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Invariant-Free Clausal Temporal Resolution

The presentation outlines an approach for invariant-free clausal temporal resolution. It introduces temporal logic and its role in modeling dynamic systems. The temporal logic PLTL is described, as well as existing techniques for clausal resolution and clausal normal forms. The presentation proposes an invariant-free approach to temporal resolution and discusses ongoing and future work.

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

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Invariant-Free Clausal Temporal Resolution

  • 1.
    Invariant- Free Clausal Temporal Resolution Invariant-Free Clausal Temporal Resolution Introduction to Temporal Logic The Temporal Logic PLTL Clausal J. Gaintzarain, M. Hermo, P. Lucio, M. Navarro, F. Orejas Resolution for PLTL Clausal to appear in Journal of Automated Reasoning Normal Form (Online from December 2th, 2011) Invariant- Free Temporal PROLE 2012, September 19th Resolution Invariant-Free Clausal Temporal Resolution
  • 2.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 3.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 4.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 5.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 6.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 7.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 8.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 9.
    Temporal Logic Invariant- Free Clausal Temporal Resolution Significant role in Computer Science. Introduction to Temporal Useful for specification and verification of dynamic systems Logic Robotics The Temporal Agent-Based Systems Logic PLTL Clausal Control Systems Resolution Dynamic Databases for PLTL Clausal etc. Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 10.
    Temporal Logic Invariant- Free Clausal Temporal Resolution Significant role in Computer Science. Introduction to Temporal Useful for specification and verification of dynamic systems Logic Robotics The Temporal Agent-Based Systems Logic PLTL Clausal Control Systems Resolution Dynamic Databases for PLTL Clausal etc. Normal Form Invariant- Also important in other fields: Philosophy, Mathematics, Free Temporal Linguistics, Social Sciences, Systems Biology, etc. Resolution Invariant-Free Clausal Temporal Resolution
  • 11.
    Temporal Logic: Example Invariant- Free Clausal Temporal Resolution Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 12.
    Temporal Logic: Specification Invariant- Free Clausal Temporal Resolution Introduction 1: Being in error means being neither available nor printing to Temporal Logic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X)) The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 13.
    Temporal Logic: Specification Invariant- Free Clausal Temporal Resolution Introduction 1: Being in error means being neither available nor printing to Temporal Logic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X)) The Temporal 2: A printer will eventually end its job or produce an error Logic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X)) Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 14.
    Temporal Logic: Specification Invariant- Free Clausal Temporal Resolution Introduction 1: Being in error means being neither available nor printing to Temporal Logic ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X)) The Temporal 2: A printer will eventually end its job or produce an error Logic PLTL ∀X(printing(X) → ◦ (available(X) ∨ error(X)) Clausal Resolution for PLTL 3: A non-available printer will not receive a new job until it Clausal becomes available Normal Form ∀X(¬available(X) → ¬new job for(X) U available(X)) Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 15.
    Temporal Logic: Verification Invariant- Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X)) Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 16.
    Temporal Logic: Verification Invariant- Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X)) Introduction to Temporal System specification Logic The 1: Being in error means being neither available nor printing Temporal Logic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X)) Clausal 2: . . . Resolution for PLTL 3: A non-available printer will not receive a new job until it Clausal Normal Form becomes available Invariant- ∀X(¬available(X) → ¬new job for(X) U available(X)) Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 17.
    Temporal Logic: Verification Invariant- Free Clausal Temporal Does the system satisfy this property? Resolution ∀X(error(X) → ¬new job for(X) U ¬error(X)) Introduction to Temporal System specification Logic The 1: Being in error means being neither available nor printing Temporal Logic PLTL ∀X(error(X) ↔ ¬available(X) ∧ ¬printing(X)) Clausal 2: . . . Resolution for PLTL 3: A non-available printer will not receive a new job until it Clausal Normal Form becomes available Invariant- ∀X(¬available(X) → ¬new job for(X) U available(X)) Free Temporal Resolution Deductive verification methods Tableaux, Sequent calculi, Resolution, etc. Invariant-Free Clausal Temporal Resolution
  • 18.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 19.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 20.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Unbounded versus bounded Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 21.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Unbounded versus bounded Temporal Logic PLTL Discrete versus dense Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 22.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Unbounded versus bounded Temporal Logic PLTL Discrete versus dense Clausal Resolution Point-based versus interval-based for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 23.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Unbounded versus bounded Temporal Logic PLTL Discrete versus dense Clausal Resolution Point-based versus interval-based for PLTL Clausal Only-future versus past-and-future Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 24.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Introduction to Temporal Logic Linear versus branching The Unbounded versus bounded Temporal Logic PLTL Discrete versus dense Clausal Resolution Point-based versus interval-based for PLTL Clausal Only-future versus past-and-future Normal Form Propositional versus first-order Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 25.
