2.3 Deontic Action Logics
2.3.4 Equivalence of Level-0: PBDL and OWP
Now I provide a correspondence from PBDL and OWP, and this result shows that PBDL does partially overlap OWP. In other words, a fragment of the language for PBDL is a translation of a fragment for OWP, and vice versa. The corresponding construction is similar to the method introduced in [77].
I start with a syntactic translation from a fragment of PBDL to a fragment of OWP in two steps. First I define a fragment of PBDL in which no dynamic sentences and weak permission occur. Let L− be the fragment of language for PBDL, which in-volves formulas containing atomic propositions, action generators, and the formulas constructed with Boolean connectives, dynamic operator, and strong permission. To
2.3. Deontic Action Logics 35
Table 2.4: The sound and complete axiom system for Castro and Maibaum’s Propo-sitional Boolean Dynamic Logic. All propoPropo-sitional tautologies, Modus Ponens, and the axioms for Boolean Algebras for action terms are also taken into account. Above γ ∈Act0, andϕ[α/β]indicates that all occurrences ofαinϕare replaced byβ.
construct the translation fragment of PBDL, I allow for dynamic operators and per-missions to scope over formulas ofL−. I simply call this fragmentL×. LetL be the language for OWP. The translation proceeds in two steps. Define σ ∶ L− → Lbe a translation as follows:
The resulting δ(¬PWα) embodies WRO. So α is a type of strategy required to be played, iff all rationally permissible strategies areα-type strategies.
Let M = ⟨W, E, R, I, RP,∣∣⋅∣∣⟩ be a PBDL model. I construct a modelM∗ =
⟨H, nP, nO,∣∣⋅∣∣∗⟩as follows:
• H =W ∪E.
• nP(h)={ {X ⊆E ∶ ∀e∈X. RPhe} ifh∈W
∅ ifh∈E
• nO =∅
• ∣∣σ(a)∣∣∗ =I(a)
• ∣∣σ(p)∣∣∗ =∣∣p∣∣
I denote M∗ as the unique model generated from a PBDL model M. It should be noticed thatM∗ is an OWP model.
2.3.11.PROPOSITION. Given aPBDLmodelM =⟨W, E, R, I, RP,∣∣⋅∣∣⟩,M∗is an OWPmodel.
Proof:
The condition fornP is easy to verify by the construction. AsnO is empty, the three conditions required for OWP models are automatically satisfied. □
I now can show one of the main results in this section.
2.3.12.THEOREM. Given a PBDL model M = ⟨W, E, R, I, RP,∣∣ ⋅∣∣⟩. Then, for eachα, ϕ∈L−, andψ ∈L×,
e∈I(α) if and only if e∈∣∣σ(α)∣∣∗ M, w ⊨ϕ if and only if M∗, w ⊨σ(ϕ) M, w ⊨ψ only if M∗, w ⊨δ(ψ) Proof:
It is not difficult to provee ∈ ∣∣σ(α)∣∣∗ iff e ∈ I(α). Now I am going to prove that M, w ⊨ϕiffM∗, w ⊨σ(ϕ)for eachϕ∈L−. I do so by induction on the complexity ofϕ∈L−.
1. The cases of atomic propositions, negation, implication are easy to verify by the construction.
2. The case ofα≐β:
M, w⊨α ≐βiffI(α)=I(β)
iffe∈∣∣σ(α)∣∣∗⇔e∈∣∣σ(β)∣∣∗ iffM∗, w ⊨□(σ(α)↔σ(β))
2.3. Deontic Action Logics 37 3. The case ofP α: According to the aforementioned construction, then
M, w ⊨P αiff∀e∈I(α). RPwe iff∣∣σ(α)∣∣∗ ∈nP(w)
Next I will verify thatM, w ⊨ ψ ⇒ M∗, w ⊨ δ(ψ)for eachψ ∈L×. I show this by induction on the complexity ofϕ∈L×.
