CSC411Artificial Intelligence 1 Chapter 3 Structures and Strategies For Space State Search Contents...

CSC411CSC411 Artificial IntelligenceArtificial Intelligence 1

Chapter 3Chapter 3

Structures and Strategies For Space Structures and Strategies For Space State SearchState Search

Contents

• Graph Theory• Strategies for Space State Search• Using the Space State to Represent

Reasoning with the Predicate Calculus


The city of KönigsbergLeonhard EulerLeonhard Euler

Problem: if there is a walk around the city Problem: if there is a walk around the city that crosses each bridge exactly once?that crosses each bridge exactly once?


RepresentationsRepresentationsPredicate calculus: connect(X, Y, Z)Predicate calculus: connect(X, Y, Z)connect(i1, i2, b1) connect(i1, i2, b1) connect(i2, i1, b1)connect(i2, i1, b1)connect(rb1, i1, b2)connect(rb1, i1, b2)connect(i1, rb1, b2)connect(i1, rb1, b2)connect(rb1, i1, b3)connect(rb1, i1, b3)connect(i1, rb1, b3)connect(i1, rb1, b3)connect(rb1, i2, b4)connect(rb1, i2, b4)connect(i2, rb1, b4)connect(i2, rb1, b4)connect(rb2, i1, b5)connect(rb2, i1, b5)connect(i1, rb2, b5)connect(i1, rb2, b5)connect(rb2, i1, b6)connect(rb2, i1, b6)connect(i1, rb2, b6)connect(i1, rb2, b6)connect(rb2, i2, b7)connect(rb2, i2, b7)connect(i2, rb2, b7)connect(i2, rb2, b7)Graph theoryGraph theory– NodesNodes– LinkesLinkes– Easy proof: the walk is impossible since all nodes Easy proof: the walk is impossible since all nodes

have odd degreeshave odd degrees


Graph of the Königsberg bridge systemGraph of the Königsberg bridge system


A labeled directed graphA labeled directed graph


A rooted tree, exemplifying family A rooted tree, exemplifying family relationshipsrelationships



Finite State Machine (FSM)Finite State Machine (FSM)


(a) The finite state graph for a flip flop and

(b) its transition matrix.

Flip Flop FSMFlip Flop FSM


Finite State Accepting MachineFinite State Accepting Machine

Deterministic FSM: transition function for any input value Deterministic FSM: transition function for any input value to a state gives a unique next stateto a state gives a unique next state

Probabilistic FSM: the transition function defines a Probabilistic FSM: the transition function defines a distribution of output states for each input to a statedistribution of output states for each input to a state


(a)The finite state graph

(b)The transition matrix for string recognition example

String RecognitionString Recognition


State Space and SearchState Space and Search


• generated by “move blank” operations

-- up

-- left

-- down

-- left

State Space of the 8-Puzzle State Space of the 8-Puzzle


The travelling salesperson problemThe travelling salesperson problemFind the Find the shortest shortest path for the path for the salesperson salesperson to travel, to travel, visiting visiting each city each city and and returning to returning to the starting the starting citycity


Search for the travelling salesperson problem. Each arc is marked with the total weight of all paths from the start node (A) to its endpoint.


An instance of the travelling salesperson problem with the nearest neighbour path in bold. Note this path (A, E, D, B, C, A), at a cost of 550, is not the shortest path. The comparatively high cost of arc (C, A) defeated the heuristic.


Strategies for State Space SearchStrategies for State Space Search

Data-driven search – forward chainingData-driven search – forward chaining– Begin with the given facts and a set of legal Begin with the given facts and a set of legal

rules for changing statesrules for changing states– Apply rules to facts to produce new factsApply rules to facts to produce new facts– Repeat rules application until finding a path that Repeat rules application until finding a path that

satisfies the goal conditionsatisfies the goal conditionGoal-driven search – backward chainingGoal-driven search – backward chaining– Begin with the goal and a set of facts and legal Begin with the goal and a set of facts and legal

rulesrules– Search rules that generate this goalSearch rules that generate this goal– Determine conditions of these rules Determine conditions of these rules subgoals subgoals– Repeat until all conditions are factsRepeat until all conditions are facts


State space in which data-directed search prunes irrelevant data and their consequents and determines one of a number of possible goals.