    The Temporal LogicPLTL Invariant- Free Clausal Temporal Resolution Different versions of Temporal Logic: Linear versus branching Introduction to Temporal Logic Unbounded versus bounded The Discrete versus dense Temporal Logic PLTL Point-based versus interval-based Clausal Resolution Only-future versus past-and-future for PLTL Clausal Propositional versus first-order Normal Form Invariant- PLTL Free Temporal Resolution Propositional Linear-time Temporal Logic Invariant-Free Clausal Temporal Resolution
  • 26.
    PLTL: minimal language Invariant- Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”) Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 27.
    PLTL: minimal language Invariant- Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”) Introduction to Temporal p Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 28.
    PLTL: minimal language Invariant- Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”) Introduction to Temporal p Logic The Temporal Logic PLTL Clausal ◦p Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 29.
    PLTL: minimal language Invariant- Free Clausal Atomic propositions: p, q, r, . . . Temporal Resolution Classical connectives: ¬, ∧ (“not”, “and”) Temporal connectives: ◦, U (“next”, “until”) Introduction to Temporal p Logic The Temporal Logic PLTL Clausal ◦p Resolution for PLTL Clausal Normal Form Invariant- qU p Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 30.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . . Introduction to Temporal -VM : SM → 2Prop where Prop is the set of all the possible Logic atomic propositions. The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 31.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . . Introduction to Temporal -VM : SM → 2Prop where Prop is the set of all the possible Logic atomic propositions. The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 32.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution PLTL-structure: M = (SM , VM ) -SM : denumerable sequence of states s0 , s1 , s2 , . . . Introduction to Temporal -VM : SM → 2Prop where Prop is the set of all the possible Logic atomic propositions. The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free M, sj |= ϕ denotes that the formula ϕ is true in the state Temporal Resolution sj of M. Invariant-Free Clausal Temporal Resolution
  • 33.
    PLTL: Model Theory Invariant- Free Clausal The connective ◦ (“next”) Temporal Resolution M, sj |= ◦ϕ iff M, sj+1 |= ϕ Introduction to Temporal Logic The Temporal M, sj |= ◦p Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 34.
    PLTL: Model Theory Invariant- Free Clausal The connective U (“until”) Temporal Resolution M, sj |= ϕ U ψ iff M, sk |= ψ for some k ≥ j and M, si |= ϕ for every i ∈ {j, . . . , k − 1} Introduction to Temporal Logic The Temporal Logic PLTL M, sj |= p U q Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 35.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution Model Introduction to Temporal M |= ψ iff M, s0 |= ψ Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 36.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution Model Introduction to Temporal M |= ψ iff M, s0 |= ψ Logic The Temporal Logical consequence Logic PLTL Clausal Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM : Resolution for PLTL if M, sj |= Φ then M, sj |= ψ Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 37.
    PLTL: Model Theory Invariant- Free Clausal Temporal Resolution Model Introduction to Temporal M |= ψ iff M, s0 |= ψ Logic The Temporal Logical consequence Logic PLTL Clausal Φ |= ψ iff for every PLTL-structure M and every sj ∈ SM : Resolution for PLTL if M, sj |= Φ then M, sj |= ψ Clausal Normal Form Satisfiability Invariant- Free Temporal ψ is satisfiable iff there exists a model of ψ Resolution Invariant-Free Clausal Temporal Resolution
  • 38.
    PLTL: Defined Connectives Invariant- Free Clausal Temporal The connective (“eventually” or “some time”) Resolution ϕ ≡ TU ϕ Introduction to Temporal M, sj |= p Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 39.
    PLTL: Defined Connectives Invariant- Free Clausal Temporal The connective (“eventually” or “some time”) Resolution ϕ ≡ TU ϕ Introduction to Temporal M, sj |= p Logic The Temporal Logic PLTL Clausal Resolution for PLTL The connective (“always”) Clausal Normal Form ϕ ≡ ¬ ¬ϕ Invariant- Free M, sj |= p Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 40.
