25:
1909:
128:(RL) agent. It maximizes the expected total rewards received from the environment. Intuitively, it simultaneously considers every computable hypothesis (or environment). In each time step, it looks at every possible program and evaluates how many rewards that program generates depending on the next action taken. The promised rewards are then weighted by the
1587:
3621:
3723:
However, AIXI does have limitations. It is restricted to maximizing rewards based on percepts as opposed to external states. It also assumes it interacts with the environment solely through action and percept channels, preventing it from considering the possibility of being damaged or modified.
1598:
1306:
1040:
2887:
3266:
899:
3376:
2746:
1904:{\displaystyle a_{t}:=\arg \max _{a_{t}}\left(\sum _{o_{t}r_{t}}\ldots \left(\max _{a_{m}}\sum _{o_{m}r_{m}}\left(\sum _{q:\;U(q,a_{1}\ldots a_{m})=o_{1}r_{1}\ldots o_{m}r_{m}}2^{-{\textrm {length}}(q)}\right)\right)\right)}
584:
361:
698:
3043:
3338:
3105:
2613:(which "models" the environment) and all actions of the AIXI agent: in this sense, the environment is "computable" (as stated above). Note that, in general, the program which "models" the
2571:
1241:
2934:
2041:
1582:{\displaystyle a_{t}:=\arg \max _{a_{t}}\sum _{o_{t}r_{t}}\ldots \max _{a_{m}}\sum _{o_{m}r_{m}}\sum _{q:\;U(q,a_{1}\ldots a_{m})=o_{1}r_{1}\ldots o_{m}r_{m}}2^{-{\textrm {length}}(q)}}
398:
306:
2650:
433:
972:
833:
2322:
1133:
3720:
that balanced Pareto optimality is subjective and that any policy can be considered Pareto optimal, which they describe as undermining all previous optimality claims for AIXI.
2974:
265:
2511:
2751:
947:
923:
3110:
2254:
2143:
2101:
2071:
1271:
1163:
132:
that this program constitutes the true environment. This belief is computed from the length of the program: longer programs are considered less likely, in line with
3371:
2197:
2170:
1298:
3710:
3686:
967:
774:
754:
718:
185:
162:
1938:
237:
211:
3616:{\displaystyle \sum _{o_{t}r_{t}}\ldots \max _{a_{m}}\sum _{o_{m}r_{m}}\sum _{q:\;U(q,a_{1}\ldots a_{m})=o_{1}r_{1}\ldots o_{m}r_{m}}2^{-{\textrm {length}}(q)}}
2670:
2611:
2591:
2471:
2451:
2431:
2404:
2382:
2362:
2342:
2274:
2217:
1998:
1978:
1958:
1087:
1060:
3724:
Colloquially, this means that it doesn't consider itself to be contained by the environment it interacts with. It also assumes the environment is computable.
841:
3712:
when the length of the agent's lifetime (not time) goes to infinity. For environment classes where self-optimizing policies exist, AIXI is self-optimizing.
58:
2675:
441:
3658:: there is no other agent that performs at least as well as AIXI in all environments while performing strictly better in at least one environment.
311:
3651:
AIXI's performance is measured by the expected total number of rewards it receives. AIXI has been proven to be optimal in the following ways.
144:
According to Hutter, the word "AIXI" can have several interpretations. AIXI can stand for AI based on
Solomonoff's distribution, denoted by
2893:(in this case, a sum) over all computable environments (which are consistent with the agent's past), each weighted by its complexity
589:
949:
is the space of all possible "percepts" that can be produced by the environment. The environment (or probability distribution)
164:(which is the Greek letter xi), or e.g. it can stand for AI "crossed" (X) with induction (I). There are other interpretations.
3855:
2979:
3752:
limited agent. Another approximation to AIXI with a restricted environment class is MC-AIXI (FAC-CTW) (which stands for
35:
3912:
Veness, Joel; Kee Siong Ng; Hutter, Marcus; Uther, William; Silver, David (2009). "A Monte Carlo AIXI Approximation".
4007:
3271:
136:. AIXI then selects the action that has the highest expected total reward in the weighted sum of all these programs.
76:
3048:
1914:
Intuitively, in the definition above, AIXI considers the sum of the total reward over all possible "futures" up to
167:
AIXI is a reinforcement learning agent that interacts with some stochastic and unknown but computable environment
2516:
1168:
102:
44:
2896:
2003:
2617:
and actual environment (where AIXI needs to act) is unknown because the current environment is also unknown.
98:
1035:{\displaystyle \mu :({\mathcal {A}}\times {\mathcal {E}})^{*}\times {\mathcal {A}}\rightarrow {\mathcal {E}}}
366:
274:
2622:
403:
790:
3761:
2281:
308:(e.g. a limb movement) and executes it in the environment, and the environment responds with a "percept"
2882:{\displaystyle \sum _{q:\;U(q,a_{1}\ldots a_{m})=o_{1}r_{1}\ldots o_{m}r_{m}}2^{-{\textrm {length}}(q)}}
1092:
4031:
2939:
242:
3261:{\displaystyle o_{1}r_{1}\ldots o_{m}r_{m}=o_{1}r_{1}\ldots o_{t-1}r_{t-1}o_{t}r_{t}\ldots o_{m}r_{m}}
2476:
4041:
928:
904:
4036:
3661:
Balanced Pareto optimality: like Pareto optimality, but considering a weighted sum of environments.
