Knowledge, Strategies, and Know-How

KR 2018, Tempe, AZ, USA Knowledge, Strategies, and Know-How Jia Tao Lafayette College Pavel Naumov Claremont McKenna College http://reasoning.eas.asu.edu/kr2018/

"!! Knowledge, Strategies, Know-How!!?!! #!!

Epistemic Logic! # Alice K A (Alice has a pumpkin mask) Bob K B (Alice has a pumpkin mask) K C K B (Alice has a pumpkin mask) Cathy K C (Alice has a pumpkin mask) K A (K B (Alice has a pumpkin mask) K C (Alice has a pumpkin mask))

Epistemic Logic!! # #! Alice Bob # Cathy K B (Alice has a pumpkin mask)

Epistemic Logic!! # #! Alice Bob # Cathy K A (K B (Alice has a pumpkin mask) K C (Alice has a pumpkin mask))

Epistemic States! # Alice Bob Cathy Cathy Alice! Alice Bob Cathy Bob Alice! # # Bob Cathy

Epistemic Model (W, { a } a A, π) W a π is a set of states is an indistinguishability equivalence relation is a valuation function w p iff w π(p) w φ iff w φ w φ ψ iff w φ or w ψ w K a φ iff u φ for all u W such that w a u

Model Example p a p b u v w u p u K a p v K a p v K b p w p w K a p w K b p w K b (K a p K a p)

Indistinguishability Relation Reflexive u a u Symmetric u a v v a u Transitive u a v v a w u a w

Multiagent S5 all propositional tautologies K a φ φ K a φ K a K a φ K a φ K a K a φ K a (φ ψ) (K a φ K a ψ) φ, φ ψ ψ φ K a φ

Proof of Positive Introspection Lemma 1 K a φ K a K a φ Proof K a K a φ K a φ K a φ K a K a φ K a K a φ K a K a K a φ K a φ K a K a K a φ K a φ K a K a φ K a K a φ K a φ K a ( K a K a φ K a φ) K a K a K a φ K a K a φ K a φ K a K a φ

Soundness and Completeness Theorems If φ, then w φ for each state w of each epistemic model. If w φ for each state w of each epistemic model, then φ.

Alice s Knowledge! #! #! Alice Bob # Cathy K A ("Either Bob or Cathy has a ghost mask.") K A ("Cathy has a ghost mask.")

Bob s Knowledge! #! #! Alice Bob K B ("Either Alice or Cathy has a ghost mask.") K B ("Cathy has a ghost mask.") # Cathy

Distributed Knowledge! #! #! #! #! Alice Bob # Cathy K A,B ("Cathy has a ghost mask.")

Epistemic States! # Alice Bob Cathy Cathy Alice! Alice Bob Cathy Bob! # # Alice Bob Cathy

Epistemic Model (W, { a } a A, π) W a π is a set of states is an indistinguishability equivalence relation is a valuation function Notation: w C u if a C(w a u). w p iff w π(p) w φ iff w φ w φ ψ iff w φ or w ψ w K C φ iff u φ for all u W such that w C u

Distributed Knowledge Axioms all propositional tautologies K C φ φ K C φ K C K C φ K C (φ ψ) (K C φ K C ψ) K C φ K D φ, where C D φ, φ ψ ψ φ K C φ

Soundness and Completeness Theorems If φ, then w φ for each state w of each epistemic model. If w φ for each state w of each epistemic model, then φ.

Group Knowledge

Forms of Group Knowledge Individual Knowledge Alice and Bob both know the value of x Distributed Knowledge Alice knows x, Bob knows y, they distributedly know the value of x+y Common Knowledge two generals problem

Logic of Coalition Power

"!! Strategies!!?!! #!!

Coalition Strategies! Alice Bob # Cathy 0/1 0/1 0/1 sum=3 p p u S a p u v sum=1,2 u S a p p 0,1 u S abc p s v S abc p 0,1,2 2,3 v S Ø p sum=0,1 p w S abc p w w S ab p s S a p u S abc S Ø p s S a

Game Definition (W, Δ, M, π) W is a set of states. Δ is a nonempty domain of actions. M W Δ A W is a mechanism. π is a function from propositional variables into subsets of W.

Formal Semantics w p iff w π(p), w φ iff w φ, w φ ψ iff w φ or w ψ, w S C φ iff there is an action profile s Δ C of coalition C such that for each complete action profile δ Δ A and each state u W if s = C δ and (w, δ, u) M, then u φ.

Examples of Statements S a p = Alice has an action that guarantees p in the next state. S Ø p = p is unavoidable in the next state. S Ø = there is no next state. S a S b p = Alice has an action after which Bob will not have an action that guarantees p (S a p S b p) S ab p = Although neither Alice nor Bob has an action that guarantees p they have a joint action that does.

