An Information Theoretic Representation of Agent Dynamics as Set Intersections

An Information Theoretic Represe n tat ion of Agen t Dynamics as Set In tersections Samuel Epstein and Marg rit Betke Boston Universit y , 11 1 Cummington S t, Boston, MA 02215 { samepst, betke } @cs .bu.edu Abstract. W e represent agen ts as sets of strings. Each string enco des a potential intera ction wi th another agen t or environmen t. W e represent the total set of dynamics b etw een tw o agents as th e intersectio n of their respective strings, w e prove complexity prop erties of play er interactions using Algorithmic Information Theory . W e show how the prop osed con- struction is compatible with U niversal Artificial Intellig ence, in that the AIXI mod el can b e seen as universal with resp ect to intera ction. 1 Keywords: Universa l Artificial Intelligence, AIX I Mod el, K olmogorov Complexit y , Algorithmic In formation Theory 1 In tro duction Whereas classical Information Theor y is c o ncerned with q uantif ying the ex- pec ted num b er of bits needed for communication, Algor ithmic Infor mation The- ory (AIT) principally studies the complex ity of individual strings. A central measure of AIT is the Ko lmogorov Complexity C ( x ) of a string x , which is the size o f the smallest progr am that will output x on a universal T uring machine. Another central definition of AIT is the universal pr io r m ( x ) that w eight s a hypothesis ( string) by the complexity of the prog rams that pr o duce it [L V08]. This univ ersal prior has many remark a ble pr op erties; if m ( x ) is used for in- duction, then a ny computable s equence can b e learned with only the minimum amount of data . Unfortunately , C ( x ) and m ( x ) are not finitely computable. Algorithmic Infor mation Theor y can b e interpreted a s a gene r alization of classi- cial Information Theory [CT91] and the Minim um Description Length principal. Some other applications include universal P AC lea rning a nd Algor ithmic Statis- tics [L V08,GTV01]. The question of whether AIT can b e used to form the foundatio n o f Artificial Int elligence was answered in the affir mative with Hutter’s Universal Artificial Int elligence (UAI) [Hut04]. This was achiev ed b y the application of the universal prior m ( x ) to the c y be r netic a g ent mo del, where a n ag ent co mmunicates with an environmen t through sequential cyc le s of a ction, p erception, a nd reward. It was shown that there exists a univ ersal age nt , the AIXI model, that inherits many universality prop erties from m ( x ). In particula r, the AIXI mo del will conv erge to achieve optimal rewards given long eno ug h time in the environment. 1 The authors are grateful to Leonid Lev in for insightful discussions and ac knowl edge partial supp ort by N SF grant 0713229. As almost all AI pro blems can b e formalized in the cyb er netic agent mo del, the AIXI mo del is a complete theoretical solution to the field of Artificial General Int elligence [GP07]. In this pap er , we represent agents as s ets o f s tr ings a nd the p otential dyna m- ics b etw een them a s the int ersection of their resp ective sets of string s (Sec. 2). W e c o nnect this interpretation of interacting agents to the cyb ernetic a gent mo del (Sec. 2.2). W e provide ba ckground o n Algorithmic Information Theory (Sec. 3) and show how age nt le a rning can b e describ ed with algor ithmic com- plexity (Sec. 4). W e apply combinatorial and algor ithmic pro of techniques [VV10] to study the dy namics b etw een agents (Sec. 5 ). In par ticular, we desc rib e the approximation of a gents (Th. 2), the co nditions for remov al of sup erfluous in- formation in the enco ding of an agent (Th. 3), and the consequences of having m ultiple pay ers achieving the s ame rewards in a n e nvironment (Th. 4). W e show how the int erpretation given in Sec. 2 is compatible with Universal Artificial Int elligence, in that the AIXI mo del has universalit y pr op erties with r esp ect to our definition of “interaction” (Sec. 6 ). 