2410:
2103:) that can generate that future, and then picks the action that maximises expected future rewards.
721:
54:
3343:
Let us now put all these components together in order to understand this equation or definition.
436:
40:
2222:
2111:
756:
is computable, that is, the observations and rewards received by the agent from the environment
3757:
2593:
is thus used to "simulate" or compute the environment responses or percepts, given the program
2407:
2076:
2046:
1246:
1138:
125:
3107:
is the sequence of actions already executed in the environment by the AIXI agent. Similarly,
2890:
3990:"Universal Algorithmic Intelligence: A mathematical top->down approach", Marcus Hutter,
3813:
3733:
3349:
2175:
2148:
1276:
106:
3954:
3695:
3671:
952:
759:
739:
703:
170:
147:
8:
3737:
1917:
216:
190:
3839:
Universal
Artificial Intelligence: Sequential Decisions Based on Algorithmic Probability
3817:
894:{\displaystyle \pi :({\mathcal {A}}\times {\mathcal {E}})^{*}\rightarrow {\mathcal {A}}}
3991:
3913:
3861:
3803:
3753:
2655:
2596:
2576:
2456:
2436:
2416:
2389:
2367:
2347:
2327:
2259:
2202:
1983:
1963:
1943:
1072:
1045:
732:(as opposed to other RL algorithms). Note again that this probability distribution is
4003:
3851:
3655:
3865:
3740:. However, there are computable approximations of it. One such approximation is AIXI
4011:
3843:
133:
129:
3776:
3837:
2741:{\displaystyle 2^{-{\textrm {length}}(q)}={\frac {1}{2^{{\textrm {length}}(q)}}}}
729:
110:
4015:
3934:
777:
3980:
3797:
2106:
Let us break this definition down in order to attempt to fully understand it.
2043:) consistent with the agent's past (that is, the previously executed actions,
50:
4025:
3799:
A Theory of
Universal Artificial Intelligence based on Algorithmic Complexity
3639:, which need to be chosen. The latter parameter can be removed by the use of
579:{\displaystyle \mu (o_{t}r_{t}|a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}a_{t})}
114:
901:, which is the function it uses to choose actions at every time step, where
117:
in 2000 and several results regarding AIXI are proved in Hutter's 2005 book
2364:, so AIXI needs to look into the future to choose its action at time step
3640:
1063:
356:{\displaystyle e_{t}\in {\mathcal {E}}={\mathcal {O}}\times \mathbb {R} }
3883:
700:
is the "history" of actions, observations and rewards. The environment
3808:
3717:
1089:(which ranges from 1 to m), AIXI, having previously executed actions
969:
can also be thought of as a stochastic policy (which is a function):
3847:
24:
3995:
3842:. Texts in Theoretical Computer Science an EATCS Series. Springer.
2219:
from the environment (which is unknown and stochastic). Similarly,
93:
3918:
2433:
ranges over all (deterministic) programs on the universal machine
3764:
3340:
is the sequence of percepts produced by the environment so far.
724:
over "percepts" (observations and rewards) which depend on the
2513:(that is, all actions), and produces the sequence of percepts
3911:
3744:, which performs at least as well as the provably best time
925:
is the space of all possible actions that AIXI can take and
693:{\displaystyle a_{1}o_{1}r_{1}...a_{t-1}o_{t-1}r_{t-1}a_{t}}
3760:), which has had some success playing simple games such as
16:
Mathematical formalism for artificial general intelligence
3942:. Proceedings of the 28th Conference on Learning Theory.
3631:
The parameters to AIXI are the universal Turing machine
1592:
or, using parentheses, to disambiguate the precedences
1273:), chooses and executes in the environment the action,
838:
The AIXI agent is associated with a stochastic policy
4002:, eds. B. Goertzel and C. Pennachin, Springer, 2007,
3698:
3674:
3379:
3352:
3274:
3113:
3051:
2982:
2942:
2899:
2754:
2678:
2658:
2625:
2599:
2579:
2519:
2479:
2459:
2439:
2419:
2392:
2370:
2350:
2330:
2284:
2262:
2225:
2205:
2178:
2151:
2114:
2079:
2049:
2006:
1986:
1980:), weighs each of them by the complexity of programs
1966:
1946:
1920:
1601:
1309:
1279:
1249:
1171:
1141:
1095:
1075:
1048:
975:
955:
931:
907:
844:
835:, that is, the sum of rewards from time step 1 to m.
793:
762:
742:
706:
592:
444:
406:
369:
314:
277:
245:
219:
193:
173:
150:
3955:"Formalizing Two Problems of Realistic World-Models"
3038:{\displaystyle a_{1}\ldots a_{t-1}a_{t}\ldots a_{m}}
2145:
is the "percept" (which consists of the observation
3704:
3680:
3615:
3365:
3332:
3260:
3099:
3037:
2968:
2928:
2881:
2740:
2672:(which is encoded as a string of bits). Note that
2664:
2644:
2605:
2585:
2565:
2505:
2465:
2445:
2425:
2398:
2376:
2356:
2336:
2316:
2268:
2248:
2211:
2191:
2164:
2137:
2095:
2065:
2035:
1992:
1972:
1952:
1932:
1903:
1581:
1292:
1265:
1235:
1157:
1127:
1081:
1054:
1034:
961:
941:
917:
893:
827:
768:
748:
712:
692:
578:
427:
392:
355:
300:
259:
231:
205:
179:
156:
3981:Playing Pacman using AIXI Approximation – YouTube
1135:(which is often abbreviated in the literature as
776:can be computed by some program (which runs on a
4023:
3411:
1679:
1622:
1377:
1330:
736:to the AIXI agent. Furthermore, note again that
267:is the lifespan of the AIXI agent. At time step
39:, potentially preventing the article from being
3333:{\displaystyle o_{1}r_{1}\ldots o_{t-1}r_{t-1}}
187:. The interaction proceeds in time steps, from
3936:Bad Universal Priors and Notions of Optimality
1165:) and having observed the history of percepts
3835:
3795:
3668:is called self-optimizing for an environment
3100:{\displaystyle a_{1}\ldots a_{t-1}=a_{<t}}
2256:is the percept received by AIXI at time step
780:), given the past actions of the AIXI agent.
