1. Speculative Computation by Consequence Finding Katsumi Inoue Kobe University Koji Iwanuma Yamanashi University

2. Overviews 1.Speculative computation for incomplete communication environments [Satoh, Inoue, Iwanuma & Sakama, ICMAS ’2000]. 2.Default theory and Consequence-finding for speculative computation [Inoue, Kawaguchi & Haneda, CLIMA ’01] 3.SOL tableaux: Skip-regularity and TCS-freeness [Iwanuma, Inoue & Satoh, FTP ’2000]. 4.Conditional answer computation in Ｓ OL as speculative computation [Iwanuma & Inoue, CLIMA ’02] 5.Skip-preference for avoiding irrational conclusions [Iwanuma & Inoue, CLIMA ’02] 6.Process maintainence for avoiding duplicate computation [Inoue, Kawaguchi & Haneda, CLIMA ’01]

3. Communication under Incomplete Information Communication between agents is guaranteed. Under incomplete communication environments (e.g., Internet), this assumption does not hold in general. Messages between agents might be lost or delayed.  [Satoh, Inoue, Iwanuma & Sakama, 2000] proposed a method of speculative computation for reasoning / question- answering under incomplete communication environments in MAS. Use default answers as expected without waiting for responses too much  Reduce suspended processes  Reduce the risk

4. Speculative Computation [Satoh, Inoue, Iwanuma & Sakama, 2000]  top-down SLDNF-like proof procedure  all literals asked by Master have their default values.  slave agents cannot change their answers, once they return answers.  Applet is used in implementation. Master agent makes planning with default answers for slave agents. When responses comes from slave agents, if the answer is the same as the default, keep the current computation process; if the answer is different from the default, recompute a plan.

5. SOL-based Speculative Computation SOL-based Speculative Computation [Inoue, Kawaguchi & Haneda, 2001] [Iwanuma & Inoue, 2002] Define a logical framework of MAS with speculative computation  default logic [Reiter, 80] Data-driven approach and bottom-up computation (reactive behavior)  consequence-finding procedure (SOL)  avoidance of duplicate computation Implementation in a distributed environment with delayed inputs  Servlet (or Java-RMI) and emails

6. Partial Default Answers and Tentative Answers Default answers can just be partially determined in advance. Answers sent from agemts are tentative, i.e., answers may often be changed later. Speculative computation must have the ability to handle not only default values but hypothetical reasoning. Here, we introduce a conditional answer format for handling both default and hypothetical reasoning, and a skip-preference rule for refining the SOL calculus to avoid irrational reasoning.

7. A (Modified) Meeting-Room Reservation Problem There are 3 persons: A, B & C. If a person is free, he/she will attend the meeting. The chair asks each person whether he/she is free or not. If only 2 persons are free, the chair reserves a small room. If all persons are free, the chair reserves a large room. If neithre A nor B is free, the chair reserves no room because A and B are key persons. Suppose that the chairperson gets no answers from A, B, C. What should/can the chair do in this situation?

8. Multi-Agent System Agent framework 〈 ∑,Δ, P, D 〉： ∑ ： slave agent identifiers Δ ： askable literals, Δ= Δ D ∪ Δ U ， Δ D : ground literals, having default answers, Δ U : ground literals, called uncertain literals, having no default truth values. D : (partial) default answer set : for every L ∈ Δ D, D contains either L or ￢ L, but not both. Note: L ∈ D means that the default answer of L ∈ Δ D is true. P ： first-order clauses, called a program.

9. Example: Agent Framework ∑ ={ a, b, c } : agent identifiers Δ = Δ D ∪ Δ U : askable literals Δ D = {free(b), free(c)} : literals having default values Δ U = {free(a)} : uncertain literals D = { free(c) }: default answers P : program ￢ free(a) ∨￢ free(b) ∨ free(c) ∨ meeting(small_room, [a,b]). free(a) ∨￢ free(b) ∨￢ free(c) ∨ meeting(small_room, [b,c]). ￢ free(a) ∨ free(b) ∨￢ free(c) ∨ meeting(small_room, [a,c]). ￢ free(a) ∨￢ free(b) ∨￢ free(c) ∨ meeting(large_room, [a,b,c]). free(a) ∨￢ free(b) ∨ meeting(no_room, []).