2 In teraction as In tersection W e define play ers A a nd B as tw o sets co ntaining strings of size n . Each str ing x in the in tersection set A ∩ B repr esents a particular “ int eraction” b etw een play ers A and B . W e will use the terms string and inter action interc hangeably . This s et r epresentation can b e used to enco de non-co op era tive games (Sec. 2.1) and instances of the cyb ernetic agent mo del (Sec. 2.2). Uncertainties in instances of b oth do ma ins ca n b e enco ded int o the s ize o f the intersections. T he amo un t of uncertaint y betw een players is equa l to | A ∩ B | . If the interaction b etw een the players is deterministic then | A ∩ B | = 1. If uncer taint y exists, then multiple int eractions are poss ible and | A ∩ B | > 1. W e say that play er A int er acts with B if | A ∩ B | > 0 . 2.1 Non-co op erativ e Games Sets c a n be use d to enco de adversaries in sequential games [RN09], where agents exchange a series o f actio ns ov er a finite num ber of plies. Each game or in- teraction consis ts of the reco rding o f actions by adversar ie s α a nd β , with x = ( a 1 , b 1 )( a 2 , b 2 )( a 3 , b 3 ) for a game of three r ounds. The play er (set) rep- resentation A of adversary α is the set of games repr esenting all p ossible actions by α ’s adversary with α ’s resp onding a ctions, and similarly for play er B r epre- senting adversary β . An exa mple game is ro ck-paper- scissors whe r e adversaries α and β play tw o sequential r ounds with an ac tio n s pace of { R , P, S } . Adversary α only plays ro ck, wher eas a dversary β first plays pap er, then copies his a dver- sary’s play of the firs t ro und. The c orresp o nding play ers (sets) A a nd B can be seen in Fig . 1a . The intersection set of A and B contains the single interaction x =“ ( R, P )( R, R ),” which is the only p ossible game (interaction) that α and β can play . Example 1 (Chess Game). W e use the exa mple o f a chess game with uncerta int y betw een t w o players: Anatoly as white and Bor is as black. An interaction x ∈ A B (R,R)(R,R) (R,P)(R,R) (R,R)(R,P) (R,P)(P , R) (R,R)(R,S) (R,P)(S,R) (R,P)(R,R) (P ,P)(R, P) (R,P)(R,P) (P ,P)(P ,P) (R,P)(R,S) (P ,P)(S ,P) (R,S)(R,R) (S,P)(R,S) (R,S)(R,P) (S,P)(P ,S) (R,S)(R,S) (S,P)(S,S ) working tape ... working tape ... Environment µ y 1 y 2 y 3 y 4 y 5 y 6 y 7  r 1 o 1 r 2 o 2 r 3 o 3 r 4 o 4 r 5 o 5 r 6 o 6 r 7 o 7  Agent p perception x: action y : Fig. 1. ( a) The set representation of play ers A and B pla ying tw o games of pa- p er, ro ck, scissors. The intersection set of A and B contai ns the single interaction x =“( R, P )( R, R ).” (b ) The cyb ernetic agent mo del. A ∩ B be tw een Ana to ly and Boris is a ga me of chess play ed for at most m plies for each player, with x = a 1 b 1 a 2 b 2 . . . a m b m = a b 1: m . The chess mov e space V ⊂ { 0 , 1 } ∗ has a short binar y enco ding, whose precise definition is not imp or tant. If the game ha s not ended a fter m rounds, then the game is c onsidered a dr aw. Both