Tableau T 0 : Zohar Manna 2 1 1 A 1 : - PowerPoint PPT Presentation

1 2 1 1 2 1 1 2 1 1 Example: ϕ 0 : p CS256/Winter 2009 Lecture #13 Tableau T ϕ 0 : Zohar Manna 2 1 1 ✘ ✛ ✤ ✜ ✤ ✜ ❄ ❄ ✲ ✛ ✘ A 1 : { p, p, p } A 2 : {¬ p, p, p } ✛ ✣ ✢ ✣ ✢ ✚✙ ✚✙ ✻ ❅ � ✻ ❅ � ❅ � ❅ � ✤ ❅ � ✜ ❅ ❘ � ✠ ✘ A 3 : { p, ¬ p } p, ✛ ✣ ✢ ✛✘ ✤ ✜ ❄ ✙ ✛ A 4 : {¬ p, ¬ p, ¬ p } ✣ ✢ 13-2

0 1 1 0 1 2 0 2 1 1 0 1 2 0 2 1 Promising Formula Example: 0 ϕ 1 : p ∧ ¬ p In T p , a path can start and stay forever in atom A 2 . 1   But A 2 includes p , i.e., A 2 promises that p will ¬ p, ϕ 1 , p, p, p, p   Φ ϕ 1 : eventually happen, but it is never fulfilled in the path. ¬ ϕ 1 , ¬ p, ¬ ¬ p, ¬ p, ¬ p, ¬ p   We want to exclude these paths. 1 0 The idea is that if a path contains an atom that in- Only 2 promising formulas in Φ ϕ cludes a promising formula, then the path should fulfill the promise. 1 1 1 ψ 1 : ¬ p promises r 1 : ¬ p ψ 2 : ¬ p promises r 2 : ¬ p A formula ψ ∈ Φ ϕ is said to promise the formula r if ψ is one of the forms: ¬ (( ¬ r ) W p ) r p U r ¬ ¬ r � �� � � �� � � �� � ≈ r ∧ ... ≈ ≈ r ∧ ... r 13-3 13-4

q ψ then ( σ, k ) q r for some k ≥ j 1 Promise Fulfillment Fulfilling Atoms 1 Property: 2 1 1 Definition: Atom A fulfills ψ ∈ Φ ϕ Let σ be an arbitrary model of ϕ , (which promises r ) 1 and ψ ∈ Φ ϕ a formula that promises r . if ¬ ψ ∈ A or r ∈ A . q ¬ ψ q r 2 1 1 If ( σ, j ) 1 Proof: Follows from the semantics of temporal Example: In T p , formulas. 2 1 1 Only one promising formula: Claim: (promise fulfillment by models) 1 1 Let σ be an arbitrary model of ϕ , ψ : p promises r : p and ψ ∈ Φ ϕ a formula that promises r . A + 2 1 1 1 : { p, p, p } Then σ contains infinitely many positions j ≥ 0 1 1 such that fulfills p since p ∈ A 1 A + ( σ, j ) or ( σ, j ) 3 : { p, ¬ p } p, Proof: fulfills p since p ∈ A 3 1. Assume σ contains infinitely many ψ -positions. A + 4 : {¬ p, ¬ p, ¬ p } Then σ must contain infinitely many r -positions, fulfills p since ¬ since ψ promises r . p ∈ A 4 2. Assume σ contains finitely many ψ -positions. But A − Then it contains infinitely many ¬ ψ -positions. 2 : {¬ p, p, p } does not fulfill p since p, ¬ p ∈ A 2 13-5 13-6

1 2 1 1 2 1 1 2 1 1 1 Tableau T Fulfilling Paths p 2 1 1 ❅ � Definition: A path π : A 0 , A 1 , . . . is fulfilling if ❅ � ✤ ✜ ✤ ✜ ❘ ❅ � ✠ ✲ A + A − ✛ ✘ for every promising formula ψ ∈ Φ ϕ 2 : {¬ p, p } 1 : { p, p, p } p, ✛ ✣ ✢ ✣ ✢ it contains infinitely many A j that fulfill ψ . ✚✙ ✚✙ ✻ ❅ � ✻ ❅ � ❅ � ❅ � ❅ � ✤ ✜ ❘ ❅ � ✠ Example: In T p , A + 3 : { p, ¬ p } p, ✣ ✢ ✒ � � A − 2 , A − 2 , A − 2 , A + 3 , A + 4 , A + � 4 , . . . ✛✘ A − 2 , A + 1 , A − 2 , A + 1 , A + 1 , A + ✤ ✜ 1 , . . . ❄ ✙ A + ✛ 4 : {¬ p, ¬ p, ¬ p } ✣ ✢ are fulfilling paths, but A − 2 , A − 2 , A − 2 , A − 2 , A − 2 , A − 2 , A − 2 , . . . is not a fulfilling path. 13-7 13-8

