Chapter 6: Propositional Dynamic Logic - Syntax and Semantics

propositional dynamic logic (PDL)

formulas φ

Every basic proposition is a formula: p,q,r,...
If φ and ψ are formulas, then the following are formulas: ¬φ, φ ∧ψ, φ ∨ψ, φ →ψ, φ ↔ ψ
If φ is a formula and α an action, then the following is a formula: 〈α〉φ

action α

Every basic action is a action a, b, c, . . .
If α and β are actions, then the following are actions: $ α; β, α ∪β, α^∗ $
If φ is a formula, then the following is an action: $?\varphi$

Symbols: (!!! Exam)

α; β sequential composition: execute α and then β
α∪β non-deterministic choice: execute α or β
$ α^* $ repetition: loop to do α
$αˇ$ Converse: reversing basic actions
$?\varphi$ test: check wheather $\varphi$ is true
$〈α〉φ $ α can be executed in such a way that, after doing it, $\varphi$ is the case
p ∨¬p as T
¬T as ⊥
$¬〈α〉¬φ $ as $ [α] φ$ After any execution of α, $\varphi$ is the case
〈α〉T : α can be executed
[α] ⊥ : α cannot be executed
〈α〉φ ∧¬[α] φ α can be executed it at least two different ways
[[p]]: interpretations of p

tips:

$〈\alpha〉ϕ$ is true in a state s if for some $s_0$ with $(s, s_0)$ in the interpretation of α it holds that ϕ is true in $s_0$
$[α]ϕ$ is true in a state s if for every $s_0$ with $(s, s_0)$ in the interpretation of α it holds that ϕ is true in $s_0$

labelled transition systems (LTS)

3 components: $M = 〈S,R_a,V〉$

S a non-empty set S of states
V a valuation function, ndicating which atomic propositions are true in each state s ∈S
R binary relation $R_a$ for each basic action a

labelled transition system 带一堆箭头圆圈标志互相指的那个

Axiom system

The valid formulas of PDL can be derived from the following principles:

All propositional tautologies(valid)
$[α] (φ →ψ) →([α] φ →[α] ψ)$ for any action α.
Modus ponens (MP): from φ and φ →ψ, infer ψ.
Necessitation (Nec): from φ infer [α] φ for any action α.

Princeples for action operations

Test: $[?ψ] φ ↔(ψ →φ)$
Sequence: $[α; β] φ ↔[α] [β] φ$
Choice: $[α ∪β] φ ↔([α] φ ∧[β] φ)$
Repetition:
- Mix: $[α∗] φ ↔(φ ∧[α] [α∗] φ)$
- Induction: $φ ∧[α^∗] (φ →[α] φ)→[α^∗] φ$

Semantics of PDL (!!!!! Exam)

Given is a labelled transition system M = <S, V, R> for P and A. S: set of states V: valuation R: set of labelled transitions ->:basic actions

$M, s |= p ⇐⇒ p ∈ V (s)$
$M, s |= ¬ϕ ⇐⇒ M, s 6|= ϕ$
$M, s |= ϕ ∨ ψ ⇐⇒ M, s |= ϕ or M, s |= ψ$
$M, s |= ϕ ∧ ψ ⇐⇒ M, s |= ϕ and M, s |= ψ$
$M, s |= <α>ϕ ⇐⇒$ there is a $t ∈S$ such that $R_αst$ and $(M,t) |= φ$
$M, s |= [α]ϕ ⇐⇒$ for all t with $(s, t) ∈ [[α]]^M$ it holds that $M, t |= ϕ$

$[[α]]^M$: a→M 图里的箭头

From exercise:

[a]p: 如果S没有任何a的关系，这句话始终成立