KARO logic (Van Linder et al.) Epistemic logic. Dynamic Logic. Dynamic Logic KARO. Dynamic Logic. knowledge. belief. Interpretation formulas

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Lynne Blankenship
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1 KARO logic (Van Linder et al.) Knowledge & Belief: epistemic logic Abilities, Results & Opportunities: dynamic logic Modalities for Desires & Goals Epistemic logic Kϕ ϕ Kϕ KKϕ Kϕ K Kϕ B Bϕ BBϕ Bϕ B Bϕ knowledge belief Dynamic Logic Dynamic Logic Syntax Operator [α] with reading: [α]ϕ : after execution of α it holds (nec.) that ϕ <α>ϕ = [α] ϕ Semantics Accessibility relation R α for every action α R α;β = R α ± R β R α+β = R α R β R α = R α * Interpretation formulas M,s ² [α]ϕ for all s with R α (s,s ): M,s ² ϕ M,s ² <α>ϕ for some s with R α (s,s ): M,s ² ϕ Dynamic Logic Basic property (K) [α](ϕ ψ) ([α]ϕ [α]ψ) Structure of actions [α 1 ; α 2 ]ϕ [α 1 ]([α 2 ]ϕ) [α 1 + α 2 ]ϕ [α 1 ]ϕ [α 2 ]ϕ [α*]ϕ ϕ [α*]ϕ [α][α*]ϕ KARO K: Knowledge (& Belief) epistemic logic: Knowledge K : the logic S5 Belief B : the logic weak S5 A: Abilities ability operator A [α*](ϕ [α]ϕ) (ϕ [α*]ϕ)

2 KARO R: Results dynamic logic ("multi-modal K"): [α]ϕ O: Opportunities dynamic logic: <α>true KARO: formal syntax (omitting agent indexes) Set A of atomic actions Set P of atomic propositions Formulas ϕ ::= p ( P) ϕ ϕ 1 ϕ 2 Kϕ Bϕ Dϕ [α]ϕ Aα Actions α ::= a ( A) α 1 ; α 2 ϕ? if ϕ then α 1 else α 2 fi while ϕ do α od KARO: model for knowledge/beliefs/desires Kripke models of the form: <W, θ, R K, R B, R D > where: W is a non-empty set of states θ truth assignment function per state R K, R B, R D accessibility relations on W KARO: constraints on models R K, R B, R D are accessibility relations on W R K is assumed to be an equivalence relation R B is assumed to be euclidean, serial and transitive R B R K No special constraints on R D KARO: modelling actions Structures of the form: < Σ, {R a a A}, C, Ag> where: Σ set of model/state pairs R a ( a A ) accessibility relations on Σ C, Ag functions yielding set of actions the agent is capable to do, and the agent s agenda, resp., per model/state pair KARO: constraints on structures R a is an accessibility relation on (model, state) pairs! R a is taken to be deterministic: {(M,w ) R a (M,w)(M,w )} 1 for all M, w If R α is deterministic we may write R α (M,w) = {(M,w )} if R α (M,w)(M,w ) = Ø otherwise