1. The case of⟨α⟩ϕ:
M, w ⊨⟨α⟩ϕiff∃(w, u, e)∈Rande∈I(α)s.t.M, u⊨ϕ
only if∃w, e, u∈W s.t.e∈∣∣σ(α)∣∣∗andu∈∣∣δ(ϕ)∣∣∗ iffM∗, w ⊨◇(σ(α)∧ ◇δ(ϕ))
2. The case of¬PWα:
M, w ⊨¬PWαiff¬∃e∈I(α)s.t.RPwe
iff∀e∈E. (RPwe⇒ e∈I(α))
only if∀β ∈Act.∀e∈E. ((e∈I(β)⇒ RPwe)
⇒ (e∈I(β)⇒ e∈I(α)))
iff∀β∈Act.[∣∣σ(β)∣∣∗ ∈nP(w)⇒ ∣∣σ(β)∣∣∗ ⊆∣∣σ(α)∣∣∗] iffM∗, w ⊨P σ(β)→□(σ(β)→σ(α)), for allβ ∈Act iffM∗, w ⊨ ⋀
β∈Act
[P σ(β)→□(σ(β)→σ(α))]
□
Now I can start with the second contribution in this section, which is a correspon-dence result from a fragment of OWP to a fragment of PBDL. Recall that OWP and MDL share the common language. So given the language L for PBDL, similarly, I simply call the fragment of OWP where no modal operator (□, O or P) occurs L0, and the fragment, the so-calledL1of OWP, is defined within the classical conjunction, negation, and modal operators based onL0. Again, sinceAct0is finite, my translation only considers finitely many atomic propositions. I show thatL1can be translated back to PBDL in a non-trivial way. Similarly, the translation runs in two steps.
LetL0be a fragment of OWP in which neither□,O, norP occur. Define a function µ∶L0 →L+as an action-translation as follows:
µ(p)∈Act0for eachp∈P rop0 µ(¬ϕ)∶=µ(ϕ)
µ(ϕ∧ψ)∶=µ(ϕ)∩µ(ψ)
I can then define a translation fromL1 toL+. The translation ∶L1 → L+is defined for complex formulas, which relies on the action-translationµdefined above.
(p)∈P rop0 for eachp∈P rop0 (¬ϕ)∶=¬(ϕ)
(ϕ∧ψ)∶=(ϕ)∧(ψ) (◇ϕ)∶=⟨µ(ϕ)⟩⊤
(P ϕ)∶=P µ(ϕ)
(Oϕ)∶=P µ(ϕ)∧ ¬PWµ(¬ϕ)
Notice that obligation in OWP can be translated by a combination of strong permission and weak permission in PBDL via the translation.
The transformation from the OWP models to PBDL models need certain restric-tions. Not every OWP model can be transformed into a PBDL model. To do so, the transformation needs to satisfy two conditions. The first is the valuation of atomic propositions in the transformed models should satisfy the required conditions (I.1)-(I.3). Second, the domain of the transformed model should be restricted to the valua-tion of atomic proposivalua-tions in OWP.
To satisfy the first restriction, I define the following “functional OWP models.” An OWP model MOWP = ⟨H, nP, nO,∣∣⋅∣∣⟩is functional, if and only if it satisfies the following three conditions:
• For eachp∈P rop0,∣∣∣p∣∣−⋃{∣∣q∣∣∶q∈(P rop0−{p})}∣ ≤1
• For eachw∈W, ifw∈∣∣p∣∣∩∣∣q∣∣such thatp≠q ∈P rop0, then
⋂{∣∣p∣∣∶p∈P rop0andw∈∣∣p∣∣} ={w}
• W =⋃p∈P rop0∣∣p∣∣
The three conditions correspond to the conditions (I.1)-(I.3) in PBDL models.