    PLTL: Defined Connectives Invariant- Free Clausal Temporal Resolution The connective R (“release”) ϕ R ψ ≡ ¬(¬ϕ U ¬ψ) Introduction to Temporal Logic M, sj |= q R p The Temporal Either Logic PLTL Clausal Resolution for PLTL Clausal Normal Form or Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 41.
    PLTL: Eventualities andInvariants Invariant- Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives: Introduction to Temporal ϕ U ψ, ψ Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 42.
    PLTL: Eventualities andInvariants Invariant- Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives: Introduction to Temporal ϕ U ψ, ψ Logic The Temporal Invariants Logic PLTL They assert that a formula is always true from some moment Clausal Resolution onwards for PLTL They are often expressed in an intricate way by means of sets Clausal Normal Form of formulas: Invariant- ψ Free Temporal {ψ, (ψ → ◦ψ)} ψ is a logical consequence Resolution {ψ, (ψ → ◦ϕ), (ϕ → ψ)} ψ is a logical consequence Invariant-Free Clausal Temporal Resolution
  • 43.
    PLTL: Eventualities andInvariants Invariant- Free Clausal Eventualities Temporal Resolution They assert that a formula will some time become true They are expressed by means of specific connectives: Introduction to Temporal ϕ U ψ, ψ Logic The Temporal Invariants Logic PLTL They assert that a formula is always true from some moment Clausal Resolution onwards for PLTL They are often expressed in an intricate way by means of sets Clausal Normal Form of formulas: Invariant- ψ Free Temporal {ψ, (ψ → ◦ψ)} ψ is a logical consequence Resolution {ψ, (ψ → ◦ϕ), (ϕ → ψ)} ψ is a logical consequence Usually, their syntactic detection is not trivial: “hidden” invariants Invariant-Free Clausal Temporal Resolution
  • 44.
    PLTL: Decidability Invariant- Free Clausal Temporal Resolution Introduction PLTL is decidable to Temporal Logic PSPACE-complete The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 45.
    PLTL: Decidability Invariant- Free Clausal Temporal Resolution Introduction PLTL is decidable to Temporal Logic PSPACE-complete The Temporal Logic PLTL Key issue in every deduction method for PLTL Clausal Resolution Given a set of formulas Φ and an eventuality ψ, how to for PLTL Clausal detect whether or not Φ contains a “hidden” invariant that Normal Form prevents the satisfaction of ψ? Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 46.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 47.
    Clausal Resolution forPLTL Invariant- Free Clausal Temporal Resolution Fisher’s Clausal Temporal Resolution for PLTL: Introduction Clauses are in the so-called Separated Normal Form. to Temporal Logic Requires invariant generation for solving eventualities. The Invariant generation is carried out by means of an Temporal Logic PLTL algorithm based on graph search. Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 48.
    Clausal Resolution forPLTL Invariant- Free Clausal Temporal Resolution Fisher’s Clausal Temporal Resolution for PLTL: Introduction Clauses are in the so-called Separated Normal Form. to Temporal Logic Requires invariant generation for solving eventualities. The Invariant generation is carried out by means of an Temporal Logic PLTL algorithm based on graph search. Clausal Resolution for PLTL Our Clausal Temporal Resolution for PLTL: Clausal Different clausal normal form. Normal Form Invariant- New rule for solving eventualities ( U ) Free Temporal that does not require invariant generation. Resolution Invariant-Free Clausal Temporal Resolution
  • 49.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 50.
    Clausal Normal Form Invariant- Free Clausal Temporal Propositional literals P ::= p | ¬p Resolution Introduction Temporal literals T ::= P1 U P2 | P1 R P2 | P | P to Temporal Logic The Literals L ::= ◦i P | ◦i T for i ∈ I N Temporal Logic PLTL Clausal Now-clauses N ::= ⊥ | L ∨ N Resolution for PLTL Clausal Clauses C ::= N | N Normal Form Invariant- Always-clauses Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 51.
    Transformation into ClausalNormal Form Invariant- Free Clausal Temporal Resolution PLTL-formula ϕ → Translation → CNF(ϕ) Conjunction of clauses Introduction to Temporal Logic Set of clauses The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 52.
