A Canonical Model Construction for Iteration-Free PDL with - PowerPoint PPT Presentation

A Canonical Model Construction for Iteration-Free PDL with Intersection Florian Bruse Daniel Kernberger Martin Lange University of Kassel, Germany September 22, 2016

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Canonical Models Tool to show completeness of proof calculus (for e.g., ML) Idea: • take set of maximally consistent sets of formulas (mcs) as underlying set of structure • atomic propositions via membership a • Φ − − → Ψ iff [ a ] ¬ ψ ∈ Φ for no ψ ∈ Ψ a [ a ] ¬ ψ ∉ ∋ ψ Φ Ψ → (via induction): ϕ true at Φ iff ϕ ∈ Φ. yields satisfiability of any consistent set of formulas, i.e., completeness. NB: presence of edge depends only on endpoints. 2 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Iteration-Free PDL with Intersection (PDL 0 ) fix propositions { P , Q ,... } = P , atomic programs { a , b ,... } = R Syntax: formulas: ϕ ∶∶ = P ∣ ϕ ∨ ϕ ∣ ϕ ∧ ϕ ∣ ¬ ϕ ∣ ⟨ α ⟩ ϕ ∣ [ α ] ϕ programs: α ∶∶ = a ∣ α ; α ∣ α ∩ α ∣ α ∪ α ∣ ϕ ? Semantics (sketch) over LTS T : • ⟨ α ⟩ ϕ true at s iff ex. t with s − − α → t and ϕ true at t a → t iff ( s , t ) ∈ a T • s − − α 1 ; α 2 • s α 1 α 2 − − − − − − → t iff ex. u with s − − − → u and u − − − → t α 1 ∩ α 2 α 1 α 2 • s − − − − − − → t iff s − − − → t and s − − − → t ϕ ? • s − − − → t iff s = t and ϕ true at s 3 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 PDL 0 in action a ⊧ P ∧ ⟨ a ⟩ P ∧ [ a ∩ ⊺ ? ]� P P → no tree model property b a P , Q ⊧ ⟨ a ; [ b ; P ? ]¬ Q ?; b ⟩( P ∧ Q ) / → convoluted and nested programs hard to conquer inductively 4 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 More complications Consider sat. set Φ (e.g. theory of dead end world) Ψ = ⋃ ϕ ∈ Φ {⟨ a ⟩ ϕ, [ a ] ϕ, ⟨ b ⟩ ϕ, [ b ] ϕ } ∪ {[ a ∩ b ]�} Ψ has model: b a Φ Ψ Φ But no model with only one instance of Φ → canonical model needs adaption Existing constructions not convincing enough 5 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 A Proof Calculus for PDL 0 Standard style proof system with derivation rules ( MP ) ϕ ϕ → ψ ϕ ϕ ( Gen ) ( USub ) ψ [ α ] ϕ ϕ � ψ / p � α ⇒ α ′ ( PSub ) ϕ ϕ �⟨ α ′ ⟩/⟨ α ⟩� ( pos ) and axioms and axiom schemes: α ∩ β ⇒ α ( p ?; α ) ∩ β ⇔ p ?; ( α ∩ β ) ... 6 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model Idea: build “free” structure, i.e., maximally tree-like, no unnecessary connections 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Φ Φ ′ start with mcs, no edges → atomic and box formulas satisfied (generation 0) 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Φ Φ ′ 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α Ψ ,α Φ Ψ ,α ′ Φ ′ add witnesses for missing diamonds 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α β Ψ ,α Φ α ′ Ψ ,α ′ Φ ′ add witnesses for missing diamonds, connect with abstract edges 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α β Ψ ,α Φ α ′ Ψ ,α ′ Φ ′ add witnesses for missing diamonds, connect with abstract edges in disjoint fashion 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α β Ψ ,α Φ α ′ Ψ ,α ′ Φ ′ 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α β Ψ ,α Φ α ′ Ψ ,α ′ Φ ′ refine iteratively α = α 1 ∩ α 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α 1 β Ψ ,α Φ α 2 α ′ Ψ ,α ′ Φ ′ refine iteratively α = α 1 ∩ α 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α 1 β Ψ ,α Φ α 2 α ′ Ψ ,α ′ Φ ′ refine iteratively α = α 1 ∩ α 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α 1 β Ψ ,α Φ α 2 α ′ Ψ ,α ′ Φ ′ refine iteratively, add intermediate nodes if necessary α = α 1 ∩ α 2 β = β 1 ; β 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 α ′ Ψ ,α ′ Φ ′ refine iteratively, add intermediate nodes if necessary α = α 1 ∩ α 2 β = β 1 ; β 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ... Ψ ′ ,α α α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 α ′ Ψ ,α ′ Φ ′ continue inductively until abstract programs converted to concrete programs α = α 1 ∩ α 2 β = β 1 ; β 2 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 Problem: New unsatisfied diamonds in generation 1 nodes 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ( X ′ ,γ ′ ) ( X ,γ ) γ ′ γ α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 Repeat Process: Add witnesses (generation 2), refine 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Construction of the Canonical Model ( X ′ ,γ ′ ) ( X ,γ ) γ ′ γ α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 Repeat Process: Add witnesses (generation 2), refine All diamonds satified in limit (generation ω ) 7 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Correctness of the Construction α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 Need to show: ϕ true at node labelled Φ iff ϕ ∈ Φ 8 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Correctness of the Construction α 1 β 2 Ψ ,α Ψ ′′ Φ β 1 α 2 In particular: If [ α ]¬ ψ ∈ Ψ ′′ and Ψ ′′ − − α → Ψ, then ψ ∉ Ψ 8 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Correctness of the Construction a b Ψ ,α Ψ ′′ Φ b c In particular: If [( b ; a ) ∩ ( b ; c )]¬ P ∈ Ψ ′′ and Ψ ′′ ( b ; a )∩( b ; c ) − − − − − − − − − − − → Ψ, then P ∉ Ψ 8 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Correctness of the Construction a b Ψ ,α Φ Ψ ′′ b c In particular: If [( b ; a ) ∩ ( b ; c )]¬ P ∈ Ψ ′′ and Ψ ′′ ( b ; a )∩( b ; c ) − − − − − − − − − − − → Ψ, then P ∉ Ψ Problem: Program unplanned: structure constructed for b ; ( a ∩ c ) 8 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 Correctness of the Construction a b Ψ ,α Ψ ′′ Φ b c In particular: If [( b ; a ) ∩ ( b ; c )]¬ P ∈ Ψ ′′ and Ψ ′′ ( b ; a )∩( b ; c ) − − − − − − − − − − − → Ψ, then P ∉ Ψ Problem: Program unplanned: structure constructed for b ; ( a ∩ c ) Can rewrite: [( b ; a ) ∩ ( b ; c )]¬ P → [ b ; ( a ∩ c )]¬ P Correctness of construction provable 8 / 9

Florian Bruse, Daniel Kernberger, Martin Lange: A Canonical Model for PDL 0 End of Talk Further work: • Extend to full PDL with intersection, i.e., with Kleene star (weak completeness only) • Compare present work to existing constructions more thoroughly Thanks for listening! 9 / 9