CPSC 322 Introduction to Artificial Intelligence

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Presentation transcript:

CPSC 322 Introduction to Artificial Intelligence October 8, 2004

Things to look for Your final exam will be from noon to 3:00pm on Friday, December 10 location to be determined

Search A physical symbol system exercises its intelligence in problem solving by search -- that is, by generating and progressively modifying symbol structures until it produces a solution structure. Allen Newell and Herbert A. Simon, “Computer Science as Empirical Inquiry: Symbols and Search”

Graph search - informal Most problem-solving tasks can be mapped onto the problem of finding a path in a graph • nodes represent partial solutions • arcs represent transformations from one partial solution to another • try to find path along arcs from a start node to a goal node

Graph search - formal A directed graph consists of a set of nodes and a set of ordered pairs called arcs (links?). A node n2 is a neighbor of n1 if there is an arc from n1 to n2. A path is an ordered sequence of nodes such that there is an arc between consecutive nodes in the path A cycle is a nonempty path with start node and end node the same. A directed graph without cycles is called a directed acyclic graph.

How do I do this graph search stuff? Given a set of start nodes, a set of goal nodes, and a graph (i.e., the nodes and arcs): make a “list” of the start nodes - let’s call it the “frontier” repeat if no nodes on the frontier then terminate with failure choose one node from the frontier and remove it if the chosen node matches the goal node then terminate with success else put next nodes (neighbors) on frontier end repeat

8-Tile Puzzle 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [a] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [b,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [e,f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [k,f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [q,r,f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [r,f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [r,f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [l,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [s,t,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [t,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [t,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [g,h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [m,h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [u,v,h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [v,h,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

Generic graph search algorithm Given a set of start nodes, a set of goal nodes, and a graph (i.e., the nodes and arcs): make a “list” of the start nodes - let’s call it the “frontier” repeat if no nodes on the frontier then terminate with failure choose one node from the frontier and remove it if the chosen node matches the goal node then terminate with success else put next nodes (neighbors) on frontier end repeat

Generic graph search algorithm Given a set of start nodes, a set of goal nodes, and a graph (i.e., the nodes and arcs): make a “list” of the start nodes - let’s call it the “frontier” repeat if no nodes on the frontier then terminate with failure choose one node from the front of frontier and remove it if the chosen node matches the goal node then terminate with success else put next nodes (neighbors) on front of frontier end repeat

Depth-first graph search algorithm Given a set of start nodes, a set of goal nodes, and a graph (i.e., the nodes and arcs): make a “list” of the start nodes - let’s call it the “frontier” repeat if no nodes on the frontier then terminate with failure choose one node from the front of frontier and remove it if the chosen node matches the goal node then terminate with success else put next nodes (neighbors) on front of frontier end repeat

Datalog graph search algorithm search(F0) <- select(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2).

Datalog graph search algorithm search(F0) <- select(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2).

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2).

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). interpreting the predicates: search(F0) is true if there is a path from one element in the list F0 - the frontier - to a goal node

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). interpreting the predicates: is_goal(N) is true if N is a goal node

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). interpreting the predicates: neighbors(N,NN) is true if NN is the list of neighbors of node N

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). interpreting the predicates: choose(N,F0,F1) means N is some element selected from F0 - the frontier - and F1 is the set of nodes remaining when N is removed. This fails if F0 is empty.

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). interpreting the predicates: add_to_frontier(NN,F1,F2) means that F2 is the new frontier made by adding the list of nodes NN to the old frontier F1

Datalog graph search algorithm search(F0) <- choose(Node,F0,F1) ^ is_goal(Node). neighbors(Node,NN) ^ add_to_frontier(NN,F1,F2) ^ search(F2). So how do you implement choose and add_to_frontier for depth-first search?

