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Arc consistency ac3, ac4, ac6/7/8

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Di = {1,2,3,4,5} V1 V3 V2 V4 V4 V3 V1 V4 1 V4 + V2 = 5 V1 V2 V2 V3 6 AR33 figure 18, page 35 What can you infer?

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Di = {1,2,3,4,5} V1 V3 V2 V4 V4 V3 V1 V4 1 V4 + V2 = 5 V1 V2 V2 V3 6 D1 = {1,2} D2 = {2,3} D3 = {4,5} D4 = {2,3} Do you agree? Was that easy?

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V1 < V2 D1 = {1,2,3,4,5} D2 = {1,2,3,4,5} D1 := {1,2,3,4} V2 > V1 D1 = {1,2,3,4} D2 = {1,2,3,4,5} D2 := {2,3,4,5} V4 ≥ V1 + 1 D4 = {1,2,3,4,5} D1 = {1,2,3,4} D4 := {2,3,4,5} V1 ≤ V4 - 1 D1 = {1,2,3,4} D4 = {2,3,4,5} no change V2 + V3 > 6 D2 = {2,3,4,5} D3 = {1,2,3,4,5} no change V3 + V2 > 6 D3 = {1,2,3,4,5} D2 = {2,3,4,5} D3 := {2,3,4,5} V2 + V4 = 5 D2 = {2,3,4,5} V4 = {2,3,4,5} D2 = {2,3} V1 < V2 D1 = {1,2,3,4} D2 = {2,3} D1 = {1,2} V2 > V1 D2 = {2,3} D1 = {1,2} no change V4 ≥ V1 + 1 D4 = {2,3,4,5} D1 = {1,2} no change V3 + V2 > 6 D3 = {2,3,4,5} D2 = {2,3} D3 = {4,5} V2 + V3 > 6 D2 = {2,3} D3 = {4,5} no change V4 + V2 = 5 D4 = {2,3,4,5} D2 = {2,3} D4 = {2,3} V1 ≤ V4 - 1 D1 = {1,2} D4 = {2,3} no change V2 + V4 = 5 D2 = {2,3} D4 = {2,3} no change V4 < V3 D4 = {2,3} D3 = {4,5} no change V3 > V4 D3 = {4,5} D4 = {2,3} no change Here’s the reasoning V4 V3 V1 V4 1 V4 + V2 = 5 V1 V2 V2 V3 6 Di = {1,2,3,4,5}

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Arc consistency: so what’s that then?

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A constraint Cij is arc consistent if for every value x in Di there exists a value y in Dj that supports x i.e. if v[i] = x and v[j] = y then Cij holds note: we are assuming Cij is a binary constraint A csp (V,D.C) is arc consistent if every constraint is arc consistent This is also called 2-consistency If (V,D,C) is arc consistent then I can choose any variable v[i] assign it a value x from its domain Di I can now choose any other variable v[j] I can find a consistent instantiation for v[j] from Dj NOTE: this is in isolation, where I have only 2 variables that I instantiate

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A constraint Cij is arc consistent if for every value x in Di there exists a value y in Dj that supports x i.e. if v[i] = x and v[j] = y then Cij holds note: we are assuming Cij is a binary constraint A csp (V,D.C) is arc consistent if every constraint is arc consistent

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Just because a problem (V,D,C) is arc consistent does not mean that it has a solution! Arc-consistency processes a problem and removes from the domains of variables values that CANNOT occur in any solution Arguably, it makes the resultant problem easier. Why? The arc-consistent problem has the same set of solutions as the original problem Note: is constraint graph is a tree, AC is a decision procedure

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So, is there 1-consistency? Yip when we have unary constraints example odd(V[i]) 1-consistency, we weed out all odd values from Di also called node-consistency (NC) 3-consistency? Given constraints Cij and Cjk, disallow all pairs (x,z) in the constraint Cik where there is no value y in Dj such that Cij(x,y) and Cjk(y,z) This adds nogood tuples to an existing constraints, or creates a new constraint! sometimes called path-consistency (PC) (Given 2 variables in isolation, we can instantiate those consistently, and pick any third variable and …)