0 1 0 1 0 2 0 2 1 2 0 2 1 1 0 1 0 1 Fig. 5.3: Tableau T ϕ 1 for formula Example: 0 1 ϕ 1 : p ∧ ¬ p ϕ 1 : p ∧ ¬ p 0 1 2 0 2 1 2 0 2 1 0 1 0 1 T ϕ 1 in Fig 5.3 0 1 � ¬ p, ¬ � p, ¬ � � ¬ p, ¬ p, p, p, A ++ A −− There are two promising formulas in Φ : : : 0 1 2 ¬ ¬ p, ¬ ϕ 1 3 ¬ ¬ p, ¬ ϕ 1 p, p, 2 0 2 1 2 0 2 1 p promises ψ 1 : ¬ r 1 : ¬ p 0 1 0 1 0 1 ψ 2 : ¬ p promises r 2 : ¬ p � ¬ p, ¬ � p, ¬ 0 1 � � p, ¬ ¬ p, p, ¬ ¬ p, A ++ A − + : : 0 ¬ p, ¬ p, ¬ ϕ 1 1 ¬ p, ¬ ¬ p, ¬ ϕ 1 A ++ : { ¬ p, ¬ ¬ p, } 0 p, 1 . . . 2 0 2 1 2 0 2 1 0 0 1 0 1 A − + : { ¬ ¬ ¬ p, } p, 0 p, 1 . . . 1 � ¬ p, � p, � � p, ¬ ¬ p, p, ¬ ¬ p, A ++ A ++ A ++ : : : { ¬ p, ¬ p, ¬ p, . . . } 4 ¬ p, ¬ p, ¬ ϕ 1 5 p, ¬ ¬ p, ¬ ϕ 1 2 A −− : { ¬ ¬ p, } p, p, . . . 3 A ++ : { ¬ p, ¬ p, ¬ p, . . . } � ¬ p, � p, � � 4 p, ¬ p, p, ¬ p, A ++ A + − : : 6 ¬ p, ¬ p, ¬ ϕ 1 7 p, ¬ p, ϕ 1 A ++ : { p, p, ¬ ¬ p, . . . } 5 A ++ : { ¬ p, ¬ ¬ p, } p, . . . 6 A + − : { ¬ p, } p, p, . . . 7 13-9 13-10

Models vs. fulfilling paths Example: (Cont’d) Claim 2 (model → fulfilling path): ) ω not fulfilling. • path ( A + − If 7 π σ : A 0 , A 1 , . . . is a path induced by a model σ of ϕ , • path ( A ++ ) ω is fulfilling. 2 then π σ is fulfilling. ) ω is fulfilling. • path ( A ++ , A −− 2 3 Claim 3 (fulfilling path → model): If • path A ++ , ( A ++ ) ω is fulfilling. 4 5 π σ : A 0 , A 1 , . . . is a fulfilling path in T ϕ , • For arbitrary m , path then there exists a model σ of ϕ that induces π σ . π : ( A ++ , A −− ) m , A ++ , ( A ++ ) ω 2 3 4 5 is fulfilling. 13-11 13-12

0 2 Proposition 1 (satisfiability by path) Examples 0 2 0 2 0 2 0 2 Formula ϕ is satisfiable In the examples below we use the following optimization: A path starting in A can only visit nodes that are reach- iff 2 2 0 able from A in T ϕ . So we only need to consider nodes q p that are reachable from nodes labeled by atoms A such the tableau T ϕ contains a fulfilling path 2 2 0 0 q ϕ and thus σ q ϕ . that ϕ ∈ A . π : A 0 , A 1 , A 2 , . . . such that ϕ ∈ A 0 0 Example: ϕ : p ∧ ¬ p Proof: q ϕ . Then by Claims 1, 2, there exists a fulfilling Φ ϕ = { ϕ, p, p, p, p, A 0 , A 1 , . . . is a fulfilling path in T ϕ with ( ⇐ ) π : q ϕ , by the definition of induced, ¬ ϕ, ¬ p, ¬ p, ¬ p, ¬ p } ϕ ∈ A 0 Then, by Claim 3, there exists model σ such that Basic formulas: { p, p, p } → 8 atoms ∀ j ≥ 0 , ∀ p ∈ Φ ϕ : ( σ, j ) iff p ∈ A j There is only one atom such that ϕ ∈ A : A : {¬ p, ϕ } Since ϕ ∈ A 0 , ( σ, 0) p, p, p, Any successor of A requires ¬ p, p , but these cannot coexist in any atom. ( ⇒ ) σ path π σ in T ϕ that is induced by σ . So the part of T ϕ reachable from A is A Since ( σ, 0) ϕ ∈ A 0 . So there is no fulfilling path (no path at all, as A does not have a successor). 13-13 13-14 Hence, ϕ is not satisfiable.

