TMS and ATMS

1. TMS and ATMS Philippe Dague and Yuhong YAN NRC-IIT Yuhong.yan@nrc.ca Philippe.dague@lipn.univ-paris13.fr

2. TMS and ATMS • TMS: Truth Maintain System • ATMS: Assumption Truth Maintain System • Why needs TMS: • Improve efficiency of reasoning system • Conventional TMS: • Justification-based • Logic-based

3. ATMS and diagnosis engine Justifications Problem Solver TMS Beliefs P(a) x P(x)->Q(x) Q(a) P(s), x ,P(x)->Q(x) =>Q(a) From Johan de Kleer, “An Assumption-based TMS”, Artificial Intelligence 28(1986) p141

4. Why TMS • Holding trace of the information • beginning from which • Dependencies to get the conclusions • Management of the reasoning contexts. • hypothetical reasoning (assumptions) • search between (alternative) contexts • Caching inferences to avoid to repeat more times some steps than reasoning • remember which roads they have carried to successes or failures in the reasoning process. manager search space like gragh and not like tree

5. Chronological Backtracking X=0 X=1 Y=0 Y=1 Y=0 Y=1 ! ! ? ? Z=0 Z=1 Z=0 Z=1 Z=0 Z=1 Z=0 Z=1 * * • X{0,1}, (2) a = e1(X) • (3)Y{0,1}, (4) b = e2(Y) • (5)Z{0,1}, (6) c = e3(Z) • (7) b  c, (8) a  b Rediscovering contradictions (!) Rediscovering inferences (?) Incorrect ordering

6. Limitation of TMS • Single state • TMS keeps consistent under single context

7. Ideas of ATMS Justifications Problem Solver TMS Beliefs P(a) x P(x)->Q(x) Q(a) P(s), x ,P(x)->Q(x) =>Q(a) It executes the reasoning steps and it passes to the TMS its assumptions and the justification of its conclusions (incrementally) It can interrogate the TMS Holds trace of the dependencies between assumptions and conclusions and of the inconsistence

8. Terminology for TMS/ATMS • Nodes:objects on which the problem solver works, e.g. data, rules, assumption • Justification: describes how a node is derivable from other nodes • Consequent: the node being justified • Antecedents: a list of nodes which imply the consequent • E.g. x1,x2,…=>n, logically it is implication: X1,x2,…->n

9. Terminology for ATMS • Environment: a set of assumptions • A node n is said to hold in environment E if n can be derived from E and a set of justification J: E,J |- n • An environment is inconsistent if false () is derivable: E,J |-  • No-good: inconsistent environment (the assumption set which causes the contradiction) (stored separately) • Context: the environment and all nodes derivable from the environment

10. Label • Label: a set L of environments E associated with a node n • Four properties • Consistency: each environment E is consistent • Soundness: E, J |- n, or J|- E->n • Completeness: E’ (J|- E’->n)->(E, EE’) • Minimality: E1, E2, ¬(E1E2)

11. Data Structure in ATMS • Structure of a node: datum: <datum, label, justifications> • A premise holds universally: <p,{{}},{()}> • An assumption is a node whose label contains only itself: <A,{{A}},{(A)}>

12. Data Structure in ATMS • Assumptions can be justified <A, {{A},{B,C}},{(A),(d)}> • An assumed node: holds under an assumption and a justification of this assumption <b,{{A}},{(A)}> • A derived node <n,{{A1,A2,…},{B1,B2,…}…},{(z1,z2,…),(y1,y2,…)…}>

13. Logic meaning A node: <n,{{A1,A2,…},{B1,B2,…}…},{(z1,z2,…),(y1,y2,…)…}> Means: (A1A2…)(B1B2…)… ->n And: (z1z2 …)  (y1 y2 …)  …->n A node: :<,{},{…}>

14. Basic Operation in ATMS To keep the label of each node is consistent, sound, complete and minimal • Union of all possible combinations of picking one environment from each antecedent node label • Remove subsumed environments • Remove inconsistent environments

15. Example x+y=1:<x+y=1,{{A,B},{B,C,D}},{…}> x=1:<x=1,{{A,C},{D,E}},{…}> y=0:<y=0,{},{}> Nogood {A,B,E} Add justification: x+y=1, x=1:=> y=0 Union: {{A,B,C},{A,B,D,E},{A,B,C,D},{B,C,D,E}} Result: y=0 :<y=0, {{A,B,C},{B,C,D,E}},{(x+y=1, x=1)}>

16. Process: When one new justification arrives • Union, remove inconsistent and subsumed environment • If the new label is as the old one, stop • If the node is :, each environment of the label is added to a nogood database, and all inconsistent environments (itself and the supersets) are removed from every node label • If the node is not , update any other influenced node (nodes whose justifications contain label changed node)

17. Example of using ATMS on Diagnosis Before output observation <x=6,{{M1}},{(A=3, B=2)}> <A=3,{{}},{()}> M1 X <F=12,{{M1,M2,A1}},..> <B=2,{{}},{()}> A1 F <C=2,{{}},{()}> M2 Y <Y=6,{{M2}},{..}> <D=3,{{}},{()}> A2 G <G=12,{{M2,M3,A2}},..> <E=3,{{}},{()}> M3 Z <Z=6,{{M3}},{..}>

18. Example of using ATMS on Diagnosis After F=10 observation <X=6,{{M1}}> <X=4,{{A1, M2}}> <F=12,{{M1,M2,A1}}> <A=3,{{}}> M1 X <F=10,{{}}> <B=2,{{}}> A1 F <C=2,{{}}> <Y=6,{{M2}}> <Y=4,{{A1,M1}}> M2 Y <D=3,{{}}> A2 G <G=12,{{M2,M3,A2}}> <G=10,{{A1,A2,M1,M3}}> <E=3,{{}}> M3 Z <Z=6,{{M3}}>

19. Example of using ATMS on Diagnosis After G=12 observation <X=6,{{M1}}> <X=4,{{A1, M2}, {A1,A2,M3}}> <A=3,{{}}> M1 X <F=10,{{}}> <B=2,{{}}> A1 F <C=2,{{}}> <Y=6,{{M2},{A2,M3}}> <Y=4,{{A1,M1}}> M2 Y <D=3,{{}}> A2 G <G=12,{{}}> <G=12,{{M2,M3,A2}}> <G=10,{{A1,A2,M1,M3}}> <E=3,{{}}> M3 Z <Z=6,{{M3}},{A2,M2}}> <Z=8,{{A1,M1,A2}}

20. Summary • ATMS remove the single-state limitation • ATMS is an important technique for MBD • Completeness and efficiency are issues for implementation

21. Assignment 2 Full adder in Reiter’s paper (figure 1), use the ATMS principle to get the conflict sets (writing down all the justifications and labels for each step) and the minimal hitting set principle to get the minimal diagnoses under the observations: Step1: Obs1: {in1(X1)=1; in2(X1)=0;in(A2)=1} Step2: Obs2: Obs1{out(x2)=1} Step3: Obs3: Obs2 {out(O1)=0}