Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Invariant-Free Clausal Temporal Resolution
J. Gaintzarain, M. Hermo, P. Lucio, M. Navarro, F. Orejas
to appear in Journal of Automated Reasoning(Online from December 2th, 2011)
PROLE 2012, September 19th
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic
2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL
3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL
4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form
5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution
6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic
� Significant role in Computer Science.
� Useful for specification and verification of dynamic systems
� Robotics� Agent-Based Systems� Control Systems� Dynamic Databases� etc.
� Also important in other fields: Philosophy, Mathematics,Linguistics, Social Sciences, Systems Biology, etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic
� Significant role in Computer Science.
� Useful for specification and verification of dynamic systems
� Robotics� Agent-Based Systems� Control Systems� Dynamic Databases� etc.
� Also important in other fields: Philosophy, Mathematics,Linguistics, Social Sciences, Systems Biology, etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Example
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: A printer will eventually end its job or produce an error� ∀X(printing(X) → ◦� (available(X) ∨ error(X))
3: A non-available printer will not receive a new job until itbecomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: A printer will eventually end its job or produce an error� ∀X(printing(X) → ◦� (available(X) ∨ error(X))
3: A non-available printer will not receive a new job until itbecomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: A printer will eventually end its job or produce an error� ∀X(printing(X) → ◦� (available(X) ∨ error(X))
3: A non-available printer will not receive a new job until itbecomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Verification
Does the system satisfy this property?
� ∀X(error(X) → ¬new job for(X)U ¬error(X))
System specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: . . .3: A non-available printer will not receive a new job until it
becomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Deductive verification methodsTableaux, Sequent calculi, Resolution, etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Verification
Does the system satisfy this property?
� ∀X(error(X) → ¬new job for(X)U ¬error(X))
System specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: . . .3: A non-available printer will not receive a new job until it
becomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Deductive verification methodsTableaux, Sequent calculi, Resolution, etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Logic: Verification
Does the system satisfy this property?
� ∀X(error(X) → ¬new job for(X)U ¬error(X))
System specification
1: Being in error means being neither available nor printing� ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X))
2: . . .3: A non-available printer will not receive a new job until it
becomes available� ∀X(¬available(X) → ¬new job for(X)U available(X))
Deductive verification methodsTableaux, Sequent calculi, Resolution, etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branching
Unbounded versus boundedDiscrete versus densePoint-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus bounded
Discrete versus densePoint-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus dense
Point-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus densePoint-based versus interval-based
Only-future versus past-and-futurePropositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus densePoint-based versus interval-basedOnly-future versus past-and-future
Propositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus densePoint-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The Temporal Logic PLTL
Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus densePoint-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order
PLTLPropositional Linear-time Temporal Logic
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: minimal language
Atomic propositions: p, q, r, . . .Classical connectives: ¬,∧ (“not”, “and”)Temporal connectives: ◦, U (“next”, “until”)
p
◦p
qU p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: minimal language
Atomic propositions: p, q, r, . . .Classical connectives: ¬,∧ (“not”, “and”)Temporal connectives: ◦, U (“next”, “until”)
p
◦p
qU p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: minimal language
Atomic propositions: p, q, r, . . .Classical connectives: ¬,∧ (“not”, “and”)Temporal connectives: ◦, U (“next”, “until”)
p
◦p
qU p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: minimal language
Atomic propositions: p, q, r, . . .Classical connectives: ¬,∧ (“not”, “and”)Temporal connectives: ◦, U (“next”, “until”)
p
◦p
qU p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
� PLTL-structure: M = (SM,VM)-SM: denumerable sequence of states s0, s1, s2, . . .-VM: SM → 2Prop where Prop is the set of all the possibleatomic propositions.
� 〈M, sj〉 |= ϕ denotes that the formula ϕ is true in the statesj of M.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
� PLTL-structure: M = (SM,VM)-SM: denumerable sequence of states s0, s1, s2, . . .-VM: SM → 2Prop where Prop is the set of all the possibleatomic propositions.
