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Constraint Satisfaction Patrick Prosser

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An Example, Exam Timetabling Someone timetables the exams We have a number of courses to examine how many? Dept has 36 Faculty? University? There are constraints if a student S takes courses Cx and Cy Cx and Cy cannot be at same time! If Cy and Cz have no students in common they can go in room R1 if there is space Temporal and resource constraints

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An Example, Exam Timetabling Represent as graph colouring vertices are courses colours are time vertices have weight (room requirements) edge connects vertices of diff colour How complex is this if we have n vertices and k times an n-digit number to the base k? How would you solve this backtracking search? Greedy? Something else GA? SA, TS, GLS, HC,...

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An Example, Exam Timetabling How does the person solve this? Is that person intelligent? Is there always a solution? If there isnt, do we want to know why? Do you think they can work out why?

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A CSP A csp has n variables each has a domain of (m) values constraints define compatible tuples of values n-ary, binary find an assignment of values to variables that satisfies the constraints or show none exists O(m n )

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1 23456 4 7 An example Make a crossword puzzle! Given the above grid and a dictionary, fill it. Then go get the clues (not my problem)

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1A 1 across 4D 4 down 2D 2 down 4A 4 across 7D 7 down 1 23456 4 7 1A4D 2D 7D 4A An example Variables

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1 23456 4 7 1A4D 2D 7D 4A An example 1A-4D: 4th of 1A equals 1st of 4D 1A-2D: 2nd of 1A equals 1st of 2D 2D-4A: 4th of 2D equals 2nd of4D 4D-4A: 4th of 4A equals 4th of 4D 4A-7D: 7th of 4A equals 2nd of 7D Constraints

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1 23456 4 7 1A4D 2D 7D 4A An example 1A: any 6 letter word 4A: any 8 letter word 4D: any 5 letter word 2D: any 7 letter word 7D: any 3 letter word Domains (also unary constraints!)

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1 23456 4 7 1A4D 2D 7D 4A An example Find an assignment of values to variables, from their domains, such that the constraints are satisfied (or show that no assignment exists) A CSP!

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Choose a variable Assign it a value Check compatibility If not compatible try a new value If no values remain re-assign previous variable 1 23456 4 7 1A4D 2D 7D 4A An example Good old fashioned BT!

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What variable should I choose? 1 23456 4 7 1A4D 2D 7D 4A Questions? Decisions, decisions! What value should I choose? What reasoning can I do when making an assignment? What reasoning can I do on a dead end?

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Wheres the AI?

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Scene Labelling David Waltz, MIT, 1975

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In a trihedral world, these are the only scenarios

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Label an edge as follows Walk along the edge in this direction and the object is on the right, and to your left is open space + This is an outside edge, with the object on both sides - This is an inside edge

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We now have the following cases - - + + + + - - - - - - + +-+ -- + + + -

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- - - + + + + + - - + ++ A consistent labelling is an interpretation The edges are the variables, labels their domains, meeting points are the constraints

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Another example: n-queens Place n non-attacking queens on an n x n chess board representations? There is a polynomial solution circa 1800That was BT

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Its all just depth first search, right?

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BT Thrashes! pastpast futurefuture current variable v[i] conflict with v[h] past variable v[h] future variable v[j]

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Another example: n-queens Forward Checking

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1 2 3 4 5 6 7 9 8 NOTE: arrows go forward!

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How to improve search use a heuristic variable and/or value ordering dynamic or static Fail First? More inferencing at each search state old trade off, knowledge versus search maintain consistency

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Consistency arc consistency whats that then? Propagate supports deduce illegal values polynomial at each search node AC can be specialised for special constraints MAC the heart of constraint programming

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Give us a demo?

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Not just binary csps! N-ary but can be mapped to binary why bother with n-ary? allDiff, sum, permutation, marriage Not just arc consistency path inverse restricted singleton When, what?

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Applications scheduling timetabeling frequency allocation transportation design layouts packing...

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Toolkits ILOG solver, scheduler, dispatcher chip choco OZ Eclipse Jsolver Screamer CSPLab

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Research Direction reactivity and explanation retraction in particular better heuristics new search algorithms complete, quasi-complete, local new levels of consistency new specialised constraints effects of representation understanding the problem its structure and why it is hard

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Constraint Satisfaction

Constraint Satisfaction

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