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Path-consistency (aka 3-consistency) Vi Vj Vk There might be no constraint Cik Therefore 3-consistency may create it! It may create nogood tuples {(i/x,k/z),…} Therefore increases size of model/problem. May result in more constraints to check! M. Singh, TAI-95

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ac3: Mackworth 1977 Alan Mackworth presented ac1, ac2, and ac3 in 1977. ac1 and ac2 were “straw men”

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ac3: revise a constraint (claire3 code) [revise(d:array,c:tuple,i:integer,j:integer) : boolean ->let revised := false in (for x in copy(d[i]) (let supported := false in (for y in d[j] (if check(c,x,y) (supported := true, return())), if not(supported) (delete(d[i],x), revised := true))), CONSISTENT := d[i] != {}, revised)]

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ac3: revise a constraint (pseudo code) revise(i,j) revised := false for x in d[i] // iterate over all values in d[i] do supported := false for y in d[j] while ¬supported // find first support in d[j] for x do supported := check(i,x,j,y) // is v[i]=x && v[j]=y consistent? if ¬supported // if no support, delete x from d[i] then d[i] := d[i] \ {x} revised := true // and set revised to true return revised // delivers true or false Given constraint C_ij remove from the domain d_i all values that have no support in d_j

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5 4 2 1 3 5 4 2 1 3 support The micro-structure of C23 revise searches for 1st support for a valueIs it a bijection? What are implications of this?

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[ac3(d:array,constraints:set) : boolean -> let Q := copy(constraints) in (while (Q != {} & CONSISTENT) let c := Q[1], i := c[2], j := c[3] in (delete(Q,c), if revise(d,c,i,j) Q := Q U {z in constraints | z[3] = i})), CONSISTENT] Ac3 (claire3 code)

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Ac3 (pseudo code) ac(v,d,c) consistent := true; q := c // enqueue all constraints while q ≠ {} & consistent do (i,j) := dequeue(q) // get a constraint if revise(i,j) // if d_i has values removed then q := q U {(k,i) | C ik in C} // need to revise all constraints C_ki consistent := d[i] ≠ {} // stop if domain wipe out return consistent

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if revise(d,c,i,j) Q := Q U {z in constraints | z[3] = i} ac3 What? if revise(i,j) then q := q U {(k,i) | (k,i) in c}

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Note: ac3 has a queue of constraints that need revision because some values in the domains of the variables may be unsupported. Remember: ac3 processes a queue of constraints But forgive me, the queue might be treated as just a set

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ac1 and ac2 (the straw men) essentially revised constraints over and over again, until no change … until reaching a fixed point

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[ac3(d:array,constraints:set) : boolean -> let Q := copy(constraints) in (while (Q != {} & CONSISTENT) let c := Q[1], i := c[2], j := c[3] in (delete(Q,c), if revise(d,c,i,j) Q := Q U {z in constraints | z[3] = i})), CONSISTENT] Complexity of ac3 proved in 1985 by Mackworth & Freuder (AIJ 25) e is number of constraints d is domain size

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[ac3(d:array,constraints:set) : boolean -> let Q := copy(constraints) in (while (Q != {} & CONSISTENT) let c := Q[1], i := c[2], j := c[3] in (delete(Q,c), if revise(d,c,i,j) Q := Q U {z in constraints | z[3] = i})), CONSISTENT] Prove it! Also look at paper by Zhang & Yap e is number of constraints d is domain size

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[ac3(d:array,constraints:set) : boolean -> let Q := copy(constraints) in (while (Q != {} & CONSISTENT) let c := Q[1], i := c[2], j := c[3] in (delete(Q,c), if revise(d,c,i,j) Q := Q U {z in constraints | z[3] = i})), CONSISTENT] The order that we revise the constraints make no difference to the outcome It reaches the same fixed point, the same set of arc-consistent domains The order that we revise the constraints may make a difference to run time. … constraint ordering heuristic, anyone?

