The expressive power of modal logic with inclusion atoms Johanna - - PowerPoint PPT Presentation

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

The expressive power of modal logic with inclusion atoms

Johanna Stumpf

Dagstuhl Seminar

• 24. Juni 2015

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Let Φ be a set of proposition symbols. The set of formulas of ML(Φ) is generated by the following grammar ϕ := p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | ♦ϕ | ϕ, where p ∈ Φ.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition A Kripke model K is a triple (W , R, V ), where W is a set, R ⊆ W × W is the accessibility relation, V : Φ → P(W ) is the valuation.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Let K = (W , R, V ) be a Kripke model.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Let K = (W , R, V ) be a Kripke model. Any subset T of W is called a team of K.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Let K = (W , R, V ) be a Kripke model. Any subset T of W is called a team of K. For any T ⊆ W we write R[T] = {v ∈ W | ∃w ∈ T : wRv} and R−1[T] = {w ∈ W | ∃v ∈ T : wRv}.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Let K = (W , R, V ) be a Kripke model. Any subset T of W is called a team of K. For any T ⊆ W we write R[T] = {v ∈ W | ∃w ∈ T : wRv} and R−1[T] = {w ∈ W | ∃v ∈ T : wRv}. For teams T, S ⊆ W we write T[R]S if S ⊆ R[T] and T ⊆ R−1[S].

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition The team semantics for ML is defined as follows: K, T | = p ⇐ ⇒ T ⊆ V (p) K, T | = ¬p ⇐ ⇒ T ∩ V (p) = ∅ K, T | = ϕ ∧ ψ ⇐ ⇒ K, T | = ϕ and K, T | = ψ K, T | = ϕ ∨ ψ ⇐ ⇒ K, T1 | = ϕ and K, T2 | = ψ for some T1, T2 such that T1 ∪ T2 = T. K, T | = ♦ϕ ⇐ ⇒ K, T ′ | = ϕ for some T ′ such that T[R]T ′. K, T | = ϕ ⇐ ⇒ K, T ′ | = ϕ where T ′ = R[T].

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Example Let T = {w, v} and T ′ = {w′, v′} be in a Kripke model K, such that K, T ′ | = ϕ K, w′ | = ψ Then K, T | = ϕ and K, T | = ♦ψ.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let (K, w) be a pointed Φ-model. The k-th Hintikka formula χk

K,w

• f (K, w) is defined recursively as follows:

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let (K, w) be a pointed Φ-model. The k-th Hintikka formula χk

K,w

• f (K, w) is defined recursively as follows:

χ0

K,w := {p | p ∈ Φ, w ∈ V (p)} ∧ {¬p | p ∈ Φ, w /

∈ V (p)}

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Definition Assume that Φ is a finite set of proposition symbols. Let k ∈ N and let (K, w) be a pointed Φ-model. The k-th Hintikka formula χk

K,w

• f (K, w) is defined recursively as follows:

χ0

K,w := {p | p ∈ Φ, w ∈ V (p)} ∧ {¬p | p ∈ Φ, w /

∈ V (p)} χk+1

K,w := χk K,w ∧ v∈R[w] ♦χk K,v ∧ v∈R[w] χk K,v.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

Proposition Let Φ be a finite set of proposition symbols, k ∈ N, and (K, w) and (K,′ w′) pointed Φ-models. Then the following holds: K, w ⇄k K ′, w′ ⇐ ⇒ K ′, w′ | = χk

K,w.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

A Φ-model with a team is a pair (K, T), where K is a Kripke model over Φ and T is a team of K. We denote by KT (Φ) the class of all Φ-models with teams. Definition Let (K, T), (K ′, T ′) ∈ KT (Φ) and k ∈ N. We say that (K, T) and (K ′, T ′) are team k-bisimilar and denote K, T[⇄k]K ′, T ′ if