10. Reply Set Reply set (at time i ) is a set of literals of the form L or ￢ L, where L is an askable literal in Δ. For any literal L ∈ Δ, L ∈ R i and ￢ L ∈ R i do not hold simultaneously. A reply set is used to store the latest answers from slave agents. Ex. R 3 = { ￢ free(b) }

11. Tentative Answer Set Tentative answer set (at time i ) T Ri is a union of a reply set R i at i and the set of default answers with respect to the askable literals that have not yet been answered at i : Ex. T R3 = { ￢ free(b), free(c) }

12. Formalization in Default Logic (1) (∑,Δ, P, D ) : agent framework R i : reply set at time i T Ri : tentative answer set at time i If P ∪ T Ri is consistent, then the default theory (D*, P ∪ R i ) has exactly one extension E s.t. T Ri = R i ∪ (E ∩ D).

13. Formalization in Default Logic (2) Suppose that the same conditions hold. E is an extension of the default theory (D*, P ∪ R i ) if and only if E = Th ( P ∪ T Ri ).  Tentative answer set T Ri can be used to compute extensions.  Extensions can be computed by consequence-finding from P ∪ T Ri.

14. Consequence Finding Given an axiom set, the task is to find out some theorems of interest. These theorems are not given in an explicit way, but are only characterized by some properties. Consequence Finding is clearly distinguished from Proof Finding or Theorem Proving. In fact, Theorem Proving is a special case of Consequence Finding.

15. Finding Interesting Consequences The set of theorems is generally infinite, even if they are restricted to be minimal wrt subsumption. Solutions: Production field and characteristic clauses plus SOL procedure (Skipping Ordered Linear resolution), a model-elimination-like calculus with Skip operation  [Inoue, 90;91;92] reformulated the problem as follows: How to find only interesting consequences?

16. Production Field Production field: P = L : the set of literals to be collected Cond : the condition to be satisfied (e.g. length) Th P (Σ) : the clauses entailed byΣ which belong to P.  P1 = : {ANS} + is the set of positive literals with the predicate ANS. Th P1 (∑) is the set of all positive clauses of the form ANS (t 1 ) ∨ … ∨ ANS (t n ) being derivable from ∑.  P2 = : L  is the set of negative literals. Th P2 (∑) is the set of all negative clauses derivable from ∑ consisting of fewer than k literals.

17. Characteristic Clauses Characteristic clause of Σ (wrt P ): A clause C such that C belongs to Th P (Σ) ; no other clause in Th P (Σ) subsumes C.  Carc(Σ, P) = μTh P (Σ), where μ represents “subsumption-minimal”. New characteristic clause of C wrtΣ (and P ) : A char. clause of Σ ∧ C which is not a char. clause of Σ.  NewCarc(Σ,C,P) = μ[Th P (Σ ∧ C) － Th (Σ) ] = Carc(Σ ∧ C, P) － Carc(Σ, P).

18. Example: Group theory [Lee, 1967] length ≦ 1 and term depth ≦ 1 〉

19. Applications in AI Nonmonotonic Reasoning Abduction Prime Implicants/Implicates Knowledge Compilation Diagnoses, Design Query Answering, Planning Inductive Logic Programming Knowledge Discovery Bioinformatics Multi-Agent Systems

20. Computing Characteristic Clauses NewCarc(Σ,C,P) (C : clause) can be directly realized by sound & complete consequence-finding procedures such as SOL resolution [Inoue, 1992] SFK resolution [del Val, 1999] NewCarc(Σ,F,P) (F : CNF formula) and Carc(Σ, P) can also be computed.

21. SOL Resolution [Inoue, 1991; 1992] (Skipping Ordered Linear resolution) Model Elimination + Skip rule Skip, Resolve, Reduce rules complete for consequence-finding in C-ordered linear resolution format complete for finding (new) characteristic clauses connection tableau format [Iwanuma, Inoue & Satoh, 2000]

22. Connection Tableaux [Letz, 94;98] Example: Σ: (1) P ∨ Q (2) ￢ P ∨ Q (3) P ∨￢ Q (4) ￢ P ∨￢ Q ￢P￢P ￢Q￢Q P ￢Q￢Q Q ￢P￢P P Q P Q closed (4) (3)(2) (1) Clausal tableau whose every non-leaf node has an immediate successor labeled with the complementary literal.

