invariant-free clausal temporal resolution

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



TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Clausal Resolution for PLTL4 Clausal Normal Form5 Invariant-Free Temporal Resolution6 Ongoing and Future Work



TemporalResolution




ClausalNormal Form


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.



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ClausalNormal Form


Temporal Logic: Example



TemporalResolution




ClausalNormal Form


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))



TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form






TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form



Different versions of Temporal Logic:Linear versus branchingUnbounded versus boundedDiscrete versus densePoint-based versus interval-basedOnly-future versus past-and-futurePropositional versus first-order



TemporalResolution




ClausalNormal Form



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



TemporalResolution




ClausalNormal Form


PLTL: minimal language

Atomic propositions: p, q, r, . . .Classical connectives: ¬,∧ (“not”, “and”)Temporal connectives: ◦, U (“next”, “until”)

p

◦p

qU p



TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form


PLTL: Model Theory

The connective ◦ (“next”)

〈M, sj〉 |= ◦ϕ iff 〈M, sj+1〉 |= ϕ

〈M, sj〉 |= ◦p



TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form


PLTL: Defined Connectives

The connective � (“eventually” or “some time”)�ϕ ≡ TU ϕ

〈M, sj〉 |= � p

The connective � (“always”)�ϕ ≡ ¬�¬ϕ

〈M, sj〉 |= � p



TemporalResolution




ClausalNormal Form


PLTL: Defined Connectives

The connective R (“release”)

ϕRψ ≡ ¬(¬ϕU ¬ψ)

〈M, sj〉 |= qR p

Either

or



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ClausalNormal Form


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



TemporalResolution




ClausalNormal Form


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?



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ClausalNormal Form






TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form






TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form






TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form


Our Rules

Clasical-like RulesResolution ruleSubsumption rule

Temporal RulesTemporal decomposition rulesThe unnext rule.



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ClausalNormal Form


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



TemporalResolution




ClausalNormal Form


Subsumption Rule

(Sbm) {� bN,� bN′} 7−→ {� bN′} if N′ ⊆ N

Required for completeness unlike in classical propositionallogic.



TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form



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))}



TemporalResolution




ClausalNormal Form



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))}



TemporalResolution




ClausalNormal Form



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) )}



TemporalResolution




ClausalNormal Form



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 ¬∆



TemporalResolution




ClausalNormal Form


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)}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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,}



TemporalResolution




ClausalNormal Form


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)}



TemporalResolution




ClausalNormal Form


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)



TemporalResolution




ClausalNormal Form


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, ⊥ }



TemporalResolution




ClausalNormal Form


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), , }




TemporalResolution




ClausalNormal Form


Example


s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p}

Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }

Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }




TemporalResolution




ClausalNormal Form


Example


s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)

Γ10 = {� (¬p ∨ ◦p),�¬a, , ,, , }

Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,,� (¬b ∨ a), , }




TemporalResolution




ClausalNormal Form


Example


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), }




TemporalResolution




ClausalNormal Form


Example


s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)


(Res)

Γ11 = {� (¬p ∨ ◦p), , p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), }




TemporalResolution




ClausalNormal Form


Example


s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)


(Res)

Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}




TemporalResolution




ClausalNormal Form


Example


s1 Γ9 = {� (¬p ∨ ◦p),�¬a, p, aU ¬p} (U Set)


(Res)

Γ11 = {� (¬p ∨ ◦p),�¬a, p,¬p ∨ a,¬p ∨ ◦(bU ¬p),� (¬b ∨ a),� (¬b ∨ ¬p), a}

(Res)




TemporalResolution




ClausalNormal Form


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



TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form



1 Introduction to Temporal Logic2 The Temporal Logic PLTL3 Invariant-Free Clausal Temporal Resolution4 Ongoing and Future Work



TemporalResolution




ClausalNormal Form


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.



TemporalResolution




ClausalNormal Form


Thank you!


invariant-free clausal temporal resolution

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