� 〈M, sj〉 |= ϕ denotes that the formula ϕ is true in the statesj of M.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
� PLTL-structure: M = (SM,VM)-SM: denumerable sequence of states s0, s1, s2, . . .-VM: SM → 2Prop where Prop is the set of all the possibleatomic propositions.
� 〈M, sj〉 |= ϕ denotes that the formula ϕ is true in the statesj of M.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
The connective ◦ (“next”)
〈M, sj〉 |= ◦ϕ iff 〈M, sj+1〉 |= ϕ
〈M, sj〉 |= ◦p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
The connective U (“until”)
〈M, sj〉 |= ϕU ψ iff 〈M, sk〉 |= ψ for some k ≥ j and〈M, si〉 |= ϕ for every i ∈ {j, . . . , k − 1}
〈M, sj〉 |= pU q
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
ModelM |= ψ iff 〈M, s0〉 |= ψ
Logical consequence
Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM:if 〈M, sj〉 |= Φ then 〈M, sj〉 |= ψ
Satisfiabilityψ is satisfiable iff there exists a model of ψ
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
ModelM |= ψ iff 〈M, s0〉 |= ψ
Logical consequence
Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM:if 〈M, sj〉 |= Φ then 〈M, sj〉 |= ψ
Satisfiabilityψ is satisfiable iff there exists a model of ψ
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Model Theory
ModelM |= ψ iff 〈M, s0〉 |= ψ
Logical consequence
Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM:if 〈M, sj〉 |= Φ then 〈M, sj〉 |= ψ
Satisfiabilityψ is satisfiable iff there exists a model of ψ
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Defined Connectives
The connective � (“eventually” or “some time”)�ϕ ≡ TU ϕ
〈M, sj〉 |= � p
The connective � (“always”)�ϕ ≡ ¬�¬ϕ
〈M, sj〉 |= � p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Defined Connectives
The connective � (“eventually” or “some time”)�ϕ ≡ TU ϕ
〈M, sj〉 |= � p
The connective � (“always”)�ϕ ≡ ¬�¬ϕ
〈M, sj〉 |= � p
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Defined Connectives
The connective R (“release”)
ϕRψ ≡ ¬(¬ϕU ¬ψ)
〈M, sj〉 |= qR p
Either
or
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Eventualities and Invariants
Eventualities� They assert that a formula will some time become true� They are expressed by means of specific connectives:
ϕU ψ, �ψ
Invariants� They assert that a formula is always true from some moment
onwards� They are often expressed in an intricate way by means of sets
of formulas:�ψ
{ψ,� (ψ → ◦ψ)} �ψ is a logical consequence{ψ,� (ψ → ◦ϕ),� (ϕ→ ψ)} �ψ is a logical consequence
� Usually, their syntactic detection is not trivial: “hidden” invariants
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Eventualities and Invariants
Eventualities� They assert that a formula will some time become true� They are expressed by means of specific connectives:
ϕU ψ, �ψ
Invariants� They assert that a formula is always true from some moment
onwards� They are often expressed in an intricate way by means of sets
of formulas:�ψ
{ψ,� (ψ → ◦ψ)} �ψ is a logical consequence{ψ,� (ψ → ◦ϕ),� (ϕ→ ψ)} �ψ is a logical consequence
� Usually, their syntactic detection is not trivial: “hidden” invariants
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Eventualities and Invariants
Eventualities� They assert that a formula will some time become true� They are expressed by means of specific connectives:
ϕU ψ, �ψ
Invariants� They assert that a formula is always true from some moment
onwards� They are often expressed in an intricate way by means of sets
of formulas:�ψ
{ψ,� (ψ → ◦ψ)} �ψ is a logical consequence{ψ,� (ψ → ◦ϕ),� (ϕ→ ψ)} �ψ is a logical consequence
� Usually, their syntactic detection is not trivial: “hidden” invariants
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Decidability
PLTL is decidablePSPACE-complete
Key issue in every deduction method for PLTLGiven a set of formulas Φ and an eventuality ψ, how todetect whether or not Φ contains a “hidden” invariant thatprevents the satisfaction of ψ?
