Problem Solving with Constraints CSCE421/821, Spring 2018 cse.unl/~ choueiry/S18-421-821

Path Consistency & Global Consistency Properties Problem Solving with Constraints CSCE421/821, Spring 2018 www.cse.unl.edu/~choueiry/S18-421-821 All questions: Piazza Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Tel: +1(402)472-5444

Lecture Sources Required reading • Algorithms for Constraint Satisfaction Problems, Mackworth and Freuder AIJ'85 • Sections 3.1, 3.2, 3.3. Chapter 3. Constraint Processing. Dechter Recommended • Sections 3.4—3.10. Chapter 3. Constraint Processing. Dechter • Networks of Constraints: Fundamental Properties and Application to Picture Processing, Montanari, Information Sciences 74 • Bartak: Consistency Techniques (link) • Path Consistency on Triangulated Constraint Graphs, Bliek & Sam-Haroud IJCAI'99

Outline • Motivation • Path consistency and its complexity • Global consistency properties • Minimality • Decomposability • When PC guarantees global consistency

= = = AC is not enough Example borrowed from Dechter Arc-consistent? Satisfiable?  seek higher levels of consistency V V 1 1 b a a b V V V V 2 3 2 a b 3 b a a b a b

V2 V1 Vm-1 Vm V0 for all y  DVm for all x  DV0 Consistency of a path A path (V0, V1, V2, …, Vm) of length m is consistent iff • for any value xDV0 and for any value yDVm that are consistent (i.e., PV0 Vm(x, y)) •  a sequence of values z1, z2, … , zm-1 in the domains of variables V1, V2, …, Vm-1, such that all constraints between them (along the path, not across it) are satisfied (i.e., PV0 V1(x, z1)  PV1 V2(z1, z2)  …  PVm-1 Vm(zm-1, zm) )

V2 V1 Vm-1 Vm V0 for all y  DVm for all x  DV0 Note  The same variable can appear more than once in the path  Every time, it may have a different value  Constraints considered: PV0,Vm and those along the path • Universal constraints can be included in path • All other constraints are neglected

V2 V3 V1 V4 {a, b, c} {a, b, c} {a, b, c} {a, b, c} All mutex constraints {a, b, c} {a, b, c} V5 V7 {a, b, c} V6 Example: consistency of a path  Check path length = 2, 3, 4, 5, 6, ....            

Path consistency: definition  A path of length m is path consistent  A CSP is path consistent Property of a CSP Definition: A CSP is path consistent (PC) iff every path is consistent (i.e., any length of path) Question: should we enumerate every path of any length? Answer: No, only length 2, thanks to [Mackworth AIJ'77]

Tools for PC-1 Two operators • Constraint composition: ( • ) R13 = R12• R23 • Constraint intersection: (  ) R13 R13, old R13, induced

Path consistency (PC-1) Achieved by composition and intersection (of binary relations expressed as matrices) over all paths of length two. Procedure PC-1: 1 Begin 2 Yn R 3 repeat 4 begin 5 Y0 Yn 6 For k 1 untilndo 7 For i 1 untilndo 8 For j 1 untilndo 9 Ylij Yl-1ij Yl-1ik• Yl-1kk• Yl-1kj 10 end 11 until Yn = Y0 12 Y  Yn 10 end

Properties of PC-1 Discrete CSPs[Montanari'74] • PC-1 terminates • PC-1 results in a path consistent CSP • PC-1 terminates. It is complete, sound (for finding PC network) • PC-2: Improves PC-1 similar to how AC3 improves AC-1 Complexity of PC-1..

Procedure PC-1: 1 Begin 2 Yn R 3 repeat 4 begin 5 Y0 Yn 6 For k 1 untilndo 7 For i 1 untilndo 8 For j 1 untilndo 9 Ylij Yl-1ij Yl-1ik• Yl-1kk• Yl-1kj 10 end 11 until Yn = Y0 12 Y  Yn 10 end Line 9: a3 Lines 6–10: n3.a3 Line 3: at most n2relations x a2elements PC-1 is O(a5n5) Complexity of PC-1 PC-2 is O(a5n3) and (a3n3) PC-1, PC-2 are specified using constraint composition Basic Consistency Methods

Enforcing Path Consistency (PC) General case: Complete graph Theorem: In a complete graph, if every path of length 2 is consistent, the network is path consistent [Mackworth AIJ'77]  PC-1: two operations, composition and intersection  Proof by induction.

