On the undecidability of probabilistic planning and related stochastic optimization problems

Madani, Omid; Hanks, Steve; Condon, Anne

doi:10.1016/s0004-3702(02)00378-8

Cited by 202 publications

(213 citation statements)

References 22 publications

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“…The proof of Paz is a reduction from an undecidable problem about free context grammars. An alternative proof was given by Madani, Hanks and Condon [10], based on a reduction from the emptiness problem for two counter machines. Since Paz was focusing on expressiveness aspects of probabilistic automata rather than on algorithmic questions, his undecidability proof is spread on the whole book [11], which makes it arguably hard to read.…”

Section: The Emptiness Problemmentioning

confidence: 99%

Probabilistic Automata on Finite Words: Decidable and Undecidable Problems

Gimbert

Oualhadj

2010

Automata, Languages and Programming

107

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Abstract. This paper tackles three algorithmic problems for probabilistic automata on finite words: the Emptiness Problem, the Isolation Problem and the Value 1 Problem. The Emptiness Problem asks, given some probability 0 ≤ λ ≤ 1, whether there exists a word accepted with probability greater than λ, and the Isolation Problem asks whether there exist words whose acceptance probability is arbitrarily close to λ. Both these problems are known to be undecidable [11,4,3]. About the Emptiness problem, we provide a new simple undecidability proof and prove that it is undecidable for automata with as few as two probabilistic transitions. The Value 1 Problem is the special case of the Isolation Problem when λ = 1 or λ = 0. The decidability of the Value 1 Problem was an open question. We show that the Value 1 Problem is undecidable. Moreover, we introduce a new class of probabilistic automata, ♯-acyclic automata, for which the Value 1 Problem is decidable.

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Section: The Emptiness Problemmentioning

confidence: 99%

Probabilistic Automata on Finite Words: Decidable and Undecidable Problems

Gimbert

Oualhadj

2010

Automata, Languages and Programming

107

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“…It is known that partially observable Markov Decision Processes are undecidable, even with finite state space (see [13]). With full observation, they become decidable; Monte-Carlo Tree Search (MCTS, [7]) is a recent well known solver for this case, with impressive results in many cases, in particular the game of Go [12].…”

Section: State Of the Art And Outline Of The Papermentioning

confidence: 99%

“…First consider (13). By Lemma 1, there is a integer p such that all children i = 1, · · · , i 0 − 1 have been selected p or p + 1 times, with p = O n 1−α .…”

Section: Consistency Of Random Nodesmentioning

confidence: 99%

Continuous Upper Confidence Trees with Polynomial Exploration – Consistency

Auger

Cou«toux

Teytaud

2013

Lecture Notes in Computer Science

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“…The impact of partial observability in games and planning is studied in several papers, showing in particular that -Just one player and a random part makes the problem undecidable, even with a finite state space with reachability criteria [10]. -With two players with or without random parts, the problem is EXP, EXPSPACE, 2EXP (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…For example, [10] has shown that with one player only, no observation, random nodes, the probability of winning, when starting in a given node, and for an optimal strategy, is not computable, and even not approximable. [14] has shown that this also holds for two players and no random node.…”

Section: Introductionmentioning

confidence: 99%