play ers are nondeterministic, where at every ply , they can choose from a selection of mov es. Anatoly’s decisions c a n be r epresented by a function f A : V ∗ → 2 V and similarly Bo ris’ decisions by f B . Anatoly can b e represe nt ed b y a set A , with A = { a b 1: m : ∀ 1 ≤ k ≤ m a k ∈ f A ( a b 1: k − 1 ) } , and simila r ly B o ris by set B . Their int ersection, A ∩ B , r epresents the se t of po ssible g a mes that Anatoly and Bor is can play together. Generally , s e ts can enco de adversaries o f non-co o pe r ative normal form g a mes, with their in teractions consisting of pure Nash equilibriums [RN09]. A normal form g ame is defined as ( p, q ) with the a dversaries represented b y nor malized pay off functions p and q of the form { 0 , 1 } n × { 0 , 1 } n → [0 , 1]. The set of pur e Nash equilibr iums is { h x, y i : p ( x, y ) = q ( y , x ) = 1 } . F or each pay off function p there is a player A = {h x, y i | p ( x, y ) = 1 } , and fo r ea ch pay off function q there is play er B . The intersection of A and B is equa l to the set of pure Nash equilibriums of p and q . 2.2 Cyb ernetic Ag en t Mo del The in terpretation of “interaction as in tersection” is also applicable to the cy- ber netic agent mo del used in Universal Ar tificial Intelligence [Hut04]. With the cyb ernetic agent mo del, there is an agent and an environment co mm unicating in a series of cycles k = 1 , 2 , . . . (Fig. 1 b). At cycle k , the a gent p erforms an action y k ∈ Y , dep endent o n the pr evious history y x 0 } . Play er B µ m,τ represents all pos sible histories of µ (how ever unlikely) wher e the re ward is at least τ . If A p m ∩ B µ m,τ = ∅ , then environment µ is “ to o difficult” for the ag ent p ; there is no interaction wher e the agent can receive a reward of at lea st τ . W e say the ag ent p inter acts with the environment µ a t time horizon m a nd difficulty τ if A p m ∩ B µ m,τ 6 = ∅ . Example 2 (Peter and Magnus). W e present a cyb ernetic ag e nt mo del in ter- pretation of c hess with r eward based players Peter and Magnus (same rules as example 1 ). Peter, the agent p , has to b e deterministic wher eas Magnus, the environmen t µ , ha s uncer taint y . A t cycle k , ea ch action y k is Peter’s mov e a nd each p er ception x k is Magnus’ mov e. At ply m in the chess game, Magnus re- turns a reward of 1 if Peter has won. In rounds where the game is unfinished o r if Peter loses or draws, the reward is 0. The player (set) A p m represents Peter’s plays for m rounds. The play er (set) for Ma g nus with difficulty thr eshold τ = 1 and m plies, B µ m, 1 is the set of all g ames that Magnus loses in m rounds or less. If A m ∩ B µ m, 1 = ∅ , then Peter cannot int er act with Magnus at difficulty level 1; Peter can never b eat Magnus at chess in m r o unds or less. If A m ∩ B µ m, 1 6 = ∅ then Peter can b eat Mag nus at a g a me o f chess in m r ounds or less. Another construction of a play er D µ m,τ with resp ect to environment µ , is D µ m,τ = { y x 1: m : ∀ k V ∗ ,µ 1: m ( y x 1: k ) /V ∗ ,µ 1: m ≥ τ } . With this in terpretation, play er D µ m,τ represents all histories where at each time k , 1 ≤ k ≤ m , an agent can po tent ially achiev e an exp ected r eward of at lea st τ times the optimal exp ected reward. I f A p m ∩ D µ m,τ = ∅ , then environment µ is “to o difficult” for the agent p ; there is no interaction where a t every cycle k the agent has the p o tent ial to receive an exp ected re ward o f at le ast τ V ∗ ,µ 1: m . 