Towards Intuitionistic Dynamic Logic

Transcription

1 Towards Intuitionistic Dynamic Logic J. W. Degen and J. M. Werner October 25, 2006

2 Outline Definitions Separation of PDL and ipdl Separation of Necessitas and Possibilitas Induction principles Definability Small model property Monotonicity Existential properties Modal translation On negative translations

3 Definitions classical Kripke models A Kripke model K (KM) is a pair K = (W K, m K ), where a nonempty set of states W K m K : is a function with 1. for every atomic propositional a: m K (a) W K 2. for every atomic program p: m K (p) W K W K

4 Definitions intuitionistic Kripke models An intuitionistic Kripke model K (ikm) is a tripel K = (W K, K, m K ), where W K a nonempty set of states K W K W K the intuitionistic (reflexive and transitiv) accessibility relation m K : is a function with 1. for every atomic propostional a: m K (a) W K monotonic in relation to K, this means: w m K (a) and w K w implies w m K (a). 2. for every atomic program p: m K (p) W K W K

5 Definitions Extension of the denotation function m K (π ρ) := m K (π) m K (ρ) m K (π; ρ) := m K (π) m K (ρ) (composition of relations) m K (π ) := m K (π) i (reflexive-transitive closure) i N m K (ϕ?) := {(w, w) w m K (ϕ)} m K ( ) := m K (ψ η) := m K (ψ) m K (η) m K (ψ η) := m K (ψ) m K (η) m K ( [ π ] ψ) := {w W K {w (w, w ) m K (π)} m K (ψ)} m K ( π ψ) := {w W K {w (w, w ) m K (π)} m K (ψ) } m K (ψ η) := {w W K {w w K w } m K (ψ) m K (η)}

6 Separation of PDL and ipdl Separation of Necessitas and Possibilitas Separation of Necessitas and Possibilitas As well known in classical PDL [ ] and can be defined from each other: [ π ] ϕ π ϕ π ϕ [ π ] ϕ

7 Separation of PDL and ipdl Separation of Necessitas and Possibilitas Counterexample At state w 1 the instances [ p ] a p a p a [ p ] a are refuted. w 1 w 2 K p w 3 p w 4 {a}

8 Separation of PDL and ipdl Separation of Necessitas and Possibilitas Counterexample At state w 1 the directions p a [ p ] a [ p ] a p a are refuted. w 1 p w 2 K w 3 {a}

9 Separation of PDL and ipdl Induction principles Induction principles The Segerberg induction schema consists of the following two axioms ϕ [ π ] (ϕ [ π ] ϕ) [ π ] ϕ π ϕ ϕ π ( ϕ π ϕ) (Necessitas form) (Possibilitas form) Theorem The Necessitas form is valid in ipdl, the Possibilitas form is not.

10 Separation of PDL and ipdl Induction principles Proof At Necessitas form: induction on k for m K (ϕ [ π ] (ϕ [ π ] ϕ)) m K ( [ π k ] ϕ) At Possibilitas form: The instance p a a p ((a ) p a) is refuted in w 1 : w 2 {a} p w 1 K w 3 {a}

11 Definability Definability Lemma In every intuitionistic Kripke model K = (W K, K, m K ) holds 1. m K ( [ ψ? ] ) = W K \ m K (ψ) 2. m K ( π ϕ) = m K ( [ ( [ π ] [ ϕ? ] )? ] ) 3. m K (ψ η) = m K ( ψ? η) = m K ( [ ( [ ψ? ] [ η? ] )? ] ) 4. m K (ψ η) = m K ( [ ( [ ψ? ] )? ] η) 5. m K ( [ π ρ ] ϕ) = m K ( [ π ] ϕ [ ρ ] ϕ) 6. m K ( π ρ ϕ) = m K ( π ϕ ρ ϕ) 7. m K ( π ρ ϕ) = m K ( π; ρ ϕ) 8. m K ( [ π ] [ ρ ] ϕ) = m K ( [ π; ρ ] ϕ)

12 Definability Consequences it is sufficient to assume that a formula ϕ contains only,,, ;,, [ ] and? as logical connectors. The formula a [ a? ] is valid, which is very counterintuitive. No significant disjunctive or existential property possible.

13 Small model property Small model property Theorem Every formula ϕ, which is satisfiable in a ipdl-model K, is also satisfiable in a model with less than 2 c ϕ states, for some constant c. short proof by simulating the behaviour of ipdl formulae in PDL Open problem: How to construct the Fisher-Ladner closures and formulate of a filtration lemma for ipdl? Validity problem is decidable.

14 Small model property Proof choose an atomic program p not included in ϕ. do following recursive translation of ϕ to ϕ T : For atomic propositions a [atomic programs q]: a T = [ p ] a [q T = q] (ϕ ψ) T = [ p ] (ϕ T ψ T ) move the T inward for the other connectors (example (η?) T = (η T )?) Define a PDL-model K T = (W K T, m K T ) by W K T := W K Let a be a propositional. Set m T K (a) := m K (a) Let q be an atomic program with p q. Set m T K (q) := m K (q). Set m T K (p) := K.