Marc Pauly s Logic of Coalitional Power all propositional tautologies S C (φ ψ) (S D φ S C D ψ), where C D = Ø φ, φ ψ ψ φ S C φ M. Pauly, A modal logic for coalitional power in games, Journal of Logic and Computation (2002)

Example of Derivation Lemma 1. S C φ S D φ, where C D. Proof. φ φ S D C (φ φ) Since C D, by the Cooperation axiom, S D C (φ φ) (S C φ S D φ) S C φ S D φ

Soundness and Completeness Theorems If φ, then w φ for each state w of each game. If w φ for each state w of each game, then φ.

Coalition Strategies! 0/1 p a,b,c p Alice u sum is odd sum is even v 0/1 s p Bob sum is even p sum is odd # 0/1 w Cathy u S abc p?

Knowledge and Strategies

Game with Imperfect Information a,b,c u S a p! Alice 0/1 0/1 u sum=0,1 s p sum=3 v u S a,b p v S a,b p u K a,b S a,b p Bob u S a,b,c p # 0/1 sum=2,3 w p sum=0,1,2 v S a,b,c p Cathy u K a,b,c S a,b,c p

Game with Imperfect Information (W, { a } a A, Δ, M, π) W is a set of states. a is an indistinguishability equivalence relation. Δ is a nonempty domain of actions. M W Δ A W is a mechanism. π is a function from propositional variables into subsets of W. Thomas Ågotnes and Natasha Alechina (2012). "Epistemic Coalition Logic: Completeness and Complexity." International Conference on Autonomous Agents and Multiagent Systems

Formal Semantics w p iff w π(p), w φ iff w φ, w φ ψ iff w φ or w ψ, w S C φ iff there is an action profile s Δ C of coalition C such that for each complete action profile δ Δ A and each state u W if s = C δ and (w, δ, u) M, then u φ, w K C φ iff u φ for all u W such that w C u.

Distributed Knowledge and Strategies all propositional tautologies K C φ φ φ, φ ψ φ φ ψ K C φ S C φ K C φ K C K C φ K C (φ ψ) (K C φ K C ψ) K C φ K D φ, where C D S C (φ ψ) (S D φ S C D ψ), where C D = Ø Completeness: Thomas Ågotnes and Natasha Alechina (2012). "Epistemic Coalition Logic: Completeness and Complexity." International Conference on Autonomous Agents and Multiagent Systems

"!! Know-How!!?!! #!!

Know-How a,b! Alice 0/1 0/1 u sum=0,1 s p sum=3,2 v u S a,b p v S a,b p u K a,b S a,b p Bob u H a,b p # 0/1 sum=2,3 w p sum=0,1 u H a,b,c p Cathy

Another Example a,b,c! Alice 0/1 0/1 u sum=0,1 s p sum=0 v u S a,b p v S a,b p u H a,b p Bob u H a,b,c p sum=2,3 p sum=1,2,3 # Cathy 0/1 w

Formal Semantics w S C φ iff there is an action profile s Δ C of coalition C such that for each complete action profile δ Δ A and each state u W, if s = C δ and (w, δ, u) M, then u φ w H C φ iff there is an action profile s Δ C of coalition C such that for each state v and each complete action profile δ Δ A and each state u W, if w C v, s = C δ, and (v, δ, u) M, then u φ

Term Know-How Strategies uniform strategies - van Benthem (2001) difference between an agent knowing that he has a suitable strategy and knowing the strategy itself - Jamroga and van der Hoek (2004) knowledge to identify and execute a strategy - Jamroga and Ågotnes (2007) knowingly doing Broersen (2008) knowing how - Wang (2015) knows how or knowledge de re - Ågotnes and Alechina (2016) executable or know-how strategy - Naumov and Tao (2017)

Knowledge and Know-How all propositional tautologies K C φ φ φ, φ ψ φ φ ψ K C φ H C φ K C φ K C K C φ K C (φ ψ) (K C φ K C ψ) K C φ K D φ, where C D H C (φ ψ) (H D φ H C D ψ), where C D = Ø H C φ K C H C φ K Ø φ H Ø φ

Strategic Negative Introspection Lemma 1. (Alechina, private communication) Proof. H C φ K C H C φ H C φ K C H C φ K C H C φ H C φ K C H C φ H C φ H C φ K C H C φ K C ( K C H C φ H C φ) K C K C H C φ K C H C φ K C H C φ K C K C H C φ H C φ K C H C φ K C H C φ K C H C φ

Strategic Monotonicity Lemma 2. H C φ H D φ, where C D. Proof. φ φ H D C (φ φ) H D C (φ φ) (H C φ H (D C) C φ) H C φ H (D C) C φ H C φ H D φ

Knowledge and Know-How all propositional tautologies K C φ φ φ, φ ψ φ φ ψ K C φ H C φ K C φ K C K C φ K C (φ ψ) (K C φ K C ψ) K C φ K D φ, where C D H C (φ ψ) (H D φ H C D ψ), where C D = Ø H C φ K C H C φ K Ø φ H Ø φ

Soundness and Completeness Theorems If φ, then w φ for each state w of each game with imperfect information. If w φ for each state w of each game with imperfect information, then φ.

