2 Update as model change. Hard information: learning P eliminates - PDF document

1 LOGICAL DYNAMICS OF INTELLIGENT INTERACTION Johan van Benthem, http://staff.science.uva.nl/~johan Monday 12 Dec 2011, ILLC Amsterdam We motivate the logical dynamics of information-driven agency, and then illustrate the resulting research program by means of a few scenarios, showing the sort of new logical systems that arise. 1 Logic as information dynamics Main idea (many sources): bring dynamic acts and events driving agency inside logic, not just their static products. Informational powers: inference, observation – argumentation: The Restaurant information from many sources, inference and questions The Cards social information and multi-agent interaction. (Three children: rwb , one card each. 2 asks 1: “Do you have the blue card?” ) Logical analysis of all basic informational processes: “Zhi: Wen, Shuo, Qin” 知 问 说 亲 Mohists ( 500 BC ): knowledge arises from questions , proof , and experience Need dynamic logics of inference, observation, questions, communication in general. 2 Pilot system: from epistemic logic to public announcement logic Observation as shrinking a semantic range of options: the Cards go from 6 to 4 to 2 . rwb 1 rbw rwb rwb 2 3 2 2 bwr wbr 3 bwr 3 3 1 3 1 1 brw 2 wrb brw wrb wrb Epistemic content of questions/answers also involves iterated and common knowledge. Static logic. Language p |¬ φ | φ∨ψ | K i φ , models M = (W, {~ i | i ∈ G}, V), relations ~ i . Knowledge as semantic information: M , s |= K i φ iff for all t ~ i s : M , t |= φ . Common knowledge M , s |= C G φ : φ true for all t reachable from s by finite sequences of ~ i steps.

2 Update as model change. Hard information: learning P eliminates worlds with P false: from M s to M |P s P ¬P Language extension: M , s |= [!P] φ iff if M , s |= P, then M |P, s |= φ Dynamics ⇒ truth value change, order-dependence: effect of !¬Kp; !p vs. !p ; !¬Kp . Also, [!p]Kp valid, but the self-refuting Moore sentence M = p ∧ ¬Kp has [!M]K¬M. Theorem PAL is axiomatized completely by epistemic logic plus recursion axioms : [!P]q P → q for atomic facts q ↔ [!P]¬ φ P → ¬[!P] φ ↔ [!P]( φ∧ψ ) ↔ [!P] φ ∧ [!P] ψ [!P]Ki φ ↔ P → Ki(P → [!P] φ ) key recursion axiom Proof Reduction dynamic logic to completeness/validity in static base language. A forgotten law? [!P][!Q] φ ↔ [!(P ∧ [!P]Q)] φ Methodology ‘Dynamify’ existing static logics, making the underlying actions explicit. Dynamic superstructure. Compositional analysis of post-conditions: recursion equations. Requires enough pre-encoding in static logic. E.g., [!P]C G φ needs epistemic extension: P ∧ [!P] ψ [!P] φ [!P]C G ψ φ ↔ C G ‘conditional common knowledge’ General points We just identified the dynamic logic of updating with hard information. Even so, new links with epistemology, philosophy of language, philosophy of science. Technical points Simple, new mathematical problems. ☛ Stanford Logical Dynamics Lab! Open Problem For which syntactic forms of proposition φ do we have [! φ ]K φ ? Open Problem Algebraization: axiomatize the schematic validities of PAL . 3 From correctness to correction: belief change and learning Beliefs are crucial in practical reasoning. Faster inference, but need correction skills. Belief and conditional belief Models order epistemic ranges by relative plausibility: 1, p, q ≤ 2, ¬p, q ≤ 3, p, ¬q M , s |= B i φ iff M , t |= φ for all worlds t minimal in the ordering λ xy. ≤ i, s xy M , s |= B i ψ φ iff M , t |= φ for all ≤ i, s -minimal worlds in {u | M , u |= ψ } (pre-encoding) Static base logic for this: the standard principles of the minimal conditional logic.

