Propositional Dynamic LogicFormal Systems II
Syntax of PDL Formulas and Programs
1. atomic formulas r(t1, . . . , tn) ∈ FmlΣ for every n-place relation symbol r ∈ Σ and terms ti ∈ TermΣ. p ∈ PFml for any propositional variable p ∈ PV ar.
2. equations t1 = t2 ∈ FmlΣ for t1, t2 ∈ TermΣ
3. closure under predicatepropositional logic operators If F1, F2 ∈ FmlΣ then also F1 ∨ F2 F1 ∧ F2, F1 → F2, ¬F1, ∀xF1 and ∃xF1. If F1, F2 ∈ PFml then also F1 ∨ F2 F1 ∧ F2, F1 → F2, ¬F1
4. modal operators [π]F, πF ∈ FmlΣ for F ∈ FmlΣ and π ∈ Π. [π]F, πF ∈ PFml for F ∈ PFml and π ∈ Π.
Prof. P.H. Schmitt PDL Summer 2009 2 / 28
Syntax of PDL Formulas and Programs (continued)
5. atomic programs (x := t) ∈ Π for t ∈ TermΣ and x ∈ V ar. a ∈ Π for every atomic program a ∈ AP
6. composite programs If π1,π2 ∈ Π then
6.1 π1; π2 ∈ Π sequential composition 6.2 π1 ∪ π2 ∈ Π nondeterministic choice 6.3 π∗ ∈ Π iteration
7. tests con? ∈ Π for every quantifierfree formula con ∈ FmlΣ.formula con ∈ PFml. rich tests
Π as defined above is called the set of regular programs.
Prof. P.H. Schmitt PDL Summer 2009 3 / 28
Semantics of PDL Propositional Kripke Structures
A propositional Kripke structure
K = (S, |=, ρ)
is determined by:
S the set of states |= ⊆ (S × PV ar) evaluation of propositional atoms in states ρ : AP → S × S the accessibility relations for atomic programs
The semantics definition will extend
I |= to a relation |= ⊆ (S × PFml) and
I ρ to a function Π → S × S.
We will use the infix notation s |= F instead of (s, F ) ∈ |=.
Prof. P.H. Schmitt PDL Summer 2009 4 / 28
Semantics of PDL Formulas and Programs
s |= r(t1, . . . , tn)s |= p, p ∈ PV ar iff (valM,u(t1), . . . , valM,u(tn)) ∈ valM(r)s(p) = true s |= t1 = t2 iff valM,u(t1) = valM,u(t2) s |= F iff F matching one of F1 ∨ F2, F1 ∧ F2,
F1 → F2,¬F1,∀xF1 or ∃xF1
as usual. s |= [π]F iff for all s′ with (s, s′) ∈ ρ(π)
s′ |= F s |= πF iff there exists s′ with (s, s′) ∈ ρ(π)
and s′ |= F
Semantics of PDL Formulas and Programs (continued)
(u, u′) ∈ ρ(x := t)(u, u′) ∈ ρ(a), a ∈ AP iff u′ = u[x/valM,u(t)](u, u′) ∈ ρ(a) (u, u′) ∈ ρ(π1; π2) iff there exists w ∈ S with
(u, w) ∈ ρ(π1) and (w, u′) ∈ ρ(π2) (u, u′) ∈ ρ(π1 ∪ π2) iff (u, u′) ∈ ρ(π1) or (u, u′) ∈ ρ(π2) (u, u′) ∈ ρ(π∗) iff there exists n and u1, . . . un ∈ S
such that u1 = u and un = u′ and (ui, ui+1) ∈ ρ(π) for 1 ≤ i < n
(u, u′) ∈ ρ(con?) iff u = u′ and u |= con
Prof. P.H. Schmitt PDL Summer 2009 6 / 28
Example of propositional tautologies
1. [π1; π2]F ↔ [π1][π2]F 2. [π1 ∪ π2]F ↔ ([π1]F ∧ [π2]F ) 3. [(π)∗]F ↔ (F ∧ [π][(π)∗]F ) 4. πF ↔ ¬[π]¬F
5. π1; π2F ↔ π1π2F 6. π1 ∪ π2F ↔ (π1F ∨ π2F ) 7. (π)∗F ↔ (F ∨ π(π)∗F ) 8. [π](F → G) → ([π]F → [π]G) 9. [(π)∗](F → [π]F ) → (F → [(π)∗]F )
Prof. P.H. Schmitt PDL Summer 2009 7 / 28
A Calculus for Propositional Dynamic Logic
Axioms
All propositional tautologies (A1) π(F ∨G) ↔ πF ∨ πG (A2) π1; π2F ↔ π1π2F (A3) π1 ∪ π2F ↔ π1F ∨ π2F (A4) π∗F ↔ F ∨ ππ∗F (A5) A?F ↔ A ∧ F (A6) [π∗](F → [π]F ) → (F → [π∗]F ) (A7) [π](F → G) → ([π]F → [π]G) (A8)
Rules
Theorem
The presented calculus is sound and complete.