I now can construct a model transformed from a functional OWP model. Let MOWP =⟨H, nP, nO,∣∣⋅∣∣⟩be a functional OWP model, I then can construct a model M∗=⟨W, E, R, I, RP,∣∣⋅∣∣∗⟩as follows:
• W =E =H
• R={(x, y, y) ∣ (x, y)∈Alt}
• RP ={(x, y) ∣∃X ∈nP(x)s.t. y∈X}
• I(µ(p))=∣∣p∣∣
• ∣∣(p)∣∣∗=∣∣p∣∣for eachp∈P rop0 in OWP-language
2.3. Deontic Action Logics 39 I denoteM∗ as the unique model generated from a functional OWP modelMOWP. 2.3.13.PROPOSITION. Given a functional OWPmodelMOWP = ⟨H, nP, nO,∣∣⋅∣∣⟩, then the modelM∗defined above is aPBDLmodel.
Proof:
I only need to check whether R is functional. BecauseMOWP is functional, it is thus
easy to check thatRinM∗is also functional. □
With this result in hand, I now can turn to the final step of my transformation.
2.3.14.THEOREM. Given a functionalOWP modelM = ⟨H, nP, nO,∣∣⋅∣∣⟩, I con-struct the modelM∗ =⟨W, E, R, I, RP,∣∣⋅∣∣∗⟩generated fromM. Then, given any OWP sentenceϕandh∈⋃p∈P ropo0∣∣p∣∣,
1. Ifϕ∈L0, thenM, h⊨ϕiffh∈I(µ(ϕ)) 2. Ifϕ∈L1, thenM, h⊨ϕiffM∗, h⊨(ϕ) Proof:
1. Supposeϕ∈L0. By induction on the complexity ofϕ.
(a) For the case ofp∈P rop0, then
M, h⊨piffh∈∣∣p∣∣iffh ∈I(µ(p))
(b) For the case ofϕ=ψ∧ϕ∈L0, then alsoψ, ϕ∈L0. Now I have M, h⊨ψ∧ϕiffM, h⊨ψandM, h⊨ϕ
iffh∈I(µ(ψ))andh∈I(µ(ϕ)) iffh∈I(µ(ψ))∩I(µ(ϕ)) iffh∈I(µ(ψ)∩µ(ϕ)) iffh∈I(µ(ψ∧ϕ))
(c) For the case of¬ϕ∈L0, then alsoϕ∈L0. We can see that M, h⊨¬ϕiffM, h⊭ϕ
iffh∈/ I(µ(ϕ)) iffh∈I(µ(ϕ)) iffh∈I(µ(¬ϕ)) 2. Supposeϕ∈L1. By induction on the complexity ofϕ.
(a) For the case ofp∈P rop0, then
M, h⊨piffh∈∣∣p∣∣iffh ∈∣∣(p)∣∣∗iffM∗, h⊨(p)
(b) For the case ofψ∧ϕ∈L1, then alsoψ, ϕ∈L1. I can then infer M, h⊨ψ∧ϕiffM, h⊨ψ andM, h⊨ϕ
iffM∗, h⊨(ψ)andM∗, h⊨(ϕ) iffM∗, h⊨(ψ)∧(ϕ)
iffM∗, h⊨(ψ∧ϕ)
(c) For the case of¬ψ ∈L1, then alsoψ ∈L1. So I infer M, h⊨¬ψiffM, h⊭ψ
iffM∗, h⊭(ψ) iffM∗, h⊨(¬ψ)
(d) For the case of◇ψ ∈L1, then alsoψ ∈L0by the definition. So M, h⊨◇ψ iff∃h′ ∈Hs.t. Alt(h, h′)andM, h′ ⊨ψ
iff∃h′ ∈I(µ(ψ))s.t.Rh′(h)=h′andM∗, h′ ⊨ ⊤ iffM∗, h⊨⟨µ(ψ)⟩⊤
iffM∗, h⊨(◇ψ)
(e) Similar to the case ofP ψ ∈L1, thenψ ∈L0. So M, h⊨P ψiff∣∣ψ∣∣∈nP(h)
iff∀h′ ∈∣∣ψ∣∣thatRPhh′ by the Construction ofRP iff∀h′ ∈I(µ(ψ))thatRPhh′
iffM∗, h⊨P µ(ψ) iffM∗, h⊨(P ψ)
2.4. Conclusion 41