3 KARO: semantics of actions R α is defined by induction on α: R ϕ? (M,w) = {(M,w)} if M, w ² ϕ = Ø otherwise R α1;α2 (M,w) = R α2 (R α1 (M,w)) Rif ϕ then α1 else α2 fi(m,w) = R α1 (M,w) if M, w ² ϕ = R α2 (M,w) otherwise KARO: semantics of actions Rwhile ϕ do αod(m,w) = {(M,w )} iff exists k 0 exists M 0,w 0,, M k,w k : (M 0,w 0 ) = (M, w) and (M k,w k ) = (M,w ) and {(M j+1,w j+1 )} = R ϕ?;α (M j,w j ) and M,w 2 ϕ. Rwhile ϕ do αod(m,w) = Ø otherwise. R α (X) = (M,w) X R α (M,w) Note R α ({(M,w)})= R α (M,w) KARO: constraints on structures C is a function of type Σ P(Actions) C(M, w) is the set of actions that the agent is capable of to perform in (M, w) One might impose conditions on C regarding the structure of actions analogous to that of R E.g. α 1 ; α 2 C(M, w) iff α 1 C(M, w) and α 2 C(R α1 (M, w)) KARO: interpretation of formulas M,w ² p θ(w)(p) = true, for p P M,w ² ϕ M,w 2 ϕ M,w ² ϕ 1 ϕ 2 M,w ² ϕ 1 and M,w ² ϕ 2 M,w ² Kϕ M,w ² ϕ for all w such that R K (w, w ) M,w ² Bϕ M,w ² ϕ for all w such that R B (w, w ) KARO: interpretation of formulas (2) M,w ² Dϕ M,w ² ϕ for all w such that R D (w, w ) M,w ² [α]ϕ M,w ² ϕ for all M,w such that R α ((M,w), (M,w )) M,w ² Aα α C(M, w) M,w ² Comα α Ag(M, w) Validities <ϕ?>ψ (ϕ ψ) <α 1 ;α 2 >ψ <α 1 >(<α 2 >ψ) <if ϕ then α 1 else α 2 fi >ψ ((ϕ <α 1 >ψ) ( ϕ <α 2 >ψ)) <while ϕ do α od>ψ (( ϕ ψ) (ϕ <α> <while ϕ do α od>ψ)) A(α 1 ;α 2 ) Aα 1 [α 1 ]Aα

4 KARO: correctness, feasibility Correct(α, ϕ) = <α>ϕ Feasible(α) = Aα PracPoss(α,ϕ) = Correct(α,ϕ) Feasible(α) Can(α,ϕ) = KPracPoss(α,ϕ) = K(<α>ϕ Aα) Cannot(α, ϕ) = K PracPoss(α, ϕ) An agent can do an action α with result ϕ iff it knows that it has the pract. poss. to do α, i.e. that α has ϕ as result and that it is able to do α. Remark Note that ² <α>ϕ <α>true I.e. correctness implies opportunity If α is deterministic: ² <α>ϕ [α]ϕ <α>true So, for deterministic actions the diamond is stronger than the box: <α>ϕ expresses both result and opportunity! KARO : properties of Can some properties: Can(ϕ?, ψ) K(ϕ ψ) Can(α 1 ; α 2, ϕ) Can(α 1, PracPoss(α 2, ϕ)) Can(α 1 ; α 2, ϕ) <α 1 > Can(α 2, ϕ), if α 1 accordant, i.e. K[α 1 ]ϕ [α 1 ]Kϕ Can(if ϕ then α 1 else α 2 fi, ψ) Kϕ Can(α 1, ψ) Kϕ Can(if ϕ then α 1 else α 2 fi, ψ) K ϕ Can(α 2, ψ) K ϕ KARO: implementability Implϕ = "PracPoss(a 1 ;...; a k, ϕ) for some atomic actions a 1,..., a k ϕ is implementable / realizable / achievable by the agent by means of a plan a 1 ;...; a k! The agent has the pract. poss. (so is able and has opportunity) to realize ϕ by executing this plan Belief types & belief revision in an agent-oriented setting: K = B k (certain) knowledge B o belief by observation B c belief by communication B d belief by default KARO: informational attitudes Agn x = B x ϕ B x ϕ (x {k, o, c, d}) Sawϕ = Agn k ϕ B o ϕ Heardϕ = Agn o ϕ B c ϕ Jumpedϕ = Agn c ϕ B d ϕ

