Symbolic and Quantitative Representations of Uncertainty: an Overview

Symbolic and quantitative representations of uncertainty: an overview

Didier Dubois

IRIT, CNRS & University of Toulouse, France.

July 2017

ECSQARU/ISIPTA 2017, Lugano, CH

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 1 / 50 Introduction

Uncertainty is present when an agent cannot assign a truth-value to a proposition, and can just express (degrees of) belief that this proposition is true or not Uncertainty modeling often appears in two forms Aleatory uncertainty originated from the representation of random phenomena. naturally measured by (limit of ) frequencies of occurrence of events in repeated experiments probability theory is used, where Belief(future event) = Frequency(past occurrences) has some objective ﬂavor

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 2 / 50 Introduction

Epistemic uncertainty originated from the lack of information of agents. more naturally qualitative than quantitative : uses sets of possible values. many proposals to represent beliefs : from subjective (imprecise) probability to logic is essentially subjective (agent-dependent)

Diﬃcult to ﬁnd one’s way in the variety of proposals.... Can have epistemic uncertainty about frequencies, and repeated imprecise observations.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 3 / 50 Why not just probabilities ?

Subjective probability is the favorite approach to represent belief Assessment methods clear for the degree of belief b(A) Analogical method : compared to the frequency of the event of drawing a ball from a known urn. Betting approach : fair price of a bet on the occurrence of A Easy to compute with, good properties Rooted in a long respectable tradition Mainstream BUT Single probabilities cannot represent ignorance Boolean probability = deterministic precise information Uniform probability (ignorance) is not stable via rescaling Cannot assign the same degree of belief to all contingent events

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 4 / 50 Information items

The basic block for representing information is the information item denoted by T that provides information about some entity x valued on some frame of discernment Ω Essential characteristics : Support : A non-empty set S(T ) ⊆ Ω that contains the set of values considered not impossible by information T .

Core A set C(T ) ⊆ Ω that contains the set of most plausible values according to T .

Plausibility ordering on Ω deﬁned by T a partial preorder T : ω1 T ω2 means that ω1 is at least as plausible as ω2 according to T .

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 5 / 50 Information items

T 7→ (C(T ), S(T ), T )

∀ω1 ∈ S(T ), ω2 6∈ S(T ) implies ω1 T ω2.

the core is made of the maximal elements of T

Extreme cases Total ignorance : T > such that C(T ) = Ω . 0 0 It represents vacuous information (ω ∼T ω , ∀ω, ω ∈ Ω). Complete knowledge : T ω such that S(T ω) = {ω} (the actual world is known).

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 6 / 50 Various representations of information items

An information item can be encoded in various settings : Boolean framework epistemic sets : Just a support S(T ) excluding impossible values Logic : Propositional logic, epistemic modal logics Qualitative framework Plausibility orderings on possible worlds or on events Comparative probabilities, etc. Qualitative possibility distributions valued on a ﬁnite scale L Many-valued logics Quantitative frameworks using set-functions Possibility distributions : fuzzy sets Probability distributions(but modeling ignorance is problematic) Belief functions (weighted epistemic sets) Credal sets (convex sets of probabilities)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 7 / 50 Aim of the talk

What is common to set-functions, modalities and truth-tables ? Similarities between qualitative and quantitative approaches Sophisticated numerical approaches often have coarse qualitative counterparts Information coming from the merging of unreliable testimonies : handling contradictions

Claim Capacities as the unifying concept for handling incomplete and inconsistent information

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 8 / 50 Outline

1 Modal logic, three-valued logics and possibility theory

2 Logics of graded belief

3 Qualitative vs. quantitative capacities : analogies

4 Handling multisource information Outline

1 Modal logic, three-valued logics and possibility theory

2 Logics of graded belief

3 Qualitative vs. quantitative capacities : analogies

4 Handling multisource information Modal logic, three-valued logics and possibility theory Reasoning about beliefs (and uncertainty) : traditions