    Transformation into ClausalNormal Form Invariant- Free Clausal Temporal Resolution PLTL-formula ϕ → Translation → CNF(ϕ) Conjunction of clauses Introduction to Temporal Logic Set of clauses The Temporal Logic PLTL a U ¬r, Clausal Resolution (¬a ∨ p), for PLTL ((p ∧ q) U ¬r) ∧ ¬◦(p ∨ q) → (¬a ∨ q), Clausal Normal Form ◦¬p, Invariant- ◦¬q Free Temporal New propositional variables. Resolution Satisfiability is preserved. Invariant-Free Clausal Temporal Resolution
  • 53.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal 1 Introduction to Temporal Logic Logic 2 The Temporal Logic PLTL The Temporal Logic PLTL 3 Clausal Resolution for PLTL Clausal 4 Clausal Normal Form Resolution for PLTL 5 Invariant-Free Temporal Resolution Clausal Normal Form 6 Ongoing and Future Work Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 54.
    Resolution Procedure Invariant- Free Clausal Temporal Derivation Resolution A derivation D for a set of clauses Γ is a sequence Introduction to Temporal Logic Γ0 → Γ1 → . . . → Γi → . . . The Temporal where Logic PLTL Clausal Γ0 = Γ Resolution and for PLTL Clausal Γi is obtained from Γi−1 by applying some of the rules Normal Form for every i ≥ 1 Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 55.
    Resolution Procedure Invariant- Free Clausal Temporal Derivation Resolution A derivation D for a set of clauses Γ is a sequence Introduction to Temporal Logic Γ0 → Γ1 → . . . → Γi → . . . The Temporal where Logic PLTL Clausal Γ0 = Γ Resolution and for PLTL Clausal Γi is obtained from Γi−1 by applying some of the rules Normal Form for every i ≥ 1 Invariant- Free Temporal Resolution Refutation If D contains the empty clause, then D is a refutation for Γ. Invariant-Free Clausal Temporal Resolution
  • 56.
    Our Rules Invariant- Free Clausal Temporal Resolution Introduction to Temporal Logic Clasical-like Rules The Resolution rule Temporal Subsumption rule Logic PLTL Clausal Temporal Rules Resolution for PLTL Temporal decomposition rules Clausal The unnext rule. Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 57.
    Resolution Rule Invariant- Free Clausal Temporal b (L ∨ N) b (L ∨ N ) Resolution (Res) where b, b ∈ {0, 1} b×b (N ∨ N ) Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 58.
    Resolution Rule Invariant- Free Clausal Temporal b (L ∨ N) b (L ∨ N ) Resolution (Res) where b, b ∈ {0, 1} b×b (N ∨ N ) Introduction to Temporal Logic The Complement of a literal: Temporal Logic PLTL Clausal p = ¬p ¬p = p Resolution for PLTL Clausal ◦L = ◦L Normal Form Invariant- Free Temporal P1 U P2 = P1 R P2 P1 R P2 = P1 U P2 Resolution P= P P= P Invariant-Free Clausal Temporal Resolution
  • 59.
    Subsumption Rule Invariant- Free Clausal Temporal Resolution Introduction to Temporal Logic (Sbm) { b N, bN } −→ { bN } if N ⊆ N The Temporal Logic PLTL Clausal Required for completeness unlike in classical propositional Resolution for PLTL logic. Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 60.
    Temporal Decomposition Rules Invariant- Free Clausal Temporal Resolution The usual inductive decomposition rule for the connective U Introduction to Temporal Logic The pU q ∨ N −→Inductive def. (q ∨ (p ∧ ◦(p U q))) ∨ N ≡ Temporal Logic PLTL Original clause Clausal −→Distribution ((q ∨ p) ∧ (q ∨ ◦(p U q))) ∨ N ≡ Resolution for PLTL Clausal Normal Form −→Distribution (q ∨ p ∨ N)∧(q ∨ ◦(p U q) ∨ N) Invariant- Two new clauses Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 61.
    Temporal Decomposition Rules Invariant- Free Clausal Temporal Usual inductive definition of U Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 62.
    Temporal Decomposition Rules Invariant- Free Clausal Temporal Usual inductive definition of U Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ))} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 63.
    Temporal Decomposition Rules Invariant- Free Clausal Usual inductive definition of U Temporal Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 64.
    Temporal Decomposition Rules Invariant- Free Clausal Usual inductive definition of U Temporal Resolution {ϕ U ψ} −→ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦(ϕ U ψ) )} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution New context-based rule for the connective U ∆ ∪ {ϕ U ψ} −→ ∆ ∪ {ψ ∨ (ϕ ∧ ¬ψ ∧ ◦((ϕ ∧ ¬∆) U ψ) )} Invariant-Free Clausal Temporal Resolution
  • 65.