3 Bac kground in Algorithmic Information Theory W e denote finite binary strings by x ∈ { 0 , 1 } ∗ and the length of string s by l ( x ). Let the pairing function h· , ·i b e the standa rd one- to-one ma pping from N × N to N , where: h x, y i = x ′ y = 1 l ( l ( x )) 0 l ( x ) xy and l ( h x, y i ) = l ( y ) + l ( x )+ 2 l ( l ( x ))+ 1. The Kolmog o rov complex it y C ( x ) is the length of the shortes t binary progr am to co mpute x on a universal T uring machine ψ , C ( x ) = min { l ( d ) : ψ ( d ) = x } . The pr efix-free Kolmo gorov complexity , K ( x ), restricts the universal machine ψ so no halting prog r am is a prop er pr efix o f another ha lting program. F or the rest of this pap er, we use plain K olmogor ov complexity . Ko lmogorov complexity is no t finitely computable. The conditional Ko lmogorov complex it y of x rela tive to y , C ( x | y ), is defined as the length of a sho rtest pro gram to co mpute x , us ing y as an a uxiliary input to the computation. The complex it y of tw o strings x and y is denoted by C ( x, y ) = C ( h x, y i ). The conditional co mplexity of tw o strings is C ( x | y , z ) = C ( x |h y , z i ). The complexity of information in x ab out y is I ( x : y ) = C ( y ) − C ( y | x ). The conditiona l mutual informatio n is I ( x : y | z ) = C ( y | z ) − C ( y | x, z ) a nd can b e interpreted as the informa tio n z r eceives ab out y when given x . The co mplexity of a function f : { 0 , 1 } ∗ → { 0 , 1 } ∗ is C ( f ) = min { C ( p ) : ∀ x ψ ( p, x ) = f ( x ) } . The Levin complexit y is defined by C t ( x ) = min p { l ( p ) + log t ( p, x ) : ψ ( p ) = x } , with t ( p, x ) b eing the num ber of steps taken by ψ until x is printed (without ψ necess ary halting). Levin co mplex it y is computable. The complexity of a finite set S is C ( S ), the length of the s ho rtest pro g ram f from which the univ ersal T uring machine ψ computes a listing of the elemen ts of S and then halts. If S = { x 1 , . . . , x n } , then ψ ( f ) = h x 1 , h x 2 , . . . , h x n − 1 , x n i . . . ii . The conditional complexity C ( x | S ) is the length of the shor tes t prog ram from which ψ , given S literally a s auxiliary information, computes x . F or every set S containing x , it must b e that C ( x | S ) ≤ log | S | + O (1). The rando mness deficiency is the lack o f typicalit y of x with resp ect to set S , with δ ( x | S ) = lo g | S | − C ( x | S ), for x ∈ S and ∞ o ther wise. If δ ( x | S ) is small enough, then x is a typical element of S ; x satisfies all simple pr o p erties that hold with hig h ma jor ities of s trings in S . Example 3 (Anatoly’s Games). Chess play er Anato ly with function f A can b e represented a s a set A (see example 1). Set A is simple relative to f A and the maximum nu m be r of plies m , with C ( A | f A , m ) = O (1), wher e O (1 ) is the length of co de required to use f A and m to enum erate all ga mes x ∈ A . The fo llowing theorem, used in section 5, shows that if a string x is contained by a large num ber of sets of a certain co mplexity , then it is contained by a simpler set [VV04]. The enumerative complexity , C E ( F ), is the Ko lmogorov complexity of a non ha lting progra m tha t enumerates all the sets F ∈ F . This theorem also holds for conditional complexity b ounds, C ( F | y ). Theorem 1 ([VV04]). L et F b e a family of su bset s of a set of strings G . If x ∈ G is an memb er of e ach of 2 k sets F ∈ F with C ( F ) ≤ r , t hen x is a memb er of a set F ′ in F with C ( F ′ ) ≤ r − k + O (log k + log r + log log |G | + C E ( F )) . 