Diagnosis through constrain propagation and dependency recording. 2 ATMS for dependency recording

Transcription

1 Diagnosis through constrain propagation and dependency recording 2 ATMS for dependency recording

2 Fundamentals of Truth Maintenance Systems, TMS Motivation (de Kleer): for most search tasks, there is a great similarity among the points in the search space efficiency may be gained by carrying results obtained in one region into other regions unless carefully designed, efficiency is achieved at the cost of coherency and completeness Valladolid 2009 ATMS for dependency recording 2

3 Fundamentals of Truth Maintenance Systems, TMS TMS allows to increase efficiency of problem solvers without giving out coherency or completeness Provides cache for all the inferences ever made allows the problem solver to make non-monotonic inferences ensures that the data base is contradiction-free Valladolid 2009 ATMS for dependency recording 3

4 Basic Architecture Problem Solver justifications beliefs TMS P(a) x P(x) Q(a) Q(x) P(a), x P(x) Q(x) Q(a) Valladolid 2009 ATMS for dependency recording 4

5 Basic definitions Assumption: decision to assume without any commitment as to what is assumed- Assumed datum: problem solver datum that has been assumed The ATMS builds a graph - of nodes An ATMS node corresponds to a problem solver datum Assumption: special kind of node Valladolid 2009 ATMS for dependency recording 5

6 Basic definitions An ATMS justification describes how a node is derivable from other nodes Consequent: the node being justified Antecedent: list of nodes Represented as: x 1, x 2,... x k x n x 1, x 2,... x k are antecedent nodes x n is the consequent node x 1, x 2,... x k, x n are atoms it is a propositional implication Horn clause only the antecedent is kept Valladolid 2009 ATMS for dependency recording 6

7 Basic definitions An ATMS environment is a set of assumptions type of node- A node n holds in environment E if E, J n E conjunction of atoms, J conjunction of implications, propositional An environment is inconsistent if E, J Inconsistent environments are called NOGOODS Valladolid 2009 ATMS for dependency recording 7

8 Basic definitions An ATMS context is the set formed by assumptions of a consistent environment all nodes derivable from those assumptions A characterizing environment for a context is set of assumptions from which every node of the context can be derived existence: at least one unique: if assumptions have no justifications Valladolid 2009 ATMS for dependency recording 8

9 ATMS goals EFFICIENTLY DETERMINE CONTEXT from set of assumptions and justification Efficiency achieved by problem solver supplies justifications and assumptions one at a time the ATMS is incremental updating only the changed context problem solvers rarely requires the content of context Optimised for two queries context consistency checking a node holds in a context Valladolid 2009 ATMS for dependency recording 9

10 ATMS Labels An ATMS Label, L, is a set of environments associate to a node, n, with properties consistent: every E L is consistent sound: E L, E, J n complete: E / E, J n, E L / E E minimal: E L, no E L / E E Valladolid 2009 ATMS for dependency recording 10

11 ATMS Labels Consequences of label s properties A node n with label L is derivable from environment E iff E L / E E A node n with label L holds in a context iff the characterizing environment of the context includes an environment of its label Valladolid 2009 ATMS for dependency recording 11

12 Basic data structures ATMS node: datum :[datum, label, justifications] usually the node is represented by its datum datum: provided by problem solver; ATMS don t even examine it justifications are provided by problem solver; ATMS examine but don t modify label: computed by the ATMS Valladolid 2009 ATMS for dependency recording 12

13 Types of nodes Premise, p [p, {{}}, {()}] true facts, holds universally, not assumed, not justified Assumption, A [A, {{A}}, {(A)}] represents the assuming act Assumed node, a [a, {{A}}, {(A)}] a holds under assumption A Derived node w=1 [w=1, {{A,B},{C},{D}} {(b), (c, d)}] any other node is derived Valladolid 2009 ATMS for dependency recording 13

14 Derived nodes [w=1, {{A,B},{C},{D}} {(b), (c, d)}] means: w=1 is derived from either the node b or the nodes c and d b (c d) w=1 w=1 holds in environments {A, B}, {C}, {D} (A B) C D w=1 Valladolid 2009 ATMS for dependency recording 14

15 Falsity node :[,, {...}] If inconsistent environments were allowed in Labels, the Label of will be made of NOGOODS must be detected by the problem solver Valladolid 2009 ATMS for dependency recording 15

