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In Praise of Belief Bases: Doing Epistemic Logic without Possible Worlds

Emiliano Lorini CNRS-IRIT, Universit´ e Paul Sabatier, Toulouse Workshop on “Doxastic Agency and Epistemic Logic” Bochum, December 2017

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Explicit beliefs, implicit beliefs and belief bases

“...a sentence is explicitly believed when it is actively held to be true by an agent and implicitly believed when it follows from what is believed” (Levesque 1984, p. 198). The concept of explicit belief is tightly connected with the concept

• f belief base (Nebel 1992; Makinson 1985; Hansson 1993; Rott

1998): belief base ≈ set of facts explicitly believed by an agent Existing logical formalizations of explicit and implicit beliefs (Levesque 1984; Fagin & Halpern 1987) do not clearly account for this connection

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Contribution of the paper

Multi-agent logic capturing the distinction between:

explicit belief, as a fact in an agent’s belief base, and implicit belief, as a fact that is deducible from the agent’s explicit beliefs, given the agents’ common ground

Main difference with Kripke-style semantics for epistemic logic:

Kripkean semantics: notions of possible world and doxastic/epistemic alternative are primitive Our semantics: notion of doxastic alternative is defined from — and more generally grounded on — the concept of belief base

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Motivation

Explicit representation of agents’ belief bases is crucial in order to facilitate the task of designing intelligent systems:

robotic agents conversational agents

Extensional semantics for epistemic logic, whose most representative example is the Kripkean semantics, is too abstract and too far from the agent specification

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Grammar

A countably infinite set of atomic propositions Atm = {p, q, . . .} A finite set of agents Agt = {1, . . . , n} Language L0(Atm): α ::= p | ¬α | α1 ∧ α2 | Eiα where p ranges over Atm and i ranges over Agt Language L(Atm): ϕ ::= α | ¬ϕ | ϕ1 ∧ ϕ2 | Iiϕ where α ranges over language L0(Atm)

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Reading of the operators

Eiα: agent i explicitly (actually) believes α Iiϕ: agent i implicity (potentially) believes ϕ

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Multi-agent belief base

Definition (Multi-agent belief base) A multi-agent belief base is a tuple B = (B1, . . . , Bn, V ) where: for every i ∈ Agt, Bi ⊆ L0 is agent i’s belief base, V ⊆ Atm is the actual state.

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Satisfaction relation

Definition (Satisfaction relation) Let B = (B1, . . . , Bn, V ) be a multi-agent belief base. Then: B | = p ⇐ ⇒ p ∈ V B | = ¬α ⇐ ⇒ B | = α B | = α1 ∧ α2 ⇐ ⇒ B | = α1 and B | = α2 B | = Eiα ⇐ ⇒ α ∈ Bi

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Doxastic alternatives

Definition (Doxastic alternatives) Let B = (B1, . . . , Bn, V ) and B′ = (B′

1, . . . , B′ n, V ′) be two multi-agent

belief bases. Then, BRiB′ if and only if, for every α ∈ Bi, B′ | = α. BRiB′: B′ is a doxastic alternative for agent i at B

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Multi-agent belief model

Definition (Multi-agent belief model) A multi-agent belief model (MAB) is a pair (B, Cxt), where B is a multi-agent belief base and Cxt is a set of multi-agent belief bases. The class of MABs is denoted by MAB. Cxt represents the agents’ common ground (or context)

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Satisfaction relation

Definition (Satisfaction relation (cont.)) Let (B, Cxt) ∈ MAB. Then: (B, Cxt) | = α ⇐ ⇒ B | = α (B, Cxt) | = Iiϕ ⇐ ⇒ ∀B′ ∈ Cxt : if BRiB′ then(B′, Cxt) | = ϕ

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Consistent multi-agent belief model

Definition (Consistent MAB) (B, Cxt) is a consistent MAB (CMAB) if and only if, for every B′ ∈ Cxt ∪ {B}, there exists B′′ ∈ Cxt such that B′RiB′′. The class of CMABs is denoted by CMAB.