23. SOL Tableaux: Connection Tableaux + Skip Complete calculus for deriving logical consequences Σ : (1) ￢ P ∨￢ Q (2) P ∨￢ R (3) Q ∨￢ R ￢P￢P ￢Q￢Q P ￢R￢R Q ￢R￢R closed (１)(１) (２)(２)(３)(３) skip merging to a skipped literal skipped

24. Soundness and Completeness 1. If a clause S is derived by an SOL deduction from Σ+C and P, then S belongs to Th(Σ ∪ {C}) and P. 2. If a clause F does not belong to Th(Σ) but belongs to Th(Σ ∪ {C}) and P, then there is an SOL deduction of a clause S from Σ+C and P such that S subsumes F.

25. Answer Completeness [Iwanuma & Inoue, JELIA-02] The completeness of SOL resolution implies the answer completeness. In particular, SOL resolution is complete for finding the minimal (length) answers. c.f. P. Baumgartner, U. Furbach and F. Stolzenburg: Computing Answers with Model Elimination, Artificial Intelligence, 90 (1997) pp.135-176. Not all answers in condensed form can be computed.

26. Meeting-Room Reservation Problem Abbreviated Form Meeting-Room Reservation Problem : Abbreviated Form ∑ ={ a, b, c }: agent identifiers Δ = Δ D ∪ Δ U : Δ D = { f(c) } : askable literals having default answers Δ U = { f(a), f(b) }: uncertain askable literals D = { f(c) }: default answers P ： ￢ f(a) ∨￢ f(b) ∨ f(c) ∨ m(s, [a,b]). (1) f(a) ∨￢ f(b) ∨￢ f(c) ∨ m(s, [b,c]). (2) ￢ f(a) ∨ f(b) ∨￢ f(c) ∨ m(s, [a,c]). (3) ￢ f(a) ∨￢ f(b) ∨￢ f(c) ∨ m(l, [a,b,c]). (4) f(a) ∨ f(b) ∨ m(no_room, []). (5)

27. 1 st Step: Speculative Computation in SOL with Answer literals Theorem: Suppose that P ∪ T R i is consistent. Let ← Q(X) be a query. If Q(X)θ 1 ∨... ∨ Q(X)θ n belongs to Th (P ∪ T R i ), there is an SOL-deduction D from (P ∪ T R i ) s.t. 1.The top clause is ￢ Q(X) ∨ ANS(X). 2.The production field P is. 3.D generates a clause ANS(X) σ 1 ∨... ∨ ANS(X)σ k which subsumes ANS(X)θ 1 ∨... ∨ ANS(X)θ n. Note: The uncertain literals are not considered here.

28. Query and Conditional Answer Query ← Q(X) : Q(X) is a conjunction of literals Conditional answer for ← Q(X) wrt a production field P : a clause in the form of A 1 ∨ … ∨ A m ∨ Q(X)θ 1 ∨... ∨ Q(X)θ n s.t. A 1 ∨ … ∨ A m belongs to P. Conditional ANS-clause (CA-clause) wrt a production field P : a clause in the form of A 1 ∨ … ∨ A m ∨ ANS(X)θ 1 ∨... ∨ ANS(X)θ n s.t. A 1 ∨ … ∨ A m belongs to P

29. Which tentative answers, partial defaults and hypotheses for uncertain literals are used to derivethe conclusion ? The dependency representation is important for avoiding duplicated computations when a new tentative answer arrives in a later stage. Why Conditional Answer Format is Valuable in Speculative Computation?  SOL tableaux can reduce redundant computation which derives irrational conclusions in the conditional answer format by means of the skip-regularity and TCS-freeness constraints. Conditional answer format can explicitly represent:

30. Constraint: Skip-Regularity No complementary literal ・・・・・・ R skipped R R R merge Any complementary literals of skipped literals can be forbidden to appear in an SOL tableau, without losing the completeness.