3907:
3905:
3903:
3932:
2566:{\displaystyle o_{1}r_{1}\ldots o_{m}r_{m}}
2276:(the last time step where AIXI is active).
1236:{\displaystyle o_{1}r_{1}...o_{t-1}r_{t-1}}
3500:
2929:{\displaystyle 2^{-{\textrm {length}}(q)}}
2766:
2199:) received by the AIXI agent at time step
2036:{\displaystyle 2^{-{\textrm {length}}(q)}}
1773:
1466:
59:reliable, independent, third-party sources
3917:
3900:
3831:
3829:
3827:
3807:
421:
349:
253:
77:Learn how and when to remove this message
3727:
3346:At time step t, AIXI chooses the action
720:is thus mathematically represented as a
53:by replacing them with more appropriate
3692:approaches the theoretical maximum for
393:{\displaystyle o_{t}\in {\mathcal {O}}}
301:{\displaystyle a_{t}\in {\mathcal {A}}}
36:too closely associated with the subject
4024:
3824:
2453:, which receives as input the program
787:goal of the AIXI agent is to maximise
92:
2645:{\displaystyle {\textrm {length}}(q)}
2324:is the sum of rewards from time step
428:{\displaystyle r_{t}\in \mathbb {R} }
363:, which consists of an "observation"
3877:
3875:
828:{\displaystyle \sum _{t=1}^{m}r_{t}}
400:(e.g., a camera image) and a reward
18:
3933:Leike, Jan; Hutter, Marcus (2015).
3884:"Universal Artificial Intelligence"
2317:{\displaystyle r_{t}+\ldots +r_{m}}
13:
3881:
2748:. Hence, in the definition above,
1128:{\displaystyle a_{1}\dots a_{t-1}}
1027:
1017:
997:
987:
934:
910:
886:
866:
856:
385:
340:
330:
293:
14:
4053:
3872:
3716:It was later shown by Hutter and
2969:{\displaystyle a_{1}\ldots a_{m}}
260:{\displaystyle m\in \mathbb {N} }
119:Universal Artificial Intelligence
2506:{\displaystyle a_{1}\dots a_{m}}
1940:time steps ahead (that is, from
34:may rely excessively on sources
23:
4000:Artificial General Intelligence
2573:. The universal Turing machine
435:, distributed according to the
103:artificial general intelligence
3974:
3946:
3926:
3789:
3608:
3602:
3536:
3504:
3486:
3454:
2921:
2915:
2874:
2868:
2802:
2770:
2731:
2725:
2700:
2694:
2639:
2633:
2028:
2022:
1881:
1875:
1809:
1777:
1754:
1722:
1574:
1568:
1502:
1470:
1452:
1420:
1022:
1003:
982:
942:{\displaystyle {\mathcal {E}}}
918:{\displaystyle {\mathcal {A}}}
881:
872:
851:
573:
472:
448:
271:, the agent chooses an action
1:
3782:
3646:
3626:
2652:is the length of the program
1243:(which can be abbreviated as
139:
113:. AIXI was first proposed by
2473:and the sequence of actions
7:
4016:10.1007/978-3-540-68677-4_8
3770:
2889:should be interpreted as a
10:
4058:
3664:Self-optimizing: a policy
2249:{\displaystyle o_{m}r_{m}}
2138:{\displaystyle o_{t}r_{t}}
111:sequential decision theory
3952:
3635:and the agent's lifetime
2096:{\displaystyle e_{<t}}
2073:, and received percepts,
2066:{\displaystyle a_{<t}}
1266:{\displaystyle e_{<t}}
1158:{\displaystyle a_{<t}}
1069:In general, at time step
2411:universal Turing machine
728:history, so there is no
722:probability distribution
2976:can also be written as
437:conditional probability
3796:Marcus Hutter (2000).
3758:Context-Tree Weighting
3706:
3688:if the performance of
3682:
3623:attains its maximum.