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Notional doxastic model

Definition (Notional doxastic model plus satisfaction relation) A notional doxastic model (NDM) is a tuple M = (W , D, N, V) where: W is a set of worlds, D : Agt × W − → 2L0 is a doxastic function, N : Agt × W − → 2W is a notional function, and V : Atm − → 2W is a valuation function, and that satisfies the following conditions for all i ∈ Agt and w ∈ W : (C1) N(i, w) =

α∈D(i,w) ||α||M,

(C2) there exists v ∈ W such that v ∈ N(i, w), with: (M, w) | = p ⇐ ⇒ w ∈ V(p) (M, w) | = ¬ϕ ⇐ ⇒ (M, w) | = ϕ (M, w) | = ϕ ∧ ψ ⇐ ⇒ (M, w) | = ϕ and (M, w) | = ψ (M, w) | = Eiα ⇐ ⇒ α ∈ D(i, w) (M, w) | = Iiϕ ⇐ ⇒ ∀v ∈ N(i, w) : (M, v) | = ϕ and ||α||M = {v ∈ W : (M, v) | = α}. The class of notional doxastic models is denoted by NDM.

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Quasi-notional doxastic model

Definition (Quasi-notional doxastic model) A quasi-notional doxastic model (quasi-NDM) is a tuple M = (W , D, N, V) where W , D, N and V are as in a NDM except that Condition C1 is replaced by the following condition, for all i ∈ Agt and w ∈ W : (C1∗) N(i, w) ⊆

α∈D(i,w) ||α||M.

The class of quasi-notional doxastic models is denoted by QNDM. A NDM/quasi-NDM M = (W , A, D, N, V) is finite if and only if W , D(i, w) and V−1(w) are finite for all i ∈ Agt and w ∈ W .

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Relations between semantics

CMABs finite NDMs NDMs finite quasi-NDMs quasi-NDMs

Theorem 3 Theorem 2 Theorem 1 Figure: An arrow means that satisfiability relative to the first class of structures implies satisfiability relative to the second class. Dotted arrows denote relations that follow straightforwardly given the inclusion between classes.

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Equivalence between quasi-NDMs and finite quasi-NDMs

Theorem 1 Let ϕ ∈ L. Then, if ϕ is satisfiable for the class of quasi-NDMs, if and

• nly if it is satisfiable for the class of finite quasi-NDMs.

Idea of the proof. Filtration argument: A (possibly infinite) quasi-NDM M = (W , D, N, V) and finite Σ ⊆ L Define filtrated model MΣ = (WΣ, DΣ, NΣ, VΣ) with:

WΣ = {|w|Σ : w ∈ W } for all p ∈ Atm:

VΣ(p) = {|w|Σ : (M, w) | = p} if p ∈ Atm(Σ) VΣ(p) = ∅ otherwise

for all |w|Σ, DΣ(i, |w|Σ) = D(i, w) ∩ Σ for all |w|Σ, NΣ(i, |w|Σ) = {|v|Σ : v ∈ N(i, w)}

Prove that MΣ is a finite quasi-NDM and that (M, w) | = ϕ iff (MΣ, |w|Σ) | = ϕ for all ϕ ∈ Σ Apply the construction to Σ = sub(ϕ)

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Equivalence between finite NDMs and finite quasi-NDMs

Theorem 2 Let ϕ ∈ L. Then, ϕ is satisfiable for the class of finite NDMs if and only if ϕ is satisfiable for the class of finite quasi-NDMs.

Idea of the proof. Finite quasi-NDM M = (W , D, N, V) satisfying ϕ Define terminology of model M, T (M) = ∪w∈W ,i∈AgtAtm(D(i, w)) Define injection (naming function): f : Agt × W − → Atm \ (T (M) ∪ Atm(ϕ)) Injection f exists since Atm is infinite while T (M) is finite (since M is finite) Define new model M′ = (W ′, D′, N ′, V′) with: W ′ = W , N ′ = N, for every i ∈ Agt and w ∈ W : D′(i, w) = D(i, w) ∪ {f (i, w)}, for every p ∈ Atm: V′(p) = V(p) if p ∈ T (M) ∪ Atm(ϕ), V′(p) = N(i, w) if p = f (i, w), V′(p) = ∅

• therwise.

