ∨ ◦w), ◦t, (◦r ∨ s)} −→
Normal Form
Invariant-
Free { ◦x ∨ w, t, (◦◦x ∨ ◦w), (◦r ∨ s)}
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

69. Example
Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p}
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

70. Example
Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set)
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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)}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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)}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic (¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic (¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic (¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p),
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p),
for PLTL
(¬a ∨ ¬p), ¬a}
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL
(¬a ∨ ¬p), ¬a}
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL
(¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a}
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL (¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL (¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)
Invariant-
Free
Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p}
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL (¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)
Invariant-
Free
Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res)
Resolution
Invariant-Free Clausal Temporal Resolution

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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The
Temporal
Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL (¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)
Invariant-
Free
Temporal 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. 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 Temporal
Logic
(¬a ∨ p), (¬a ∨ ¬p)}
The Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
Temporal
Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a}
Clausal
Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm)
for PLTL (¬a ∨ ¬p), ¬a}
Clausal
Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res)
Invariant-
Free
Temporal 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. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)}
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

86. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

87. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p}
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

88. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

89. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

90. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

91. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,
Clausal
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

92. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Clausal
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

93. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Clausal
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}
Normal Form
Invariant-
Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,
Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ }
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

94. Example
Invariant-
Free Clausal
Temporal
Resolution
Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext)
Introduction
to Temporal
Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set)
The
Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Logic PLTL
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)}
Clausal
Resolution
for PLTL
Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res)
Clausal
¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a}
Normal Form
Invariant-
Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a,
Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ }
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

95. Systematic resolution: Decision procedure
Invariant-
Free Clausal
Temporal
Resolution
Soundness: If a refutation is obtained for Γ then Γ
is unsatisfiable.
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

96. Systematic resolution: Decision procedure
Invariant-
Free Clausal
Temporal
Resolution
Soundness: If a refutation is obtained for Γ then Γ
is unsatisfiable.
Introduction
to Temporal
Logic
The
Refutational completeness: If Γ is unsatisfiable then
Temporal
Logic PLTL
there exists a systematic refutation for Γ.
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

97. Systematic resolution: Decision procedure
Invariant-
Free Clausal
Temporal
Resolution
Soundness: If a refutation is obtained for Γ then Γ
is unsatisfiable.
Introduction
to Temporal
Logic
The
Refutational completeness: If Γ is unsatisfiable then
Temporal
Logic PLTL
there exists a systematic refutation for Γ.
Clausal
Resolution
for PLTL
Completeness: If Γ is satisfiable then there exists a
Clausal
systematic cyclic derivation for Γ that yields a
Normal Form
model for Γ.
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

98. Systematic resolution: Decision procedure
Invariant-
Free Clausal
Temporal
Resolution
Soundness: If a refutation is obtained for Γ then Γ
is unsatisfiable.
Introduction
to Temporal
Logic
The
Refutational completeness: If Γ is unsatisfiable then
Temporal
Logic PLTL
there exists a systematic refutation for Γ.
Clausal
Resolution
for PLTL
Completeness: If Γ is satisfiable then there exists a
Clausal
systematic cyclic derivation for Γ that yields a
Normal Form
model for Γ.
Invariant-
Free
Temporal
Resolution Resolution-based decision procedure for PLTL
Invariant-Free Clausal Temporal Resolution

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 between
Introduction
to Temporal
two consecutive applications of unnext.
Logic
The
Temporal New rule for U : applied just after unnext.
Logic PLTL
Clausal
Resolution The usual rule is applied to the other eventualities.
for PLTL
Clausal
Normal Form The selection process of eventualities must be fair.
Invariant-
Free
Temporal The new eventualities generated by the new rule for U
Resolution
have priority for being selected.
Invariant-Free Clausal Temporal Resolution

100. Systematic resolution: Termination
Invariant-
Free Clausal
Temporal
Resolution
Eventualities and definitions generated from p U q
pU q
Introduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 )))
to Temporal
Logic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 )))
The ... Finite sequence?
Temporal
Logic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 )))
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

101. Systematic resolution: Termination
Invariant-
Free Clausal
Temporal
Resolution
Eventualities and definitions generated from p U q
pU q
Introduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 )))
to Temporal
Logic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 )))
The ... Finite sequence?
Temporal
Logic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 )))
Clausal
Resolution
for PLTL Always-clauses: not in the negation of the context.
Clausal The new variables a1 , a2 , . . . only appear in
Normal Form
always-clauses.
Invariant-
Free The number of possible contexts is always finite.
Temporal
Resolution Repetition of contexts produces a refutation.
Invariant-Free Clausal Temporal Resolution

102. Outline of the presentation
Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic 1 Introduction to Temporal Logic
The
Temporal 2 The Temporal Logic PLTL
Logic PLTL
Clausal
3 Invariant-Free Clausal Temporal Resolution
Resolution
for PLTL 4 Ongoing and Future Work
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution

103. Ongoing and Future Work
Invariant-
Free Clausal
Temporal
Resolution
Implementation (from preliminary prototypes to ...)
Tableau system:
Introduction
to Temporal
http://www.sc.ehu.es/jiwlucap/TTM.html
Logic Resolution method:
The http://www.sc.ehu.es/jiwlucap/TRS.html
Temporal
Logic PLTL
TeDiLog: Resolution-based Declarative Temporal Logic
Clausal
Resolution Programming Language (to appear)
for PLTL
Clausal
Application to CTL (Full Computation Tree Logic)
Normal Form
Decidable fragments of First-Order Linear-time
Invariant-
Free Temporal Logic (FLTL)
Temporal
Resolution etc.
Invariant-Free Clausal Temporal Resolution

104. Invariant-
Free Clausal
Temporal
Resolution
Introduction
to Temporal
Logic
The
Temporal
Logic PLTL
Thank you!
Clausal
Resolution
for PLTL
Clausal
Normal Form
Invariant-
Free
Temporal
Resolution
Invariant-Free Clausal Temporal Resolution