Sequent Calculi for the Modal µ-calculus over S5 Luca Alberucci, University of Berne Logic Colloquium Berne, July 4th 2008
Introduction Koz: Axiomatisation for the modal µ-calculus over K Axioms: All classical propositional tautologies ( (α β) α) β ηx.α α[x/ηx.α], η {µ, ν} Inference Rules: Modus Ponens Necessitation Induction α (Distribution Axiom) α β β α α ϕ α(ϕ) ϕ νx.α (Fixpoint Axioms) (Nec) (MP) (Ind)
Introduction Koz S5 : Axiomatisation for the modal µ-calculus over S5 Axioms: All classical propositional tautologies, Distribution and Fixpoint Axiom and S5-axioms α α (T) α α (4) α α (5) Inference Rules: Modus Ponens, Necessitation and Induction
Introduction Koz S5 : Axiomatisation for the modal µ-calculus over S5 Axioms: All classical propositional tautologies, Distribution and Fixpoint Axiom and S5-axioms α α (T) α α (4) α α (5) Inference Rules: Modus Ponens, Necessitation and Induction, Σ, ϕ α(ϕ), Σ, ϕ νx.α (IndS5 )
Introduction Correctness and completeness Theorem (Walukiewicz(2000)) For all formulae ϕ we have that Koz ϕ = ϕ.
Introduction Correctness and completeness Theorem (Walukiewicz(2000)) For all formulae ϕ we have that Koz ϕ = ϕ. Theorem For all formulae ϕ we have that Koz S5 ϕ = S5 ϕ.
Introduction Overview Introduction The modal µ-calculus The Tait-Style calculus T 1 S5 The Tait-Style calculus T 2 S5 Embedding T 1 S5 into T2 S5 Conclusion
The modal µ-calculus The modal µ-calculus
The modal µ-calculus Syntax of the modal µ-calculus L µ is defined as follows (p P): ϕ :: p p (ϕ ϕ) (ϕ ϕ) ϕ ϕ µx.ϕ νx.ϕ where x P appears only positively in ϕ. ϕ is defined by using de Morgan dualities for boolean connectives, the usual modal dualities for and, and by using µx.ϕ νx. ϕ[x/ x] and νx.ϕ µx. ϕ[x/ x].
The modal µ-calculus Transition Systems A transition system T is a triple (S, T, λ) consisting of a set S of states, a binary relation T S S called transition relation, the valuation λ : P (S) assigning to each propositional variable p a subset λ(p) of S. T is a S5-model if the transition relation is reflexive, transitive and symmetric.
The modal µ-calculus Denotation and validity of a formula ϕ T is defined as usual by induction on the complexity of ϕ L µ. Simultaneously for all transition systems T we set:... νx.α T = {S S S α(x) T [x S ]} µx.α T = {S S α(x) T [x S ] S } Validity: (T, s) = ϕ if we have s ϕ T, T = ϕ if we have that ϕ T = S, = ϕ if for all T we have that T = ϕ, = S5 ϕ if for all S5-models T we have that T = ϕ.
The modal µ-calculus Well-bounded formulae A formula ϕ is well-bounded if for all subformulae of the form ηx.α (η {µ, ν}) we have that x appears free at most once in α. For all ϕ(x, y) and all T νx.νy.ϕ(x, y) T = νx.ϕ(x, x) T µx.µy.ϕ(x, y) T = µx.ϕ(x, x) T. Any formula ϕ is equivalent to a well-bounded formula wb(ϕ). If x appears only not guarded in ϕ then we have that νx.ϕ(x) T = ϕ( ) T and µx.ϕ(x) T = ϕ( ) T.
The modal µ-calculus An important equivalence Theorem (AF08) If x appears only once and guarded in α(x) then for all S5 models T we have that µx.α T = α 2 ( ) T and νx.α T = α 2 ( ) T. where α 2 (x) α[x/α]
The Tait-Style calculus T 1 S5 The Tait-Style calculus T 1 S5
The Tait-Style calculus T 1 S5 The closure of Γ For all sets of formulae Γ we define sub(γ) to be the smallest set such that Γ sub(γ), if α β, α β sub(γ) then α, β sub(γ), if α, α, α sub(γ) then α sub(γ). if x appears at most once and guarded in α then µx.α sub(γ) implies α 2 ( ) sub(γ), and νx.α sub(γ) then α 2 ( ) sub(γ). if x appears at most once and not guarded in α then µx.α sub(γ) implies α( ) sub(γ), and νx.α sub(γ) then α( ) sub(γ).
The Tait-Style calculus T 1 S5 The closure of Γ (cont.) The closure, C(Γ), is defined as sub(γ) { α; α sub(γ)} { α; α sub(γ)}...... { α; α sub(γ)} { α; α sub(γ)}. = for any finite Γ we have that C(Γ) is also finite.
The Tait-Style calculus T 1 S5 The calculus T 1 S5 Γ, p, p (Ax) Γ, α Γ, β Γ, α β ( ) Γ, α, β Γ, α β ( ), Γ, α, Γ, α, Σ ( ) Γ, ϕ Γ, ϕ ( ) If x appears at most once and guarded in α(x): Γ, α 2 ( ) Γ, νx.α (ν2 ) Γ, α 2 ( ) Γ, µx.α (µ2 ) If x appears at most once and not guarded in α(x): Γ, α( ) Γ, νx.α (ν) Γ, α( ) Γ, µx.α (µ)
The Tait-Style calculus T 1 S5 The calculus T 1 S5 (cont.) We need the (analytical) cut: If α C(Γ, ) then we have Γ, α, α (Ccut) Γ,
The Tait-Style calculus T 1 S5 Completeness of T 1 S5 Theorem For all finite sequents Γ L µ we have that = S5 Γ = T 1 S5 wb(γ).