Three main traditions

The probabilistic tradition : subjective probability (De Finetti, Ramsey), possibility theory, belief function, imprecise probability using set-functions The multiple-valued logic tradition (Łukasiewicz, Kleene, Belnap) : reasoning with incomplete and contradictory information using truth-tables The modal logic tradition : (von Wright, Hintikka, Halpern...) epistemic and doxastic modal logics, based on Kripke accessibility relations Possibility theory is instrumental to bridge these approaches.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 11 / 50 Modal logic, three-valued logics and possibility theory Boolean approach to incomplete information : 2-valued possibility theory

Based on an epistemic state described by a set ∅= 6 E ⊆ Ω, we can deﬁne possibility and necessity degrees N(A) and Π(A) by Π(A) = 1 if and only if E ∩ A 6= ∅ and 0 otherwise N(A) = 1 if and only if E ⊆ A and 0 otherwise. N is called a necessity measure and Π a possibility measure. N(A) = 1 means that A is certainly true, and Π(A) = 0 that A is certainly false, In particular, if N(A) = 0 and Π(A) = 1 it means that the truth of A is unknown in epistemic state E.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 12 / 50 Modal logic, three-valued logics and possibility theory Modal Epistemic logic and possibility theory

This framework ﬁts the one of modal logic, where φ means N(φ) = 1

Boolean possibility theory KD Modal logic Tools set functions N, Π modalities , ♦ Scale {0, 1} {0, 1} Adjunction N(φ ∧ ψ) = min(N(φ), N(ψ)) (φ ∧ ψ) ≡ φ ∧ ψ Duality Π(φ) = 1 − N(¬φ) φ ≡ ¬♦¬φ Axiom D Π(φ) ≥ N(φ) φ → ♦φ

but if N is a probability then E is a singleton !

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 13 / 50 Modal logic, three-valued logics and possibility theory A minimal two-tiered epistemic logic (MEL)

(Banerjee, Dubois, 2009)

1 Standard propositional Boolean logic language L Ontic propositional variables V = {a, b, c,..., p,...} α, β, . . . propositional formulae of L built using conjunction, disjunction, and negation (∧, ∨, ¬)

2 Modal level : A propositional language L Epistemic propositional variables : V = { α : α ∈ L} L propositional language based on V MEL is the minimal language to express partial knowledge about the truth of propositions. (you can write “the agent ignores α” as ¬α ∧ ¬¬α) ⇒ The "subjective" fragment of KD (or S5) without modality nesting.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 14 / 50 Modal logic, three-valued logics and possibility theory The MEL axioms

MEL is based on the language L with the propositional axioms (PL) Axioms of PL for L -formulas (K) (α → β) → (α → β) (D) α → ♦α (Nec) α, for each α ∈ L that is a PL tautology, i.e. if Mod(α) = Ω. the inference rule is modus ponens.

It is a two-tiered propositional logic, not a full-ﬂedged modal logic :

B `MEL Φ ⇐⇒ B ∪ {K, D, Nec} `PL Φ

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 15 / 50 Modal logic, three-valued logics and possibility theory Possibilistic semantics

The semantics does not require accessibility relations

A classical interpretation of MEL is equivalent to an epistemic state, a non-empty set E ⊆ Ω of Boolean interpretations, or equivalently a Boolean necessity measure.

t(α) = 1 ⇐⇒ Nt (α) = 1 ⇐⇒ Et ⊆ [α] ⇐⇒ Et |= α

Satisﬁability E |= α means that α is true in all worlds compatible with the epistemic state E (as usual in epistemic logic)

Ω The models of α are {E 6= ∅ : E ⊆ [α]} ⊆ 2 MEL is sound and complete with respect to this semantics

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 16 / 50 Modal logic, three-valued logics and possibility theory Is MEL a modal logic in the usual sense ?