3617:
3367:
3334:
3262:
3101:
3039:
2970:
2930:
2883:
2742:
2666:
2646:
2607:
2587:
2567:
2507:
2467:
2447:
2427:
2400:
2378:
2358:
2338:
2318:
2270:
2250:
2213:
2193:
2166:
2139:
2097:
2067:
2037:
1994:
1974:
1954:
1934:
1905:
1583:
1300:, defined as follows:
1294:
1267:
1237:
1159:
1129:
1083:
1056:
1036:
963:
943:
919:
895:
829:
814:
770:
750:
714:
694:
580:
429:
394:
357:
302:
261:
233:
207:
181:
158:
126:reinforcement learning
99:mathematical formalism
3728:Computational aspects
3707:
3683:
3618:
3368:
3366:{\displaystyle a_{t}}
3335:
3263:
3102:
3040:
2971:
2931:
2884:
2743:
2667:
2647:
2608:
2588:
2568:
2508:
2468:
2448:
2428:
2401:
2379:
2359:
2339:
2319:
2271:
2251:
2214:
2194:
2192:{\displaystyle r_{t}}
2167:
2165:{\displaystyle o_{t}}
2140:
2098:
2068:
2038:
1995:
1975:
1955:
1935:
1906:
1584:
1295:
1293:{\displaystyle a_{t}}
1268:
1238:
1160:
1130:
1084:
1057:
1037:
964:
944:
920:
896:
830:
794:
771:
751:
715:
695:
581:
430:
395:
358:
303:
262:
234:
208:
182:
159:
4010:, pp. 227–290,
3762:partially observable
3734:Solomonoff induction
3705:{\displaystyle \mu }
3696:
3681:{\displaystyle \mu }
3672:
3377:
3350:
3272:
3111:
3049:
2980:
2940:
2897:
2752:
2676:
2656:
2623:
2597:
2577:
2517:
2477:
2457:
2437:
2417:
2390:
2368:
2348:
2328:
2282:
2260:
2223:
2203:
2176:
2149:
2112:
2077:
2047:
2004:
1984:
1964:
1944:
1918:
1599:
1307:
1277:
1247:
1169:
1139:
1093:
1073:
1046:
973:
962:{\displaystyle \mu }
953:
929:
905:
842:
791:
769:{\displaystyle \mu }
760:
749:{\displaystyle \mu }
740:
713:{\displaystyle \mu }
704:
590:
442:
404:
367:
312:
275:
243:
217:
191:
180:{\displaystyle \mu }
171:
157:{\displaystyle \xi }
148:
107:Solomonoff induction
3818:2000cs........4001H
3373:where the function
1933:{\displaystyle m-t}
232:{\displaystyle t=m}
206:{\displaystyle t=1}
94:['ai̯k͡siː]
3702:
3678:
3613:
3586:
3453:
3426:
3406:
3363:
3330:
3258:
3097:
3035:
2966:
2926:
2879:
2852:
2738:
2662:
2642:
2603:
2583:
2563:
2503:
2463:
2443:
2423:
2396:
2374:
2354:
2334:
2314:
2266:
2246:
2209:
2189:
2162:
2135:
2093:
2063:
2033:
1990:
1970:
1950:
1930:
1901:
1859:
1721:
1694:
1669:
1637:
1579:
1552:
1419:
1392:
1372:
1345:
1290:
1263:
1233:
1155:
1125:
1079:
1052:
1032:
959:
939:
915:
891:
825:
766:
746:
710:
690:
576:
425:
390:
353:
298:
257:
229:
203:
177:
154:
4032:Optimal decisions
3857:978-3-540-22139-5
3656:Pareto optimality
3599:
3489:
3427:
3410:
3380:
2912:
2865:
2755:
2736:
2722:
2691:
2665:{\displaystyle q}
2630:
2606:{\displaystyle q}
2586:{\displaystyle U}
2466:{\displaystyle q}
2446:{\displaystyle U}
2426:{\displaystyle q}
2399:{\displaystyle U}
2377:{\displaystyle t}
2357:{\displaystyle m}
2337:{\displaystyle t}
2269:{\displaystyle m}
2212:{\displaystyle t}
2019:
1993:{\displaystyle q}
1973:{\displaystyle m}
1953:{\displaystyle t}
1872:
1762:
1695:
1678:
1643:
1621:
1565:
1455:
1393:
1376:
1346:
1329:
1082:{\displaystyle t}
1055:{\displaystyle *}
730:Markov assumption
130:subjective belief
97:is a theoretical
87:
86:
79:
4049:
4042:Machine learning
3983:
3978:
3972:
3971:
3969:
3968:
3962:Intelligence.org
3959:
3950:
3944:
3943:
3941:
3930:
3924:
3923:
3921:
3909:
3898:
3897:
3895:
3894:
3882:Hutter, Marcus.
3879:
3870:
3869:
3836:— (2005).
