Speculative Computation by Consequence Finding

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

2. Overviews • Speculative computation for incomplete communication environments [Satoh, Inoue, Iwanuma & Sakama, ICMAS ’2000]. • Default theoryand Consequence-finding for speculative computation[Inoue, Kawaguchi & Haneda, CLIMA ’01] • SOL tableaux:Skip-regularity and TCS-freeness[Iwanuma, Inoue & Satoh, FTP ’2000]. • Conditional answercomputation in ＳOL as speculative computation[Iwanuma & Inoue, CLIMA ’02] • Skip-preferencefor avoiding irrational conclusions[Iwanuma & Inoue, CLIMA ’02] • Process maintainencefor 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 expectedwithout 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[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 andTentative 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 formatfor 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= ΔD ∪ ΔU　，ΔD: ground literals, having default answers, ΔU : ground literals, called uncertain literals, having no default truth values. • D : (partial) default answer set : for every L ∈ΔD ,D contains either L or ￢L , but not both.Note: L ∈ D means that the default answer of L ∈ΔD is true. • P： first-order clauses, called a program.

9. Example: Agent Framework ∑={a, b, c} : agent identifiers Δ=ΔD∪ΔU:askable literals ΔD= {free(b), free(c)} : literals having default valuesΔU= {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∈ Ri and ￢L∈ Ri do not hold simultaneously. • A reply set is used to store the latest answers from slave agents. Ex.R3={￢ free(b) }

11. Tentative Answer Set • Tentative answer set (at time i )TRi is a union of a reply set Ri at iand the set of default answers with respect to the askable literals that have not yet been answered at i: Ex.TR3= {￢free(b),free(c) }

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

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

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

16. Production Field • Production field: P = <L, Cond > • L : the set of literals to be collected • Cond : the condition to be satisfied (e.g. length) • ThP(Σ): the clauses entailed byΣ which belong to P. • P1 = <{ANS}+, none> : • {ANS}+is the set of positive literals with the predicate ANS. • ThP1 (∑) is the set of all positive clauses of the form ANS (t1) ∨ … ∨ ANS (tn)being derivable from ∑. • P2 = <L, length is fewer than k >: • L is the set of negative literals. • ThP2 (∑) 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 ThP(Σ) ; • no other clause in ThP(Σ) subsumes C. • Carc(Σ, P) = μThP(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[ThP(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 tableauformat [Iwanuma, Inoue & Satoh, 2000]

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

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

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 ∑={a, b, c }: agent identifiers Δ=ΔD∪ΔU:ΔD= {f(c)} : askable literals having default answersΔU= {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. 1st Step: Speculative Computation in SOL with Answer literals • Theorem:Suppose that P ∪TR i is consistent. Let ← Q(X) be a query. If Q(X)θ1∨... ∨Q(X)θn belongs to Th (P ∪TR i ), there is an SOL-deduction Dfrom (P ∪T R i) s.t. • The top clause is ￢Q(X)∨ANS(X). • The production field P is <ANS+, none>. • D generates a clause ANS(X) σ1∨... ∨ANS(X)σk which subsumes ANS(X)θ1∨... ∨ANS(X)θn . • Note:The uncertain literals are not considered here.

28. Query← Q(X):Q(X)is a conjunction of literals Conditional answer for← Q(X)wrt a production fieldP: a clause in the form ofA1∨…∨Am∨Q(X)θ1∨... ∨Q(X)θn s.t. A1∨…∨Am belongs to P . Conditional ANS-clause (CA-clause) wrt a production fieldP :a clause in the form of A1∨…∨Am∨ANS(X)θ1∨... ∨ANS(X)θn s.t. A1∨…∨Am belongs to P Query and Conditional Answer

29. Which tentative answers, partial defaults and hypotheses for uncertain literals are used to derive the 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? Conditional answer format can explicitly represent: • SOL tableaux can reduce redundant computation which derives irrational conclusions in the conditional answer format by means of the skip-regularity andTCS-freenessconstraints.

