This paper presents Generalized Proof-Number Monte-Carlo Tree Search: a generalization of recently proposed combinations of Proof-Number Search (PNS) with Monte-Carlo Tree Search (MCTS), which use (dis)proof numbers to bias UCB1-based Selection strategies towards parts of the search that are expected to be easily (dis)proven. We propose three core modifications of prior combinations of PNS with MCTS. First, we track proof numbers per player. This reduces code complexity in the sense that we no longer need disproof numbers, and generalizes the technique to be applicable to games with more than two players. Second, we propose and extensively evaluate different methods of using proof numbers to bias the selection strategy, achieving strong performance with strategies that are simpler to implement and compute. Third, we merge our technique with Score Bounded MCTS, enabling the algorithm to prove and leverage upper and lower bounds on scores - as opposed to only proving wins or not-wins. Experiments demonstrate substantial performance increases, reaching the range of 80% for 8 out of the 11 tested board games.

This paper proposes a new game-search algorithm, PN-MCTS, which combines Monte-Carlo Tree Search (MCTS) and Proof-Number Search (PNS). These two algorithms have been successfully applied for decision making in a range of domains. We define three areas where the additional knowledge provided by the proof and disproof numbers gathered in MCTS trees might be used: final move selection, solving subtrees, and the UCB1 selection mechanism. We test all possible combinations on different time settings, playing against vanilla UCT on several games: Lines of Action ($7$$\times$$7$ and $8$$\times$$8$ board sizes), MiniShogi, Knightthrough, and Awari. Furthermore, we extend this new algorithm to properly address games with draws, like Awari, by adding an additional layer of PNS on top of the MCTS tree. The experiments show that PN-MCTS confidently outperforms MCTS in all tested game domains, achieving win rates up to 96.2\% for Lines of Action.

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic. Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player. We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte-Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha-Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4min/move.

The production of any new medicine requires solutions to many planning problems. The most fundamental of these is determining the sequence of chemical reactions necessary to physically create the drug. Surprisingly, these organic syntheses can be modeled as branching paths in a discrete, fully-observable state space, making the construction of new medicines an application of heuristic search. We describe a model of organic chemistry that is amenable to traditional AI techniques from game tree search, regression, and automatic assembly sequencing. We demonstrate the applicability of AND/OR graph search by developing the first chemistry solver to use proof-number search. Finally, we construct a benchmark suite of organic synthesis problems collected from undergraduate organic chemistry exams, and we analyze our solvers performance both on this suite and in recreating the synthetic plan for a multibillion dollar drug.

Most modal logics such as S5, LTL, or ATL are extensions of Modal Logic K. While the model checking problems for LTL and to a lesser extent ATL have been very active research areas for the past decades, the model checking problem for the more basic Multi-agent Modal Logic K (MMLK) has important applications as a formal framework for perfect information multi-player games on its own. We present Minimal Proof Search (MPS), an effort number based algorithm solving the model checking problem for MMLK. We prove two important properties for MPS beyond its correctness. The (dis)proof exhibited by MPS is of minimal cost for a general definition of cost, and MPS is an optimal algorithm for finding (dis)proofs of minimal cost. Optimality means that any comparable algorithm either needs to explore a bigger or equal state space than MPS, or is not guaranteed to find a (dis)proof of minimal cost on every input. As such, our work relates to A* and AO* in heuristic search, to Proof Number Search and DFPN+ in two-player games, and to counterexample minimization in software model checking.