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Examples/Demos Run ac3 with the following csp6 Barbara’s instance in AR33 csp7 the crystal maze

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Revise ignored any semantics of the constraint Is that dumb, or what? Could we get round this? Use OOP? A class of constraint? revise as a specialised method?

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AC4, AC6, AC7, …. “Optimal” support counting algorithms

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AC4, AC6, AC7,... 5 4 2 1 3 5 4 2 1 3 Associate with each value in Di a counter supportCount[x,i,j] the number of values in Dj that support x a boolean supports[x,y,j] true if x supports y in Dj 1st stage of the algorithm builds up the supportCount and support flags 2nd stage if supportCount[x,i,j] = 0 (x has no support in Dj over constraint Cij) delete(Di,x) decrement supportCount[y,k,i] (where supports[x,y,k] is true) continue this till no change i.e. propagate If x supports y in Dk and x is deleted from Di Then support count for y in Dk over constraint Cki is decremented

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Best case and worst case performance of ac4 is the same Ac6, 7, and 8 exploit symmetries, and lazy evaluation if x supports y over constraint Cij then y supports x over Cji find the 1st support for x, and only look for more when support is lost Ac3 worst case performance rarely occurs (experimental evidence due to Rick Wallace)

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Why is best case and worst case performance of ac4 ?

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ac1/2/3 due to Alan Mackworth 1977 ac4 Mohr & Henderson AIJ28 1986 ac6, 7, 8 due to Freuder, Bessiere, Regin, and others in AIJ, IJCAI, etc Downside of ac4, ac6, ac7, and ac8 algorithms is “hard to code” ac3 is easy! History Lesson

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AC5 A generic arc-consistency algorithm and its specializations AIJ 57 (2-3) October 1992 P. Van Hentenryck, Y. Deville, and C.M. Teng

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ac5 Ac5 is a “generic” ac algorithm and can be specialised for special constraints (i.e. made more efficient when we know something about the constraints) Ac5 is at the heart of constraint programming It can be “tweaked” to become ac3, or “tweaked” to become ac4 But, it can exploit the semantics of constraints, and there’s the win

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ac5 ac5 processes a queue of tuples (i,j,w) w has been removed from Dj we now need to reconsider constraint Cij to determine the unsupported values in Di A crucial difference between ac3 and ac5 ac3 revise(i,j) because Dj has lost some values seek support for each and every value in Di ac5 process(i,j,w) because Dj has lost the value w (very specific) seek support, possibly, for select few (1?) value in Di Therefore, by having a queue of (i,j,w) we might be able to do things much more efficiently

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ac5, the intuition take an OOP approach constraint is a class that can then be specialised have a method to revise a constraint object allow specialisation have basic methods such as revise when lwb increases revise when upb decreases revise when a value is lost revise when variable instantiated revise initially methods take as arguments the variable in the constraint that has changed possibly, what values have been lost

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ac5, the algorithm [arcCons(d:array,c:tuple) : set -> let s := {}, i := c[2], j := c[3] in (for x in d[i] let supported := false in (for y in d[j] (supported := check(c,x,y), if supported return()), if not(supported) s := s U set(x)), s)] This is just like revise, but delivers the set s of unsupported values in d[i] over the constraint C_i,j s is the set of disallowed values in d[i] remember c is a tuple, the relation r between v[i] and v[j]

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ac5, the algorithm [localArcCons(d:array,c:tuple,w:integer) : set -> arcCons(d,c)] // // opportunity to specialize this with respect to c // We can specialise this method if we know something about the semantics of c We might then do much better that arcCons in terms of complexity What we have above is the dummy’s version, or when we know nothing about the constraint (and we get AC3!) localArcCons(d,c,w) - c is the tuple - d[j] lost the value w - return the set of values unsupported in d[i]