3 Belief change under hard information Axiom for belief change under hard facts: P [!P] φ [!P]B i φ ↔ P → B i Theorem The logic of belief change under public announcements is axiomatized by (a) static logic of B i ψ φ , (b) PAL , (c) recursion axiom for conditional belief P ∧ [!P] ψ [!P] φ [!P]B i ψ φ ↔ P → B i Further scenarios, richer logics ‘Misleading true information’ creates false beliefs: from 1, p, q ≤ 2, ¬p, q ≤ 3, p, ¬q to 1, p, q ≤ 2, ¬p, q Motivates a new notion of ‘safe belief’ (as truth in all more plausible worlds) in between knowledge and belief. The optimal epistemic-doxastic repertoire is still under discussion. Belief change under soft information Beijing 1912 : where to leave Qing dynasty sign? Soft versus hard: the new information is believed, but might still be false… Modeling: ‘soft information’ does not eliminate, but changes the plausibility order of the worlds. Radical upgrade ⇑ P changes the current belief model M to M ⇑ P : P -worlds now better than all ¬P -worlds; within zones, old order remains . P M , s |= [ ⇑ P] φ iff M ⇑ P, s |= φ ¬P Theorem The dynamic logic of radical upgrade is axiomatized by base logic plus recursion axioms, with key principle ( E is the epistemic existential modality) [ ⇑ P] B ψ φ ↔ ( Ε ( P ∧ [ ⇑ P] ψ ) ∧ B P ∧ [ ⇑ P] ψ [ ⇑ P] φ ) ∨ ( ¬ Ε ( P ∧ [ ⇑ P] ψ ) ∧ B [ ⇑ P] ψ [ ⇑ P] φ ) Variations Other revision policies: ‘conservative revision’ ↑ P just raises the ‘ best P ’. General logic: one ‘Priority Update Rule’ plus richer format of input signals. General points From guaranteed correctness to tandem of inference, observation and acts of correction (learning from errors). Talent for revising theories instead of foundations. Technical points Logics of model change. Dealing with common belief, ‘belief merge’.

4 4 From local to global: temporal behavior and procedural information ‘Making sense’ of local actions has to bring in procedures of inquiry over time. This happens even in some standard logical systems, such as intuitionistic logic. Program operations Sequential composition ; , guarded choice IF THEN ELSE , iteration WHILE DO . Even parallel composition || . Typical example: Puzzle of the Muddy Children : “At least one of you is dirty. Now keep stating whether you know your status.” Common knowledge is achieved in the limit models #( M , ϕ ) of iterated updates ! ϕ : DDD DDD 3 1 3 1 * CDD 2 DDC * CDD 2 DDC DCD 1 3 DCD 2 1 2 3 2 2 CDC 1 3 CDC 2 CCD DCC 1 3 3 1 CCD DCC CCC Much richer structure in temporal information flow: complex actions, limit behavior. Limit models. If #( M , ϕ ) is non-empty, then ϕ is common knowledge (‘self-fulfilling’), if #( M , ϕ ) is empty, ¬ ϕ is common knowledge (‘self-refuting’): how to define this? Logics of programs are useful such as propositional dynamic logic PDL , µ –calculus: Fact Limit update models for special ‘positive-existential’ formulas ϕ are definable in the modal µ –calculus. Arbitrary formulas require inflationary µ –calculus. Open Problem Characterize the self-fulfilling and self-refuting formulas syntactically. Complexity may explode in this temporal setting (tiling problems become encodable): Theorem PAL plus Kleene iteration of updates is Π 1 1 - complete. Epistemic temporal logic on trees, epistemic links between nodes. Histories constrained by a protocol for process of communication or inquiry. Generalizes the earlier updates: Theorem PAL is axiomatizable and decidable on epistemic-temporal trees. The crucial recursion axioms change to allow for ‘procedural information’: <!P>q ↔ (<!P>T ∧ q), <!P>Ki φ ↔ <!P>T ∧ Ki(<!P>T → [!P] φ ) If one adds temporal operators and common knowledge, Π 1 1 -completeness strikes again. Belief change over time Links with Learning Theory, Game Theory (social strategic interaction). More complex: possible cycles in relation change, richer fixed-point logics.