Data-driven SearchData-driven Search


State space in which goal-directed search effectively prunes extraneous search paths.

Goal-driven SearchGoal-driven Search


Search and BacktrackSearch and Backtrack

Search – find a pathSearch – find a path

Backtrack – when the path is dead, Backtrack – when the path is dead, try otherstry others– Backtrack to the most recent node on Backtrack to the most recent node on

the path having unexamined siblingsthe path having unexamined siblings– Continue toward to a new pathContinue toward to a new path– Like a recursionLike a recursion– Implemented in Prolog as an internal Implemented in Prolog as an internal

mechanismmechanism


Backtrack algorithmBacktrack algorithm


Backtracking search of a hypothetical Backtracking search of a hypothetical state space space.state space space.


A trace of backtrack on the previous graphA trace of backtrack on the previous graph


Depth-First and Breadth-First SearchDepth-First and Breadth-First SearchDetermine the order of nodes (states) to Determine the order of nodes (states) to be examinedbe examinedDepth-first search Depth-first search – When a state is examined, all of its children When a state is examined, all of its children

and their descendants are examined before and their descendants are examined before any of its siblingsany of its siblings

– Go deeper into the search space where Go deeper into the search space where possiblepossible

Breadth-first searchBreadth-first search– When a state is examined, all of its children When a state is examined, all of its children

are examined after any of its siblingsare examined after any of its siblings– Explore the search space in a level-by-level Explore the search space in a level-by-level

fashionfashion


Graph for search examplesGraph for search examples


The breadth-first search algorithmThe breadth-first search algorithm


A trace of breadth-first searchA trace of breadth-first search


The graph at iteration 6 of breadth-first search. States on open and closed are highlighted


Breadth-first search of the 8-puzzle, showing order in which states were removed from open


The depth-first search algorithmThe depth-first search algorithm


A trace of depth-first searchA trace of depth-first search


The graph at iteration 6 of depth-first search. States on open and closed are highlighted


Depth-first search of 8-puzzle with a depth bound of 5


Comparison between breadth- and Comparison between breadth- and depth-first searchdepth-first search

Breadth-firstBreadth-first– Always find the shortest path to a goalAlways find the shortest path to a goal– High branching factor -- Combinatorial High branching factor -- Combinatorial

explosionexplosion

Depth-firstDepth-first– More efficient More efficient – May get lost May get lost


State Space Representation of State Space Representation of Logical SystemsLogical Systems

RepresentationRepresentation– Logical expressions as statesLogical expressions as states– Inference rules as linksInference rules as links

CorrectnessCorrectness– Soundness and completeness of Soundness and completeness of

predicate calculus inference rules predicate calculus inference rules guarantee the correctness of guarantee the correctness of conclusionsconclusions

Theorem ProofTheorem Proof– State space search State space search


State space graph of the State space graph of the propositional calculuspropositional calculus

• Letters as nodes• Implications as links

•qp•rp•vq•sr•tr•su


And/or graphAnd/or graph

• Or – separate

• And -- connected

• And/or graph of expression q r p

• And/or graph of the expression q r → p



And/or graph of a set of propositional And/or graph of a set of propositional calculus expressions.calculus expressions.


And/or graph of part of the state space for And/or graph of part of the state space for integrating a functionintegrating a function


The facts and rules of this example are given as English sentences followed by their predicate calculus equivalents:


The solution subgraph showing that Fred is at the museum.


Rules for a simple subset of English grammar are:


And/or graph for the grammar. Some of the nodes (np, art, etc) have been written more than once to simplify drawing the graph.


And/or graph searched by the financial advisor.


Parse tree for the sentence “The dog bites the man.”

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