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[enqueue(i:integer,delta:set,q:set,constraints:set) : set -> for c in {z in constraints | z[3] = i} for w in delta q := q U set(tuple(c,w)), q] // // d[i] has lost the values delta // add to the q tuples (c,w) where // - the constraint c is the constraint C_k,i // - w is in delta // ac5, the algorithm The domain d[i] has lost values delta Add to the queue of “things to do” the tuple (c,w) where - c is the constraint C_k,i and - w is in delta This is similar to the step in AC3 Q := Q U {z in constraints | z[3] = i}

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[ac5(d:array,constraints:set) : boolean -> let Q := {} in (for c in constraints let delta := arcCons(d,c), i := c[2] in (d[i] := difference(d[i],delta), Q := enqueue(i,delta,Q,constraints)), while (Q != {}) (let t := Q[1], c := t[1], w := t[2], i := c[2], delta := localArcCons(d,c,w) in (delete(Q,t), Q := enqueue(i,delta,Q,constraints), d[i] := difference(d[i],delta)))), CONSISTENT] ac5, the algorithm ac5 has 2 phases - revise all the constraints and build up the Q, using arcCons - process the queue and propagate using localArcCons, exploiting knowledge on deleted values and the semantics of constraints

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Complexity of ac5 If arcCons is O(d 2 ) and localArcCons is O(d) then ac5 is O(e.d 2 ) If arcCons is O(d) and localArcCons is O( ) then ac5 is O(e.d) See AIJ 57 (2-3) page 299 Remember - ac3 is O(e.d 3 ) - ac4 is O(e.d 2 )

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Functional constraints

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If c is functional We can specialise arcCons and localArcCons for functional constraints A constraint C_i,j is functional, with respect to a domain D, if and only if for all x in D[i] there exists at most one value y in D[j] such that C_i,j(x,y) holds and for all y in D[j] there exists at most one value in D[i] such that C_j,i(y,x) holds The relation is a bijection 5 4 2 1 3 5 4 2 1 3 An example: 3.x = y

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If c is functional Let f(x) = y and f’(y) = x (f’ is the inverse of f) [arcCons(d:array,c:tuple) : set -> let s := {}, f := c[1], i := c[2], j := c[3] in (for x in d[i] if not(f(x) % d[j]) s := s U set(x), s)] // // functional arcCons // the relation is a bijection // 5 4 2 1 3 5 4 2 1 3 The % operator is

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Claire manual, section 4.5, inverse of a relation!

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If c is functional Let inverse(f) deliver the f’, the inverse of function f [localArcCons(d:array,c:tuple,w:integer) : set -> let f := c[1], i := c[2], j := c[3], g := inverse(f) in if (g(w) % d[i]) set(g(w)) else {}] // // assume inverse delivers the inverse of a function // 5 4 2 1 3 5 4 2 1 3 In our picture, assume d[j] looses the value 2 g(2) = 3 localArcCons(d,c12,2) = {3} localArcCons(d,c,w) - c is the tuple - d[j] lost the value w - return the set of values unsupported in d[i]

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Anti-functional constraints

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If c is anti-functional The value w in d[j] conflicts with only one value in d[i] an example: x + 4 ≠ y [localArcCons(d:array,c:tuple,w:integer) : set -> let f := c[1], i := c[2], j := c[3], g := inverse(f) in if (size(d[i]) = 1 & g(w) = d[i][1]) set(d[i][1]) else {}] // // assume inverse delivers the inverse of a function // 5 4 2 1 3 5 4 2 1 3 If D[i] has more than one value, no problem!