Proof See e.g.,pp. 559-560 in David Harel’s article Dynamic Logic in the Handbook of Philosophical Logic, Volume II, published by D.Reidel in 1984. or D. Harel, D. Kozen and J. Tiuryn Dynamic Logic in Handbook of Philosophical Logic, 2nd edition , volume 4 by Kluwer Academic Publisher, 2001.
Prof. P.H. Schmitt PDL Summer 2009 9 / 28
Is
Fischer-Ladner Closure
Let S0 be a set of formulas in PFml. The Fischer-Ladner closure of S0 is the smallest subset S ⊆ PFml satisfying:
1 S0 ⊆ S 2 ¬G ∈ S ⇒ G ∈ S 3 (G1 ∨G2) ∈ S ⇒ G1 ∈ S and G2 ∈ S 4 πG ∈ S ⇒ G ∈ S 5 π1; π2G ∈ S ⇒ π1π2G ∈ S 6 π1 ∪ π2G ∈ S ⇒ π1G ∈ S and π2G ∈ S 7 π∗1G ∈ S ⇒ π1π∗1G ∈ S 8 G1?G2 ∈ S ⇒ G1 ∈ S and G2 ∈ S
For F ∈ PFml we denote by FL(F ) the Fischer-Ladner closure of {F}. We assume that F does not contain [ ], ∧, →.
Prof. P.H. Schmitt PDL Summer 2009 11 / 28
Fischer-Ladner Closure A Tableau Procedure
F ∈ PFml cl♦(F ) is smallest set C with F ∈ C and if πG ∈ C then G ∈ C. Notation: cl♦(F ) = {F1, . . . Fk}, cl♦(G) = {G1, . . . Gm}.
¬F
π1; π2F π1π2F π2F
π∗F ππ∗F
F?G F1 . . . Fk
Fischer-Ladner Closure First Step in Tableau Procedure
When constructing the tableau for a formula F with
cl♦(F ) = {F1, . . . Fk}
the first step is
After every rule appliction during tableau construction it is true:
if there is a node labeled πG, then there is also a node labeled G.
Prof. P.H. Schmitt PDL Summer 2009 13 / 28
Fischer-Ladner Closure Example
p → (q?; a)∗;¬q?r ¬p (q?; a)∗;¬q?rr
p (q?; a)∗¬q?r ¬q?r
¬qq?; a(q?; a)∗¬q?r qq?a(q?; a)∗¬q?r a(q?; a)∗¬q?r
q
Prof. P.H. Schmitt PDL Summer 2009 14 / 28
Fischer-Ladner Closure Properties of the Tableau Procedure
1. The procedure terminates
2. The set of all formulas generated by the procedure starting with the formula(s) cl♦(F ) is the Fischer-Ladner closure of F .
3. In particular, we now know that a finite Fischer-Ladner closure exists for every F .
Comment It can be shown that the cardinality of FL(F ) is not greater than the size of F (i.e., the number of symbols in F ). But, this is not strictly needed for the decidability result.