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

A Φ-model with a team is a pair (K, T), where K is a Kripke model over Φ and T is a team of K. We denote by KT (Φ) the class of all Φ-models with teams. Definition Let (K, T), (K ′, T ′) ∈ KT (Φ) and k ∈ N. We say that (K, T) and (K ′, T ′) are team k-bisimilar and denote K, T[⇄k]K ′, T ′ if for every w ∈ T there exists some w′ ∈ T ′ such that K, w ⇄k K ′, w′,

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

A Φ-model with a team is a pair (K, T), where K is a Kripke model over Φ and T is a team of K. We denote by KT (Φ) the class of all Φ-models with teams. Definition Let (K, T), (K ′, T ′) ∈ KT (Φ) and k ∈ N. We say that (K, T) and (K ′, T ′) are team k-bisimilar and denote K, T[⇄k]K ′, T ′ if for every w ∈ T there exists some w′ ∈ T ′ such that K, w ⇄k K ′, w′, for every w′ ∈ T ′ there exists some w ∈ T such that K, w ⇄k K ′, w′.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic and team semantics

A class K is closed under team k-bisimulation if (K, T) ∈ K and K, T[⇄k]K ′, T ′ imply that (K ′, T ′) ∈ K. A class K is closed under unions if (K, Ti) ∈ K for i ∈ I implies (K,

i∈I Ti) ∈ K.

A class K has the empty team property if (K, ∅) ∈ K for each model K.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Nonempty disjunction: ϕ

⊲ ⊳

∨ ψ

Definition K, T | = ϕ

⊲ ⊳

∨ ψ if and only if

T = ∅ or there exist T1, T2 = ∅ such that T = T1 ∪ T2 and K, T1 | = ϕ and K, T2 | = ψ.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Example Let T in propositional logic be the set of assignments {{01}, {10}}. The assignment {01} satisfies q and the assignment {10} satisfies p. Thus, T | = p

⊲ ⊳

∨ q.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Theorem ML(

⊲ ⊳

∨) has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Theorem ML(

⊲ ⊳

∨) has the empty team property.

Theorem ML(

⊲ ⊳

∨) is closed under unions.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Theorem ML(

⊲ ⊳

∨) has the empty team property.

Theorem ML(

⊲ ⊳

∨) is closed under unions.

Theorem ML(

⊲ ⊳

∨) is closed under k-bisimulation.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Theorem Let Φ be a finite set of propositional symbols and let K ⊆ KT (Φ). The class K is definable in ML(

⊲ ⊳

∨) if and only if K closed under

unions, closed under k-bisimulation for some k ∈ N and has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with nonempty disjunction

Proof idea: ⇒ ⇐ Lemma Let Φ := {p1, . . . , pn} be a set of propositional symbols. For any pair (K, X) ∈ KT (Φ) there exists a formula φK,X ∈ ML(

⊲ ⊳

∨)(Φ)

such that K ′, T ′ | = φK,X if and only if K, X[⇄k]K ′, T ′ or T ′ = ∅. X = ∅, then φK,X = ⊥. Else: φK,X = ⊲ ⊳

s∈X χk K,s.

Then the formula

(K,X)∈K φK,X defines the class K.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Inclusion atom: ϕ1 . . . ϕk ⊆ ψ1 . . . ψk, where ϕi, ψi ∈ ML Definition The semantics for MINC is given by the semantics for ML and the following additional clause: K, T | = ϕ1 . . . ϕk ⊆ ψ1 . . . ψk ⇐ ⇒ ∀w ∈ T ∃w′ ∈ T :

n

• i=1

(K, w | = ϕi ⇐ ⇒ K, w′ | = ψi)

