Download - 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



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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



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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



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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



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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



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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



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



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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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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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Temporal Logic: Example



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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))



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Temporal Logic: Verification

Does the system satisfy this property?

� ∀X(error(X) → ¬new job for(X)U ¬error(X))

System specification


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.



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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



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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



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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



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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



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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



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



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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



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PLTL: minimal language

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

p

◦p

qU p



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p

◦p

qU p



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p

◦p

qU p



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p

◦p

qU p



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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.



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PLTL: Model Theory





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PLTL: Model Theory





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PLTL: Model Theory

The connective ◦ (“next”)

〈M, sj〉 |= ◦ϕ iff 〈M, sj+1〉 |= ϕ

〈M, sj〉 |= ◦p



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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



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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



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PLTL: Model Theory


Logical consequence





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PLTL: Model Theory


Logical consequence





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PLTL: Defined Connectives

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

〈M, sj〉 |= � p

The connective � (“always”)�ϕ ≡ ¬�¬ϕ

〈M, sj〉 |= � p



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

〈M, sj〉 |= � p

The connective � (“always”)�ϕ ≡ ¬�¬ϕ

〈M, sj〉 |= � p



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The connective R (“release”)

ϕRψ ≡ ¬(¬ϕU ¬ψ)

〈M, sj〉 |= qR p

Either

or



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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



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ϕU ψ, �ψ



of formulas:�ψ





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ϕU ψ, �ψ



of formulas:�ψ





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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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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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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.



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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.



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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



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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.



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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.



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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.



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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.



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Our Rules

Clasical-like RulesResolution ruleSubsumption rule

Temporal RulesTemporal decomposition rulesThe unnext rule.



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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



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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



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Subsumption Rule

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

Required for completeness unlike in classical propositionallogic.



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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



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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))}



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Usual inductive definition of U{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ))}




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Usual inductive definition of U

{ϕU ψ} 7−→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕU ψ) )}




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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) )}



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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 ¬∆



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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 ¬∆



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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)}



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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)}



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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,}



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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,}



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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,}



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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,}



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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,}



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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,}



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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)


(Res)




TemporalResolution




ClausalNormal Form


Example


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


(Res)


(Res)




TemporalResolution




ClausalNormal Form


Example


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


(Res)


(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









TemporalResolution




ClausalNormal Form









TemporalResolution




ClausalNormal Form









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


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!
