LOGIC IN PLAY
Two Perspectives on Logic World description: tracing the structure of reality. Structured social activity: conversation, argumentation,...!!!
Compatible and Interacting Views Process Product ambiguity in natural language: e.g., Logische Aufbau der Welt Our purpose today: explore the process perspective. Processes and activities bring actors to the fore. Games are a natural prism for this purpose.
I Logic and Games: A Natural Combination
Evaluation Games Dispute about truth and falsity of ϕ in model M played between Verifier and Falsifier., Verifier s choice,, for Falsifier, role switch General: logical constants mirror spectrum of game operations.
Truth, Falsity and Winning Strategies Fact A first-order formula ϕ is true in model M ( ) iff Verifier has a winning strategy in game(ϕ, M) Excluded Middle ϕ ϕ Evaluation games are determined: one of the two players has a winning strategy.
Zermelo s Theorem Fact Two-player zero-sum games with finite depth are determined. Proof: Excluded Middle unpacked, x y Rxy x y Rxy, Computable solution: coloring algorithm. Game equilibrium definable in modal fixed-point logic: WIN i <=>!
Logic and Game Equivalence When are two games the same? Distribution says this is so if players have the same powers. Natural alternative looks at how : players actions and choices. Then game invariance is like modal process bisimulation that can simulate strategies step by step on each side.
Invariances and Zoom Levels Invariance under transformations and origins of language. Different natural levels of identity, different logics of games. Zoom levels in describing structure: logicians can dig deep into detail, but also soar in the sky seeing patterns from far above.
Entanglement in Two Directions Logic and games form a natural combination. Two directions: Logic => Games Games => Logic logics of games logics as games Iterations are possible, too: L(G(L)), G(L(G)), Main topic for today: logic of games.
More On Logic as Games Evaluation games are just one example. Argumentation is a game. Computation is a game. Proofs are strategies for winning such games. Dual reading: laws of logic as game-theoretic principles.!
II Logic of Intelligent Agency Games are played by agents: These display a wide range of abilities, that can all be studied in logic. We now explore a few basic features of interactive agency. The resulting logics apply far beyond games.
Many Information Sources The Restaurant Three people order drinks: water, beer, wine. A new waiter comes with 3 glasses. There are 6 ways these could be distributed. Solved by 2 questions, 1 inference zhi wen shuo qin Knowledge comes from: hearing from others, proof, or experience.!!!
Social Knowledge About Others Intelligence seldom comes alone. Asking a question: knowledge about others. Logical structure abounds, with new notions. For instance, common knowledge in groups: C G ϕ (ϕ E G C G ϕ)
Three Card Game John, Mary, Paul get one card each: John Red, Mary White, Paul Blue Mary asks John: Do you have the blue card? Who knows what now? John answers: No. Who knows what now?
Information Change and Updates John and Mary know the cards, Paul does not. But Paul knows that they know, and in fact, this is common knowledge in the group.
Dynamic Logics of Information All informational acts satisfy precise logical axioms. Learning laws describe what happens to one s existing knowledge through an informational event: [!ϕ]k i ψ (ϕ K i (ϕ [!ϕ]ψ)) Cooperation of philosophy, linguistics, and computer science.!!!
Proven Methods, Broader Scope!!!!! Mathematical methods of logic still apply.! Information dynamics has laws extending our usual repertoire.
From Information Processing to Agency Agency and games are more than information processing: Reasoning toward the bold-face (or any other) strategy involves knowledge, action, but also preferences and beliefs. Messy, irrational? To the contrary.
Benchmark: Backward Induction Backward Induction algorithm Iteratively remove strictly dominated nodes, until we reach the root. Logical form: rationality-in-belief, definable in fixed-point logics, or even high-level modal logic of preference and best action.
Players Can Do Much More Logical activities: before, during, and after a game. Pregame deliberation Postgame rationalization During the game: observing informational events, but also Dealing with surprises, belief revision, hypotheses about variety of player types. Game-theoretic methods such as Forward Induction.