31. f(a) ∧ f(c) →m(l,[a,b,c]) ∨ m(s,[a,c]) ∨ m(no_room,[]) Irrational Answers Violating Skip-Regularity The tableau violates the skip-regularity wrt f(a). Skip-regluarity violation

32. Constraint: TCS (Tableau Clause Subsumption)-Freeness Any tableau clause C (i.e., a disjunction of sibling literals in a tableau) is not subsumed by any clause in an axiom theory ∑ other than origin clauses of C. R L1L1 L2L2 LnLn  a tableau clause C ∑: a clausal set as an axiom theory

33. Irrational Answers Violating TCS-Freeness Skip-regular but not TCS-free for the new underlying theory P ∪ { f(b) }. The tableau clause (3) is subsumed by newly added clause f(b). f(a) ∧ f(c) →m(l,[a,b,c]) ∨ m(s,[a,c])

34. Rational Answers Satisfying Skip-Regularity and TCS-Freeness f(a) ∧ f(c) → m(l,[a,b,c])

35. 2 nd step: Speculative Computation in SOL with Conditional Answer Format Theorem: Suppose that P ∪ T Ri is consistent. Let ← Q(X) be a query. If A 1 ∨ … ∨ A m ∨ Q(X)θ 1 ∨... ∨ Q(X)θ n is a member of Th(P ∪ T Ri ) and A 1 ∨ … ∨ A m belongs to, then there is an SOL-deduction D from P s.t. 1.The top clause is ￢ Q(X) ∨ ANS(X). 2.The production field P is. 3.D generates a CA-clause B 1 ∨ … ∨ B s ∨ C 1 ∨ … ∨ C t ∨ ANS(X) σ 1 ∨... ∨ ANS(X)σ k : B 1 ∨ … ∨ B s belongs to. C 1 ∨ … ∨ C t belongs to. C 1 ∨ … ∨ C t ∨ ANS(X)σ 1 ∨... ∨ ANS(X)σ k subsumes A 1 ∨ … ∨ A m ∨ ANS(X)θ 1 ∨... ∨ ANS(X)θ n ．

36. Problems Not Solved Yet Answers are often tentative. These tentative answers should not be considered as newly added axioms. 1.The extension (Resolve) with tentative answers as newly added unit clauses becomes impossible. 2.TCS-subsumption by tentative answers as newly added unit clauses becomes inapplicable to tableaux. Hence, many irrational tableaux cannot be pruned. ⇒ Skip-preference rule ⇒ Γ-subumption rule

37. SOL-S(Γ) calculus: SOL + Skip-Preference + Γ-subsumption 1.Skip-preference : Apply Skip as much as possible by ignoring the possibility of other inference rules. The extension (Resove) with tentative answers can completely be simulated. 2.Γ-subsumption check : Check whether a selected subgoal is subsumed by a tentative answer or not. Γ-subsumption check only partially simulates TCS-subsumption, but is enough for speculative computation.

38. Irrational Tableaux Example Tentative answer: f(b). ￢ f(a) ∧ f(c) → ANS(no_room,[]) ∨ ANS(s,[b,c]) f (a) ∧￢ f(c) → ANS(l,[a,b,c]) ∨ ANS(s,[a,c])

39. Survived Rational Tableaux in SOL with Skip-Preference and Γ-subsumption

40. 3 rd step: Speculative Computation in SOL with Skip-Preference and Γ-subsumption Theorem: Suppose that P ∪ T Ri is consistent. Let ← Q(X) be a query. If A 1 ∨ … ∨ A m ∨ Q(X)θ 1 ∨... ∨ Q(X)θ n is a member of Th(P ∪ T Ri ) and A 1 ∨ … ∨ A m belongs to, then there is an SOL-S(Γ) deduction D from P s.t. 1.The top clause is ￢ Q(X) ∨ ANS(X). 2. Γ is (T Ri ) －. 3.The production field P is. 4.D generates a CA-clause B 1 ∨ … ∨ B s ∨ C 1 ∨ … ∨ C t ∨ ANS(X) σ 1 ∨... ∨ ANS(X)σ k : B 1 ∨ … ∨ B s belongs to. C 1 ∨ … ∨ C t belongs to. C 1 ∨ … ∨ C t ∨ ANS(X)σ 1 ∨... ∨ ANS(X)σ k subsumes A 1 ∨ … ∨ A m ∨ ANS(X)θ 1 ∨... ∨ ANS(X)θ n ．