CILOG graph search neighbors([2,8,3,1,0,4,7,6,5],[[2,8,3,0,1,4,7,6,5],[2,0,3,1,8,4,7,6,5], [2,8,3,1,4,0,7,6,5]]). neighbors([2,8,3,0,1,4,7,6,5],[[0,8,3,2,1,4,7,6,5],[2,8,3,7,1,4,0,6,5]]). neighbors([2,0,3,1,8,4,7,6,5],[[0,2,3,1,8,4,7,6,5],[2,3,0,1,8,4,7,6,5]]). neighbors([2,8,3,1,4,0,7,6,5],[[2,8,0,1,4,3,7,6,5],[2,8,3,1,4,5,7,6,0]]). neighbors([0,8,3,2,1,4,7,6,5],[[8,0,3,2,1,4,7,6,5]]). : neighbors([2,3,4,1,0,8,7,6,5],[]). neighbors([2,3,4,1,8,5,7,6,0],[]). neighbors([0,2,8,1,4,3,7,6,5],[]). neighbors([2,4,8,1,0,3,7,6,5],[]). neighbors([2,8,3,1,4,5,0,7,6],[]). neighbors([2,8,3,1,0,5,7,4,6],[]). is_goal([1,2,3,8,0,4,7,6,5]).

CILOG graph search search(F0) <- choose(Node,F0,F1) & neighbors(Node,NN) & add_to_frontier(NN,F1,F2) & search(F2). search(F0) <- choose(Node,F0,F1) & is_goal(Node). /* choose(N,Flist0,Flist1) <- append([N],Flist1,Flist0). */ choose(N,[N|Flist],Flist). add_to_frontier(Nodelist,Flist1,Flist2) <- append(Nodelist,Flist1,Flist2).

Time for a CILOG break! look at ‘puzzle.ci’ in Word load ‘puzzle.ci’ in CILOG cilog: ask search( [ [2,8,3,1,0,4,7,6,5] ] ).

How would you change it all... ...to get breadth-first search instead of depth-first search?

Breadth-first search 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [a] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [b,c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [c,d,e,f] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [d,e,f] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [d,e,f,g,h] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [e,f,g,h] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [e,f,g,h,i,j] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,g,h,i,j] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [f,g,h,i,j,k] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [g,h,i,j,k] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [g,h,i,j,k,l] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [h,i,j,k,l] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [h,i,j,k,l,m] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [i,j,k,l,m] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [i,j,k,l,m,n] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [j,k,l,m,n] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [j,k,l,m,n,o] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [k,l,m,n,o] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [k,l,m,n,o,p] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [l,m,n,o,p] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [l,m,n,o,p,q,r] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [m,n,o,p,q,r] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [m,n,o,p,q,r,s,t] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [n,o,p,q,r,s,t] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [n,o,p,q,r,s,t,u,v] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [o,p,q,r,s,t,u,v] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [o,p,q,r,s,t,u,v,w,x] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [p,q,r,s,t,u,v,w,x] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [p,q,r,s,t,u,v,w,x,y,z] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [q,r,s,t,u,v,w,x,y,z] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [q,r,s,t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [r,s,t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [r,s,t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [s,t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [s,t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [t,u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [u,v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

frontier: [v,w,x,y,z,aa,bb] 2 8 3 1 2 3 1 _ 4 goal: 8 _ 4 7 6 5 7 6 5 a 2 8 3 2 _ 3 2 8 3 _ 1 4 1 8 4 1 4 _ 7 6 5 7 6 5 7 6 5 b c d _ 8 3 2 8 3 _ 2 3 2 3 _ 2 8 _ 2 8 3 2 1 4 7 1 4 1 8 4 1 8 4 1 4 3 1 4 5 7 6 5 _ 6 5 7 6 5 7 6 5 7 6 5 7 6 _ e f g h i j 8 _ 3 2 8 3 1 2 3 2 3 4 2 _ 8 2 8 3 2 1 4 7 1 4 _ 8 4 1 8 _ 1 4 3 1 4 5 7 6 5 6 _ 5 7 6 5 7 6 5 7 6 5 7 _ 6 k l m n o p 8 3 _ 8 1 3 2 8 3 2 8 3 1 2 3 1 2 3 2 3 4 2 3 4 _ 2 8 2 4 8 2 8 3 2 8 3 2 1 4 2 _ 4 7 _ 4 7 1 4 8 _ 4 7 8 4 1 _ 8 1 8 5 1 4 3 1 _ 3 1 4 5 1 _ 5 7 6 5 7 6 5 6 1 5 6 5 _ 7 6 5 _ 6 5 7 6 5 7 6 _ 7 6 5 7 6 5 _ 7 6 7 4 6 q r s t u v w x y z aa bb

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