3833:
3822:
3821:
3811:
3793:
3711:
3709:
3708:
3703:
3687:
3685:
3684:
3679:
3622:
3620:
3619:
3614:
3612:
3611:
3601:
3600:
3597:
3585:
3584:
3583:
3574:
3573:
3561:
3560:
3551:
3550:
3535:
3534:
3522:
3521:
3485:
3484:
3466:
3465:
3452:
3451:
3450:
3441:
3440:
3425:
3424:
3423:
3405:
3404:
3403:
3394:
3393:
3372:
3370:
3369:
3364:
3362:
3361:
3339:
3337:
3336:
3331:
3329:
3328:
3313:
3312:
3294:
3293:
3284:
3283:
3267:
3265:
3264:
3259:
3257:
3256:
3247:
3246:
3234:
3233:
3224:
3223:
3214:
3213:
3198:
3197:
3179:
3178:
3169:
3168:
3156:
3155:
3146:
3145:
3133:
3132:
3123:
3122:
3106:
3104:
3103:
3098:
3096:
3095:
3080:
3079:
3061:
3060:
3044:
3042:
3041:
3036:
3034:
3033:
3021:
3020:
3011:
3010:
2992:
2991:
2975:
2973:
2972:
2967:
2965:
2964:
2952:
2951:
2935:
2933:
2932:
2927:
2925:
2924:
2914:
2913:
2910:
2888:
2886:
2885:
2880:
2878:
2877:
2867:
2866:
2863:
2851:
2850:
2849:
2840:
2839:
2827:
2826:
2817:
2816:
2801:
2800:
2788:
2787:
2747:
2745:
2744:
2739:
2737:
2735:
2734:
2724:
2723:
2720:
2709:
2704:
2703:
2693:
2692:
2689:
2671:
2669:
2668:
2663:
2651:
2649:
2648:
2643:
2632:
2631:
2628:
2612:
2610:
2609:
2604:
2592:
2590:
2589:
2584:
2572:
2570:
2569:
2564:
2562:
2561:
2552:
2551:
2539:
2538:
2529:
2528:
2512:
2510:
2509:
2504:
2502:
2501:
2489:
2488:
2472:
2470:
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2464:
2452:
2450:
2449:
2444:
2432:
2430:
2429:
2424:
2405:
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2402:
2397:
2383:
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2375:
2363:
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2355:
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2323:
2321:
2320:
2315:
2313:
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2294:
2293:
2275:
2273:
2272:
2267:
2255:
2253:
2252:
2247:
2245:
2244:
2235:
2234:
2218:
2216:
2215:
2210:
2198:
2196:
2195:
2190:
2188:
2187:
2171:
2169:
2168:
2163:
2161:
2160:
2144:
2142:
2141:
2136:
2134:
2133:
2124:
2123:
2102:
2100:
2099:
2094:
2092:
2091:
2072:
2070:
2069:
2064:
2062:
2061:
2042:
2040:
2039:
2034:
2032:
2031:
2021:
2020:
2017:
1999:
1997:
1996:
1991:
1979:
1977:
1976:
1971:
1959:
1957:
1956:
1951:
1939:
1937:
1936:
1931:
1910:
1908:
1907:
1902:
1900:
1896:
1895:
1891:
1890:
1886:
1885:
1884:
1874:
1873:
1870:
1858:
1857:
1856:
1847:
1846:
1834:
1833:
1824:
1823:
1808:
1807:
1795:
1794:
1753:
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1734:
1733:
1720:
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1718:
1709:
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1693:
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1657:
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1636:
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1634:
1611:
1610:
1588:
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1585:
1580:
1578:
1577:
1567:
1566:
1563:
1551:
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1549:
1540:
1539:
1527:
1526:
1517:
1516:
1501:
1500:
1488:
1487:
1451:
1450:
1432:
1431:
1418:
1417:
1416:
1407:
1406:
1391:
1390:
1389:
1371:
1370:
1369:
1360:
1359:
1344:
1343:
1342:
1319:
1318:
1299:
1297:
1296:
1291:
1289:
1288:
1272:
1270:
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1264:
1262:
1261:
1242:
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1234:
1232:
1231:
1216:
1215:
1191:
1190:
1181:
1180:
1164:
1162:
1161:
1156:
1154:
1153:
1134:
1132:
1131:
1126:
1124:
1123:
1105:
1104:
1088:
1086:
1085:
1080:
1061:
1059:
1058:
1053:
1041:
1039:
1038:
1033:
1031:
1030:
1021:
1020:
1011:
1010:
1001:
1000:
991:
990:
968:
966:
965:
960:
948:
946:
945:
940:
938:
937:
924:
922:
921:
916:
914:
913:
900:
898:
897:
892:
890:
889:
880:
879:
870:
869:
860:
859:
834:
832:
831:
826:
824:
823:
813:
808:
775:
773:
772:
767:
755:
753:
752:
747:
719:
717:
716:
711:
699:
697:
696:
691:
689:
688:
679:
678:
663:
662:
647:
646:
622:
621:
612:
611:
602:
601:
585:
583:
582:
577:
572:
571:
562:
561:
546:
545:
530:
529:
505:
504:
495:
494:
485:
484:
475:
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469:
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459:
434:
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431:
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416:
415:
399:
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391:
389:
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378:
362:
360:
359:
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352:
344:
343:
334:
333:
324:
323:
307:
305:
304:
299:
297:
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287:
286:
266:
264:
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258:
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238:
236:
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230:
212:
210:
209:
204:
186:
184:
183:
178:
163:
161:
160:
155:
96:
82:
75:
71:
68:
62:
27:
19:
4057:
4056:
4052:
4051:
4050:
4048:
4047:
4046:
4037:Decision theory
4022:
4021:
3987:
3986:
3979:
3975:
3966:
3964:
3957:
3951:
3947:
3939:
3931:
3927:
3910:
3901:
3892:
3890:
3888:www.hutter1.net
3880:
3873:
3858:
3848:10.1007/b138233
3834:
3825:
3794:
3790:
3785:
3773:
3730:
3697:
3694:
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3673:
3670:
3669:
3649:
3629:
3596:
3595:
3591:
3587:
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3575:
3569:
3565:
3556:
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3546:
3542:
3530:
3526:
3517:
3513:
3493:
3480:
3476:
3461:
3457:
3446:
3442:
3436:
3432:
3431:
3419:
3415:
3414:
3399:
3395:
3389:
3385:
3384:
3378:
3375:
3374:
3357:
3353:
3351:
3348:
3347:
3318:
3314:
3302:
3298:
3289:
3285:
3279:
3275:
3273:
3270:
3269:
3252:
3248:
3242:
3238:
3229:
3225:
3219:
3215:
3203:
3199:
3187:
3183:
3174:
3170:
3164:
3160:
3151:
3147:
3141:
3137:
3128:
3124:
3118:
3114:
3112:
3109:
3108:
3088:
3084:
3069:
3065:
3056:
3052:
3050:
3047:
3046:
3029:
3025:
3016:
3012:
3000:
2996:
2987:
2983:
2981:
2978:
2977:
2960:
2956:
2947:
2943:
2941:
2938:
2937:
2909:
2908:
2904:
2900:
2898:
2895:
2894:
2862:
2861:
2857:
2853:
2845:
2841:
2835:
2831:
2822:
2818:
2812:
2808:
2796:
2792:
2783:
2779:
2759:
2753:
2750:
2749:
2719:
2718:
2717:
2713:
2708:
2688:
2687:
2683:
2679:
2677:
2674:
2673:
2657:
2654:
2653:
2627:
2626:
2624:
2621:
2620:
2598:
2595:
2594:
2578:
2575:
2574:
2557:
2553:
2547:
2543:
2534:
2530:
2524:
2520:
2518:
2515:
2514:
2497:
2493:
2484:
2480:
2478:
2475:
2474:
2458:
2455:
2454:
2438:
2435:
2434:
2418:
2415:
2414:
2391:
2388:
2387:
2369:
2366:
2365:
2349:
2346:
2345:
2329:
2326:
2325:
2308:
2304:
2289:
2285:
2283:
2280:
2279:
2261:
2258:
2257:
2240:
2236:
2230:
2226:
2224:
2221:
2220:
2204:
2201:
2200:
2183:
2179:
2177:
2174:
2173:
2156:
2152:
2150:
2147:
2146:
2129:
2125:
2119:
2115:
2113:
2110:
2109:
2084:
2080:
2078:
2075:
2074:
2054:
2050:
2048:
2045:
2044:
2016:
2015:
2011:
2007:
2005:
2002:
2001:
1985:
1982:
1981:
1965:
1962:
1961:
1945:
1942:
1941:
1919:
1916:
1915:
1869:
1868:
1864:
1860:
1852:
1848:
1842:
1838:
1829:
1825:
1819:
1815:
1803:
1799:
1790:
1786:
1766:
1761:
1757:
1748:
1744:
1729:
1725:
1714:
1710:
1704:
1700:
1699:
1687:
1683:
1682:
1677:
1673:
1662:
1658:
1652:
1648:
1647:
1642:
1638:
1630:
1626:
1625:
1606:
1602:
1600:
1597:
1596:
1562:
1561:
1557:
1553:
1545:
1541:
1535:
1531:
1522:
1518:
1512:
1508:
1496:
1492:
1483:
1479:
1459:
1446:
1442:
1427:
1423:
1412:
1408:
1402:
1398:
1397:
1385:
1381:
1380:
1365:
1361:
1355:
1351:
1350:
1338:
1334:
1333:
1314:
1310:
1308:
1305:
1304:
1284:
1280:
1278:
1275:
1274:
1254:
1250:
1248:
1245:
1244:
1221:
1217:
1205:
1201:
1186:
1182:
1176:
1172:
1170:
1167:
1166:
1146:
1142:
1140:
1137:
1136:
1113:
1109:
1100:
1096:
1094:
1091:
1090:
1074:
1071:
1070:
1047:
1044:
1043:
1026:
1025:
1016:
1015:
1006:
1002:
996:
995:
986:
985:
974:
971:
970:
954:
951:
950:
933:
932:
930:
927:
926:
909:
908:
906:
903:
902:
885:
884:
875:
871:
865:
864:
855:
854:
843:
840:
839:
819:
815:
809:
798:
792:
789:
788:
761:
758:
757:
741:
738:
737:
705:
702:
701:
684:
680:
668:
664:
652:
648:
636:
632:
617:
613:
607:
603:
597:
593:
591:
588:
587:
567:
563:
551:
547:
535:
531:
519:
515:
500:
496:
490:
486:
480:
476:
471:
465:
461:
455:
451:
443:
440:
439:
420:
411:
407:
405:
402:
401:
384:
383:
374:
370:
368:
365:
364:
348:
339:
338:
329:
328:
319:
315:
313:
310:
309:
292:
291:
282:
278:
276:
273:
272:
252:
244:
241:
240:
218:
215:
214:
192:
189:
188:
172:
169:
168:
149:
146:
145:
142:
83:
72:
66:
63:
48:
28:
17:
12:
11:
5:
4055:
4045:
4044:
4039:
4034:
4020:
4019:
3985:
3984:
3973:
3953:Soares, Nate.