Doxastic logic KD45 MEL Syntax Nested modal formulas No nesting Semantics Accessibility relations Sets and induced set-functions Axioms S4, S5 None Scope Introspection Information from an agent

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 17 / 50 Modal logic, three-valued logics and possibility theory Previous related works relating uncertainty and modal logic

Petruszczak (2009) showed that Kripke semantics of K45 and similar logics can be drastically simpliﬁed. Existence of various approaches to relate probability and modal logics where α stands for P([α]) ≥ λ. Hamblin (1959), Burgess (1969), Walley and Fine (1979), Logic of Risky Knowledge (Kyburg and Teng, 2002) Smets (1988) noticed that KD45 modalities are related to Shafer belief functions : Bel(α) = P(α). Fuzzy logic approach (Hajek, Godo...) : α is many-valued : the truth-value of a modal proposition is the degree of belief in the proposition

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 18 / 50 Modal logic, three-valued logics and possibility theory Kleene logic : a logic of incomplete information

1 1 Truth values T3 = {0 < 2 < 1}, where 2 is a third truth value referring to unknown. Syntax : the same connectives as classical logic : (∧, ∨, ¬) Negation : t(¬α) = n(t(α)), where n(1) = 0, n(0) = 1 and 1 1 n( 2 ) = 2 ; Conjunction : t(α ∧ β) = min(t(α), t(β)) ; Disjunction : t(α ∨ β) = max(t(α), t(β)), by De Morgan laws

Implication : p →K q ≡ ¬p ∨ q.

Paradox ? There are no tautologies, in particular 1 1 t(α ∧ ¬α) = t(α ∨ ¬α) = 2 when t(α) = 2

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 19 / 50 Modal logic, three-valued logics and possibility theory “Unknown” is not a truth value in the usual ontic sense

1 In practice, 2 is used to model the idea that the truth-value of a Boolean proposition is unknown. “‘Unknown" is in conﬂict with “Known to be true” and “Known to be false, not with “true” and “false". One must distinguish between two levels : Ontic values : true (T ), false (F ) Epistemic values are possibility distributions on {F , T } : 1 certainly true 1 = {T }, certainly false 0 = {F }, 2 = {F , T } The three-valued t(α) encodes knowledge about the truth t(α) of a Boolean proposition α.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 20 / 50 Modal logic, three-valued logics and possibility theory Translating 3-valued propositional atoms

(with D. Ciucci) T (t(a) ∈ T ) : translation into MEL of the assertion t(a) ∈ T ⊆ 3

T (t(a) = 1) = a (certainty of a) T (t(a) = 0) = ¬a 1 T (t(a) = 2 ) = ♦a ∧ ♦¬a = ¬a ∧ ¬¬a (ignorance) 1 T (t(a) ≥ 2 ) = ♦a (possibility of a) 1 T (t(a) ≤ 2 ) = ♦¬a Formally, “t(a) ∈ T ” stands for {t : t(a) ∈ T } so the translation is from V3 2 to L.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 21 / 50 Modal logic, three-valued logics and possibility theory Translation of Kleene logic into MEL

Elementary connectives on atoms : Conjunction : T (t(a ∧ b) = 1) = a ∧ b Disjunction : T (t(a ∨ b) = 1) = a ∨ b Negation : T (t(¬a) = 1) = ¬a Kleene Implication :T (t(¬a ∨ b) = 1) = ¬a ∨ b = ♦a → b A knowledge base B in Kleene logic : The translation T (B) = {T (t(β) = 1): β ∈ B} in MEL consists in the same conjunction of clauses as B, with modality in front of literals.