4 Pla y er Strategy Learning Play ers A and B ca n lear n information ab out each o ther’s strateg ies from a single int eraction (game) x ∈ A ∩ B or from their entire interaction set (all po s sible games) A ∩ B . The c ap acity of a play er A is the maximum amount o f information that A ca n re ceive ab out ano ther player through a ll p oss ible interactions, i.e . their int eraction set. It is equal to the log of the num ber of p ossible subsets that it can hav e, log 2 | A | = | A | . W e define the lack of typicalit y of a subset S with resp ect to A to b e δ ( S | A ) = | A | − C ( S | A ), for S ⊆ A and ∞ other wise. Example 4 (Cap acity). B o ris B use s a range of black o pe nings wherea s Bill B ′ uses o nly the Sicilia n defence. So Boris has a higher capacity , | B | ≫ | B ′ | , and can po tent ially lear n more than Bill. Example 5 (R andomness Deficiency). Let A b e the chess games play ed by Ana- toly . Bo b is a simple pla yer B ′ , who only mov es his knight back and forth. Set S = A ∩ B ′ represents a ll A ’s games with B ′ . The rando mnes s deficiency of these games, δ ( S | A ), is high, as S is e a sily computable from A , with C ( S | A ) ≪ | A | . Let T ⊆ A , in which T = A ∩ B are ga mes play ed b etw een Anatoly and Boris, who uses a range of chess strategies unknown to Anatoly . Then δ ( T | A ) is low and C ( T | A ) is high. If A views every interaction in A ∩ B , the amount of informatio n B r e veals ab out itself is, I ( A ∩ B : B | A ), the mutual infor ma tion b etw een B and A ∩ B , given A . This term can b e r educed to C ( A ∩ B | A ) − C ( A ∩ B | A, B ) = C ( A ∩ B | A ) + O (1 ). W e define the amo unt of k nowledge that A received ab out B fr om the interaction setp as: R ( B | A ) = C ( A ∩ B | A ) . (1) The higher the r andomness deficiency , δ ( A ∩ B | A ), o f an interaction se t, A ∩ B , with resp e ct to play e r A , the less information, R ( B | A ), play er A can rece ive ab out its opp onent B , with R ( B | A ) + δ ( A ∩ B | A ) = | A | . (2) Play er A receives the most informa tion ab out its opp onent when the ra ndomness deficiency is δ ( A ∩ B | A ) ≈ 0. Example 6. Let Anatoly , A , and Bob, B ′ , b e the play ers of example 5. Bob has a simple strategy and has a low er capacity | B ′ | ≪ | A | , but he learns a lot fr o m Anatoly , with δ ( A ∩ B ′ | B ′ ) ≈ 0 a nd R ( A | B ′ ) ≈ | B ′ | . Anatoly lear ns very little from Bob, with R ( B ′ | A ) ≈ 0 and δ ( A ∩ B ′ | A ) ≈ | A | . Play ers c an reveal informatio n a bo ut themse lves through a s ing le interaction. The amount of information that A rec e ived a b out B from their interaction x is I ( x : B | A ) = C ( x | A ) − C ( x | A, B ) . (3) A graphica l depiction of the complexities relating to A , B , and x c an b e seen in Fig. 2. W e define the lack of typicalit y of an interaction x with res pe ct to bo th play ers to b e δ ( x | A, B ) = log | A ∩ B | − C ( x | A, B ) (4) A (Anatoly) B (Boris) C(A|x,B) C(B|x,A) C(x|A,B) I(x:B|A) I(x:A|B) x C(B|A) C(A|B) C(x|B) C(x|A) ∩ ∩ ∩ ∩ Fig. 2. The complex ities and information of A , B , and th eir interaction x . The rela- tionships hold up t o logarithmic precision. for x ∈ A ∩ B and ∞ otherw is e. If δ ( x | A, B ) is small, then x repr esents a typical int eraction. T he infor ma tion pa ssed fro m player B to player A through a single int eraction is represe n ted by I ( x : B | A ) + δ ( x | A ) = log | A | / | A ∩ B | + δ ( x | A, B ) . (5) The informatio n passe d b etw een players through a single in teraction