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
PLTL: Decidability
PLTL is decidablePSPACE-complete
Key issue in every deduction method for PLTLGiven a set of formulas Φ and an eventuality ψ, how todetect whether or not Φ contains a “hidden” invariant thatprevents the satisfaction of ψ?
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Clausal Resolution for PLTL
� Fisher’s Clausal Temporal Resolution for PLTL:� Clauses are in the so-called Separated Normal Form.� Requires invariant generation for solving eventualities.� Invariant generation is carried out by means of an
algorithm based on graph search.
� Our Clausal Temporal Resolution for PLTL:� Different clausal normal form.� New rule for solving eventualities (U )
that does not require invariant generation.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Clausal Resolution for PLTL
� Fisher’s Clausal Temporal Resolution for PLTL:� Clauses are in the so-called Separated Normal Form.� Requires invariant generation for solving eventualities.� Invariant generation is carried out by means of an
algorithm based on graph search.
� Our Clausal Temporal Resolution for PLTL:� Different clausal normal form.� New rule for solving eventualities (U )
that does not require invariant generation.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Clausal Normal Form
Propositional literals P ::= p | ¬p
Temporal literals T ::= P1 U P2 | P1 RP2 | �P | � P
Literals L ::= ◦iP | ◦iT for i ∈ IN
Now-clauses N ::= ⊥ | L ∨ N
Clauses C ::= N | � N︸ ︷︷ ︸Always-clauses
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Transformation into Clausal Normal Form
PLTL-formula ϕ → Translation → CNF(ϕ)Conjunction of clauses
9Set of clauses
((p ∧ q)U ¬r) ∧ ¬◦(p ∨ q) →
aU ¬r,� (¬a ∨ p),� (¬a ∨ q),◦¬p,◦¬q
New propositional variables.Satisfiability is preserved.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Transformation into Clausal Normal Form
PLTL-formula ϕ → Translation → CNF(ϕ)Conjunction of clauses
9Set of clauses
((p ∧ q)U ¬r) ∧ ¬◦(p ∨ q) →
aU ¬r,� (¬a ∨ p),� (¬a ∨ q),◦¬p,◦¬q
New propositional variables.Satisfiability is preserved.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Resolution Procedure
DerivationA derivation D for a set of clauses Γ is a sequence
Γ0 7→ Γ1 7→ . . . 7→ Γi 7→ . . .
whereΓ0 = Γ
andΓi is obtained from Γi−1 by applying some of the rulesfor every i ≥ 1
RefutationIf D contains the empty clause, then D is a refutation for Γ.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Resolution Procedure
DerivationA derivation D for a set of clauses Γ is a sequence
Γ0 7→ Γ1 7→ . . . 7→ Γi 7→ . . .
whereΓ0 = Γ
andΓi is obtained from Γi−1 by applying some of the rulesfor every i ≥ 1
RefutationIf D contains the empty clause, then D is a refutation for Γ.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Our Rules
Clasical-like RulesResolution ruleSubsumption rule
Temporal RulesTemporal decomposition rulesThe unnext rule.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Resolution Rule
(Res)� b(L ∨ N) � b′
(L̃ ∨ N′)
� b×b′(N ∨ N′)
where b, b′ ∈ {0, 1}
Complement of a literal:
p̃ = ¬p ¬̃p = p
◦̃L = ◦L̃
P̃1 U P2 = P̃1 R P̃2 P̃1 RP2 = P̃1 U P̃2
�̃P = � P̃ �̃ P = � P̃
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Resolution Rule
(Res)� b(L ∨ N) � b′
(L̃ ∨ N′)
� b×b′(N ∨ N′)
where b, b′ ∈ {0, 1}
Complement of a literal:
p̃ = ¬p ¬̃p = p
◦̃L = ◦L̃
P̃1 U P2 = P̃1 R P̃2 P̃1 RP2 = P̃1 U P̃2
�̃P = � P̃ �̃ P = � P̃
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Subsumption Rule
(Sbm) {� bN,� bN′} 7−→ {� bN′} if N′ ⊆ N
Required for completeness unlike in classical propositionallogic.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
The usual inductive decomposition rule for the connective U
pU q ∨ N 7−→Inductive def. (q ∨ (p ∧ ◦(pU q))) ∨ N ≡︸ ︷︷ ︸Original clause
7−→Distribution ((q ∨ p) ∧ (q ∨ ◦(pU q))) ∨ N ≡
7−→Distribution (q ∨ p ∨ N)∧(q ∨ ◦(pU q) ∨ N)︸ ︷︷ ︸ ︸ ︷︷ ︸Two new clauses
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
Usual inductive definition of U{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ))}
New context-based rule for the connective U∆ ∪ {ϕU ψ} 7−→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆)U ψ))}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
Usual inductive definition of U{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ))}
New context-based rule for the connective U∆ ∪ {ϕU ψ} 7−→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆)U ψ))}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
Usual inductive definition of U