T ({¬a ∨ b, c ∨ ¬c}) = {¬a ∨ b, c ∨ ¬c}

A theorem-preserving translation : B `Kleene α ⇐⇒ T (B) `MEL T (α)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 22 / 50 Modal logic, three-valued logics and possibility theory The Kleene fragment of MEL

The Kleene fragment of MEL : in front of literals. LK := a| ¬a|α ∨ β|α ∧ β ⊂ L`

In this fragment, formulas of the form ¬a are not allowed. We reconcile Boolean logic and the lack of tautologies in Kleene logic a is always true or false a ∨ ¬a is not a tautology ♦a ∧ ♦¬a is not a contradiction This translation into MEL shows the meaning of Kleene implication.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 23 / 50 Modal logic, three-valued logics and possibility theory Kleene valuations as partial Boolean models

Three-valued interpretations t can be mapped to partial Boolean models where only some literals are known. ^ Et = [ ∧t(a)=1 a ∧t(b)=0 ¬b ]

Partial models : one can only express information on each variable independently of other ones. ( Cannot express information on disjunction of literals.) Kleene logic is to MEL and possibility theory what reasoning with product of marginal probabilities is to Bayes nets. Kleene logic, more generally all three-valued logics, is possibilistic reasoning with a very limited expressive power

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 24 / 50 Outline

1 Modal logic, three-valued logics and possibility theory

2 Logics of graded belief

3 Qualitative vs. quantitative capacities : analogies

4 Handling multisource information Logics of graded belief Graded possibility theory

Basic properties in the ordinal case 0 0 A ≥Π B ⇐⇒ ∀w ∈ B, ∃w ∈ A, w ≥π w and A ≥N B ⇐⇒ B ≥Π A Axioms

A ≥Π B ⇒ A ∪ C ≥Π B ∪ C (comparative possibility, Lewis) A ≥N B ⇒ A ∩ C ≥N B ∩ C (epistemic entrenchment, Gärdenfors)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 26 / 50 Logics of graded belief Possibility theory

Basic properties in the qualitative case

Π(A) = maxw∈A π(x) ; N(A) = 1 − Π(A) Axioms : Π(A ∪ B) = max(Π(A), Π(B)) ; N(A ∩ B) = min(N(A), N(B)) Characterized by the ordinal axioms. can express absolute notions of full possibility and impossibility.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 27 / 50 Logics of graded belief Possibility theory

At the bridge between numerical and symbolic representations of uncertainty : expressing that some state of aﬀairs are more plausible than other ones. A possibility distribution can be : Quantitative : a mapping π :Ω → [0, 1] A membership function pertaining to a quantitative linguistic category : “John is tall" 7→ π = µTall . A nested family of set Ei with P(Ei ) ≥ ai (probabilistic inequalities, conﬁdence intervals) A nested random set (contour function of a consonant belief function) A likelihood function P(B|x) since P(B|A) ≤ maxx∈B P(B|x) a Spohn kappa function κ :Ω → N letting π(w) = 2−κ(w)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 28 / 50 Logics of graded belief

Generalized possibilistic logic (D, Prade, Schockaert)

GPL is based on the language LL with atoms α, α ∈ L, λ ∈ L \{0} λ (PL) Axioms of PL for L-formulas (K) λ(α → β) → (λα → λβ) (D) λα → ♦1α (Nec) λα, for each α ∈ L that is a PL tautology, i.e. if Mod(α) = Ω. (W) λα → µα if λ ≥ µ (weakening) the inference rule is modus ponens. Same axioms as MEL for each λ + Weakening

π |= λα ⇐⇒ N([α]) ≥ λ Possibilistic logic : λα ≡ (α, λ), only conjunctions. λα ∧ β ≡ λα ∧ λβ

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 29 / 50 Logics of graded belief Graded belief

To each proposition A (subset of Ω), assign a conﬁdence degree γ(A) ∈ L (a totally ordered scale)

A q-capacity (or fuzzy measure) is a mapping γ : 2Ω → L such that γ(∅) = 0; γ(S) = 1; and if A ⊆ B then γ(A) ≤ γ(B).