Prove that M′ is a finite NDM and that M′ satisfies ϕ

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Equivalence between CMABs and NDMs

Theorem 3 Let ϕ ∈ L. Then, ϕ is satisfiable for the class of CMABs if and only if ϕ is satisfiable for the class of NDMs.

Idea of the proof. Left-to-right direction: Start with a CMAB (B, Cxt) satisfying ϕ Build the NDM M containing one world per multi-agent belief base in Cxt ∪ {B} Prove that M satisfies ϕ Right-to-left direction: Start with a NDM M satisfying ϕ From M build a non-redundant NDM M′ and world w in M′ such that (M′, w) | = ϕ Define a multi-agent belief base for every world in M′ and take all multi-agent belief bases to define Cxt Prove (B, Cxt) | = ϕ, where B is the element in Cxt corresponding to w

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Logic of Doxastic Attitudes LDA

Definition LDA is the logic defined by the following principles: All tautologies of propositional calculus (Iiϕ ∧ Ii(ϕ → ψ)) → Iiψ ¬(Iiϕ ∧ Ii¬ϕ) Eiα → Iiα ϕ, ϕ → ψ ψ ϕ Iiϕ

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Soundness and completeness

Theorem 4 The logic LDA is sound and complete for the class of quasi-NDMs. Idea of the proof. Canonical model argument.

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Soundness and completeness (cont.)

Theorem 5 The logic LDA is sound and complete for the class of NDMs. Idea of the proof. Equivalence between satisfiability relative to NDMs and satisfiability relative to quasi-NDMs (Theorems 1 and 2).

• Theorem 6

The logic LDA is sound and complete for the class of CMABs. Idea of the proof. Equivalence between satisfiability relative to CMABs and satisfiability relative to NDMs (Theorem 3) and Theorem 5.

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Decidability

Theorem 7 The satisfiability problem of LDA is decidable. Idea of the proof. Based on the finite model property of LDA (Theorems 1 and 2).

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Outline

1 A language for explicit and implicit beliefs 2 Belief base semantics 3 Notional doxastic semantics 4 Equivalences between semantics 5 Axiomatics and decidability 6 Research agenda

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Complexity of SAT and model checking

Conjecture: existence of polynomial reduction of SAT-problem for LDA to SAT-problem for KDn (PSPACE-complete problem) Model checking algorithm for LDA:

MABs are compact models As shown by van Benthem et al. (2015), compact models can be useful for model checking

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Introspection, distributed belief and justification

KD45 logic of introspective implicit belief. Principles of LDA plus the following axioms: Iiϕ → IiIiϕ (1) ¬Iiϕ → Ii¬Iiϕ (2) S5 logic of introspective implicit knowledge. Principles of LDA, (1), (2) plus the following: Iiϕ → ϕ (3) Extension by distributed (implicit) belief (Fagin et al., 1995) Connection with justification logic (Artemov, 2008):

implicitly believing something ≈ something is provable explicit beliefs ≈ reasons for believing

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Belief change and theory of mind

Multi-agent belief revision using syntactic approach based on maximal consistent sets: partial meet base revision (Hansson, 1991) Formalization of theory of mind (ToM)

An agent is identified with her maximal level k of reasoning about

• thers’ minds

≈ Her belief base only contains explicit belief of depth at most k (syntactic definition) ⇒ Maximal uncertainty for implicit beliefs of depth at most k (semantic definition)

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Deontic logic

Merging the ideality-based (or value-based) view of norms and the imperative view of norms:

Ideality-based view: ϕ is obligatory iff ϕ is true in all ideal worlds Imperative view: obligations as injunctions and permissions as concessions

Useful analogies:

Explicit beliefs ≈ imperatives Belief base ≈ normative system (Alchourr´

• n & Bulygin, 1971)

Implicit beliefs ≈ derived obligations

More technically: ideality relation of standard deontic logic (SDL) not given as a primitive but grounded on normative system

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Thank you for your attention!

For more details ⇓

• E. Lorini (forthcoming). In Praise of Belief Bases: Doing Epistemic Logic

without Possible Worlds. In Proceedings of the The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18), AAAI Press.

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