15 Small model property Proof (cont d) Lemma: w m K (ϕ) if and only if w m K T (ϕ T ) By the small model theorem of the PDL one can choose a (PDL-) model L = (W L, m L ), which satisfies: There is a state u W L, such that u m L (ϕ T ) WL 2 ϕt 2 c ϕ Transform L = (W L, m L ) back to an ipdl-model by defining W LP := W L For each atomic propositional a: m LP (a) := m L ( [ p ] a) For each atomic program q: m LP (q) := m L (q) LP := (m L (p)) Lemma: w m LP (ϕ) if and only if w m L (ϕ T )

16 Monotonicity Monotonicity A formula ϕ is called monotonic if in every ipdl-model K and any of its worlds u, v W K : u K v, u m K (ϕ) implies v m K (ϕ). Counterexample: Consider the formula p a at states w 1 and w 3 w 2 {a} p w 1 K w 3 {a}

17 Monotonicity Consequences of nonmonotonic formulae Failure of the rule in Gentzen style systems. Example: p, p is valid, while p p is not. w 2 w 3 p K w 1 Failure of substitution: The valid formula from ipdl are not closed under substitution of atomic propositions for arbitrary formulae. (a (b a) is valid, while p ( p ) is not)

18 Existential properties iterative property: If π ψ is valid, then there is a natural number k such that π k ψ is also valid. disjunctive property: If ϕ ψ is valid, then one of ϕ or ψ is valid. The disjunctive property fails for ipdl: Consider the formula p [ p ] (both parts of the formula are nonmonotonic) extended version: Restriction to subclasses of ipdl-models given by sets of (Harrop-) formulae (F π ψ then there is a k N s.t. F π k ψ) no nontrivial form of the both versions.

19 Existential properties Harrop formulae Definition The set H of Harrop formulae is recursive defined as follows: every atomic proposition is in H if ψ, η H then is also ψ η H if p is an atomic program and ϕ H then is also [ p ] ϕ H if [ π ] ϕ, [ ρ ] ϕ H then is also [ π ρ ] ϕ H if [ π ] [ ρ ] ϕ H then is also [ π; ρ ] ϕ H if ϕ H and ψ is a monotonic formula, then is also ψ ϕ H

20 Existential properties The definition of H is very strong with respect to the iterative property: Lemma Let F H be a set of Harrop formulae, π a program and η a monotonic formula, then holds: if F π η, then F η. Also a disjunctive property can be proven: Lemma Let F H be a set of Harrop formulae and η and ψ monotonic formulae. If F η ψ, then we have F η or F ψ.

21 Existential properties Open problems Can the definition of H be extended to get nontrivial results? When is the monotonicity condition not required? Is there a valid formula π ψ such that ψ is not monotonic?

22 Modal translation Definition The modal translation ϕ of a formula ϕ, resp. π of a program π, is recursively defined, as follows: Let ϕ, η, ψ be formulae and π, ρ, σ be programs for atomic propositions a: a := a ( := ) for atomic programs p: p := p let ϕ = ψ η: then ϕ := [ ψ? ] η let ϕ = ψ η: then ϕ := ψ η ( out of, ) let π = ρ σ: then π := ρ σ ( out of ;, ) let π = ρ : then π := (ρ) let π = η?: then π := η? let ϕ = [ ρ ] η: then ϕ := [ ρ ] η let ϕ = ρ η: then ϕ := ρ η

23 Modal translation Properties conceptual inpure: modalities can be introduced in nonmodal formulae unusual since the is replaced while everything else is untouched. Theorem Let ϕ be a formula. Then ϕ is valid in PDL iff ϕ is valid in ipdl.

24 On negative translations so far no usefull translation for the testoperator. different behaviour when occuring in [ ] or undeterminable behaviour in complex programs The tried out negative translations preserve only classical tautologoids The translated Segerberg axioms have mostly instances which are not valid.

25 On negative translations Example (test free) Consider the following negative translation of usual brand: For atomic propositions a: a := a ( = ) For atomic programs p: p := p (ϕ ψ) = ϕ ψ (ϕ ψ) = ( ϕ ψ ) (ϕ ψ) = ϕ ψ (π; ρ) = π ; ρ (π ρ) = π ρ (π ) = (π ) ( [ π ] ϕ) = [ π ] ϕ ( π ϕ) = [ π ] ϕ

26 Towards Intuitionistic Dynamic Logic On negative translations Example (cont d) Segerberg axiom 6: [ π; ρ ] ϕ [ π ] [ ρ ] ϕ (valid in ipdl!) Translation: [ π; ρ ] ϕ [ π ] [ ρ ] ϕ The instance [ p; q ] a [ p ] [ q ] a is not valid: w 4 q w 5 K w 1 p w 2 q w 3 {a}

27 On negative translations Thanks, for your attention.