Prof. P.H. Schmitt PDL Summer 2009 15 / 28
Filtration Equivalent States
Let K = (S, |=, ρ) be a propositional Kripke structure, Γ ⊆ PFml. The relation ∼Γ on S is defined by:
s1 ∼Γ s2 iff s1 |= F ⇔ s2 |= F for all F ∈ Γ
It is not hard to see that ∼Γ is an equivalence relation.
Prof. P.H. Schmitt PDL Summer 2009 16 / 28
Filtration Quotient Structure
The quotient structure KΓ = (SΓ, |=Γ, ρΓ) for K = (S, |=, ρ) with respect to the equivalence relation ∼Γ is defined by:
[s] = {s′ | s ∼Γ s′} equiv. class of s SΓ = {[s] | s ∈ S} [s] |=Γ p ⇔ s |= p for p ∈ Γ [s] |=Γ p arbitrary otherwise ([s1], [s2]) ∈ ρΓ(a) iff for all aF ∈ Γ a ∈ AP
if s1 |= ¬aF then s2 |= ¬F
To guarantee that this definition is independent of the choice of representatives for equivalence classes we assume that πF ∈ Γ implies F ∈ Γ. The given definition of ρΓ is equivalent to
([s1], [s2]) ∈ ρΓ(a) iff for all [a]F ∈ Γ if s1 |= [a]F then s2 |= F
Prof. P.H. Schmitt PDL Summer 2009 17 / 28
Filtration Properties
Let F be PFml formula, Γ = FL(F ) the Fischer-Ladner closure of F K = (S, |=, ρ) a propositional Kripke structure KΓ = (SΓ, |=Γ, ρΓ) its quotient modulo ∼Γ,
then the following is true for all G ∈ Γ, π ∈ Π and s1, s2 ∈ S
1. The relation ∼Γ can have at most 2card(Γ) equivalence classes. Since Γ is finite, SΓ is also finite.
2. ([s1], [s2]) ∈ ρΓ(π) implies for all πB ∈ Γ s1 |= ¬πB ⇒ s2 |= ¬B
3. (s1, s2) ∈ ρ(π) entails ([s1], [s2]) ∈ ρΓ(π)
4. s |= G iff [s] |= G
Prof. P.H. Schmitt PDL Summer 2009 18 / 28
A Taste of the Proof Item 4 s |= G iff [s] |= G
Proof by induction on the complexity of G. We consider the step from B to G = πB. Implication from left to right If s1 |= πB, the there is s2 with (s1, s2) ∈ ρ(π) and s2 |= B. By induction hypothesis also [s2] |= B and by part 3 also ([s1], [s2]) ∈ ρΓ(π) thus [s1] |=< π > B. Implication from right to left From [s1] |= πB we get [s2], ([s1], [s2]) ∈ ρΓ(π) and [s2] |= B By induction hypothesis also s2 |= B. Assume s1 |= ¬πB. Part 2 yields s2 |= ¬B A contradiction. Thus s1 |= πB.
Prof. P.H. Schmitt PDL Summer 2009 19 / 28
A Taste of the Proof ([s1], [s2]) ∈ ρΓ(π) ∧ s1 |= ¬πB ⇒ s2 |= ¬B for all πB ∈ Γ
Proof by induction on the complexity of π. We consider the step from π to π∗. ([s1], [s2]) ∈ ρΓ(π∗) yields by definition states u0, . . . , uk such that [s1] = [u0], [s2] = [uk] and for all 0 ≤ i < k ([ui], [ui+1]) ∈ ρΓ(π). By induction hypothesis ui |= ¬πC ⇒ ui+1 |= ¬C for all πC ∈ Γ, all 0 ≤ i < k We need to show s1 |= ¬π∗B ⇒ s2 |= ¬B for all π∗B ∈ Γ. Observe that ¬π∗B ↔ ¬B ∧ ¬ππ∗B is a tautology. From s1 |= ¬π∗B we thus get s1 |= ¬ππ∗B. From ππ∗B ∈ Γ = FL(F ) and s1 ∼Γ u0 we know u0 |= ¬ππ∗B Induction hypothesis with C = π∗B yields u1 |= ¬π∗B Repeat this argument to obtain uk |= ¬π∗B uk |= ¬B by the tautology. s2 |= ¬B via uk ∼Γ s2
Prof. P.H. Schmitt PDL Summer 2009 20 / 28
Theorem
The satisfiability problem for PFml is decidable.
