Definition B.10 LetAandLA0 be as earlier. The languageLA3 hasAas its non-logical vocabulary, and in its logical vocabulary one additional binary operator and one additional unary operator
presumable. A string of symbols φ is a sentence ofLA3 iff there are sentences ψ andχ of LA0 such thatφ=ψ, or φ=ψ χ, or φ= presumablyψ.
Definition B.11 (i) Let W be as before. A frame onW is a function π assigning to every subset dof W a pattern πd ond.
(ii) Let π be a frame on W and d, e ⊆W. The proposition e is a default in πd iff d∩e6= ∅ and πd·e=πd.
Definition B.12 Let π be a frame onW, and d⊆W.
(i) wis a normal world inπd iffw∈dand for every d0⊆dsuch thatw∈d0 it holds thatw≤πd0 v
for everyv∈d0;
(ii) nπd is the set of all normal worlds in πd;
(iii) π is coherent iff for every non-empty d⊆W, nπd6=∅.
Definition B.13 Let W be as before.
(i) σ is an information state iffσ =hπ, si and one of the following conditions is fulfilled: (a) π is a coherent frame onW, and sis a non-empty subset of W;
(b) π is the frame hι,∅i, where ιd={hw, wi|w∈d} for everyd⊆W. (ii) 0=hυ, Wi, where υd=d×d for everyd⊆W.
1=hι,∅i.
(iii) Let σ=hπ, si and σ0 =hπ0, s0i be states.
Let π00 be the frame such that for every d, π00d=πd∩π0d. Then σ+σ0 =hπ00, s∩s0i, if hπ00, s∩s0i is coherent;
σ+σ0 =1, otherwise.
Definition B.14
(i) Let π and π0 be frames, both based on W.
The frame π is a refinement of π0 iff πd⊆π0d for everyd⊆W. (ii) Let π be a frame and d, e⊆W. πd·e is the refinement of π given by
(a) if d0 =d, then πd·ed0=πd0;
(b) πd·ed=πd·e.
The frame πd·e is the result of refiningπd inπ withe.
Definition B.15 Let σ=hπ, si be an information state.
·σ[φ ψ] =1if kφk ∩ kψk=∅ or πkφk·kψk is incoherent.
Definition B.16 Let σ = hπ, si be a coherent information state and assume that e1, . . . , en are
defaults in πd1, . . . , πdn respectively.
(i) A world wcomplies with {e1, . . . , en} iff w∈ei for every isuch that w∈di (1≤i≤n).
(ii) The set of defaults {e1, . . . , en} applies withinsiff for every d⊇sthere is somew∈nπd such
thatw complies with {e1, . . . , en}. In this case, we also say thate1, . . . , en jointly apply withins.
Definition B.17 Let σ = hπ, si be a coherent information state and assume that e1, . . . , en are
defaults in πd1, . . . , πdn.
(i) Then {e1, . . . , en} is a maximal applicable set in σ iff e1, . . . , e2 jointly apply within s, and for
every en+1 and dn+1 such thaten+1 is a default in πdn+1, and e1, . . . , en+1 jointly apply within s
it holds that en+1 =ei anddn+1=di for some i≤n.
(ii) A world w is optimal inσ iff w∈sand w complies with a maximal applicable set of defaults. The set of optimal worlds is denoted by mσ.
(iii) σ[presumably ψ]is determined as follows:
· If mσ∩ kψk=mσ, then σ[presumably ψ] =σ.
C
Recap: The Dynamic-Epistemic Upgrade Logic
The Dynamic-Epistemic Upgrade Logic DEUL is introduced in the paper Dynamic Logic for Prefer- ence Upgrade[10]. We will use the version with world-eliminating announcement !ϕand use slightly different notation.
Outside this appendix we will also omit the subscripts. These are used in DEUL to deal with multi-agent situations, but for our purposes we can do fine effectively having only one of them.
Definition C.1 (Language) Take a set of propositional variables P and a set of agents I, with p ranging over P andi over I. The dynamic epistemic preference language is given by:
ϕ ::= ⊥|p|¬ϕ|ϕ∧ψ|iϕ|≤i ϕ|U ϕ|hπiϕ π ::= !ϕ|#ϕ
(Furthermore we will use ∨, ♦i, ♦≤i and E being the duals of∧, i,
≤
i and U, respectively.)