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Other kinds of constraints that can be specialised monotonic localArcCons tops and tails domains but domains must be ordered piecewise functional piecewise anti-functional piecewise monotonic See: AIJ 57 & AR33 section 7

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The AIJ57 paper by van Hentenryck, Deville, and Teng is a remarkable paper they tell us how to build a constraint programming language they tell us all the data structures we need for representing variables, their domains, etc they introduce efficient algorithms for arc-consistency they show us how to recognise special constraints and how we should process them they show us how to incorporate ac5 into the search process they explain all of this in easy to understand, well written English It is a classic paper, in my book! AIJ57

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AC2001 Taken from IJCAI01 “Making AC-3 an Optimal Algorithm” by Y Zhang and R. Yap and “Refining the basic constraint propagation algorithm” by Christian Bessiere & Jean-Charles Regin

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The complexity of ac3A beautiful proof A constraint C_i,j is revised iff it enters the Q C_i,j enters the Q iff some value in d[j] is deleted C_i,j can enter Q at most d times (the size of domain d[j]) A constraint can be revised at most d times There are e constraints in C (the set of constraints) revise is therefore executed at most e.d times the complexity of revise is O(d 2 ) the complexity of ac3 is then O(e.d 3 )

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[revise(d:array,c:tuple,i:integer,j:integer) : boolean ->let revised := false in (for x in copy(d[i]) (let supported := false in (for y in d[j] // this is naive (if check(c,x,y) (supported := true, return())), if not(supported) (delete(d[i],x), revised := true))), CONSISTENT := d[i] != {}, revised)] When searching for a support for x in d[j] we always start from scratch This is naïve.

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Assume domains are totally ordered Let resumePoint[i,j,x] = y where y is the first value in d[j] that supports x in d[i] i.e. C_i,j is satisfied by the pair (x,y) Assume succ(y,d) delivers the successor of y in domain d (and nil if no successor exists) When we revise constraint C_i,j, we look for support for x in d[j] get the resumePoint for x in domain d[j] over the constraint call this value y if y is in d[j] then x is supported! Do nothing! If y is not in d[j] then we search through the remainder of d[j] looking for support we use a successor function to get next y in d[j] if we find a y that satisfies the constraint set resumePoint[i,j,x] = y

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[succ(x:integer,l:list) : integer -> if (l = nil) -1 else if (l[1] > x) l[1] else succ(x,cdr(l))] // // find the successor of x in ordered list l // NOTE: in the ac2001 algorithms we assume // we can do this in O(1) //

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[existsY(d:array,c:tuple,i:integer,j:integer,x:integer) : boolean -> let y := RP[i][j][x] in (if (y % d[j]) true else let supported := false in (y := succ(y,list!(d[j])), while (y > 0 & not(supported)) (if check(c,x,y) (RP[i][j][x] := y, supported := true) else y := succ(y,list!(d[j]))), supported))] // // find the first value y in d[j] that supports x in d[i] // if found return true otherwise false //

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[revise(d:array,c:tuple,i:integer,j:integer) : boolean -> let revised := false in (for x in copy(d[i]) (if not(existsY(d,c,i,j,x)) (delete(d[i],x), revised := true)), revised)]

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In the two papers, the algorithm is shown to be optimal, for arbitrary binary csp’s

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arc-consistency is at the heart of constraint programming it is the inferencing step used inside search it has to be efficient data structures and algorithms are crucial to success ac is established possibly millions of times when solving it has to be efficient we have had an optimal algorithm many times ac4, ac6, ac7, ac2001 ease of implementation is an issue we like simple things but we might still resort to empirical study!

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MAC What’s that then

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Maintain arc-consistency Instantiate a variable v[i] := x impose unary constraint d[i] = {x} make future problem ac if domain wipe out backtrack and impose constraint d[i] x make future ac and so on

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Maintain arc-consistency why use instantiation? Domain splitting? resolve disjunctions first for example (V1 < V2 OR V2 < V1)

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You now know enough to go out and build a reasonably efficient and useful CP toolkit

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Now its time to start using a CP toolkit

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