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 : !1 T !2 means that !1 is at least as plausible as !2 according to T . Didier Dubois (IRIT) Symbolic vs quantitative uncertainty July 2017 5 / 50 Information items T 7! (C(T ); S(T ); T ) 8!1 2 S(T );!2 62 S(T ) implies !1 T !2. 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 (! ∼T ! ; 8!; ! 2 Ω). Complete knowledge : T ! such that S(T !) = f!g (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} f0; 1g Adjunction N(φ ^ ) = min(N(φ); N( )) (φ ^ ) ≡ φ ^ 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 = fa; b; c;:::; p;:::g α; β; : : : propositional formulae of L built using conjunction, disjunction, and negation (^; _; :) 2 Modal level : A propositional language L Epistemic propositional variables : V = f α : α 2 Lg 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 2 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 [ fK; D; Necg `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 j= α Satisfiability E j= 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 fE 6= ; : E ⊆ [α]g ⊆ 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