Suppose one interprets belief in A (λα with A = [α]) as γ(A) ≥ λ for a sufficiently high confidence threshold. To extend the simplified epistemic logic setting to graded belief, one needs the following axiom

Adjunction : If γ(α) ≥ λ and γ(β) ≥ λ then γ(α ∧ β) ≥ λ, ∀λ ∈ L

Then γ(α ∧ β) = min(γ(α), γ(β)), so that γ is a necessity measure.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 30 / 50 Logics of graded belief Logical accounts of graded uncertainty

Three ways of reasoning based on non-Boolean capacities

Graded modalities : λp stands for γ([p]) ≥ λ ∈ L Probabilistic logic (Nilsson, Halpern...), GPL, logic of risky knowledge.

No adjunction axiom : non-regular modal logics.

Fuzzy logic approach, e.g. Łukasiewicz logic

p is many-valued : truth-degree(α) = belief-degree(α) Hajek et. al. (from 1995 on) for possibility, probability, belief functions. Comparative/ conditional approaches

Atomic formula p > q stands for g([p]) > g([q])

Lewis comparative possibility, Halpern (1997) on partial order...

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 31 / 50 Outline

1 Modal logic, three-valued logics and possibility theory

2 Logics of graded belief

3 Qualitative vs. quantitative capacities : analogies

4 Handling multisource information Qualitative vs. quantitative capacities : analogies Analogies between quantitative and qualitative capacities

A capacity (or fuzzy measure) is a mapping γ : 2Ω → [0, 1] such that g(∅) = 0; g(S) = 1; and if A ⊆ B then g(A) ≤ g(B).

Many concepts making sense for numerical capacities (viz. Belief functions, lower probabilities and their duals) have counterparts for qualitative capacities Möbius transforms Contour functions Core (dominating probabilities) and extreme points Choquet integrals

However the role played by probability measures in the numerical setting is now played by possibility measures.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 33 / 50 Qualitative vs. quantitative capacities : analogies Möebius transforms

Numerical Qualitative probability distribution p :Ω → [0, 1] possibility distribution π :Ω → L P P(A) = w∈A p(w) Π(A) = maxw∈A π(w) Capacity g q-capacity γ Möebius transform Qualitative Möebius transform ( P |A\B| γ(A) if > γ(A \{w}) mg (A) = (−1) g(A) γ#(A) = B⊆A 0 otherwise. P g(A) = B⊆A mg (A) γ(A) = maxB⊆A γ#(B) mg ≥ 0 : g = belief function All γ have positive focals singleton focals : probability possibility measures nested focals : necessity necessity

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 34 / 50 Qualitative vs. quantitative capacities : analogies Möebius transforms in the qualitative setting

Focal sets can be viewed as imprecise information from sources (like in belief function theory) Necessity measures : consonant imprecise sources Possibility measures : precise focal sets as conﬂicting and precise sources Capacities : imprecise and conﬂicting sources This view of possibility measure is at odds with the usual view :

N(A) = maxF ⊆A γ#(F ) = minw6∈A 1 − π(w) with π(w) = 1 − N(Ω \ w) = minF :w6∈F 1 − γ#(F ) represents imprecise information

Π(A) = maxw∈A γ#({w}) where π(w) = γ#({w}) represents conﬂicting information.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 35 / 50 Qualitative vs. quantitative capacities : analogies Dual vs. upper set-functions

Numerical Qualitative Dual capacity g c (A) = 1 − g(Ac ) γc (A) = 1 − γ(Ac ) P c ∗ Pl(A) = B∩A6=∅ mg (B) = g (A) γ (A) = maxB∩A6=∅ γ#(B) g = Bel ⇒ g c = Pl ≥ g γ∗ ≥ γ but γ∗ 6= γc , γc 6≥ γ P Contour funct. : Pl(w) = w∈B mg (B) πγ(w) = maxw∈B γ#(B) ∗ Pl(A) 6= maxw∈A Pl(w) γ (A) = maxw∈A πγ(w) In the quantitative case : the dual of a belief function is the expected consistency and is not the possibility measure based on the contour function In the qualitative case : the dual of a capacity is not expressing consistency with information sources, but coincides with the possibility measure based on the contour function