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

31. Skip-regluarity violation 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).

32. Constraint: TCS (Tableau Clause Subsumption)-Freeness Any tableau clauseC (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 ∑: a clausal set as an axiom theory  L1 L2 Ln a tableau clause C

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

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

35. 2nd step: Speculative Computation in SOL with Conditional Answer Format • Theorem:Suppose that P ∪TRi is consistent. Let ← Q(X) be a query. If A1∨…∨Am∨Q(X)θ1∨... ∨Q(X)θn is a member of Th(P ∪TRi ) and A1∨…∨Am belongs to <(ΔU)±, none>, then there is an SOL-deduction Dfrom Ps.t. • The top clause is ￢Q(X)∨ANS(X). • The production field P is < (TRi)－∪ANS+∪(ΔU)±, none>. • D generates a CA-clause • B1∨…∨Bs∨C1∨…∨Ct ∨ANS(X) σ1∨... ∨ANS(X)σk : • B1∨…∨Bs belongs to< (TRi)－, none>. • C1∨…∨Ct belongs to< (ΔU)±, none>. • C1∨…∨Ct ∨ANS(X)σ1∨... ∨ANS(X)σk subsumes A1∨…∨Am∨ANS(X)θ1∨... ∨ANS(X)θn．

36. Problems Not Solved Yet Answers are often tentative. These tentative answers should not be considered as newly added axioms. • The extension (Resolve) with tentative answers as newly added unit clauses becomes impossible. • 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 • 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. • 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. 3rd step: Speculative Computation in SOL with Skip-Preference and Γ-subsumption • Theorem: Suppose that P ∪TRi is consistent. Let ← Q(X) be a query. If A1∨…∨Am∨Q(X)θ1∨... ∨Q(X)θn is a member of Th(P ∪TRi ) and A1∨…∨Am belongs to <(ΔU)±, none>, then there is an SOL-S(Γ) deduction Dfrom P s.t. • The top clause is ￢Q(X)∨ANS(X) . 2. Γ is (TRi)－. • The production field P is <(TRi)－∪ANS+∪(ΔU)±, none>. • D generates a CA-clause • B1∨…∨Bs∨C1∨…∨Ct ∨ANS(X) σ1∨... ∨ANS(X)σk : • B1∨…∨Bs belongs to< (TRi)－, none>. • C1∨…∨Ct belongs to< (ΔU)±, none>. • C1∨…∨Ct ∨ANS(X)σ1∨... ∨ANS(X)σk subsumes A1∨…∨Am∨ANS(X)θ1∨... ∨ANS(X)θn．

41. Computation Process • Pri = <Ri , TRi , Si , Hi > • Ri : reply set at i • TRi : tentative answer set at i • Si : tentative solution set at i • Hi : history set at i (i≧1): Si = { (Ai1, Oi1), …, (Ain, Oin)} Hi=Hi-１∪ {Aik → Oik | (Aik, Oik) ∈ Si } • Aik: assumption set at i (TRi ∪Aik is consistent) • Oik: solution set at i (ANS-clause) • Pro0 = <φ, φ, φ, φ> • Pro1 = <φ, D, S1, H1 >

42. Updating Computation Processes (1/2) • InputPri = <Ri , TRi , Si , Hi > Rnew : new replies from slave agents • OutputPri+1= <Ri+1, TRi+1 , Si+1 , Hi+1 > • Step1Rold = { ￢L∈ Ri | L∈ Rnew} Ri+1 =Rnew ∪ (Ri＼ Rold) • Step2 Told = Rold ∪ { ￢L∈ TRi | L∈ Rnew} TRi+1 =Rnew∪ (TRi＼ Told) • Step3 If TRi+1 =TRi,then Si+1 =Si and Hi+1 =Hi

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

44. Process Example (1/2) Pro0 = ＜ φ,φ,φ,φ＞ Pro1 = ＜φ,{ f(b),f(ｃ) }, S1, H1＞ where S1 = {({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 H1 = {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) Pro2 = ＜ {f(b)}, {f(b),f(c)},S1, H1 ＞

45. Process Example (2/2) • Agent B changes the answer into ￢free(b) • Pro3 = ＜{￢f(b)} , {￢f(b), f(c)},S3, H3＞ where S3 = { ({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 H3 = H1 ∪ {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) Pro4 = ＜{f(a),f(b)}, {f(a),f(b),f(c)},S4,H3 ＞ where S4 = {({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