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CN and PN Tree Search Algorithms

CN and PN Tree Search Algorithms. Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms Expand one node at a time, at any point on the fringe of a game tree , Generate all child nodes - consider all moves - as part of node expansion.

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CN and PN Tree Search Algorithms

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  1. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  2. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  3. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st 2nd http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  4. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st 2nd 3rd http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  5. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st 2nd 4th 3rd http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  6. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st 2nd 4th 3rd 5th http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  7. CN and PN Tree Search Algorithms • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation 1st 2nd 4th 6th 3rd 5th http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  8. Deeper Search gives better results • Experience with computer chess shows that deeper search gives better play. • Programs that can search one extra ply of game tree do gain advantage from it. • A static evaluator gives an estimate of a position’s worth. • Evaluation of a parent node should be not very unlike the backed-up minimax evaluation of child nodes: the correlation should be good. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  9. Deeper Search gives better results • Experience with computer chess shows that deeper search gives better play. • Programs that can search one extra ply of game tree do gain advantage from it. • A static evaluator gives an estimate of a position’s worth. • Evaluation of a parent node should be not very unlike the backed-up minimax evaluation of child nodes: the correlation should be good. • But the child nodes will be a little closer to the game’s end, • A static evaluator gets to see the resulting position(s) of playing out any immediate threat(s) http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  10. Deeper Search gives better results • Experience with computer chess shows that deeper search gives better play. • Programs that can search one extra ply of game tree do gain advantage from it. • A static evaluator gives an estimate of a position’s worth. • Evaluation of a parent node should be not very unlike the backed-up minimax evaluation of child nodes: the correlation should be good. • But the child nodes will be a little closer to the game’s end, • A static evaluator gets to see the resulting position(s) of playing out any immediate threat(s) • Minimaxing one or more levels of game subtree can and frequently does give a result different to static evaluation of the node at the root of the subtree. • That is of course why search and minimaxing are performed! http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  11. Conspiracy Numbers • An interior node of a game tree derives its current value from minimax of the part of the game tree beneath it. • The subtree beneath an interior node need not be expanded to uniform depth: alpha-beta, null-move quiescence, B* … all expand trees nonuniformly. • If a leaf node is expanded, its backed-up minimax value might not be the same as its statically evaluated value. • i.e. its value might change • … and if its value does change, that might or might not be enough to cause a change of its parent’s value, and so on up the tree. • So: how many leaf nodes would have to change together - have to conspire - to cause the value of an interior node to change to some particular other value? http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  12. A: (max) 4 B: (min) 4 C: (min) 2 D: (max) 7 E: (max) 4 F: (max) 2 G: (max) 3 Conspiracy Numbers - Simple Case • For A to get the value 5 6 or 7, just 1 node - E - would have to change. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  13. A: (max) 4 B: (min) 4 C: (min) 2 D: (max) 7 E: (max) 4 F: (max) 2 G: (max) 3 Conspiracy Numbers - Simple Case • For A to get the value 5 6 or 7, just 1 node - E - would have to change. • For A to get the value 8 (… +), requires conspiracy of size 2: D&E, or F&G. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  14. A: (max) 4 B: (min) 4 C: (min) 2 D: (max) 7 E: (max) 4 F: (max) 2 G: (max) 3 Conspiracy Numbers - Simple Case • For A to get the value 5 6 or 7, just 1 node - E - would have to change. • For A to get the value 8 (… +), requires conspiracy of size 2: D&E, or F&G. • For A to get the value 4, 0 nodes have to change. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  15. A: (max) 4 B: (min) 4 C: (min) 2 D: (max) 7 E: (max) 4 F: (max) 2 G: (max) 3 Conspiracy Numbers - Simple Case • For A to get the value 5 6 or 7, just 1 node - E - would have to change. • For A to get the value 8 (… +), requires conspiracy of size 2: D&E, or F&G. • For A to get the value 4, 0 nodes have to change. • For A to get the value 1 (… -), 2 nodes (D or E, & F or G) have to change. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  16. Range of likely values • For a root node, and indeed for any interior node generally, • possible values increasinglygreaterthan the current value require progressively larger conspiracies likewise, • possible values increasinglylessthan the current value require progressively larger conspiracies • With a limit on the size of conspiracy, there follow bounds on the range of values that the root node could have. • Conspiracies beyond a certain size may be regarded as not likely. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  17. Conspiracies of unlikely size • Values for A of less than 1, or more than 3, require two or more conspirators. • Values for A of less than 0, or more than 4, require three or more conspirators. • For a given threshold, we derive bounds on the range of likely values for A. A:+2 Max-ing level B:+2 C:+0 D:+1 Min-ing level E:+3 F:+2 G:+4 H:+0 I:+2 J:+2 K:+1 L:+2 M:+2 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  18. CN Tree Growth Procedure • Pick a desired accuracy, , to limit the accepted range (VMAX-VMIN) of likely values of the root node; The smaller , the bigger the tree will get. • Pick a conspiracy threshold, NT, to determine what size of conspiracies are deemed sufficiently likely; The greater NT, the bigger the tree will get. • The procedure seeks to narrow the range of sufficiently-likely values of the root until that range has the desired accuracy. • This can be attempted • By ruling out the largest of the current likely values, VMAX • Or by ruling out the smallest of the current likely values, VMIN • Nondeterministically, select any node in (one of) the minimal conspiracy set(s) for ruling out either VMAX or VMIN, whichever is further from current value V. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  19. How big is the minimal conspiracy set? • Upsize (node, target) • if node.value ≥ target • then return 0 • elseif [node is terminal in the game] • then return + • elseif [node is leaf node] • then return 1 • elseif [node is maxing node] • then return [min value of Upsize(child, target) • where child in node.children] • else return [sum values of Upsize(child, target) • where child in node.children] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  20. How big is the minimal conspiracy set? • Downsize (node, target) • if node.value ≤ target • then return 0 • elseif [node is terminal in the game] • then return + • elseif [node is leaf node] • then return 1 • elseif [node is mining node] • then return [min value of Downsize(child, target) • where child in node.children] • else return [sum values of Downsize(child, target) • where child in node.children] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  21. Any node in a minimal conspiracy set will do. • Consider hoping to rule out VMAX, which must be > current V. • MIN-LEAF(node,Vmax) • if [node is a leaf node] • then Return node • elseif [node is a MAX-ing node] • then [Return MIN-LEAF(c,Vmax) • where c is child minimising Upsize(child, Vmax)] • else [Quals := children of node with Value<Vmax; • pick one, call it C1; • Return MIN-LEAF(C1,Vmax)] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  22. Any node in a minimal conspiracy set will do. • Consider hoping to rule out VMIN, which must be < current V. • MAX-LEAF(node,Vmin) • if [node is a leaf node] • then Return node • elseif [node is a MIN-ing node] • then [Return MAX-LEAF(c,Vmin) • where c is child minimising Downsize(child, Vmin)] • else [Quals := children of node with Value>Vmin; • pick one, call it C1; • Return MAX-LEAF(C1,Vmin)] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  23. Tree Growth example: =0 NT=2 name A 0 [-∞..] max-ing http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  24. Tree Growth example: =0 NT=2 name A 1 (0) [0..] Static evaluation score max-ing C 0 B 1 D 0 min-ing http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  25. Tree Growth example: =0 NT=2 A 0 (0) [0..] Static evaluation score max-ing C 0 B 0 (1) [-..1] D 0 min-ing G 1 E 0 F 1 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  26. K 0 L 0 M 0 N 0 H -1 J 0 Tree Growth example: =0 NT=2 A 0 (0) [-1..+1] Static evaluation score max-ing C -1(0) [-..0] B 0 (1) [-..1] D 0 (0) [-..0] min-ing G 1 E 0 F 1 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  27. K 0 L 0 M 0 N 0 H -1 J 0 Tree Growth example: =0 NT=2 A 0 (0) [-1..+1] Static evaluation score max-ing C -1(0) [-..0] B 0 (1) [-..1] D 0 (0) [-..0] min-ing G 1 E 0 F 1 Bringing any of {E F G} and any of {L M N} to -1 or less would bring B & D to -1 or less, and A to -1 Raising E to 1 or more would bring B & A to 1 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  28. K 0 L 0 M 0 N 0 H -1 J 0 Tree Growth example: =0 NT=2 A 0 (0) [-1..+1] Static evaluation score max-ing C -1(0) [-..0] B 0 (1) [-..1] D 0 (0) [-..0] min-ing G 1 E 0 F 1 range.max and range.min are equally far from current value, arbitrarily pick range.min to rule out next. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  29. K 0 L 0 M 0 N 0 H -1 J 0 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [-1..3] Static evaluation score max-ing C -1(0) [-..0] B 1 (1) [-..3] D 0 (0) [-..0] min-ing G 1 E 3 (0) [0…] F 1 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  30. K 0 L 0 M 0 N 0 H -1 J 0 R 2 S 1 T 1 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [-1..3] Static evaluation score max-ing C -1(0) [-..0] B 1 (1) [-..3] D 0 (0) [-..0] min-ing G 1 E 3 (0) [0…] F 2 (1) [1…] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  31. K 0 L 0 M 0 N 0 H -1 J 0 U 1 V 1 W 1 R 2 S 1 T 1 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [0..3] Static evaluation score max-ing C -1(0) [-..0] B 1 (1) [0..3] D 0 (0) [-..0] min-ing G 1 (1) [1..] E 3 (0) [0…] F 2 (1) [1…] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  32. K 0 L 0 M 0 N 0 H -1 J 0 U 1 V 1 W 1 R 2 S 1 T 1 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [0..3] Static evaluation score max-ing C -1(0) [-..0] B 1 (1) [0..3] D 0 (0) [-..0] min-ing G 1 (1) [1..] E 3 (0) [0…] F 2 (1) [1…] Raising any of {R S T} and any of {U V W} to 3 or beyond allows max value 3 for A Reducing to 0 or beyond allows min value 0 for A http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  33. K 0 L 0 M 0 N 0 H -1 J 0 U 1 V 1 W 1 R 2 S 1 T 1 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [0..3] Static evaluation score max-ing C -1(0) [-..0] B 1 (1) [0..3] D 0 (0) [-..0] min-ing G 1 (1) [1..] E 3 (0) [0…] F 2 (1) [1…] http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  34. K 0 L 0 M 0 N 0 H -1 J 0 U 1 V 1 W 1 R 2 S 1 T 1 O 3 P 0 Q 0 Tree Growth example: =0 NT=2 A 1 (0) [1..1] max-ing C -1(0) [-..] B 1 (1) [..] D 0 (0) [..] min-ing G 1 (1) [..] E 3 (0) […] F 2 (1) […] The tree grows non-uniformly until eventually the range of likely values at the root node is -or-less in width http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  35. Calculating ranges of likely values • Each node in an expanded tree is either • A Game-Terminal node, whose value cannot be changed • A Non-Game-Terminal node (hereinafter a NGT node), which is either • an expanded interior node, whose value could be changed by a conspiracy among nodes in its subtree • an unexpanded leaf node, whose value could be changed without limit by expanding the node and finding the value of just one child. • NGT nodes can be associated with two bounds sequences: • An ascending bounds sequence, • being a sequence of nondecreasing limiting values which could be achieved with successively larger conspiracies, starting at the node’s current value and going perhaps to infinity • A descending bounds sequence, … nonincreasing … minus infinity http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  36. Bounds Sequences contain descendant node values & perhaps ± • A NGT leaf node, whether a max or a min node, with current value V0, • needs no conspiracy to achieve its current value V0, • needs only itself to change to have value + itself • its ascending bounds sequence is therefore {V0, +} • A NGT interior node, if a max node with current value V0, • needs no conspiracy to achieve its current value V0, (which must be the current value of at least one of the children) • can achieve any greater value that any of its child nodes may achieve with the same conspiracy as that child node needs • its ascending bounds sequence {V0, V1 … Vn} consists of those Vj which are maximal for its children’s ascending bounds sequences, or Vj-1 if greater http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  37. Bounds Sequences contain descendant node values & perhaps ± • A NGT interior node, if a min node with current value V0, • needs no conspiracy to achieve its current value V0, (which must be the current value of at least one of the children) • can achieve any greater value that any of its child nodes may achieve only when all its child nodes have achieved at least that value • its ascending bounds sequence {V0, V1 … Vn} consists of all Vj which are elements of its children’s ascending bounds sequences, sorted into ascending order, with duplicates retained, but stopping at the lowest maximum value. • A game terminal node with current value V0, cannot have any other value • its ascending bounds sequence is {V0} • Descending bounds sequences can be calculated in very similar fashion • greater  lesser, min  max, +  -, ascending  descending http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  38. Calculating ascending bounds sequences (top-down) • (defun AscSeq (node) • (cond • ((GameTerminalP node) • (list (NodeValue node))) • ((UnExpandedP node) • (list (NodeValue node) +)) • ((MaxP node) • (AscSeqMax • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))) • ((MinP node) • (AscSeqMin • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))))) http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  39. Calculating ascending bounds sequences (top-down) • (defun AscSeq (node) • (cond • ((GameTerminalP node) • (list (NodeValue node))) • ((UnExpandedP node) • (list (NodeValue node) +)) • ((MaxP node) • (AscSeqMax • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))) • ((MinP node) • (AscSeqMin • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))))) (defun AscSeqMax (minimum sequences) (when sequences (let ((new (reduce #’max (mapcar #’first sequences) :initial-value minimum))) (cons new (AscSeqMax new (remove nil (mapcar #’rest sequences))))))) http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  40. Calculating ascending bounds sequences (top-down) • (defun AscSeq (node) • (cond • ((GameTerminalP node) • (list (NodeValue node))) • ((UnExpandedP node) • (list (NodeValue node) +)) • ((MaxP node) • (AscSeqMax • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))) • ((MinP node) • (AscSeqMin • (NodeValue node) • (mapcar #’AscSeq (NodeChildren node)))))) (defun AscSeqMin (maximum sequences) (let ((leastbest (reduce #'min (mapcar #'(lambda (seq) (apply #'max seq)) sequences) :initial-value maximum))) (sort #'< (cons maximum (mapcan #'(lambda (seq) (remove leastbest seq :test #'>)) sequences))))) http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  41. Conspiracy Numbers: Summary • Search can terminate