Break

Multi-Step Strategies a q 0 1 0 p! 0/1 w u Alice 0 1 1 a w S a p u S a q w H a p u H a q

Single-Player Multi-Step Strategies to Achieve a Goal all propositional tautologies Kφ φ Kφ K Kφ K(φ ψ) (Kφ Kψ) Hφ KHφ HHφ Hφ Hφ HKφ Kφ Hφ H (due to verifiability) φ, φ ψ φ φ ψ φ(p) ψ Kφ Hφ Hψ φ[ψ/p] Completeness: Raul Fervari, Andreas Herzig, Yanjun Li, Yanjing Wang (2017), Strategically Knowing How, International Joint Conference on Artificial Intelligence (IJCAI)

Coalitional Multi-Step Strategies to Maintain a Goal all propositional tautologies H C (φ ψ) (H D φ H C D ψ), where C D = Ø K C φ φ H C φ K C φ K C φ K C K C φ H C φ H C H C φ K C (φ ψ) (K C φ K C ψ) K Ø φ H Ø φ K C φ K D φ, where C D φ, φ ψ ψ φ H C φ Completeness: Pavel Naumov and Jia Tao (2017), Coalition Power in Epistemic Transition Systems, International Conference on Autonomous Agents and Multiagent Systems (AAMAS)

Blind Date Failure! # K a ("date is at 6pm") K b ("date is at 6pm") K a K b ("date is at 6pm") K b K a ("date is at 6pm") K a K b K a ("date is at 6pm") K b K a K b ("date is at 6pm") K a K b K a K b ("date is at 6pm") K b K a K b K a ("date is at 6pm")

Common Knowledge! # C a,b,c ("date is at 6pm") 6pm, enjoy Cathy

Common Knowledge and Strategies all propositional tautologies S A (φ ψ) (S B φ S A B ψ), where A B = Ø K a φ φ K a φ K a K a φ φ, φ ψ ψ φ C A φ φ S A φ K a (φ ψ) (K a φ K a ψ) C A φ a A K a (φ C A φ) ψ a A K a (φ ψ) ψ C A φ Completeness: Thomas Ågotnes and Natasha Alechina (2012). "Epistemic Coalition Logic: Completeness and Complexity." International Conference on Autonomous Agents and Multiagent Systems

Common-Know-How? CH A (φ ψ) (CH B φ CH A B ψ), where A B = Ø

Distributed Knowledge, Strategies, and Know-How all propositional tautologies H C (φ ψ) (H D φ H C D ψ), where C D = Ø K C φ φ H C φ K C H C φ K Ø φ H Ø φ K C φ K C K C φ φ, φ ψ φ φ φ K C (φ ψ) (K C φ K C ψ) ψ K C φ S C φ H C φ K C φ K D φ, where C D H C φ S C φ S C S C (φ ψ) (S D φ S C D ψ), where C D = Ø H C (φ ψ) (K C S Ø φ H C ψ) Completeness: Pavel Naumov and Jia Tao (2018), Together We Know How to Achieve: An Epistemic Logic of Know-How, Journal of Artificial Intelligence

No Perfect Recall! Alice 0/1 w 0 u p a v 1 w S a p u K a p w S a K a p w H a p w H a K a p

Perfect Recall Semantics (w 0, δ 1, w 1,, δ n, w n ) p iff w π(p), (w 0, δ 1, w 1,, δ n, w n ) φ iff (w 0, δ 1, w 1,, δ n, w n ) φ, (w 0, δ 1, w 1,, δ n, w n ) φ ψ iff (w 0, δ 1, w 1,, δ n, w n ) φ or (w 0, δ 1, w 1,, δ n, w n ) ψ, (w 0, δ 1, w 1,, δ n, w n ) K C φ iff (w 0, δ 1, w 1,, δ n, w n ) φ for all (w 0, δ 1, w 1,, δ n, w n ) such that (w 0, δ 1, w 1,, δ n, w n ) C (w 0, δ 1, w 1,, δ n, w n ) (w 0, δ 1, w 1,, δ n, w n ) H C φ iff there is an action profile s Δ C of coalition C such that for each history (w 0, δ 1, w 1,, δ n, w n ), each complete action profile δ Δ A, and each state u W if (w 0, δ 1, w 1,, δ n, w n ) C (w 0, δ 1, w 1,, δ n, w n ), s = C δ, and (w n, δ, u) M, then (w 0, δ 1, w 1,, δ n, w n, δ, u) φ

