24.2 Naive Backtracking

Transcription

1 Foundations of Atificial Intelligence Apil 16, Constaint Satisfaction Polems: Backtacking Foundations of Atificial Intelligence 24. Constaint Satisfaction Polems: Backtacking Malte Helmet Univesity of Basel Apil 16, CSP Algoithms 24.2 Naive Backtacking 24.3 Vaiale and Value Odes 24.4 Summay M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 Constaint Satisfaction Polems: Oveview 24. Constaint Satisfaction Polems: Backtacking CSP Algoithms Chapte oveview: constaint satisfaction polems: Intoduction Basic Algoithms 24. Backtacking 25. Ac Consistency 26. Path Consistency Polem Stuctue 24.1 CSP Algoithms M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21

2 24. Constaint Satisfaction Polems: Backtacking CSP Algoithms CSP Algoithms In the following chaptes, we conside algoithms fo solving constaint netwoks. asic concepts: seach: check patial assignments systematically acktacking: discad inconsistent patial assignments infeence: deive equivalent, ut tighte constaints to educe the size of the seach space 24.2 Naive Backtacking M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 Naive Backtacking (= Without Infeence) Is This a New Algoithm? function NaiveBacktacking(C, α): V, dom, (R uv ) := C if α is inconsistent with C: if α is a total assignment: etun α select some vaiale v fo which α is not defined fo each d dom(v) in some ode: α := α {v d} α := NaiveBacktacking(C, α ) if α inconsistent: etun α input: constaint netwok C and patial assignment α fo C (fist invocation: empty assignment α = ) esult: solution of C o inconsistent M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 We have aleady seen this algoithm: Backtacking coesponds to depth-fist seach (Chapte 12) with the following state space: states: consistent patial assignments initial state: empty assignment goal states: consistent total assignments actions: assign v,d assigns value d dom(v) to vaiale v action costs: all 0 (all solutions ae of equal quality) tansitions: fo each non-total assignment α, choose vaiale v = select(α) that is unassigned in α assign v,d tansition α α {v d} fo each d dom(v) M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21

3 Why Depth-Fist Seach? Naive Backtacking: Example Conside the constaint netwok fo the following gaph coloing polem: Depth-fist seach is paticulaly well-suited fo CSPs: v 1 v 7 path length ounded (y the nume of vaiales) solutions located at the same depth (lowest seach laye) state space is diected tee, initial state is the oot no duplicates (Why?) v 2, g, g,, v 6 g,, y Hence none of the polematic cases fo depth-fist seach occus.,, g v 3, v 5 v 4 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 Naive Backtacking: Example Naive Backtacking: Discussion seach tee fo naive acktacking with fixed vaiale ode v 1, v 7, v 4, v 5, v 6, v 3, v 2 alphaetical ode of the values (without inconsistent nodes; continued at goal nodes) v 1 v 7 v 4 v 5 v 6 g g y g y Naive acktacking often has to exhaustively exploe simila seach paths (i.e., patial assignments that ae identical except fo a few vaiales). Citical vaiales ae not ecognized and hence consideed fo assignment (too) late. Decisions that necessaily lead to constaint violations ae only ecognized when all vaiales involved in the constaint have een assigned. moe intelligence y focusing on citical decisions and y infeence of consequences of pevious decisions v 3 v 2 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21

4 24. Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes 24. Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes Naive Backtacking 24.3 Vaiale and Value Odes function NaiveBacktacking(C, α): V, dom, (R uv ) := C if α is inconsistent with C: if α is a total assignment: etun α select some vaiale v fo which α is not defined fo each d dom(v) in some ode: α := α {v d} α := NaiveBacktacking(C, α ) if α inconsistent: etun α M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes Vaiale and Value Odes vaiale odes: Backtacking does not specify in which ode vaiales ae consideed fo assignment. Such odes can stongly influence the seach space size and hence the seach pefomance. example: execises Geman: Vaialenodnung value odes: Backtacking does not specify in which ode the values of the selected vaiale v ae consideed. This is not as impotant ecause it does not matte in sutees without a solution. (Why not?) If thee is a solution in the sutee, then ideally a value that leads to a solution should e chosen. (Why?) Geman: Weteodnung M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes Static vs. Dynamic Odes we distinguish: static odes (fixed pio to seach) dynamic odes (selected vaiale o value ode depends on the seach state) compaison: dynamic odes oviously moe poweful static odes no computational ovehead duing seach The following odeing citeia can e used statically, ut ae moe effective comined with infeence ( late) and used dynamically. M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21

5 24. Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes Vaiale Odes 24. Constaint Satisfaction Polems: Backtacking Vaiale and Value Odes Value Odes two common vaiale odeing citeia: minimum emaining values: pefe vaiales that have small domains intuition: few sutees smalle tee exteme case: only one value foced assignment most constaining vaiale: pefe vaiales contained in many nontivial constaints intuition: constaints tested ealy inconsistencies ecognized ealy smalle tee comination: use minimum emaining values citeion, then most constaining vaiale citeion to eak ties Definition (conflict) Let C = V, dom, (R uv ) e a constaint netwok. Fo vaiales v v and values d dom(v), d dom(v ), the assignment v d is in conflict with v d if d, d / R vv. value odeing citeion fo patial assignment α and selected vaiale v: minimum conflicts: pefe values d dom(v) such that v d causes as few conflicts as possile with vaiales that ae unassigned in α M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / Constaint Satisfaction Polems: Backtacking Summay 24. Constaint Satisfaction Polems: Backtacking Summay Summay: Backtacking 24.4 Summay asic seach algoithm fo constaint netwoks: acktacking extends the (initially empty) patial assignment step y step until an inconsistency o a solution is found is a fom of depth-fist seach depth-fist seach paticulaly well-suited ecause state space is diected tee and all solutions at same (known) depth M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21 M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21

6 24. Constaint Satisfaction Polems: Backtacking Summay Summay: Vaiale and Value Odes Vaiale odes influence the pefomance of acktacking significantly. goal: citical decisions as ealy as possile Value odes influence the pefomance of acktacking on solvale constaint netwoks significantly. goal: most pomising assignments fist M. Helmet (Univesity of Basel) Foundations of Atificial Intelligence Apil 16, / 21