without a value for the root node being fully determined: therefore, more cheaply. • Game trees are grown selectively, in such a way as to maximise the size of conspiracy necessary to change the root node’s value. The trees often turn out to be deep and narrow. • Node values (derived from static evaluator results) may be real numbers. • A related algorithm, Proof Number Search, to be dealt with next, is useful in situations where an evaluator returns boolean values rather than numeric ones. • Reference: McAllester, D.A. Conspiracy Numbers for Min-Max Search • Artificial Intelligence 35 (1988) 287-310 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  42. CN and PN Tree Search Algorithms Reprise • Like B*, the “Conspiracy Number” (CN) and “Proof Number” (PN) algorithms • Expand one node at a time, at any point on the fringe of a game tree, • Generate all child nodes - consider all moves - as part of node expansion. • Selective node expansion  Selective move generation “Proof Number” (PN) algorithms 1st 2nd 4th 6th 3rd 5th http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  43. Deeper Search gives better results • Experience with computer chess shows that deeper search gives better play. • Programs that can search one extra ply of game tree do gain advantage from it. • A static evaluator gives an estimate of a position’s worth. • These might be used in chess, for example to determine whether opponent Queen can be captured for less than the cost of Rook+Pawn; • Or in Go, to determine for example whether a group can be given two eyes in fewer than 15 moves. But game trees with numeric valuations are not the only game trees in town: There are also AND-OR trees with boolean valuations. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  44. And/Or Tree - Simple Case A: (or) T B: (and) T C: (and) ? D: (or) T E: (or) T F: (or) T G: (or) ? If all children of a parent AND node are true, then the parent is true. If any child of a parent OR node is true, then the parent is true. red indicates a leaf node whose value can be known directly H: (and) F I: (and) T J: (or) F K: (or) ? http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  45. Adaptation of Conspiracy Number Search • The essential purpose of conspiracy number search is to reduce the search effort necessary to establish the value of a root node. It does this by • selectively expanding leaf nodes at the frontier of a game tree • focusing effort on those nodes most likely to make a difference • therefore not necessarily expanding some or all other nodes at a level • optionally, allowing some uncertainty in the value computed • Proof number search adapts the first of these to the boolean-value case: • Selectively expanding those leaf nodes likely to give an answer most cheaply http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  46. Proof and Disproof numbers • Each node in an and/or tree is adorned with two numbers: • its proof number • the minimum number of dominated leaf nodes that would have to be evaluated to provide the value true • A node already known to have value true needs 0 more leaf nodes evaluated to be shown to be true: its proof number is therefore 0 • A node already known to have value false cannot be shown to be true no matter how many leaf nodes are evaluated: its proof number is therefore  • its disproof number • the minimum number of dominated leaf nodes that would have to be evaluated to provide the value false • similarly, already-false nodes have disproof number 0 • already-true nodes have disproof number  http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  47. Calculating proof & disproof numbers For a leaf node: if the value is known to be true, then its P# := 0 D# :=  if the value is known to be false, then its P# :=  D# := 0 if the value is not known, then its P# := 1 D# := 1 http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  48. Calculating proof & disproof numbers For a leaf node: if the value is known to be true, then its P# := 0 D# :=  if the value is known to be false, then its P# :=  D# := 0 if the value is not known, then its P# := 1 D# := 1 For an interior AND node: its P# := sum of its children’s P#s its D# := min of its children’s D#s http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  49. Calculating proof & disproof numbers For a leaf node: if the value is known to be true, then its P# := 0 D# :=  if the value is known to be false, then its P# :=  D# := 0 if the value is not known, then its P# := 1 D# := 1 For an interior AND node: its P# := sum of its children’s P#s its D# := min of its children’s D#s For an interior OR node: its P# := min of its children’s P#s its D# := sum of its children’s D#s http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

  50. Calculating proof & disproof numbers For a leaf node: if the value is known to be true, then its P# := 0 D# :=  if the value is known to be false, then its P# :=  D# := 0 if the value is not known, then its P# := 1 D# := 1 For an interior AND node: its P# := sum of its children’s P#s its D# := min of its children’s D#s For an interior OR node: its P# := min of its children’s P#s its D# := sum of its children’s D#s When a leaf node with unknown value is expanded, calculate its new P# and D# according to the above recursively, when a node’s P# or D# changes, recompute its parent’s P# and/or D# too. http://csiweb.ucd.ie/Staff/acater/comp30260.htmlArtificial Intelligence for Games and Puzzles

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