Propositional Dynamic Logic Contents 1 Introduction 1 2 Syntax - PDF document

Propositional Dynamic Logic Contents 1 Introduction 1 2 Syntax and Semantics 2 2.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Hilbert-style axiom system for PDL 3 4 Soundness and Completeness; Decidability 3 4.1 Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.2 Completeness: Proof idea . . . . . . . . . . . . . . . . . . . . . . 4 4.3 Completeness: Detailed proof . . . . . . . . . . . . . . . . . . . . 5 4.3.1 Consistent sets; maximal consistent sets . . . . . . . . . . 6 4.3.2 An intermediate structure . . . . . . . . . . . . . . . . . . 7 4.3.3 Constructing a (finite) structure and a state at which ¬ F holds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3.4 End of the completeness proof . . . . . . . . . . . . . . . 10 4.3.5 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1 Introduction Propositional dynamic logic (PDL) is a multi-modal logic with structured modal- ities. For each program α , there is: – a box-modality [ α ] and – a diamond modality � α � . PDL was developed from first-order dynamic logic by Fischer-Ladner (1979) and has become popular recently. Here we consider regular PDL. 1

2 Syntax and Semantics 2.1 Syntax Prog : set of programs Prog 0 ⊆ Prog : set of atomic programs Π: set of propositional variables The set of formulae Fma PDL Prog , Π of (regular) propositional dynamic logic and the set of programs P 0 are defined by simultaneous induction as follows: Formulae: F, G, H ::= ⊥ (falsum) | ⊤ (verum) | p p ∈ Π 0 (atomic formula) | ¬ F (negation) | ( F ∧ G ) (conjunction) | ( F ∨ G ) (disjunction) | ( F → G ) (implication) | ( F ↔ G ) (equivalence) | [ α ] F if α ∈ Prog | � α � F if α ∈ Prog Programs: α, β, γ ::= α 0 α 0 ∈ Prog 0 (atomic program) | F ? F formula (test) | α ; β (sequential composition) | α ∪ β (non-deterministic choice) α ∗ | (non-deterministic repetition) 2.2 Semantics A PDL structure K = ( S, R () , I ) is a multimodal Kripke structure with an accessibility relation for each atomic program. That is, it consists of: • a non-empty set S of states • an interpretation R () : Prog 0 → S × S of atomic programs that assigns a transition relation R ( α ) to each atomic program α • an interpretation I : Π × S → { 0 , 1 } The interpretation of PDL relative to a PDL structure K = ( S, R () , I ) is de- fined by extending R () to Prog and extending I to Fma PDL Prop 0 by the following simultaneously inductive definition: 2

val K ( p, s ) = I ( p, s ) if p ∈ Π val K ( ¬ F, s ) = ¬ Bool val K ( F, s ) val K ( F ∧ G, s ) = val K ( F, s ) ∧ Bool val K ( G, s ) val K ( F ∨ G, s ) = val K ( F, s ) ∨ Bool val K ( G, s ) val K ([ α ] F, s ) = 1 iff for all t ∈ S with ( s, t ) ∈ R ( α ) , val K ( F, t ) = 1 val K ( � α � F, s ) = 1 iff for some t ∈ S with ( s, t ) ∈ R ( α ) , val K ( F, t ) = 1 R ([ F ?]) = { ( s, s ) | val K ( F, s ) = 1 } ( F ? has the same meaning as: if F then skip else do not terminate) R ( α ∪ β ) = R ( α ) ∪ R ( β ) R ( α ; β ) = { ( s, t ) | there exists u ∈ S s.t. ( s, u ) ∈ R ( α ) and ( u, t ) ∈ R ( β ) } R ( α ∗ ) = { ( s, t ) | there exists n ≥ 0 and there exist u 0 , . . . , u n ∈ S with s = u 0 , y = u n , ( u 0 , u 1 ) , . . . , ( u n − 1 , u n ) ∈ R ( α ) } • ( K , s ) satisfies F (notation ( K , s ) | = F ) iff val K ( F, s ) = 1. • F is valid in K (notation K | = F ) iff ( K , s ) | = F for all s ∈ S . • F is valid (notation | = F ) iff K | = F for all PDL-structures K . 3 Hilbert-style axiom system for PDL Axioms ( D 1) All propositional logic tautologies ( D 2) [ α ]( A → B ) → ([ α ] A → [ α ] B ) ( D 3) [ α ]( A ∧ B ) ↔ [ α ] A ∧ [ α ] B ( D 4) [ α ; β ] A ↔ [ α ][ β ] A ( D 5) [ α ∪ β ] A ↔ [ α ] A ∧ [ β ] A ( D 6) [ A ?] B ↔ ( A → B ) [ α ∗ ] A ↔ A ∧ [ α ][ α ∗ ] A, ( D 7) [ α ∗ ]( A → [ α ] A ) → ( A → [ α ∗ ] A ] ( D 8) Inference rules P, P → Q MP Q F Gen [ α ] F 4 Soundness and Completeness; Decidability We will show that PDL is determined by PDL structures, and has the finite model property (the last result is due to Fischer and Ladner (1979)). 4.1 Soundness Theorem. If the formula F is provable in the inference system for PDL then F is valid in all PDL structures. Proof: Induction of the length of the proof, unsing the following facts: 3