with the same capacity is I ( x : B | A ) + δ ( x | A ) = I ( x : A | B ) + δ ( x | B ) + O (1) . (6) Example 7. Anatoly A plays a game x with Boris B who has the same capacity with | A | = | B | . Anatoly tricks Boris with a King’s gambit and the game x follows a series of mov es extremely familia r to Anato ly . Boris reac ts with the most obvious mov e at every turn. In this c ase the game is simple to Anatoly , with δ ( x | A ) being la r ge and I ( x : B | A ) be ing sma ll. The game is new to Boris with δ ( x | B ) be ing small and I ( x : A | B ) b eing lar ge. Thus Boris learns mo re than Anatoly from x . If the play ers hav e a deter ministic interaction, then A ∩ B = { y } a nd the information A received from B reduces to I ( y : B | A ) + δ ( y | A ) = log | A | . 5 Pla y er Approxima tion and Interaction Complexit y W e s how that, g iven an interaction x betw een play ers A and B , A c an “ c on- struct” an approximate player B ′ that ha s int eraction x using a small num b er of extra bits ǫ , where C ( B ′ | A, x ) = ǫ . W e als o show that the conditional co mplex it y C ( B ′ | A ) o f the appr oximate player B ′ is no t gre ater than the amount of infor- mation I ( x : B | A ) that A obtains ab out B (up to logar ithmic pr e cision). W e use the simplified notation log A = log | A | . W e also use the play er spac e notation, B , to denote a set o f sets of s trings. Theorem 2. Given ar e a player sp ac e B and players A and B ∈ B over strings of size n with x ∈ A ∩ B and C ( B ) = O (lo g n ) . Then t her e is a player B ′ ∈ B with x ∈ B ′ , C ( B ′ | A, x ) = O ( s ) , and C ( B ′ | A ) ≤ I ( x : B | A ) + O ( s ) , with s = log C ( B | A ) + log n . Pr o of. Let r = C ( B | A ). W e define G as the s et of strings of size n , with log log |G | = log n . W e set F = B , a nd so C E ( F ) = O (log n ). Let N b e the nu m be r of s ets S ∈ B , with C ( S | A ) ≤ r a nd x ∈ S . W e first show that C ( B | A, x ) ≤ log N + O (log nr ). There is a pro gram, that when given x , A , B , and r , with C ( B , r ) = O (log nr ), can enumerate a ll sets in B containing x with conditional co mplexity to A b eing less than or equa l to r . Thus B ca n b e created using such a pr o gram and an index of siz e ⌈ lo g N ⌉ . By the application of Theorem 1, conditional on A , with k = ⌊ log N ⌋ , there is a set B ′ ∈ F with x ∈ B ′ and C ( B ′ | A ) ≤ r − k + O (log nr ) ≤ C ( B | A ) − C ( B | A, x ) + O (log nr ) = I ( x : B | A ) + O (log nr ). T o prove C ( B ′ | A, x ) = O ( s ), a ssume B ′ is the set satisfying the ab ov e prop er ties tha t minimizes C ( B ′ | A ) up to pr ecision O ( s ). It must b e tha t C ( B ′ | A, x ) = O ( s ). Otherwise C ( B ′ | A, x ) = ω ( s ) and there is a set B ′′ satisfying prop erties ab ov e and C ( B ′′ | A ) ≤ C ( B ′ | A ) − C ( B ′ | A, x ) + O ( s ) = C ( B ′ | A ) − ω ( s ), causing a contradiction. Example 8 (Opp onent R e c onstru ction). Anatoly , A , plays a chess game x with Boris, B , with x ∈ A ∩ B . The play ers use a r a ndom string b of size C ( x | A, B ) to help decide their mov es. Without using b , Anatoly can “constr uc t” B ob, B ′ , a n imper sonator of Boris, using information from the game x and O (log C ( B | A ) + log l ( x )) bits. Bob ca n play the same game x with Anatoly . Given are play ers A and B who inter ac t , in that A ∩ B 6 = ∅ . W e show that there exists a n interacting play er B ′ that has complexity b ounded by the m utual infor mation of A and B . If theorem 1 can b e streng thened such