    Temporal Decomposition Rules Invariant- Free Clausal Temporal Resolution New context-based rule for the connective U ∆ ∪ {p U q ∨ N} −→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆) U q)) ∨ N} Introduction to Temporal Logic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧ The Temporal Logic PLTL CNF( (a → (p ∧ ¬∆))) Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 66.
    Temporal Decomposition Rules Invariant- Free Clausal Temporal Resolution New context-based rule for the connective U ∆ ∪ {p U q ∨ N} −→ ∆ ∪ {q ∨ (p ∧ ◦((p ∧ ¬∆) U q)) ∨ N} Introduction to Temporal Logic −→ ∆ ∪ (q ∨ p ∨ N)∧(q ∨ ◦(a U q) ∨ N)∧ The Temporal Logic PLTL CNF( (a → (p ∧ ¬∆))) Clausal Resolution for PLTL Clausal p ∧ ¬∆ is not a propositional literal: Normal Form New propositional variable for replacing p ∧ ¬∆ Invariant- Free Temporal New clauses to define the meaning of the new variable Resolution Always-clauses in ∆ are excluded from ¬∆ Invariant-Free Clausal Temporal Resolution
  • 67.
    The unnext rule Invariant- Free Clausal Temporal Resolution (unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ} Introduction to Temporal ∪ { N | N ∈ Γ} Logic where b ∈ {0, 1} The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 68.
    The unnext rule Invariant- Free Clausal Temporal Resolution (unnext) Γ −→ {L0 ∨ · · · ∨ Ln | b (◦L0 ∨ · · · ∨ ◦Ln ) ∈ Γ} Introduction to Temporal ∪ { N | N ∈ Γ} Logic where b ∈ {0, 1} The Temporal Logic PLTL Example Clausal Resolution for PLTL Clausal {p ∨ ◦q, (◦◦x ∨ ◦w), ◦t, (◦r ∨ s)} −→ Normal Form Invariant- Free { ◦x ∨ w, t, (◦◦x ∨ ◦w), (◦r ∨ s)} Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 69.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} Free Clausal Temporal Resolution Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 70.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 71.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (¬a ∨ p), (¬a ∨ ¬p)} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 72.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 73.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 74.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 75.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Resolution Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 76.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 77.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 78.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 79.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 80.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res) Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 81.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res) Invariant- Free Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} Resolution Invariant-Free Clausal Temporal Resolution
  • 82.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res) Invariant- Free Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res) Resolution Invariant-Free Clausal Temporal Resolution
  • 83.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Temporal Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res) Invariant- Free Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res) Resolution Γ7 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p, ◦(a U ¬p)} Invariant-Free Clausal Temporal Resolution
  • 84.
    Example Invariant- s0 Γ0 = {p, (¬p ∨ ◦p), p U ¬p} ( U Set) Free Clausal Temporal Γ1 = {p, (¬p ∨ ◦p), ¬p ∨ p, ¬p ∨ ◦(a U ¬p), (Sbm) Resolution (¬a ∨ p), (¬a ∨ ¬p)} Introduction Γ2 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Res) to Temporal Logic (¬a ∨ p), (¬a ∨ ¬p)} The Γ3 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) Temporal Logic PLTL (¬a ∨ p), (¬a ∨ ¬p), ¬a} Clausal Resolution Γ4 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), (Sbm) for PLTL (¬a ∨ ¬p), ¬a} Clausal Normal Form Γ5 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a} (Res) Invariant- Free Temporal Γ6 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p} (Res) Resolution Γ7 = {p, (¬p ∨ ◦p), ¬p ∨ ◦(a U ¬p), ¬a, ◦p, (Sbm) ◦(a U ¬p)} Invariant-Free Clausal Temporal Resolution
  • 85.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 86.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 87.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 88.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 89.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 90.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 91.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a} Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 92.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a} Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 93.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a} Normal Form Invariant- Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ } Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 94.
    Example Invariant- FreeClausal Temporal Resolution Γ8 = {p, (¬p ∨ ◦p), ¬a, ◦p, ◦(a U ¬p)} (unnext) Introduction to Temporal Logic s1 Γ9 = { (¬p ∨ ◦p), ¬a, p, a U ¬p} ( U Set) The Temporal Γ10 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Logic PLTL ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p)} Clausal Resolution for PLTL Γ11 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, (Res) Clausal ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a} Normal Form Invariant- Γ12 = { (¬p ∨ ◦p), ¬a, p, ¬p ∨ a, Free ¬p ∨ ◦(b U ¬p), (¬b ∨ a), (¬b ∨ ¬p), a, ⊥ } Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 95.
