Speculative Computation by Consequence Finding

Katsumi Inoue Kobe University

Koji IwanumaYamanashi University

Overviews1. Speculative computation for incomplete communication enviro

nments [Satoh, Inoue, Iwanuma & Sakama, ICMAS ’2000].

2. Default theory and Consequence-finding for speculative co

mputation [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 comp

utation [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]

Communication under Incomplete InformationCommunication under Incomplete Information

Communication between agents is Communication between agents is guaranteed. 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 processesReduce the risk

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.

SOL-based Speculative ComputationSOL-based Speculative Computation [Inoue, Kawaguchi & Haneda, 2001]

[Iwanuma & Inoue, 2002] Define a logical framework of MAS with speculative comp

utation 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

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 format for handling both default and hypothetical reasoning, and a skip-preference rule for refining the SOL calculus to avoid irrational reasoning.

A (Modified) Meeting-Room Reservation ProblemA (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?

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.

Example: Agent Framework

∑ ={ a, b, c } : agent identifiers

Δ=ΔD∪ΔU : askable literals

ΔD = {free(b), free(c)} : literals having default values

ΔU = {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, []).

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) }

Tentative Answer Set

　 Tentative answer set (at time i ) TRi

is a union of a reply set Ri at i and 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) }

}R R{ iii LL |DLRTR andi

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).

DLLL:

D*

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 TRi can be used to compute extensions.

Extensions can be computed by consequence-finding from P ∪ TRi .

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.

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?

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.

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) .

Example: Group theory [Lee, 1967]

),,(),,( VZYpUYXpC

),,),(( ),,,( { eXXipXXepΣ

,}{ pP

)),(,,( ),,,( ),),(,( { eieepXeXpeXiXpN

),,(),,( VZUpWVXp

),,(),,( VZYpUYXp } ),,(),,( WVXpWZUp

length ≦ 1 and term depth ≦ 1 〉

} )),(),(( ),,),(( )),(,),(( eeieipXXeipeiXXip

Applications in AI

Nonmonotonic ReasoningAbductionPrime Implicants/ImplicatesKnowledge CompilationDiagnoses, DesignQuery Answering, Planning Inductive Logic ProgrammingKnowledge DiscoveryBioinformaticsMulti-Agent Systems

Computing Characteristic Clauses

NewCarc(Σ,C,P) (C : clause) can be directly realized by sound & complete cons

equence-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.

SOL Resolution [Inoue, 1991; 1992]

(Skipping Ordered Linear resolution)

Model Elimination + Skip ruleSkip, 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]

Connection Tableaux [Letz, 94;98]

Example: Σ: (1) P∨Q (2) ￢ P∨Q (3) P∨ ￢ Q (4) ￢ P∨ ￢ Q

￢ P ￢ Q

P ￢ Q Q ￢ P

P Q P Q

closed

closed closed closed

closed

closed

(4)

(3) (2)

(1)(1)

Clausal tableau whose every non-leaf node has an immediate successor labeled with the complementary literal.

SOL Tableaux:Connection Tableaux + Skip

Complete calculus for deriving logical consequences

Σ : (1) ￢ P∨ ￢ Q (2) P∨ ￢ R (3) Q∨ ￢R

￢P

￢Q

P ￢ R Q ￢ R

closed closed

( １ )

( ２ ) ( ３ )skipskip

merging toa skipped literal

R

skipped skipped

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.

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.

Meeting-Room Reservation ProblemMeeting-Room Reservation Problem:

Abbreviated FormAbbreviated Form

∑ ={ a, b, c }: agent identifiersΔ=ΔD∪ΔU :

ΔD = {f(c)} : askable literals having default answers

ΔU = {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)

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 D from (P ∪T R i) s.t.

1. The top clause is ￢ Q(X)∨ANS(X).

2. The production field P is <ANS +, none>.

3. 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.

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

A1 …∨ ∨Am∨Q(X)θ1 ... ∨ ∨Q(X)θn

s.t. A1 …∨ ∨Am belongs to P .

Conditional ANS-clause (CA-clause) wrt a production field P : a clause in the form of

A1 …∨ ∨Am∨ANS(X)θ1 ... ∨ ∨ANS(X)θn

s.t. A1 …∨ ∨Am belongs to P

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?

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:

Constraint: Skip-Regularity

No complementaryliteral

・ ・ ・

R skipped

R R

merge

Any complementary literals of skipped literals can be forbidden to appear in an SOL tableau, without losing the completeness.

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-regluarityviolation

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

L1 L2 Ln

a tableau clause C

∑: a clausal set as an axiom theory

Irrational Answers Violating TCS-Freeness

Skip-regular but notTCS-free for the new underlying theoryP ∪{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])

Rational Answers Satisfying Skip-Regularity and TCS-Freeness

f(a) ∧ f(c) → m(l,[a,b,c])

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 D from P s.t.

1. The top clause is ￢ Q(X)∨ANS(X).

2. The production field P is < (TRi) － ∪ ANS + ∪(ΔU)±, none>.

3. 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 ．

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

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.

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])

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

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 ∪TR

i ) and A1 …∨ ∨Am belongs to <(ΔU)±, none>, 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 (TRi) － .

3. The production field P is <(TRi) － ∪ ANS + ∪(ΔU)±, none>.

4. 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 ．

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 >

Updating Computation Processes (1/2)

Input 　 　　 Pri = <Ri , TRi , Si , Hi >

　　　　　　　　 Rnew : new replies from slave agentsOutput 　　 Pri+1= <Ri+1, TRi+1 , Si+1 , Hi+1 >

Step1 　　　 Rold = { ￢ 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

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 a

ll pairs (A, O) for such new A → O.

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 ＞

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])})}.

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.

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 a

rrived during previous computation of SOL-deductions (using lemmas)

Extension of speculative computation in more general frameworks of MAS