Definition C.2 (Models) An epistemic preference model is a tuple M = (S,{∼i |i∈I},{≤i |i∈I}, V), withS a set of possible worlds,∼ithe usual equivalence relation of epistemic accessibility
for agent i, and V a valuation for proposition letters. Moreover, ≤i is a reflexive and transitive relation over the worlds.
Definition C.3 (Model Upgrades) Let M = (S,{∼i |i∈I},{≤i |i∈ I}, V). Then the upgraded
models M!ϕ andM#ϕ are defined as follows:
M!ϕ := (S0,{∼i |i∈I} ∩(S0×S0),{≤i |i∈I} ∩(S0×S0), V0), where
S0 = {w∈S|M, w|=ϕ}
V0(p) = V(p)∩S0
M#ϕ := (S,{∼i |i∈I},{≤∗i |i∈I}, V), where
≤∗i = ≤i −{(s, t)|M, s|=ϕand M, t|=¬ϕ}
Definition C.4 (Semantics) Given an epistemic preference model M = (S,{∼i |i∈ I},{≤i |i∈
ϕ:
M, s|=p iff s∈V(p) (1)
M, s|=¬ϕ iff notM, s|=ϕ (2)
M, s|=ϕ∧ψ iff M, s|=ϕandM, s|=ψ (3)
M, s|=iϕ iff for allt:s∼i timpliesM, t|=ϕ (4)
M, s|=≤i ϕ iff for allt:s≤i timpliesM, t|=ϕ (5)
M, s|=U ϕ iff for allt:M, t|=ϕ (6)
M, s|=hπiϕ iff Mπ, s|=ϕ (7)
(With convention that if sis eliminated by the updateπ then it is not the case thatMπ, s|=ϕ.)
One of the interesting things aboutDEU Lis the fact it hasreduction axioms eliminating the model- upgrade modalities, making it self-embed into regular modal logic. For more on the importance of these and what to do with them, consult [10].
Theorem C.5 (Soundness) The following formulas are valid:
h!ϕip ↔ ϕ∧p h!ϕi¬ψ ↔ ϕ∧ ¬h!ϕiψ h!ϕi(ψ∧φ) ↔ h!ϕiψ∧ h!ϕiφ h!ϕi♦iψ ↔ ϕ∧♦ih!ϕiψ h!ϕi♦≤i ψ ↔ ϕ∧♦≤i h!ϕiψ h!ϕiEψ ↔ ϕ∧Eh!ϕiψ h#ϕip ↔ p h#ϕi¬ψ ↔ ¬h#ϕiψ h#ϕi(ψ∧φ) ↔ h#ϕiψ∧ h#ϕiφ h!ϕi♦iψ ↔ ♦ih#ϕiψ h#ϕi♦≤i ψ ↔ (¬ϕ∧♦≤i h#ϕiψ)∨(♦i≤(ϕ∧ h!ϕiψ)) h!ϕiEψ ↔ Eh!ϕiψ
D
Recap: Circumscription
This definition taken directly from Lifschitz ([5]), the simplest form of circumscription is as follows: First off, for any predicates of the same arity,
P =Q stands for ∀x(P(x)≡Q(x))
P ≤Q stands for ∀x(P(x)⊇Q(x))
P < Q stands for (P ≤Q)∧ ¬(P =Q)
Now if A(P) is a sentence containing the predicate constant P,the circumscription of P in A(P) is simply the following second-order sentence:
A(P)∧ ¬∃p[A(p)∧p < P]
This added clause in this sentence can be seen as meaning that you cannot reduce the extension of
P while retaining the truth ofA(P). However, it is problematic in only talking about reducing the extension ofP while leaving all other predicates the same.
Hence there is also such a thing asthe circumscription ofP in A(P) with varied Z1, . . . , Zn, which is the following second-order sentence:
A(P, Z1, . . . , Zn)∧ ¬∃p, z1, . . . , zn[A(p, z1, . . . , zn)∧p < P]
A circumscription can be further circumscribed in, and varying a predicate that was circumscribed before can be used as a prioritizing tool.
Examples involving default rules are typically handled using abnormality predicates which are circumscribed against while varying the other predicates.
References
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[7] J. McCarthy. Applications of circumscription to formalizing common-sense knowledge. In M. L. Ginsberg, editor,Readings in Nonmonotonic Reasoning, pages 153–166. Kaufmann, Los Altos, CA, 1987.
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Readings in Nonmonotonic Reasoning, pages 145–151. Kaufmann, Los Altos, CA, 1987. [9] Johan van Benthem. Situation calculus meets modal logic.
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