3945:
3925:
3899:
3871:
3856:
3823:
3787:
3786:
3784:
3781:
3780:
3779:
3772:
3769:
3729:
3726:
3714:
3713:
3701:
3677:
3662:
3659:
3648:
3645:
3628:
3625:
3610:
3607:
3604:
3594:
3590:
3582:
3578:
3572:
3568:
3564:
3559:
3555:
3549:
3545:
3541:
3538:
3533:
3529:
3525:
3520:
3516:
3512:
3509:
3506:
3503:
3499:
3496:
3492:
3488:
3483:
3479:
3475:
3472:
3469:
3464:
3460:
3456:
3449:
3445:
3439:
3435:
3430:
3422:
3418:
3413:
3409:
3402:
3398:
3392:
3388:
3383:
3360:
3356:
3327:
3324:
3321:
3317:
3311:
3308:
3305:
3301:
3297:
3292:
3288:
3282:
3278:
3255:
3251:
3245:
3241:
3237:
3232:
3228:
3222:
3218:
3212:
3209:
3206:
3202:
3196:
3193:
3190:
3186:
3182:
3177:
3173:
3167:
3163:
3159:
3154:
3150:
3144:
3140:
3136:
3131:
3127:
3121:
3117:
3094:
3091:
3087:
3083:
3078:
3075:
3072:
3068:
3064:
3059:
3055:
3032:
3028:
3024:
3019:
3015:
3009:
3006:
3003:
2999:
2995:
2990:
2986:
2963:
2959:
2955:
2950:
2946:
2923:
2920:
2917:
2907:
2903:
2876:
2873:
2870:
2860:
2856:
2848:
2844:
2838:
2834:
2830:
2825:
2821:
2815:
2811:
2807:
2804:
2799:
2795:
2791:
2786:
2782:
2778:
2775:
2772:
2769:
2765:
2762:
2758:
2733:
2730:
2727:
2716:
2712:
2707:
2702:
2699:
2696:
2686:
2682:
2661:
2641:
2638:
2635:
2602:
2582:
2560:
2556:
2550:
2546:
2542:
2537:
2533:
2527:
2523:
2500:
2496:
2492:
2487:
2483:
2462:
2442:
2422:
2395:
2373:
2353:
2333:
2311:
2307:
2303:
2300:
2297:
2292:
2288:
2265:
2243:
2239:
2233:
2229:
2208:
2186:
2182:
2159:
2155:
2132:
2128:
2122:
2118:
2090:
2087:
2083:
2060:
2057:
2053:
2030:
2027:
2024:
2014:
2010:
1989:
1969:
1949:
1929:
1926:
1923:
1912:
1911:
1899:
1894:
1889:
1883:
1880:
1877:
1867:
1863:
1855:
1851:
1845:
1841:
1837:
1832:
1828:
1822:
1818:
1814:
1811:
1806:
1802:
1798:
1793:
1789:
1785:
1782:
1779:
1776:
1772:
1769:
1765:
1760:
1756:
1751:
1747:
1743:
1740:
1737:
1732:
1728:
1724:
1717:
1713:
1707:
1703:
1698:
1690:
1686:
1681:
1676:
1672:
1665:
1661:
1655:
1651:
1646:
1641:
1633:
1629:
1624:
1620:
1617:
1614:
1609:
1605:
1590:
1589:
1576:
1573:
1570:
1560:
1556:
1548:
1544:
1538:
1534:
1530:
1525:
1521:
1515:
1511:
1507:
1504:
1499:
1495:
1491:
1486:
1482:
1478:
1475:
1472:
1469:
1465:
1462:
1458:
1454:
1449:
1445:
1441:
1438:
1435:
1430:
1426:
1422:
1415:
1411:
1405:
1401:
1396:
1388:
1384:
1379:
1375:
1368:
1364:
1358:
1354:
1349:
1341:
1337:
1332:
1328:
1325:
1322:
1317:
1313:
1287:
1283:
1260:
1257:
1253:
1230:
1227:
1224:
1220:
1214:
1211:
1208:
1204:
1200:
1197:
1194:
1189:
1185:
1179:
1175:
1152:
1149:
1145:
1122:
1119:
1116:
1112:
1108:
1103:
1099:
1078:
1051:
1029:
1024:
1019:
1014:
1009:
1005:
999:
994:
989:
984:
981:
978:
958:
936:
912:
888:
883:
878:
874:
868:
863:
858:
853:
850:
847:
822:
818:
812:
807:
804:
801:
797:
778:Turing machine
765:
745:
709:
687:
683:
677:
674:
671:
667:
661:
658:
655:
651:
645:
642:
639:
635:
631:
628:
625:
620:
616:
610:
606:
600:
596:
575:
570:
566:
560:
557:
554:
550:
544:
541:
538:
534:
528:
525:
522:
518:
514:
511:
508:
503:
499:
493:
489:
483:
479:
474:
468:
464:
458:
454:
450:
447:
423:
419:
414:
410:
387:
382:
377:
373:
351:
347:
342:
337:
332:
327:
322:
318:
295:
290:
285:
281:
255:
251:
248:
228:
225:
222:
202:
199:
196:
176:
153:
141:
138:
105:. It combines
85:
84:
67:September 2023
31:
29:
22:
15:
9:
6:
4:
3:
2:
4054:
4043:
4040:
4038:
4035:
4033:
4030:
4029:
4027:
4017:
4013:
4009:
4008:9783540237334
4005:
4001:
3997:
3993:
3989:
3988:
3982:
3977:
3963:
3956:
3949:
3938:
3937:
3929:
3920:
3915:
3908:
3906:
3904:
3889:
3885:
3878:
3876:
3867:
3863:
3859:
3853:
3849:
3845:
3841:
3840:
3832:
3830:
3828:
3819:
3815:
3810:
3809:cs.AI/0004001
3805:
3801:
3800:
3792:
3788:
3778:
3777:Gödel machine
3775:
3774:
3768:
3766:
3763:
3759:
3755:
3751:
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3743:
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3660:
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3644:
3642:
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3592:
3588:
3580:
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2344:to time step
2331:
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2116:
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2088:
2085:
2081:
2058:
2055:
2051:
2025:
2012:
2008:
2000:(that is, by
1987:
1967:
1947:
1927:
1924:
1921:
1897:
1892:
1887:
1878:
1865:
1861:
1853:
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1843:
1839:
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1763:
1758:
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1726:
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1711:
1705:
1701:
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1627:
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1497:
1493:
1489:
1484:
1480:
1476:
1473:
1467:
1463:
1460:
1456:
1447:
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1439:
1436:
1433:
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1409:
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1394:
1386:
1382:
1373:
1366:
1362:
1356:
1352:
1347:
1339:
1335:
1326:
1323:
1320:
1315:
1311:
1303:
1302:
1301:
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1258:
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1228:
1225:
1222:
1218:
1212:
1209:
1206:
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1195:
1192:
1187:
1183:
1177:
1173:
1150:
1147:
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1114:
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1106:
1101:
1097:
1076:
1067:
1065:
1049:
1012:
1007:
992:
979:
976:
956:
876:
861:
848:
845:
836:
820:
816:
810:
805:
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799:
795:
786:
781:
779:
763:
743:
735:
731:
727:
723:
707:
685:
681:
675:
672:
669:
665:
659:
656:
653:
649:
643:
640:
637:
633:
629:
626:
623:
618:
614:
608:
604:
598:
594:
568:
564:
558:
555:
552:
548:
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539:
536:
532:
526:
523:
520:
516:
512:
509:
506:
501:
497:
491:
487:
481:
477:
466:
462:
456:
452:
445:
438:
417:
412:
408:
380:
375:
371:
345:
335:
325:
320:
316:
288:
283:
279:
270:
249:
246:
226:
223:
220:
200:
197:
194:
174:
165:
151:
137:
135:
134:Occam's razor
131:
127:
122:
120:
116:
115:Marcus Hutter
112:
108:
104:
100:
95:
91:
81:
78:
70:
60:
56:
52:
46:
42:
38:
37:
32:This article
30:
26:
21:
20:
3999:
3976:
3965:. Retrieved
3961:
3948:
3935:
3928:
3891:. Retrieved
3887:
3838:
3798:
3791:
3749:
3745:
3741:
3738:incomputable
3731:
3722:
3715:
3689:
3665:
3650:
3636:
3632:
3630:
3345:
3342:
2936:. Note that
2619:
2614:
2386:
2278:
2108:
2105:
1913:
1591:
1068:
1042:, where the
837:
784:
782:
733:
725:
268:
166:
143:
123:
118:
89:
88:
73:
64:
49:Please help
33:
3754:Monte Carlo
3641:discounting
2172:and reward
1066:operation.
1064:Kleene star
4026:Categories
3998:; also in
3996:cs/0701125
3967:2015-07-19
3893:2024-09-21
3783:References
3748:and space
3736:, AIXI is
3647:Optimality
3627:Parameters
2406:denotes a
140:Definition
124:AIXI is a
51:improve it
41:verifiable
3919:0909.0801
3756:AIXI FAC-
3718:Jan Leike
3700:μ
3676:μ
3593:−
3563:…
3524:…
3491:∑
3471:…
3429:∑
3408:…
3382:∑
3323:−
3307:−
3296:…
3236:…
3208:−
3192:−
3181:…
3135:…
3074:−
3063:…
3023:…
3005:−
2994:…
2954:…
2906:−
2859:−
2829:…
2790:…
2757:∑
2685:−
2541:…
2491:…
2299:…
2013:−
1925:−
1866:−
1836:…
1797:…
1764:∑
1739:…
1697:∑
1671:…
1645:∑
1619:
1559:−
1529:…
1490:…
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1437:…
1395:∑
1374:…
1348:∑
1327:
1226:−
1210:−
1118:−
1107:…
1050:∗
1023:→
1013:×
1008:∗
993:×
977:μ
957:μ
882:→
877:∗
862:×
846:π
796:∑
764:μ
744:μ
708:μ
673:−
657:−
641:−
556:−
540:−
524:−
446:μ
418:∈
381:∈
346:×
326:∈
289:∈
250:∈
175:μ
152:ξ
55:citations
3866:33352850
3771:See also
2408:monotone
586:, where
239:, where
3814:Bibcode
3765:Pac-Man
2891:mixture
2615:current
1062:is the
734:unknown
45:neutral
4006:
3864:
3854:
3598:length
3268:, and
3045:, and
2911:length
2864:length
2721:length
2690:length
2629:length
2413:, and
2018:length
1871:length
1564:length
3992:arXiv
3958:(PDF)
3940:(PDF)
3914:arXiv
3862:S2CID
3804:arXiv
3732:Like
109:with
4004:ISBN
3852:ISBN
3090:<
2086:<
2056:<
1256:<
1148:<
785:only
783:The
726:full
101:for
90:AIXI
43:and
4012:doi
3844:doi
3412:max
1960:to
1680:max
1623:max
1616:arg
1378:max
1331:max
1324:arg
213:to
57:to
4028::
3960:.
3902:^
3886:.
3874:^
3860:.
3850:.
3826:^
3812:.
3802:.
3767:.
3742:tl
3643:.
2384:.
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1321::=
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4018:.
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3868:.
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74:(
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47:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