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 36 / 50 Qualitative vs. quantitative capacities : analogies Core and qualitative core

Numerical Qualitative Core : C(g) = {P ≥ g} QC(γ) = {π :Π ≥ γ} Sometimes empty Never empty Convex set Sup-semi lattice Extreme points C ∗ minimal elements QC ∗ among among g i i+1 γ i ∀i, pσ (sσ(i)) = g(Sσ) − g(Sσ ) ∀i, πσ(sσ(i)) = γ(Sσ) g(A) = min{P(A): P ∈ C ∗(g)} γ(A) = min{Π(A): π ∈ QC ∗(γ)} if coherence always g c (A) = sup{P(A): P ∈ C(g)} γ(A) = max{N(A): π ∈ QC(γc )} i Sσ = {sσ(i),..., sσ(n)} for a permutation σ of the n = |Ω| elements in S

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 37 / 50 Outline

1 Modal logic, three-valued logics and possibility theory

2 Logics of graded belief

3 Qualitative vs. quantitative capacities : analogies

4 Handling multisource information Handling multisource information Handling inconsistent information

Not only information items are imprecise, but sources of information may conﬂict Two approaches Merging conﬂicting information (fusion methods for information items of various nature) Reasoning under inconsistency (paraconsistent logics, Belnap setting, argumentation)

Capacities may play a role in both approaches.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 39 / 50 Handling multisource information General principles for information fusion

Focus : Merging information items Ti , supplied by sources whose reliability levels are not known, yields an information item : f (T1,..., Tn). A symmetric process : the sources play the same role and supply information of the same kind ; Information items are considered reliable insofar as it is possible, in order to be useful The result should not be arbitrarily precise. It should be faithful to the level of informativeness of the inputs. Information fusion should solve conﬂicts between sources, while neither dismissing nor favoring any of them without a reason. These principles are embodied in a set of 8 basic postulates.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 40 / 50 Handling multisource information Information ordering

One must be able to compare information items in terms of relative informativeness. Information ordering If T is consistent, T v T 0 expresses that T provides at least as much information as T 0. In particular, T v T 0 should imply S(T ) ⊆ S(T 0) and C(T ) ⊆ C(T 0). Sets : inclusion. Probability : should go beyond comparing entropies, i.e. majorization ordering Belief functions, credal sets, conﬁdence relations, capacities : less easy to deﬁne T (pignistic transform, contour function), v (specialization, Bel1 ≤ Bel2...)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 41 / 50 Handling multisource information Fusion of epistemic sets : postulates

Fusion axioms in the set-valued Boolean setting : Unanimity : Values considered impossible (resp. plausible) by all sources should be impossible (plausible).

Monotonicity : if information items Ti are consistent, and 0 0 0 Ti v Ti , ∀i = 1,..., n then f (T1,..., Tn) v f (T1,..., Tn). Consistency enforcement : the core of the result should not be empty

Optimism : if information items Ti are consistent, then f (T1,..., Tn) v Ti , ∀i.

Fairness : ∀i = 1,..., n, f (T1,..., Tn) is consistent with Ti . Insensitivity to vacuous information : non-informative sources can be deleted.