that the enum erative complexity ter m C E ( F ) is re placed by C E E ( F ), the co mplex it y o f enum erating b o th the sets and the elements of the sets of F , then the precisio n of theorems 3 and 4 can be strengthened with the replace ment of the Lev in complexity term C t ( A ) with Kolmo gorov complexity C ( A ). Theorem 3. Given ar e a player sp ac e B and players A and B ∈ B with A ∩ B 6 = ∅ . Then ther e exists a player B ′ ∈ B with A ∩ B ′ 6 = ∅ , and C ( B ′ ) ≤ I ( A : B ) + O ( s ) , with s = log C ( B ) + log C t ( A ) + C ( B ) . Pr o of. Let r = C ( B ), h = C t ( A ), and q = 2 C ( B ) . W e define G = {h S i : C t ( S ) ≤ r } , with h S i being an enco ding o f set S . This implies log log |G | = O (log h ). W e define F with a r ecursive function λ : B → F , with λ ( S ) = { h T i | C t ( T ) ≤ h, S ∩ T 6 = ∅} . It m ust b e C ( λ ) = O (log h ). The enumeration complexity of F requires the enco ding of B a nd λ , and s o C E ( F ) = O (log hq ). Thus if h T i ∈ λ ( S ), then T ∩ S 6 = ∅ . Let N b e the num ber of s ets S ∈ B , with C ( S ) ≤ r and S ∩ A 6 = ∅ . Thu s C ( B | A ) ≤ log N + O (log hq r ), as ther e is a pr ogra m, when given A , r , B , and an index of siz e ⌈ lo g N ⌉ , that c a n r eturn any such S . By the application of Theorem 1, with x = h A i and k = ⌊ log N ⌋ , there is a set F ∈ F with x ∈ F and C ( F ) ≤ r − k + O (log hq r ) ≤ C ( B ) − C ( B | A ) + O (lo g hq r ) = I ( A : B ) + O (lo g hq r ). A set B ′ ∈ B , with λ ( B ′ ) = F , can b e easily recovered fro m F by enumerating all sets in B , applying λ to each one, and selecting the first one which pro duce s F . So C ( B ′ ) ≤ C ( F ) + O (log q ) ≤ I ( A : B ) + O (log hq r ). Since h A i ∈ λ ( B ′ ), it m ust b e that A ∩ B ′ 6 = ∅ . W e show that if a play er A int er acts with num erous play ers of a given co m- plexity and uncertaint y , then there exists a simple play er B ′ who in teracts with A with the same uncertaint y . Theorem 4. Given ar e player sp ac e B , player A and 2 k players B ∈ B wher e for e ach B , 0 < | A ∩ B | ≤ c and C ( B ) ≤ r . Ther e is a player B ′ ∈ B such that 0 < | A ∩ B ′ | ≤ c and C ( B ′ ) ≤ r − k + O ( s ) , with s = log C t ( A ) + log c + log k + log r + C ( B ) . Pr o of. Let h = C t ( A ) and q = 2 C ( B ) . W e ca n define G ⊆ { 0 , 1 } ∗ as a set of strings, each enco ding a set (play e r ) S whose Levin complexity is less than or equal to h . This implies log log |G | = O (log h ). W e represent the enco ding of S with h S i . W e define F with a re cursive function λ : B → F , with λ ( S ) = {h T i | C t ( T ) ≤ h, 0 < | S ∩ T | ≤ c } . Th us it must b e C ( λ ) = O (lo g ch ). The enum eration complex ity of F r equires the enco ding of c , h , and B , with C E ( F ) = O (log chq ). Th us if h T i ∈ λ ( S ), then player T and pla yer S ha v e a no n empty int ersection of size at most c . F rom the a ssumptions of this theor em, h A i is cov ered by a t least 2 k sets λ ( B ) ∈ F of complexity C ( λ ( B )) ≤ r + O (log chq ). By the application of Theor em 1 , with x = h A i , ther e is a s et F ∈ F with x ∈ F , C ( F ) ≤ r − k + O (log ( chk q r )). A set B ′ ∈ B , with λ ( B ′ ) = F can be recov ered fro m F by enumerating all s ets in B , applying λ to each one, a nd selecting the first o ne which pro duces F . Therefore C ( B ′ | F ) ≤ O (lo g chq ) and so C ( B ′ ) ≤ C ( F ) + O (log chq ) ≤ r − k + O (log ( chk q r )). Since h A i ∈ λ ( B ′ ), it m ust b e that 0 < | A ∩ B ′ | ≤ c , thus the theorem is prov en. Example 