16 Inconsistent environments: example Problem solver rule base nodes [b, {{B}}, {(B)}] [d, {{B}}, {(b)}] [p, {{B}}, {(d)}] [c, {{C}}, {(C)}] [ p, {{C}}, {(c)}] r-1 b d r-2 d p r-3 c p problem solver steps assumption B r-1 fire r-2 fire assumption C r-3 fire the problem solver must inform the ATMS ATMS updates :[,, {(p, p}] ATMS registers {B, C} as NOGOOD Valladolid 2009 ATMS for dependency recording 16

17 Inconsistent environments: example Problem solver rule base nodes [b, {{B}}, {(B)}] [d, {{B}}, {(b)}] [p, {{B}}, {(d)}] [c, {{C}}, {(C)}] [ p, {{C}}, {(c)}] r-1 b d r-2 d p r-3 c p problem solver steps assumption B r-1 fire r-2 fire assumption C r-3 fire nodes [d, {{D}}, {(D)}] [d, {{B}, {D}}, {(b), (D)}] [p, {{B}, {D}}, {(d)}] [,, {(p, p}] problem solver assumption D ATMS new justification, label d new label p update NOGOODS Valladolid 2009 ATMS for dependency recording 17

18 The environment lattice Every consistent environment characterizes a context potentially,2 n context for n assumptions {A, B, C, D} n k {A, B, C} {A, B, D} {A, C, D} {B, C, D} {A, B} {A, C} {A, D} {B, C} {B, D} {C, D} {A} {B} {C} {D} Valladolid 2009 ATMS for dependency recording 18

19 Ruling out inconsistent environments via NOGOOD NOGOOD for example rules: {B, C} & {D, C} {A, B, C, D} {A, B, C} {A, B, D} {A, C, D} {B, C, D} {A, B} {A, C} {A, D} {B, C} {B, D} {C, D} {A} {B} {C} {D} Valladolid 2009 ATMS for dependency recording 19

20 Does a node holds in a context? Iff context includes an environment of its label p label: {{B}, {D}} {A, B, C, D} {A, B, C} {A, B, D} {A, C, D} {B, C, D} {A, B} {A, C} {A, D} {B, C} {B, D} {C, D} {A} {B} {C} {D} Valladolid 2009 ATMS for dependency recording 20

21 Basic operations creating node datum creating node assumption adding justification to a node Labels update after every justification Valladolid 2009 ATMS for dependency recording 21

22 Basic algorithm for labels update adding justification to node n, a consistent, solid, complete and minimal label may be obtained L i,k label i-th node, k-th justification of n L = k {x / x = i x i, x i L i,k } solid, complete label eliminating subsumed and inconsistent environments, a minimal, consistent label is obtained if new and former label are equal, end if node, all label environments are NOGOOG and must be registered and eliminated from every label, end if no node, update labels of every node which justifications includes node n Valladolid 2009 ATMS for dependency recording 22

23 Diagnosis application example [3] [2] [2] [3] [3] A B C D E M1 M2 M3 X Z Y A1 A2 F G [10] [12] Prediction If M1 OK, X=6 If M2 OK, Y=6 If A1 OK, F=12 Inconsistency: F=12 & F=[10] [M1, M2, A1] is a NOGOOD Valladolid 2009 ATMS for dependency recording 23

24 Inference recording via ATMS Graphical representation ATMS nodes facts Inference recording if A=3, C=2 and A1 OK, X=6 A=3 assumed node {{M1}} derived node C=2 X=6 M1 {{M1}} Valladolid 2009 ATMS for dependency recording 24

25 Inference recording via ATMS With current observations, F=12 in every context which includes its label: {M1, M2, A1} A=3 C=2 X=6 {{M1}} M1 {{M1}} B=2 A1 {{A1}} F=12 {{M1, M2, A1}} D=3 M2 {{M2}} Y=6 {{M2}} Valladolid 2009 ATMS for dependency recording 25

26 Inference recording via ATMS Adding F=10 generate a NOGOOD A=3 C=2 X=6 {{M1}} M1 {{M1}} B=2 A1 {{A1}} F=12 {{M1, M2, A1}} NOGOOD: {{M1, M2, A1}} D=3 M2 {{M2}} Y=6 {{M2}} F=10 Valladolid 2009 ATMS for dependency recording 26