{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ) )}
New context-based rule for the connective U∆ ∪ {ϕU ψ} 7−→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆)U ψ))}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
Usual inductive definition of U
{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ) )}
New context-based rule for the connective U
∆ ∪ {ϕU ψ} 7−→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆)U ψ) )}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
New context-based rule for the connective U∆ ∪ {pU q ∨ N} 7−→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆)U q)) ∨ N}
7−→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(aU q) ∨ N)∧
CNF(� (a → (p ∧ ¬∆)))
p ∧ ¬∆ is not a propositional literal:New propositional variable for replacing p ∧ ¬∆New clauses to define the meaning of the new variableAlways-clauses in ∆ are excluded from ¬∆
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Temporal Decomposition Rules
New context-based rule for the connective U∆ ∪ {pU q ∨ N} 7−→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆)U q)) ∨ N}
7−→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(aU q) ∨ N)∧
CNF(� (a → (p ∧ ¬∆)))
p ∧ ¬∆ is not a propositional literal:New propositional variable for replacing p ∧ ¬∆New clauses to define the meaning of the new variableAlways-clauses in ∆ are excluded from ¬∆
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The unnext rule
(unnext) Γ 7−→ {L0 ∨ · · · ∨ Ln | � b(◦L0 ∨ · · · ∨ ◦Ln) ∈ Γ}∪ {� N | � N ∈ Γ}
where b ∈ {0, 1}
Example
{p ∨ ◦q,� (◦◦x ∨ ◦w), ◦t,� (◦r ∨ s)} 7−→
{ ◦x ∨ w, t,� (◦◦x ∨ ◦w),� (◦r ∨ s)}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
The unnext rule
(unnext) Γ 7−→ {L0 ∨ · · · ∨ Ln | � b(◦L0 ∨ · · · ∨ ◦Ln) ∈ Γ}∪ {� N | � N ∈ Γ}
where b ∈ {0, 1}
Example
{p ∨ ◦q,� (◦◦x ∨ ◦w), ◦t,� (◦r ∨ s)} 7−→
{ ◦x ∨ w, t,� (◦◦x ∨ ◦w),� (◦r ∨ s)}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p}
Γ1 = {,� (¬p ∨ ◦p), , ,, }
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {,� (¬p ∨ ◦p), , ,, }
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),,� (¬a ∨ ¬p), }
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),, }
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {, ,¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a}
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a} (Res)
Γ6 = {,� (¬p ∨ ◦p), ,�¬a, }
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a} (Res)
Γ6 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p}
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a} (Res)
Γ6 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p} (Res)
Γ7 = {p,� (¬p ∨ ◦p), ,�¬a,◦p,}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a} (Res)
Γ6 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p} (Res)
Γ7 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p,◦(aU ¬p)}
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
s0 Γ0 = {p,� (¬p ∨ ◦p), pU ¬p} (U Set)
Γ1 = {p,� (¬p ∨ ◦p),¬p ∨ p,¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Sbm)
Γ2 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p)}
(Res)
Γ3 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ4 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),� (¬a ∨ ¬p),�¬a}
(Sbm)
Γ5 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a} (Res)
Γ6 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p} (Res)
Γ7 = {p,� (¬p ∨ ◦p),¬p ∨ ◦(aU ¬p),�¬a,◦p,◦(aU ¬p)}
(Sbm)
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)}
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, }
Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, }
Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p}
Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
(Res)
Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), }
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
(Res)
Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
(Res)
Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}
(Res)
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
(Res)
Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}
(Res)
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Example
Γ8 = {p,� (¬p ∨ ◦p),�¬a,◦p,◦(aU ¬p)} (unnext)
s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)
Γ10 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p)}
(Res)
Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}
(Res)
Γ12 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a, ⊥ }
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Decision procedure
� Soundness: If a refutation is obtained for Γ then Γis unsatisfiable.