1. The axioms are valid in every PDL structure. Easy computation. 2. If the premises of an inference rule are valid in a structure K , the conclu- sion is also valid in K . (MP) If K | = F, K | = F → G then K | = G (follows from the fact that for every state s of L if ( K , s ) | = F, ( K , s ) | = F → G then ( K , s ) | = G ). (Gen) Assume that K | = F . Then ( K , s ) | = F for every state s of K . = [ α ] F if for all t ′ with ( t, t ′ ) ∈ R ( α ) we have Let t be a state of K . ( K , t ) | ( K , t ′ ) | = F . But under the assumption that K | = F the latter is always the case. This shows that ( K , t ) | = [ α ] F for all t . 4.2 Completeness: Proof idea Theorem. If the formula F is is valid in all PDL structures then F is provable in the inference system for PDL. Idea of the proof: Assume that F is not provable in the inference system for PDL. We show that: (1) ¬ F is “consistent” with the set L of all theorems of PDL (2) We can construct a “canonical” PDL structure K and a state w in this PDL structure such that ( K , w ) | = ¬ F . Contradiction! We construct the PDL structure K as follows: 1. We know that if F is not provable then ¬ F must be consistent with the set L of all theorems of PDL. 2. This means that L ∪ {¬ F } is consistent. 3. We show that every consistent set of formulae is contained in a maximal consistent set of formulae. 4. We choose a set S of states, in which every state is a maximal consistent set W of PDL formulae (a “possible world”). 5. We define suitable relations R ( α ) on S as explained in the detailed proof. 6. Let K P be the Kripke model defined this way. We prove that ( K P , W ) | = φ iff φ ∈ W . Thus if W ¬ F is the maximal consistent set containing ¬ F then ( K P , W ¬ F ) | = ¬ F . 7. The model constructed this way is not exactly what we are looking for, because R ( α ∗ ) � = R ( α ) ∗ . We need to change this model such that this property holds, but have to take care that it is still the case that the new model has a state at which ¬ F is true. 4

8. We change it as follows: • We define a family Γ of subformulae of our formula. • We show that this family of formulae is finite and closed under sub- formulae. • We now say that two states W, W ′ ∈ S are equivalent (and can be merged) if for every G ∈ Γ, ( K P , W ) | = G iff ( K P , W ′ ) | = G (i.e. if they satisfy the same subformulae of F , in other words if we cannot distinguish these states if we only look where the subformulae of F in Γ are true or false). • We merge equivalent states in S (i.e. we partition S into equivalence classes and define a new set of states S/ ∼ , in which a state is the representative of an equivalence class of states in S ). • We define relations R ′ ( α ) such that if sR ( α ) s ′ then [ s ] R ′ ( α )[ s ′ ]. The labelling is defined similarly. • We now show that this new structure K = ( S/ ∼ , R ′ () , I ) is a PDL structure, and that ( K , [ W ¬ F ]) | = ¬ F . This ends the completeness proof. Decidability If we analyse the structure K = ( S/ ∼ , R ′ () , I ), we note that every state in S/ ∼ is the representative of a set of states in S at which certain subformulae of F are true. If we have two different states s 1 , s 2 in S/ ∼ : • s 1 is the representative of a set of states in S at which a set Γ 1 ⊆ Γ are true • s 2 is the representative of a set of states in S at which a set Γ 2 ⊆ Γ are true. Clearly, Γ 1 � = Γ 2 (otherwise s 1 and s 2 would be representatives for the same set of formulae, hence equal). We can now think of the states in S/ ∼ as being labelled with the sets of formulae in Γ which are true in them. The number of states in S/ ∼ is therefore smaller than or equal to the number of subsets of Γ. Since Γ is finite, the number of states in S/ ∼ is therefore finite (at most 2 | Γ | ). 4.3 Completeness: Detailed proof In order to complete the proof we need the following definitions and results: 5