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

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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 3 wbr 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 | Kiφ, models M = (W, {~i | i∈G}, V), relations ~i. Knowledge as semantic information: M, s |= Kiφ iff for all t ~i s: M, t |= φ. Common knowledge M, s |= CGφ: 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]Kiφ ↔ 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]CGφ needs epistemic extension: [!P]CG

ψ φ ↔ CG

P ∧ [!P]ψ [!P]φ

‘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]Kφ? 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 |= Biφ iff M, t |= φ for all worlds t minimal in the ordering λxy. ≤i, s xy M, s |= Bi

ψφ 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]Biφ ↔ P → Bi

P[!P]φ

Theorem The logic of belief change under public announcements is axiomatized by (a) static logic of Bi

ψφ, (b) PAL, (c) recursion axiom for conditional belief

[!P]Bi

ψφ ↔ P → Bi

P ∧ [!P]ψ [!P]φ

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] Bψφ ↔ (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

• f 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 CDD DDC DCD CCD CDC DCC CCC 1 3 2 2 2 2 3 1 1 3 1 3 *

DDD CDD DDC DCD CCD CDC DCC 1 3 2 2 2 1 3 1 3 *

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 Π1

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>Kiφ ↔ <!P>T ∧ Ki(<!P>T → [!P]φ) If one adds temporal operators and common knowledge, Π1

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.

5 5 Digression: update by general events with partial observation Partial observation of events, differential information flow. Email: epistemic-dynamic function of cc, bcc. Computer security. Parlor games manipulate information (Clue). Event models A = (E, {~i | i∈G}, {PREe | e∈E}). Relevant events, relations ~i encode what agents cannot distinguish. Events e have preconditions PREe for their execution. For any epistemic model (M, s) and event model (E, e), the product model (MxE, (s, e)) has Domain {(s, e) | s a world in M, e an event in E, (M, s) |= PREe}, Accessibility: (s, e) ~i (t, f) iff both s ~i t and e ~i f, Valuation for atoms p at (s, e) is that at s in M. (can be generalized to world change) Product update deals with misleading actions as well as truthful ones, and with belief as well as knowledge. Epistemic models can even get larger as update proceeds (bcc)! Dynamic-epistemic logic LEA: p | ¬φ | φ∨ψ | Kiφ | CGφ | [E, a]φ : (E, e) any event model with actual event e. Semantics: M, s |= [E, e]φ iff M x E, (s, e) |= φ. Theorem LEA is effectively axiomatizable and decidable. The key recursion axiom is the one extending that for public announcement: [E, e]Kiφ ↔ PREe → ∧ { Ki[E, f]φ)) | f ~i e in E} Challenge: axiomatizing common knowledge or belief in subgroups with secrets: 1 2 1 e f 2 Theorem DEL with recursion axiom for [E, e]CGφ can be axiomatized in epistemic PDL. Proof via Kleene’s Theorem, or alternatively, ‘product closure’ of the modal µ–calculus. Temporal setting Product update iterates to form tree models M x E1, (M x E1) x E2, …. Theorem An ETL-tree is representable via iterated local DEL update iff agents have Perfect Recall, No Miracles, and event domains are bisimulation invariant. But PR and UNM are costly and richer epistemic tree logics can be highly complex. Agent-dependent information flow is crucial to cognition, and DEL with product update is best current mathematical paradigm for understanding many types of information flow.

6 6 Inference and variety of information What does the final inference step do for the waiter? Set of options does not change. Need more fine-grained syntactic notion of information in addition to the semantic one. Logic manipulates many different kinds of information: correlation, range, code, … One solution: dynamic logic of awareness Influential proposal: give each world a set of formulas that the agent is aware of (Aϕ). Epistemic axiom and the gap for an action: Ex(ϕ → ψ) → (Exϕ → [… ] Exψ)

with Ex for ‘explicit knowledge’

Actions #ϕ of realization ‘raise awareness’: adding the conclusion to the formulas associated with the world. Other cases: acts of introspection, explicit seeing, etc. Theorem The dynamic logic of awareness raising is completely axiomatizable. Key recursion axioms: [!ϕ]Aψ ↔ (ϕ → Aψ) and [#ϕ]Aψ ↔ Kϕ → ( Aψ ∨ ψ = ϕ). General points Different notions of information in logic require different dynamic acts. Just a first step toward richer dynamic logics of evidence change and argumentation. Technical point How to merge dynamic logics with standard systems in Proof Theory? 7 Digression: questions Questions direct inquiry by setting the current issue. Static semantics: ‘issue’ is equiva- lence relation. Dotted black lines epistemic links, cells of issue relation are rectangles: p, q p, -q p, q p, -q p, q p, -q ?q !p

• p, q
• p, -q
• p, q -p, -q
• p, q
• p, -q

Acts of issue management M = (W, ~, ≈, V) is an epistemic issue model. s =ϕ t if s, t agree on ϕ. The issue update M?ϕ is (W, ~, ≈?ϕ, V) with s ≈?ϕ t iff s ≈ t and s =ϕ t. Resolve action ! resets epistemic relation to ~ ∩ ≈, refine action ? resets issue relation to ~ ∩ ≈. A complete dynamic logic Some basic recursion axioms for asking and resolving: [?ϕ]Kψ ↔ K[?ϕ]ψ, [!]Kϕ ↔ R[!]ϕ [!ϕ]Kψ ↔ (ϕ ∧ K(ϕ → [!ϕ]ψ)) ∨ (¬ϕ ∧ K(¬ϕ → [!ϕ]ψ)) General perspective: asking questions is as important to inquiry as giving answers. Now we have dealt with all actions of the waiter – maybe not natural border yet:

7 8 From correctness to being reasonable: harmony with evaluation Rational agency: harmony of information & evaluation. Information flow is ‘kinematics’, preference explanatory ‘dynamics’. Logics of preference change use earlier mechanisms. Links with Social Choice and Game Theory. To these areas, our dynamic logics add a ‘Theory of Play’ with fine-structure of deliberation, moves, and different types of agents. 9 Where we stand, changes and invariants Dynamic systems for information-driven action extend the classical scope of the field, combining philosophical and computational logic, methods from mathematical logic. Our themes interface with philosophy, linguistics, agency, game theory and social sciences. External repercussions New contacts with epistemology: informational action, dynamic definitions of knowledge, new attitudes, social epistemology. Philosophy of science: science as informational agency, belief revision and the foundations of mathematics. How this fits with logic itself Two historical faces: pursuing logic in mathematical depth, and expanding the agenda of the field. Two sets of logicians in 19th century Facebook:

Bolzano Boole Frege Peirce

Not a divide. Dynamic ideas in foundations, key role of mathematics in our program. Tarski Beth 10 Main sources

1991, Language in Action, Elsevier Amsterdam & MIT Press Cambridge. 1996, Exploring Logical Dynamics, CSLI Publications, Stanford. 2010, Modal Logic for Open Minds, CSLI Publ’s, Stanford, 2011, Dynamic Logic of Information and Interaction, Cambridge UP.