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Example Let T in propositional logic be the set of assignments {{01}, {10}}. Then for every value of p in T exists the same value for q in T. Thus, T | = p ⊆ q.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Theorem MINC has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Theorem MINC has the empty team property. Theorem MINC is closed under unions.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Theorem MINC has the empty team property. Theorem MINC is closed under unions. Theorem MINC is closed under k-bisimulation.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Theorem Let Φ be a finite set of propositional symbols and let K ⊆ KT (Φ). The class K is definable in MINC if and only if it is closed under unions, closed under k-bisimulation for some k ∈ N and has the empty team property.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Proof idea: ⇒ ⇐ Lemma Let Φ := {p1, . . . , pn} be a set of propositonal symbols. For any pair (K, X) ∈ KT (Φ) there exists a formula ψK,X ∈ MINC(Φ) such that K ′, T ′ | = ψK,X if and only if K, X[⇄k]K ′, T ′ or T ′ = ∅. X = ∅, then ψK,X = ⊥. Else: ψK,X =

s,s′∈X

• χk

K,s ⊆ χk K,s′

• ∧

s∈X χk K,s.

Then the formula

(K,X)∈K ψK,X defines the class K.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

Modal logic with inclusion atoms

Corollary ML(

⊲ ⊳

∨) ≡ MINC.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Definition Let K = (W , R, V ) be a Kripke model and T a team in W . Let ϕ ∈ ML(

⊲ ⊳

∨). Let F be a function mapping nodes ψ in the syntax

tree of ϕ to subsets of W . We define F as a winning strategy if it satsifies the following conditions:

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

i) F(ϕ) = T. ii) If ψ = p then F(p) is such that K, F(p) | = p. iii) If ψ = ¬p then F(¬p) is such that K, F(¬p) | = ¬p. iv) If ψ = θ ∧ η then F(ψ) = F(θ) = F(η). v) If ψ = θ ∨ η then F(ψ) = F(θ) ∪ F(η). vi) If ψ = θ

⊲ ⊳

∨ η then F(ψ) = F(θ) ∪ F(η) and if F(ψ) = ∅ then

F(θ), F(η) = ∅. vii) If ψ = ♦θ then F(ψ)[R]F(θ). viii) If ψ = θ then F(θ) = R[F(ψ)].

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Lemma A winning strategy F exists if and only if K, T | = ϕ.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Let F be a winning strategy. If a sufficiently large set T satisfies a formula ϕ, it is possible to remove some element a from T such that a modified version of F still satisfies the winning conditions. Therefore, the formula ϕ is still satisfied by T\{a}. Remark The condition for the nonempty disjunction ψ = θ

⊲ ⊳

∨ η gives a

restriction: The nonemptyness property is not satisfied if one of the sets F(θ), F(η) consists only of {a}. Thus, there are at most 2 · occ⊲

⊳ ∨(ϕ) elements which cannot be removed from the team in

which occ⊲

⊳ ∨(ϕ) is the number of occurrences of the disjunction ⊲ ⊳

∨ in

the formula ϕ.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

A formula ϕ in modal logic with nonempty disjunction, which defines the inclusion atoms, needs to satisfy

• cc⊲

⊳ ∨(ϕ) ≥ 1

2|{a ∈ T | T\{a} | = p1 . . . pn ⊆ q1 . . . qn}|.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Example The set T = {{0001}, {0110}, {1011}, {1100}}

• f assignments satisfies p1p2 ⊆ q1q2, but no pro-

per nonempty subset of T satisfies the inclusion atom: |{a ∈ T | T\{a} | = p1 . . . pn ⊆ q1 . . . qn}| = 4

• cc⊲

⊳ ∨(ϕ) ≥ 1

2 · 4

⇒ At least 2 occurrences of

⊲ ⊳

∨ needed to describe

this inclusion atom in ML(

⊲ ⊳

∨).

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Theorem A formula ϕ in modal logic with nonempty disjunction which describes the inclusion atom p1 . . . pn ⊆ q1 . . . qn contains at least 2n−1 symbols.

Modal logic Team Semantics Nonempty disjunction Inclusion atoms Translation

A lower bound for the size of the translation

Theorem A formula ϕ in modal logic with nonempty disjunction which describes the inclusion atom p1 . . . pn ⊆ q1 . . . qn contains at least 2n−1 symbols. Thank you!