It is well-known that any admissible unidirectional heuristic search algorithm must expand all states whose f -value is smaller than the optimal solution cost when using a consistent heuristic. Such states are called "surely expanded" (s.e.). A recent study characterized s.e. pairs of states for bidirectional search with consistent heuristics: if a pair of states is s.e. then at least one of the two states must be expanded. This paper derives a lower bound, VC, on the minimum number of expansions required to cover all s.e. pairs, and present a new admissible front-to-end bidirectional heuristic search algorithm, Near-Optimal Bidirectional Search (NBS), that is guaranteed to do no more than 2VC expansions. We further prove that no admissible frontto-end algorithm has a worst case better than 2VC. Experimental results show that NBS competes with or outperforms existing bidirectional search algorithms, and often outperforms A* as well.
In this paper we study bidirectional state space search with consistent heuristics, with a focus on obtaining sufficient conditions for node expansion, that is, conditions characterizing nodes that must be expanded by any admissible bidirectional search algorithm. We provide such conditions for front-to-front and front-to-end bidirectional search. The sufficient conditions are used to prove that the front-to-front bidirectional search algorithm BDS1 is optimally efficient, in terms of node expansion, among a broad class of bidirectional search algorithms, for a specific class of problem instances. Dechter and Pearl's well-known result on sufficient conditions for node expansion by unidirectional algorithms such as A* is shown to be a special case of our results.
Abstract. This paper is concerned with the combinatorial structure of concept classes that can be learned from a small number of examples. We show that the recently introduced notion of recursive teaching dimension (RTD, reflecting the complexity of teaching a concept class) is a relevant parameter in this context. Comparing the RTD to self-directed learning, we establish new lower bounds on the query complexity for a variety of query learning models and thus connect teaching to query learning. For many general cases, the RTD is upper-bounded by the VC-dimension, e.g., classes of VC-dimension 1, (nested differences of) intersection-closed classes, "standard" boolean function classes, and finite maximum classes. The RTD thus is the first model to connect teaching to the VC-dimension. The combinatorial structure defined by the RTD has a remarkable resemblance to the structure exploited by sample compression schemes and hence connects teaching to sample compression. Sequences of teaching sets defining the RTD coincide with unlabeled compression schemes both (i) resulting from Rubinstein and Rubinstein's corner-peeling and (ii) resulting from Kuzmin and Warmuth's Tail Matching algorithm.
In the past 40 years, research on inductive inference has developed along different lines, e.g., in the formalizations used, and in the classes of target concepts considered. One common root of many of these formalizations is Gold's model of identification in the limit. This model has been studied for learning recursive functions, recursively enumerable languages, and recursive languages, reflecting different aspects of machine learning, artificial intelligence, complexity theory, and recursion theory. One line of research focuses on indexed families of recursive languages -classes of recursive languages described in a representation scheme for which the question of membership for any string in any of the given languages is effectively decidable with a uniform procedure. Such language classes are of interest because of their naturalness. The survey at hand picks out important studies on learning indexed families (including basic as well as recent research), summarizes and illustrates the corresponding results, and points out links to related fields such as grammatical inference, machine learning, and artificial intelligence in general.
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