41. Computation Process Pr i = R i : reply set at i T Ri : tentative answer set at i S i : tentative solution set at i H i : history set at i (i ≧ 1): S i = { (A i1, O i1 ), …, (A in, O in ) } H i = H i- １ ∪ {A ik → O ik | (A ik, O ik ) ∈ S i } A ik : assumption set at i (T Ri ∪ A ik is consistent ) O ik : solution set at i (ANS-clause ) Pro 0 = Pro 1 =

42. Updating Computation Processes (1/2) Input Pr i = R new : new replies from slave agents Output Pr i+1 = Step1 R old = { ￢ L ∈ R i | L ∈ R new } R i+1 = R new ∪ (R i ＼ R old ) Step2 T old = R old ∪ { ￢ L ∈ T Ri | L ∈ R new } T Ri+1 = R new ∪ (T Ri ＼ T old ) Step3 If T Ri+1 = T Ri, then S i+1 =S i and H i+1 =H i

43. Updating Computation Processes (2/2) Step4 Check if there is a CA-clause A jk → O jk (j ≦ i) in H i such that T Ri+1 does not contradict A jk : if exists, then H i+1 = H i and collect all such pairs (A ik, O ik ) as S i+1 ; else recompute SOL-deductions to obtain new CA-clauses, which is added to H i+1. S i+1 is the set of all pairs (A, O) for such new A → O.

44. Process Example (1/2) Pro 0 = ＜ φ, φ, φ, φ ＞ Pro 1 = ＜ φ, { f(b),f( ｃ ) }, S 1, H 1 ＞ where S 1 = {({f(a),f(b),f(c)}, {ans(l,[a,b,c])}), ({ ￢ f(a),f(b),f(c)}, {ans(s,[b,c])}), ({f(b),f(c)}, {ans(l,[a,b,c]), ans(s,[b,c])})} and H 1 = { f(a) ∧ f(b) ∧ f(c) → ans(l,[a,b,c]), ￢ f(a) ∧ f(b) ∧ f(c) → ans(s,[b,c]), f(b) ∧ f(c) → ans(l,[a,b,c]) ∨ ans(s,[b,c]) }  Agent B returns the answer free(b) Pro 2 = ＜ {f(b)}, {f(b),f(c)}, S 1, H 1 ＞

45. Process Example (2/2)  Agent B changes the answer into ￢ free(b) Pro 3 = ＜ { ￢ f(b)}, { ￢ f(b), f(c)}, S 3, H 3 ＞ where S 3 = { ({f(a), ￢ f(b),f(c)}, {ans(s,[a,c])}), ({ ￢ f(a), ￢ f(b),f(c)}, {ans(no_room,[])}), ({ ￢ f(b),f(c)}, {ans(s,[a,c]), ans(no_room,[])})} and H 3 = H 1 ∪ { f(a) ∧￢ f(b) ∧ f(c) → ans(s,[a,c]), ￢ f(a) ∧￢ f(b) ∧ f(c) → ans(no_room,[]), ￢ f(b) ∧ f(c) → ans(s,[a,c]) ∨ ans(no_room,[]) }  B again changes the answer into free(b), and Agent A returns the answer free(a) Pro 4 = ＜ {f(a),f(b)}, {f(a),f(b),f(c)}, S 4, H 3 ＞ where S 4 = {({f(a),f(b),f(c)}, {ans(l,[a,b,c])})}.

46. Summary Speculative computation at each time is formalized in default logic. Default computation is significantly simplified using the notion of tentative answer sets. An agent can derive new conclusions according to incoming new information. This is easily realized using a consequence-finding procedure. Conditional answer format is useful for representing speculative computation. Skip-preference and Γ-subsumption prevents generating irrational consequences. The history set is used for updating computation processes without recomputing the same goals.

47. Future Work Efficient implementation of SOL and SOL-S(Γ) More appropriate incremental computation (Integration of top-down and bottom-up approaches) Avoidance of recomputation when updating requests are arrived during previous computation of SOL-deductions (using lemmas) Extension of speculative computation in more general frameworks of MAS