The Tait-Style calculus T 2 S5 The Tait-Style calculus T 2 S5
The Tait-Style calculus T 2 S5 The calculus T 2 S5 Γ, νx.ϕ, νx.ϕ (Axν ) Γ, p, p (Ax) Γ, α Γ, β Γ, α β ( ) Γ, α, β Γ, α β ( ), Γ, α, Γ, α, Σ ( ) Γ, ϕ Γ, ϕ ( )
The Tait-Style calculus T 2 S5 The calculus T 2 S5 (cont.), Γ, ϕ, α(ϕ), Γ, ϕ, νx.α, Σ (IndS5 ) Γ, ϕ(µx.ϕ) Γ, µx.ϕ (unf µ ) Γ, ϕ(νx.ϕ) Γ, νx.ϕ (unf ν ) Γ, α, α (cut) Γ,
The Tait-Style calculus T 2 S5 Correctness of T 2 S5 Proposition For all finite sets of formulae Γ L µ we have that T 2 S5 Γ = = S5 Γ.
The Tait-Style calculus T 2 S5 Equivalence to Koz S5 Proposition For all finite sets of formulae Γ L µ we have that T 2 S5 Γ KozS5 Γ.
Embedding T 1 S5 into T2 S5 Embedding T 1 S5 into T2 S5
Embedding T 1 S5 into T2 S5 Well-bounding in T 2 S5 For all α(x, y) we can prove that 1. T 2 S5 σx.α(x, x) σx.σy.α(x, y) where σ {ν, µ}. 2. T 2 S5 σx.α(x, x) σx.σy.α(x, y) where σ {ν, µ}.
Embedding T 1 S5 into T2 S5 Well-bounding in T 2 S5 For all α(x, y) we can prove that 1. T 2 S5 σx.α(x, x) σx.σy.α(x, y) where σ {ν, µ}. 2. T 2 S5 σx.α(x, x) σx.σy.α(x, y) where σ {ν, µ}. Therefore, we have that T 2 S5 Γ T2 S5 wb(γ).
Embedding T 1 S5 into T2 S5 Non guarded fixpoints in T 2 S5 If x appears not guarded, positive and only once in ϕ then we have 1. T 2 S5 ϕ( ) νx.ϕ, and 2. T 2 S5 ϕ( ) µx.ϕ.
Embedding T 1 S5 into T2 S5 Non guarded fixpoints in T 2 S5 If x appears not guarded, positive and only once in ϕ then we have 1. T 2 S5 ϕ( ) νx.ϕ, and 2. T 2 S5 ϕ( ) µx.ϕ. Therefore, T 2 S5 proves the rules (µ) and (ν) of T1 S5 If x appears at most once and not guarded in α(x): Γ, α( ) Γ, νx.α (ν) Γ, α( ) Γ, µx.α (µ)
Embedding T 1 S5 into T2 S5 Guarded fixpoints in T 2 S5 Proposition The following facts hold 1. For all formulae ϕ(x) we have T 2 S5, Γ, α, β = T2 S5, Γ, ϕ(α), ϕ(β). 2. If x appears guarded and at most once in α then we have T 2 S5 α2 ( ), α 3 ( ).
Embedding T 1 S5 into T2 S5 Guarded fixpoints in T 2 S5 (cont.) Proof of 2. Assume α(x) of the form β( γ(x)) ( {, }). We have γ(β( γ(x))), γ(β( γ(x))), γ(x) and with part 1 we get that T 2 S5 γ(β( γ(x))), γ(β( γ(β( γ(x))))), γ(β( γ(x))) or T 2 S5 γ(β( γ(x))), γ(β( γ(β( γ(x))))). and by applying part 1 again we have that T 2 S5 β( γ(β( γ(x)))), β( γ(β( γ(β( γ(x)))))).
Embedding T 1 S5 into T2 S5 Guarded fixpoints in T 2 S5 (cont.) T 2 S5 proves the rules (µ2 ) and (ν 2 ) of T 1 S5 If x appears at most once and guarded in α(x): Γ, α 2 ( ) Γ, νx.α (ν2 ) Γ, α 2 ( ) Γ, µx.α (µ2 )
Embedding T 1 S5 into T2 S5 Embedding T 1 S5 into T2 S5 Theorem For all sequents Γ we have that T 1 S5 wb(γ) T2 S5 Γ.
Embedding T 1 S5 into T2 S5 Embedding T 1 S5 into T2 S5 Theorem For all sequents Γ we have that T 1 S5 wb(γ) T2 S5 Γ. Corollary Let Γ be any sequent. The following are equivalent: 1. = S5 Γ, 2. T 1 S5 wb(γ), 3. T 2 S5 Γ, and 4. Koz S5 Γ.
Conclusion Conclusion Completeness for Koz + S5-Axioms + remains an open question. ϕ α(ϕ) ϕ νx.α (IndS5 )