Knowledge and Know-How with Perfect Recall all propositional tautologies H C (φ ψ) (H D φ H C D ψ), where C D = Ø K C φ φ H C φ K C H C φ K Ø φ H Ø φ K C φ K C K C φ φ, φ ψ φ φ K C (φ ψ) (K C φ K C ψ) ψ K C φ H C φ K C φ K D φ, where C D H D φ H D K C φ, where D C Ø H C Completeness: Pavel Naumov and Jia Tao (2018), Strategic Coalitions with Perfect Recall, AAAI Conference on Artificial Intelligence (AAAI 18)

Linear Plans 010 q 0 1 0 p! 0/1 w u Alice 0 1 1 H(q, p) H(p, q)

Linear Plans 000 p p q q! Alice 0/1 0 1 0 0 1 1 0 1 H(p, q)

Axioms for Single-Agent Linear Plans all propositional tautologies H(φ, ψ) NH(φ, ψ) Nφ φ H(φ, ψ) N H(φ, ψ) N(φ ψ) (Nφ Nψ) H(φ, ψ) (H(ψ, χ) H(φ, χ)) φ, φ ψ ψ φ Nφ N(φ ψ) H(φ, ψ) Completeness: Yanjing Wang (2016), A logic of goal-directed knowing how, Synthese

Linear Plans with Intermediate Constraints 10 p 0 1 0 r! 0/1 w u Alice q 0 1 1 1 H(p, q, r)

Axioms for Plans with Intermediate Constraints all propositional tautologies Nφ φ H(φ, τ, ψ) NH(φ, τ, ψ) H(φ, τ, ψ) N H(φ, τ, ψ) N(φ ψ) (Nφ Nψ) φ, φ ψ ψ φ Nφ H(φ, τ, ψ) (H(ψ, τ, χ) (N(ψ τ) H(φ, τ, χ))) N(φ ψ) H(φ,, ψ) H(φ, τ, ψ) ( H(φ,, ψ) H(φ,, τ)) N(φ φ) (H(φ, τ, ψ) H(φ, τ, ψ)) N(ψ ψ ) (H(φ, τ, ψ) H(φ, τ, ψ )) N(τ τ ) (H(φ, τ, ψ) H(φ, τ, ψ)) Completeness: Yanjun Li and Yanjing Wang (2017), Achieving while maintaining: A logic of knowing how with intermediate constraints, Indian Conference on Logic and Its Applications

Second-Order Know-How a,b! Alice 0/1 0/1 u sum=0,1 s p sum=3,2 v u S a,b p v S a,b p u K a,b S a,b p Bob u H a,b p # 0/1 sum=2,3 w p sum=0,1 u H a,b,c p Cathy u H a,b c p

Second-Order Know-How w H C φ iff there is an action profile s Δ C of coalition C such that for each state v and each complete action profile δ Δ A and each state u W if w C v, s = C δ, and (v, δ, u) M, then u φ w H D C φ iff there is an action profile s ΔD of coalition D such that for each state v and each complete action profile δ Δ A and each state u W if w C v, s = D δ, and (v, δ, u) M, then u φ

Axioms for Second-Order Know-How all propositional tautologies H D 1 C 1 (φ ψ) (H D 2 C 2 φ H D 1 D 2 C 1 C 2 ψ), where D 1 D 2 = Ø K C φ φ K C φ K C K C φ K C (φ ψ) (K C φ K C ψ) H D C φ K C HD C φ K C H Ø D φ HØ C φ K Ø φ H Ø Ø φ K C φ K D φ, where C D φ, φ ψ ψ φ K C φ φ H D C φ Completeness: Pavel Naumov and Jia Tao (2018), International Conference on Autonomous Agents and Multiagent Systems (AAMAS 18)

Strategies and Responsibility R a φ agent a used strategy that made φ unavoidable. Nφ statement φ is always true N a φ N a φ Nφ φ Nφ N Nφ N(φ ψ) (Nφ Nψ) R a φ φ R a φ R a R a φ R a (φ ψ) (R a φ R a ψ) Nφ R a φ NR a1 φ 1 NR a2 φ 2 NR an φ n N(R a1 φ 1 R a2 φ 2 R an φ n ) Belnap N., Perloff M., Xu M., Facing the Future: Agents and Choices in our Indeterminist World, Oxford, 2001

Group Responsibility R C φ coalition C used strategy that made φ unavoidable. R Ø φ Nφ N a φ N a φ R C φ φ R C R C φ R Ø φ R C φ R C R C φ R C (φ ψ) (R C φ R C ψ) R C φ R D φ, where C D Jan Broersen, Andreas Herzig, and Nicolas Troquard. What groups do, can do, and know they can do: an analysis in normal modal logics. Journal of Applied Non-Classical Logics, 19(3):261 289, 2009

Responsibility and Knowledge

Blameworthiness

Thank you!