Some improvements • Mohr & Henderson (AIJ 86) • PC-2 O(a5n3) PC-3 O(a3n3) • Open question: PC-3 optimal? • Han & Lee (AIJ 88) • PC-3 is incorrect • PC-4 O(a3n3) space and time • Singh (ICTAI 95) • PC-5 uses ideas of AC-6 (support bookkeeping) • Also: • PC8: iterates over domains, not constraints [Chmeiss & Jégou 1998] • PC2001: an improvement over PC8, not tested [Bessière et al. 2005] Note: PC is seldom used in practical applications unless in presence of special type of constraints (e.g., bounded difference)

B B A < B A < B A A B < C C A < C C Path consistency as inference of binary constraints Path consistency corresponds to inferring a new constraint (alternatively, tightening an existing constraint) between every two variables given the constraints that link them to a third variable  Considers all subgraphs of 3 variables  3-consistency B < C

V1 V1 a b a b     = V2 V2 V3 V3 a b a b a b a b a b =     a b a b V4 V4 Path consistency as inference of binary constraints Another example:

B A < B A B < C A < C C A + 3 > C Question Adapted from Dechter Given three variables Vi, Vk, and Vj and the constraints CVi,Vk, CVi,Vj, and CVk,Vj, write the effect of PC as a sequence of operations in relational algebra. B B A < B A < B A A B < C B < C C C -3 < A –C < 0 A + 3 > C Solution: CVi,Vj CVi,Vjij(CVi,Vk CVk,Vj)

Partial Path Consistency • Formal definition: Same as PC except that • Universal constraints cannotbe included • Defined over cycles • Algorithm: Same as PC-i except that • We triangulate the graph • We run the closure loops over the triangles only O(n3) • (Correction: Careful for articulation points in graph) Theorem: In a triangulated graph, if every path of length 2 is consistent, the network is partial path consistent [Bliek & Sam-Haroud ‘99]  PPC (partially path consistent)

PPC versus PC Arbitrary binary constraints • PC property is strictly stronger than PPC property • Open question: Can PC detect insatisfiability when PPC does not? Yes! Example found by Chris Reeson [TBP]

( 0, 1 ) ( 0, 1 ) ( 0, 1 ) ( 0, 1 ) ( 0, 1 ) ( 0, 1 )     1 2 1 2 3 1 2 3 2 3 Constraint propagation courtesy of Dechter After Arc-consistency: After Path-consistency: • Are these CSPs the same? • Which one is more explicit? • Are they equivalent? • The more propagation, • the more explicit the constraints • the more search is directed towards a solution

PC can detect unsatisfiability Arc-consistent? Path-consistent? V1 a b    V2 V3  a b a b a b   a b V4

Warning:Does 3-consistency guarantee 2-consistency? B • Question: • Is this CSP 3-consistent? • is it 2-consistent? • Lesson: • 3-consistency does not guarantee 2-consistency {red, blue} {red, blue}   A C { red } { red }

V2 V3 V1 V4 {a, b, c} {a, b, c} {a, b, c} {a, b, c} All mutex constraints {a, b, c} {a, b, c} V5 V7 {a, b, c} V6 PC is not enough Arc-consistent? Path-consistent? Satisfiable?  we should seek (even) higher levels of consistency • k-consistency, k = 1, 2, 3, …. …following lecture                 

Minimality • PC tightens the binary constraints • The tightest possible binary constraints yield the minimal network • Minimal network a.k.a. central problem • Given two values for two variables, if they are consistent, then they appear in at least one solution. • Note: • Minimal  path consistent • The definition of minimal CSP is concerned with binary CSPs, but it need not be

Minimal CSP • Minimal network a.k.a. central problem • Given two values for two variables, if they are consistent, then they appear in at least one solution. • Informally • In a minimal CSP the remainder of the CSP does not add any further constraint to the direct constraint CVi, Vj between the two variables Vi and Vj[Mackworth AIJ'77] • A minimal CSP is perfectly explicit: as far as the pair Vi and Vj is concerned, the rest of the network does not add any further constraints to the direct constraint CVi, Vj [Montanari'74] • The binary constraints are explicit as possible. [Montanari'74]