Dynamic Logic of Belief Revision Our actions are driven by belief as much as knowledge. [!ϕ]β ι ψ (ϕ Β ι ϕ [!ϕ]ψ) + " [ ϕ]β ψ χ (Ε(ϕ [ ϕ]ψ) Β ϕ [ ϕ]ψ [ ϕ]χ) ( Ε(ϕ [ ϕ]ψ) Β [ ϕ]ψ [ ϕ]χ)" Logical laws describe new beliefs as new information comes in. This can come in a great variety of formats/policies: either in hard totally reliable form!ϕ or as soft plausibility ϕ. From always being correct to intelligent correction.
Logicians/Philosophers of Science
Long Term Phenomena Sui Generis Single informational actions form longer histories of games. Strategies are plans enforcing histories, obeying logical laws: {G, i}ϕ {G, j}always {G, i}ϕ) Histories, finite or infinite, show new structure of their own, such as success or failure in converging to a fixed-point when running an update or deliberation procedure. Surprising scenarios: self-fulfilling, self-refuting assertions. Can be dealt with in dynamic temporal logics of agency. The same in formal learning theory with limit learning, or in many social scenarios other than games. Current challenge: interfacing with dynamical systems.
Theory of Play Logic + Game Theory * = Theory of Play. Challenges include: * describe relevant variety of logical tasks, * find the underlying (static and) dynamic laws, * determine natural kinds of agent diversity. Logic also works outside of pure settings, but the right laws may be more subtle to find. * + a bit of computer science.
III Illustrations All Around The agency perspective can be taken to virtually any topic at this congress or beyond to find new issues or connect old ones. We merely give a few examples.
Epistemology: Foundation or Correction Traditional emphasis on secure cumulative knowledge claims. But mistakes and recoveries show logical ability at its best. Correction is the more exciting goal, not just correctness. Old ideal : foundations, safe haven once and for all. But in games and agency, also more dynamic view: Logic is the immune system of the mind.
Natural Language The medium with which we describe the world, but also, and perhaps primarily, communicate with each other.!!! Language as agency: all the earlier topics apply. Language of agency. Above our dynamic logics, higher zoom level for reasoning about beliefs, decisions, actions. Natural logic with cue words like hope, fear, ought,
Game Theory!! A new look at many game-theoretic themes, such as information-dynamic scenarios for game solution. One example. Even the very notion of game identity at stake. It becomes player-dependent, e.g., because of agents preferences and beliefs, and this shows in the logic.
Game Equivalence for Rational Players is not Backward Induction-equivalent to More general equivalence between games + play styles?
Agents Inside Logic Itself Agent types inside logic. Logical systems as used. Major challenge. Give up hidden uniformity assumptions. Find parameters for different agents using the same system. Example: memory, automata and pebble games. Example: agents with inferential resources. What if agents are very different (say, human versus machine)? Related. Reconsider the current diversity of logics as diversity of cooperating or competing agents.
IV Logic Meets Reality Logic as source of computational devices that change our world. Logic and the empirical facts of human cognition. Logic and the needs of society.
Strategic Interaction and New Games Original motivation: sabotaging algorithmic tasks. Rational agents should also cope with such changes. Teaching Game Try to escape!! NEMO children At same time: new logic with model changing in evaluation.
Games and Cognition Use (new) games as a free lab to see how humans behave. More generally, mixed forms of natural and designed behavior. Turing extended: games a universal model for cognition? (Helsinki 1990)
Normative versus Descriptive Cognitive psychology, neuroscience: rivals, or partners? The traditional challenges to logic: mistakes, emotion. Recently also: mindless animal models perform just as well. Dividing line: logic is normative, cognitive science descriptive. But * All our logics were normative, but to create relevant logic, normative and descriptive perspectives must come together. Logic of agency goes beyond Is, Ought to study Improve. Requires interaction of descriptive and normative stances.
Agent-Rich Conclusion There is a lot of logical content to agency, with games as a striking example. The perspective of agency is fun to explore in itself. And it suggest new perspectives on many existing things.
Agent-Free Conclusion