    Systematic resolution: Decisionprocedure Invariant- Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable. Introduction to Temporal Logic The Temporal Logic PLTL Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 96.
    Systematic resolution: Decisionprocedure Invariant- Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable. Introduction to Temporal Logic The Refutational completeness: If Γ is unsatisfiable then Temporal Logic PLTL there exists a systematic refutation for Γ. Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 97.
    Systematic resolution: Decisionprocedure Invariant- Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable. Introduction to Temporal Logic The Refutational completeness: If Γ is unsatisfiable then Temporal Logic PLTL there exists a systematic refutation for Γ. Clausal Resolution for PLTL Completeness: If Γ is satisfiable then there exists a Clausal systematic cyclic derivation for Γ that yields a Normal Form model for Γ. Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 98.
    Systematic resolution: Decisionprocedure Invariant- Free Clausal Temporal Resolution Soundness: If a refutation is obtained for Γ then Γ is unsatisfiable. Introduction to Temporal Logic The Refutational completeness: If Γ is unsatisfiable then Temporal Logic PLTL there exists a systematic refutation for Γ. Clausal Resolution for PLTL Completeness: If Γ is satisfiable then there exists a Clausal systematic cyclic derivation for Γ that yields a Normal Form model for Γ. Invariant- Free Temporal Resolution Resolution-based decision procedure for PLTL Invariant-Free Clausal Temporal Resolution
  • 99.
    Systematic Resolution Invariant- Free Clausal Temporal unnext: only when no other rule can be applied. Resolution New rule for U : only to one selected eventuality between Introduction to Temporal two consecutive applications of unnext. Logic The Temporal New rule for U : applied just after unnext. Logic PLTL Clausal Resolution The usual rule is applied to the other eventualities. for PLTL Clausal Normal Form The selection process of eventualities must be fair. Invariant- Free Temporal The new eventualities generated by the new rule for U Resolution have priority for being selected. Invariant-Free Clausal Temporal Resolution
  • 100.
    Systematic resolution: Termination Invariant- Free Clausal Temporal Resolution Eventualities and definitions generated from p U q pU q Introduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 ))) to Temporal Logic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 ))) The ... Finite sequence? Temporal Logic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 ))) Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 101.
    Systematic resolution: Termination Invariant- Free Clausal Temporal Resolution Eventualities and definitions generated from p U q pU q Introduction a1 U q, CNF( (a1 → (p ∧ ¬∆0 ))) to Temporal Logic a2 U q, CNF( (a2 → (a1 ∧ ¬∆1 ))) The ... Finite sequence? Temporal Logic PLTL aj U q, CNF( (aj → (aj−1 ∧ ¬∆j−1 ))) Clausal Resolution for PLTL Always-clauses: not in the negation of the context. Clausal The new variables a1 , a2 , . . . only appear in Normal Form always-clauses. Invariant- Free The number of possible contexts is always finite. Temporal Resolution Repetition of contexts produces a refutation. Invariant-Free Clausal Temporal Resolution
  • 102.
    Outline of thepresentation Invariant- Free Clausal Temporal Resolution Introduction to Temporal Logic 1 Introduction to Temporal Logic The Temporal 2 The Temporal Logic PLTL Logic PLTL Clausal 3 Invariant-Free Clausal Temporal Resolution Resolution for PLTL 4 Ongoing and Future Work Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution
  • 103.
    Ongoing and FutureWork Invariant- Free Clausal Temporal Resolution Implementation (from preliminary prototypes to ...) Tableau system: Introduction to Temporal http://www.sc.ehu.es/jiwlucap/TTM.html Logic Resolution method: The http://www.sc.ehu.es/jiwlucap/TRS.html Temporal Logic PLTL TeDiLog: Resolution-based Declarative Temporal Logic Clausal Resolution Programming Language (to appear) for PLTL Clausal Application to CTL (Full Computation Tree Logic) Normal Form Decidable fragments of First-Order Linear-time Invariant- Free Temporal Logic (FLTL) Temporal Resolution etc. Invariant-Free Clausal Temporal Resolution
  • 104.
    Invariant- Free Clausal Temporal Resolution Introduction to Temporal Logic The Temporal Logic PLTL Thank you! Clausal Resolution for PLTL Clausal Normal Form Invariant- Free Temporal Resolution Invariant-Free Clausal Temporal Resolution