Decomposability • Any combination of values for k variables that satisfy the constraints between them can be extended to a solution. • Decomposability generalizes minimality • Minimality: any consistent combination of values for • any 2 variables is extendable to a solution • Decomposability: any consistent combination of values for • any k variables is extendable to a solution Minimal  Decomposable  Path Consistent     n-consistent  Strong n-consistent  Solvable

Terminology • Minimal Globally consistent • Decomposable strongly n-consistent

PC approximates.. • In general: • Decomposability minimality path consistent • PC is used to approximate minimality (which is the central problem) • When is the approximation the real thing? • Special cases: • When composition distributes over intersection, [Montanari'74] • PC-1 on the completed graph guarantees minimality and decomposability • When constraints are convex [Bliek & Sam-Haroud 99] • PPC on the triangulated graph guarantees minimality and decomposability (and the existing edges are as tight as possible)

PPC versus PC

PC: Special Case • Distributivity property • Outer loop in PC-1 (PC-3) can be ignored • Exploiting special conditions in temporal reasoning • Temporal constraints in the Simple Temporal Problem (STP): composition & intersection • Composition distributes over intersection • PC-1 is a generalization of the Floyd-Warshall algorithm (all pairs shortest path) • Convex constraints • PPC

Distributivity property Intersection,  Composition, • • In PC-1, two operations: • RAB• (RBC R'BC) = (RAB• RBC)  (RAB• R’BC) • When ( • ) distributes over (  ), then [Montanari'74] • PC-1 guarantees that CSP is minimal and decomposable • The outer loop of PC-1 can be removed B R’BC RAB RBC A C

Condition does not always hold Constraint composition does not always distribute over constraint intersection • R12= R23= R’23= • ⋅( ∩ ) = ⋅ = • ( ⋅ ) ∩ ( ⋅ )= ∩ = 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 0 0

Temporal Reasoningconstraints of bounded difference Variables: X, Y, Z, etc. Constraints: a  Y-X  b, i.e. Y-X = [a, b] = I Composition: I 1•I2 = [a1, b1] • [a2, b2] = [a1+ a2, b1+b2] Interpretation: • intervals indicate distances • composition is triangle inequality. Intersection: I1I2 = [max(a1, a2), min(b1, b2)] Distributivity: I1• (I2 I3) = (I1• I2)  (I1• I3) Proof: left as an exercise

V2 V0 R01=[2,5] R12=[3,4] R23=[1,8] V1 V3 R’13 R13=[3,5] Example: Temporal Reasoning Composition of intervals + : R’13 = R12 + R23 = [4, 12] R01 + R13 = [2,5] + [3, 5] = [5, 10] R01 + R'13 = [2,5] + [4, 12] = [6, 17] Intersection of intervals: R13 R'13 = [4, 12]  [3, 5] = [4, 5] R01 + (R13 R'13) = (R01 + R13)  (R01 + R'13) R01 + (R13 R'13) = [2, 5] + [4, 5] = [6, 10] (R01 + R13)  (R01 + R'13) = [5, 10]  [6,17] = [6, 10] Here, path consistency guarantees minimality and decomposability

Composition Distributes over  • PC-1 generalizes Floyd-Warshall algorithm (all-pairs shortest path), where • composition is ‘scalar addition’ and • intersection is ‘scalar minimal’ • PC-1 generalizes Warshall algorithm (transitive closure) • Composition is logical OR • Intersection is logical AND

Convex constraints: temporal reasoning (again!) • Thanks to Xu Lin (2002) • Constraints of bounded difference are convex • We triangulate the graph (good heuristics exist) • Apply PPC: restrict propagations in PC to triangles of the graph (and not in the complete graph) • According to [Bliek & Sam-Haroud 99] PPC becomes equivalent to PC, thus it guarantees minimality and decomposability

Summary • Alert: Do not confuse a consistency property with the algorithms for reinforcing it • Local consistency methods • Remove inconsistent values (node, arc consistency) • Remove Inconsistent tuples (path consistency) • Get us closer to the solution • Reduce the ‘size’ of the problem & thrashing during search • Are ‘cheap’ (i.e., polynomial time) • Global consistency properties are the goal we aim at • Sometimes (special constraints, graphs, etc) local consistency guarantees global consistency • E.g., Distributivity property in PC, row-convex constraints, special networks • Sometimes enforcing local consistency can be made cheaper than in the general case • E.g., functional constraints for AC, triangulated graphs for PC

Problem Solving with Constraints CSCE421/821, Spring 2018 cse.unl/~ choueiry/S18-421-821