� Refutational completeness: If Γ is unsatisfiable thenthere exists a systematic refutation for Γ.
� Completeness: If Γ is satisfiable then there exists asystematic cyclic derivation for Γ that yields amodel for Γ.
Resolution-based decision procedure for PLTL
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Decision procedure
� Soundness: If a refutation is obtained for Γ then Γis unsatisfiable.
� Refutational completeness: If Γ is unsatisfiable thenthere exists a systematic refutation for Γ.
� Completeness: If Γ is satisfiable then there exists asystematic cyclic derivation for Γ that yields amodel for Γ.
Resolution-based decision procedure for PLTL
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Decision procedure
� Soundness: If a refutation is obtained for Γ then Γis unsatisfiable.
� Refutational completeness: If Γ is unsatisfiable thenthere exists a systematic refutation for Γ.
� Completeness: If Γ is satisfiable then there exists asystematic cyclic derivation for Γ that yields amodel for Γ.
Resolution-based decision procedure for PLTL
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Decision procedure
� Soundness: If a refutation is obtained for Γ then Γis unsatisfiable.
� Refutational completeness: If Γ is unsatisfiable thenthere exists a systematic refutation for Γ.
� Completeness: If Γ is satisfiable then there exists asystematic cyclic derivation for Γ that yields amodel for Γ.
Resolution-based decision procedure for PLTL
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic Resolution
� unnext: only when no other rule can be applied.
� New rule for U : only to one selected eventuality betweentwo consecutive applications of unnext.
� New rule for U : applied just after unnext.
� The usual rule is applied to the other eventualities.
� The selection process of eventualities must be fair.
� The new eventualities generated by the new rule for Uhave priority for being selected.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Termination
Eventualities and definitions generated from pU q
pU qa1 U q,CNF(� (a1 → (p ∧ ¬∆0)))a2 U q,CNF(� (a2 → (a1 ∧ ¬∆1))). . . Finite sequence?aj U q,CNF(� (aj → (aj−1 ∧ ¬∆j−1)))
� Always-clauses: not in the negation of the context.� The new variables a1, a2, . . . only appear inalways-clauses.� The number of possible contexts is always finite.� Repetition of contexts produces a refutation.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Systematic resolution: Termination
Eventualities and definitions generated from pU q
pU qa1 U q,CNF(� (a1 → (p ∧ ¬∆0)))a2 U q,CNF(� (a2 → (a1 ∧ ¬∆1))). . . Finite sequence?aj U q,CNF(� (aj → (aj−1 ∧ ¬∆j−1)))
� Always-clauses: not in the negation of the context.� The new variables a1, a2, . . . only appear inalways-clauses.� The number of possible contexts is always finite.� Repetition of contexts produces a refutation.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Outline of the presentation
1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Invariant-Free Clausal Temporal Resolution4 Ongoing and Future Work
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Ongoing and Future Work
Implementation (from preliminary prototypes to ...)Tableau system:http://www.sc.ehu.es/jiwlucap/TTM.html
Resolution method:http://www.sc.ehu.es/jiwlucap/TRS.html
TeDiLog: Resolution-based Declarative Temporal LogicProgramming Language (to appear)Application to CTL? (Full Computation Tree Logic)Decidable fragments of First-Order Linear-timeTemporal Logic (FLTL)etc.
Invariant-Free Clausal Temporal Resolution
Invariant-Free Clausal
TemporalResolution
Introductionto TemporalLogic
TheTemporalLogic PLTL
ClausalResolutionfor PLTL
ClausalNormal Form
Invariant-FreeTemporalResolution
Thank you!
Invariant-Free Clausal Temporal Resolution
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