0 1 0 1 0 1 2 0 2 1 0 1 2 0 2 1 1 0 1 2 0 2 1 2 0 2 1 0 1 0 1 ϕ 1 : p ∧ ¬ p Fig. 5.3: Tableau T ϕ 1 for formula Example: 2 0 2 1 ϕ 1 : p ∧ ¬ p Φ ϕ 1 = 2 0 2 1 2 0 2 1 0 1 0 1 2 0 2 1 0 1 { ϕ 1 , p, ¬ p, p, p, ¬ p, 0 1 � ¬ p, ¬ � p, ¬ ¬ ϕ 1 , ¬ p , ¬ ¬ p, ¬ p, ¬ p, ¬ ¬ p } � � ¬ p, ¬ p, p, p, A ++ A −− : : � �� � 2 ¬ ¬ p, ¬ ϕ 1 3 ¬ ¬ p, ¬ ϕ 1 p, p, ¬ p 2 0 2 1 2 0 2 1 0 1 0 1 ¬ p and ¬ p promise ¬ p . � ¬ p, ¬ � p, ¬ � � p, ¬ ¬ p, p, ¬ ¬ p, A ++ A − + : : Basic formulas: 0 ¬ p, ¬ p, ¬ ϕ 1 1 ¬ p, ¬ ¬ p, ¬ ϕ 1 { p, ¬ p } → 8 atoms p, 2 0 2 1 2 0 2 1 0 1 0 1 There is only one atom s.t. ϕ 1 ∈ A : � ¬ p, � p, A 7 : { p, p, ¬ p, p, ¬ p, ϕ 1 } � � p, ¬ ¬ p, p, ¬ ¬ p, A ++ A ++ : : 4 ¬ p, ¬ p, ¬ ϕ 1 5 p, ¬ ¬ p, ¬ ϕ 1 Any successor of A 7 requires ¬ p, and therefore p, ϕ 1 . 1 So the only successor is A 7 itself, and the part of T ϕ 1 � ¬ p, � p, � � reachable from A 7 is p, ¬ p, p, ¬ p, A ++ A + − : : 6 ¬ p, ¬ p, ¬ ϕ 1 7 p, ¬ p, ϕ 1 A + − 7 which has the infinite path A ω 7 . However, A + − does not fulfill the promising 13-15 13-16 7 ¬ p , and thus A ω formula 7 is not a fulfilling path. Hence, ϕ 1 is not satisfiable.

1 Strongly Connected Subgraphs ( scs ’s) • scs S is ϕ -reachable 1 if there exist a path and k ≥ 0 Definitions B 0 , B 1 , . . . , B k , . . . • A subgraph S ⊆ T ϕ is called such that ϕ ∈ B 0 and B k ∈ S . strongly connected subgraph ( scs ) if for every 2 distinct atoms A, B ∈ S , Example: In T p , there exists a path from A to B { A + 1 } , { A + 1 , A − 2 } , { A + 4 } are fulfilling which only passes through atoms of S { A − 2 } is not fulfilling All scs s are ( p ) -reachable. Note: a single-node subgraph is an scs A 3 is a transient scs . All others are good scs s. Example: In T ϕ 1 (Fig. 5.3), • A single-node scs is called transient (“bad”) if it is not connected to itself { A 4 } transient scs { A 5 } good scs { A 7 } is the only ϕ 1 -reachable scs • A non-transient (“good”) scs S is fulfilling { A ++ } { A ++ if every promising formula ψ ∈ Φ ϕ is , A −− } fulfilling scs ’s 2 3 5 fulfilled by some atom A ∈ S , i.e. { A − + } { A + − } scs ’s but not fulfilling 1 7 ¬ ψ ∈ A or r ∈ A 13-17 13-18