9. An example applica tion of theorem 4 is a g ame of the sa me form as example 2. Magnus, represented by set B , plays 2 k games of against 2 k young play ers A ∈ A . F ur thermore the play ers and Ma gnus ar e deter ministic with for each A ∈ A , | A ∩ B | = 1. The difficulty threshold τ , is set to 1, so every one of the young pla y ers b ea t Ma gnus. By theorem 4, if all play ers A ∈ A hav e complexity at most C ( A ) ≤ r , then there is a simpler play er A ′ ∈ A that will win a g ainst Magnus, with C ( A ′ ) ≤ r − k + ǫ (with ǫ b eing of loga rithmic order) and | A ′ ∩ B | = 1. 6 F uture W ork: Unive rsal In t eraction Since the a gents a nd environmen ts of the cyb ernetic agent mo del o f Sectio n 2.2 can b e transla ted into set representations, ther e is po tent ial a pplication o f the pro of techn iques used in Section 5 to Artificial Universal Intelligence [Hut04], and in particular to describ e prop erties of the AIXI mo del. The univ ersal en viron- men t, ξ , is defined using a form of the universal pr ior, m ( x ) = P p : ψ ( p )= x 2 − l ( p ) , representing a s emimeasure (degenerate pr obability) ov er a ll infinite strings , with ξ ( y x 1: k ) = P ρ 2 − K ( ρ ) ρ ( y x 1: n ). The universal environment ξ is the weigh ted sum- mation ov er all chronological en vironment s ρ . The ter m K ( ρ ) repres ents the pr efix fr e e Kolmogo rov co mplexit y o f ρ . The AIXI mo del p ξ m is the optimal agent for the environment ξ with hor izon m , in that p ξ m = arg max p V p,ξ 1: m . The sequence o f self optimizing AIXI a gents for each time horiz o n is { p ξ i } i =1 , 2 ,... . Let M b e a set of environments where a sequenc e of self-optimizing p olicies ˜ p m ex- ists. The s e quence conv erges to re c eive the o ptimal av erage for a ll environments with ∀ ν ∈ M : 1 m V p m ,ν 1: m m →∞ − → 1 m V ∗ ,ν 1: m . By theorem 5.29 from [Hut04], it must b e that the seq ue nc e of AIXI a gents is optimal for M with 1 m V p ξ m ,ν 1: m m →∞ − → 1 m V ∗ ,ν 1: m . W e use the conv ersion of ag ents p and environmen ts µ to sets A p m and D µ m,τ as int ro duced at the end Se c tio n 2.2. The s equence of self o ptimizing AIXI agents, { p ξ i } i =1 , 2 ,... , is universal w ith regar d to interaction with res pec t to M . It is easy to see that fo r all τ and a ll environmen ts ν ∈ M , there is a num ber m ν τ where for all m > m ν,τ , A p ξ m m ∩ D µ m,τ 6 = ∅ . This implies a set re pr esentation of agent dynamics can b e used to des crib e further prop erties of the AIXI mo del. There is p otential fo r a deep connection, roughly analo gously to how prefix-free K ol- mogorov complexity a nd the universal prior are rela ted with the Co ding Theorem K ( x ) = − log m ( x ) + O (1) [L V08]. 7 Conclusions W e used Algo rithmic Informa tion Theory to quantify the information exchanged betw een agents that interact in non- c o op erative games (Sec. 4). W e hav e shown that a n ag ent A can constr uct an approximation of his o ppo nent B us ing infor- mation from a single interaction (game) with B (Th. 2). W e hav e shown that if an agent B with sup erfluous information in teracts with a n en vironment A and achiev es a certain reward, then there exists another a gent B ′ without this information tha t can achieve the s ame reward (Th. 3). W e hav e als o shown that if multiple agents in teract with an environment to achiev e a certain r eward, then there exists a simple a gent who can achiev e the same reward (Th. 4 ). 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