Proof Given a PDF formula F we need to find out wether F is satisifable. Try simultaneously to derive ¬F using Harel’s calculus and to find a finite model for F by exhaustive search. If F is satisfiable we will find a finite model for it. If F is not satisfiable we will find a finite derivation for ¬F .
If you do not wish to use the completeness result of Harel’s calculus, you can use the finite bound nF on the size of the Fischer-Ladner closure and exhaustively search through all Kripke structures upto size nF .
Prof. P.H. Schmitt PDL Summer 2009 21 / 28
Related Results
The problem to decide for F,G ∈ PFml wether G ` F holds is decidable.
Proof Use the deduction theorem G ` F iff ` G → F .
The problem to decide for F,G ∈ PFml wether G `g F holds is undecidable.
Meyer, Strett, and Mirowska 1981.
Prof. P.H. Schmitt PDL Summer 2009 22 / 28
Alternatives in PDL
Nonstandard Propositional Kripke Structures
(u, u′) ∈ ρ(a), a ∈ AP iff (u, u′) ∈ ρ(a) (u, u′) ∈ ρ(π1; π2) iff there exists w ∈ S with
(u, w) ∈ ρ(π1) and (w, u′) ∈ ρ(π2) (u, u′) ∈ ρ(π1 ∪ π2) iff (u, u′) ∈ ρ(π1) or (u, u′) ∈ ρ(π2) (u, u′) ∈ ρ(π∗) iff there exists n and u1, . . . un ∈ S
such that u1 = u and un = u′ and (ui, ui+1) ∈ ρ(π) for 1 ≤ i < n
(u, u′) ∈ ρ(con?) iff u = u′ and u |= con
replace by ρ(π∗) is reflexive and transitive and ρ(π) ⊆ ρ(π∗) and satisfies s |= [a∗]B ⇔ s |= B ∧ [a; a∗]B s |= [a∗]B ⇔ s |= B ∧ [a∗](B → [a]B)
Prof. P.H. Schmitt PDL Summer 2009 24 / 28
Difference between Standard and Nonstandard Kripke Structures
Consider the following set of modal formulas
Γ = {a∗¬p} ∪ {[an]p | n ≥ 0}
.
Γ has a nonstandard Kripke model, but Γ is inconsistent in standard Kripke semantics.
Prof. P.H. Schmitt PDL Summer 2009 25 / 28
Propositional Kripke Structures Alternatives
A propositional Kripke structure K = (S, |=, ρ) is determined by:
S the set of states |=⊆ (S × PV ar) evaluation of propositional atoms in states ρ : AP → S × S the accessibility relations for atomic programs
S ⊆ 2PV ar the set of states
I Strictly larger set of tautologies.
I Obviously decidable.
Lemma
Let
I Let AP = {a1, . . . , ak} I πall stand for the program (a1 ∪ . . . ∪ ak)∗. I U ⊆ PVar be a subset of the set of propositional atoms.
I stateU abbreviate ∧
p∈U p ∧ ∧
Then πall(stateU ∧ F ) → [πall](stateU → F )
is true in all state vector Kripke structure.
Prof. P.H. Schmitt PDL Summer 2009 27 / 28
Theorem
πall(stateU ∧ F ) → [πall](stateU → F )
with the notation from the previous slide.
Then:
1. {F} ∪H is satisfiable iff F is state vector satisfiable.
2. H ` F iff `sv F .
Prof. P.H. Schmitt PDL Summer 2009 28 / 28