Minimal commitment : f (T1,..., Tn) should be the as little informative as possible while obeying the other postulates.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 42 / 50 Handling multisource information

Fusion of n epistemic sets T1, T2,... Tn

Optimistic Fairness For any subset I of consistent sources, \ f (T1,...,Tn) ∩ Ti 6= ∅. i∈I

Let I ⊂{1,..,n} be a maximal consistent subset (MCS) of sources, i.e., I I T =∩i Ti∈I 6= ∅ and T ∩ ∪j6∈I Tj = ∅. MCS({1, . . ., n}) is the set of maximal consistent subsets of sources. A known combination rule from logic (Rescher and Manor, 1970) is [ \ f (T1, ..., Tn) = Ti I ∈MCS({1,...,n}) i∈I

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 43 / 50 Handling multisource information Optimistic fusion is destructive

Merging set-based information items A and B yields A ∩ B or A ∪ B. Original information is lost ! One cannot retrieve A nor B, if we only have A ∩ B or A ∪ B Try non destructive fusion using a capacity.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 44 / 50 Handling multisource information Non-destructive fusion : set-valued case

Given n information items A1,... An, deﬁne a basic assignment β : 2Ω → {0, 1} such that

β(Ai ) = 1, ∀i = 1,... n; β(A) = 0 otherwise.

Consider the capacity γ(A) = maxAi ⊆A β(Ai ). From γ, one can recover all non redundant information items Ai , provided that none is less speciﬁc than any other : they are minimal sets B such that γ(B) = 1

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 45 / 50 Handling multisource information Question-answering using Belnap truth-values

We can describe the epistemic status of each statement x ∈ A in view of the information provided by the sources : γ(A), γ(A) Interpretation Belnap truth-value 1 0 x ∈ A is supported “true” 0 1 x ∈ A is negated “false” 0 0 x ∈ A is unknown “unknown” 1 1 x ∈ A is conﬂicting “contradictory” These four states form a bilattice for two orderings The truth ordering : 10 > 00 >01 ; 10 > 11 >01 The information ordering : 11 > 01 > 00 ; 11 > 10 > 00

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 46 / 50 Handling multisource information

The logic of Boolean capacities BC (D, Prade, Rico)

The MEL language fragment of the monotonic modal logic EMN (no nesting of modalities) (PL) Axioms of PL for L -formulas (RM): p → q, whenever ` p → q. (N): >. (P): ♦>, where ♦p stands for ¬¬p. Modus ponens This modal logic is the natural logical account of Boolean capacities Semantics : γ |= p stands for γ([p]) = 1 This logic is sound and complete wrt Boolean capacities. No adjunction : p ∧ q 6|= p ∧ q Its usual semantics is neighbourhood semantics.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 47 / 50 Handling multisource information Embedding reasoning under inconsistency in the logic of capacities (Ciucci, D, this conf.)

In BC, p ≡ ∨i i p, where i p is a KD modality γ |= p ⇐⇒ (E1,..., En) |= p where E1,..., En are focal sets of γ, understood as “p is supported by one source” :

E1 ⊆ [p] or ... or En ⊆ [p].

Belnap truth-values and Belnap logic can be captured in this logic. If we restrict to capacities such that γ(A ∩ B) = min(γ(A), γ(B)) when A ∩ B 6= ∅, then the maximal consistent subset approach is retrieved.

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 48 / 50 Handling multisource information Non-destructive fusion : possibility distributions

Given n information items π1, . . . πn, build the graded capacity

n γ(A) = max Ni (A). i=1

From γ, one can recover all information items πi , provided that none is less speciﬁc than any other : they are the most speciﬁc elements of the core {π : N ≤ γ} Any capacity is of this form. We can describe the epistemic status of each statement x ∈ A in view of the information provided by the sources using an extension of Belnap bilattice by comparing pairs (γ(A), γ(A))

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 49 / 50 Handling multisource information Conclusion

Capacities as a unifying framework for reasoning under incomplete and conﬂicting information Bridge between symbolic and numerical information : 3 and 4-valued logics, epistemic logic, possibility theory, belief functions, imprecise probabilities Future work : Link with argumentation : viewing argument supports as sources, and using argument evaluation methods to build capacities Link with imprecise probabilities : beyond coherence

Representing all numerical capacities by conﬂicting probability sets

g(A) = maxπ∈QC(g) infP∈C(Π) P(A) (Brüning and Denneberg)

Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 50 / 50