5 KARO: informational actions observe, inform, try_jump KARO: semantics update_belief M Semantics (written as functions): R observeϕ (M, w) = update_belief o (ϕ, (M, w)) R inform(ϕ,i) (M, w) = update_belief ic (ϕ, (M, w)) R try_jumpϕ (M, w) = update_belief d (ϕ, (M, w)) R x ϕ ϕ w KARO: semantics update_belief Update_belief x (ϕ, (M,w)) ϕ ϕ delete R x w validities: <observe ϕ> Agn o ϕ ϕ <observe ϕ> B o ϕ ϕ <observe ϕ> B o ϕ ϕ Agn k ϕ <observe ϕ>sawϕ ϕ Agn k ϕ <observe ϕ>saw ϕ ϕ (Heard ϕ Jumped ϕ) <observe ϕ>sawϕ B j d ϕ D i,j ϕ (<do j (inform (ϕ, i)>χ χ) D i,j ϕ B jo ϕ <do j (inform (ϕ, i)> B ic ϕ D i,j ϕ B jo ϕ Agn io ϕ <do j (inform (ϕ, i))>heard i ϕ D i,j ϕ Heard j ϕ Agn id ϕ <do j (inform (ϕ, i))> Heard i ϕ D i,j ϕ Heard j ϕ Agn id ϕ (<do j (inform (ϕ, i))>χ χ) [D i,j ϕ : dependency of agent i on agent j w.r.t. ϕ] Further validities: Default(ϕ) Agn d ϕ <try_jump ϕ>jumpedϕ Default(ϕ) Agn d ϕ (<try_jump ϕ>χ χ)

6 Goal(ϕ) = Dϕ ϕ Implϕ A goal is desired, not yet true and implementable/realizable NOT: ² ϕ ψ ² Goal(ϕ) Goal(ψ) CanG(α, ϕ) = Can(α, ϕ) Goal(ϕ) PossIntend(α,ϕ) = Can(α, ϕ) KGoal(ϕ) An agent possibly intends to do α to achieve ϕ iff it knows that ϕ is its goal and can do α with ϕ as result. 162 An agent that possibly intends to do action α to achieve result ϕ may choose to commit to action α, which means that the agent puts α on its agenda How an agent uses its agenda to determine the actual next action that it will perform is dependent on the type of agent and is not specified in KARO 163 An agent that is committed to an action α may discover later that PossIntend(α, ϕ) does not hold any more for any ϕ. This may happen if (for all ϕ) Can(α, ϕ) holds: the agent cannot do α (with ϕ as result) any longer, or KGoal(ϕ) holds: ϕ isn t (known to be) a goal any more In this case the commitment to α has become useless and the agent may choose to uncommit to it. 164 KARO : motivational actions Special actions: commitα, uncommitα Semantics: R commitα (M, w) = update_agenda + (α, (M, w)) if M,w ² PossIntend(α, ϕ) for some ϕ, and =, otherwise R uncommitα (M, w) = update_agenda (α, (M, w)) if M,w ² Com α, and =, otherwise uncommitα C(M, w) iff M,w ² PossIntend(α, ϕ) for all ϕ 165 PossIntend(α, ϕ) <commitα> Com(α) PossIntend(α, ϕ) Auncommitα Com(α) <uncommitα> Com(α) Com(α) KCom(α) Com(α 1 ; α 2 ) Com(α 1 ) K[α 1 ]Com(α 2 ) Com(α) Can(α,true) Can(uncommitα, Com(α)) 166 One may specify specific agent types that have certain policies of committing and uncommitting, for example: Agents that may commit to one action only, i.e. the agenda C(M, s) contains at most 1 element ( single-minded agent ) Agents that never uncommit (only implicitly after completing the execution of agenda items) Agents that uncommit to an agenda item as soon as it detects that the item serves no purpose any more (i.e. PossIntend has ceased to hold: the agent cannot do the action any more or it does not 167 lead to achieving any of the known goals) 6

Modal logic. Benzmüller/Rojas, 2014 Artificial Intelligence 2

Modal logic. Benzmüller/Rojas, 2014 Artificial Intelligence 2 Modal logic Benzmüller/Rojas, 2014 Artificial Intelligence 2 What is Modal Logic? Narrowly, traditionally: modal logic studies reasoning that involves the use of the expressions necessarily and possibly.