Speculative Computation by Consequence Finding

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 SOL 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

• 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.

• 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.

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

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Multi-Agent System

• Agent framework lt?,?, P, D gt
• ? slave agent identifiers
• ? askable literals, ? ?D ? ?U , ?D
  ground literals, having default answers,
  ?U ground literals, called uncertain
  literals, having no default truth
  values.
• D (partial) default answer set for every L
  ??D , D contains either L or ?L , but not
  both. Note L ? D means that the default
  answer of L ??D is true.
• P first-order clauses, called a program.

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Example Agent Framework

• ? a, b, c agent identifiers
• ??D??U askable literals
• ?D free(b), free(c) literals having
  default values ?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, ).

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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 ?.
• For any literal L??, 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)
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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)

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Formalization in Default Logic (1)

• (?,?, 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 nD).

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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 .

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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.
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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
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Production Field

• Production field P ltL, Cond gt
• L the set of literals to be collected
• Cond the condition to be satisfied (e.g.
  length)
• ThP(S) the clauses entailed byS which belong
  to P.
• P1 ltANS, nonegt
• 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 ltL?, length is fewer than k gt
• L? is the set of negative literals.
• ThP2 (?) is the set of all negative clauses
  derivable from ? consisting of fewer than k
  literals.

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Characteristic Clauses

• Characteristic clause of S (wrt P )
• A clause C such that
• C belongs to ThP(S)
• no other clause in ThP(S) subsumes C.
• Carc(S, P) µThP(S) ,
• where µ represents subsumption-minimal.
• New characteristic clause of C wrtS (and P )
• A char. clause of S?C which is not a char.
  clause of S.
• NewCarc(S,C,P) µThP(S?C) - Th (S)
• Carc(S?C, P) -
  Carc(S, P) .

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Example Group theory Lee, 1967
length ? 1 and term depth ? 1gt
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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

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Computing Characteristic Clauses

• NewCarc(S,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(S,F,P) (F CNF formula)
• and Carc(S, P) can also be computed.

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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

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Connection Tableaux Letz, 9498
Clausal tableau whose every non-leaf node has an
immediate successor labeled with the
complementary literal.
Example S (1) P?Q (2) ?P?Q (3) P??Q
(4)?P??Q
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SOL TableauxConnection Tableaux Skip
Complete calculus for deriving logical
consequences

S (1) ?P??Q (2) P??R (3) Q??R
(1)
?Q
?P
skip
(2)
(3)
?R
P
?R
Q
closed
skipped
closed
skipped
merging to a skipped literal
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Soundness and Completeness

• 1. If a clause S is derived by an SOL
• deduction from SC and P, then
• S belongs to Th(S?C) and P.
• 2. If a clause F does not belong to Th(S)
• but belongs to Th(S?C) and P, then
• there is an SOL deduction of a clause S
• from SC and P such that S subsumes F.

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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.
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Meeting-Room Reservation Problem Abbreviated
Form

• ? a, b, c agent identifiers
• ??D??U ?D f(c) askable literals
  having default answers ?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)

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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)?1?... ?Q(X)?n
  belongs to Th (P ?TR i ), there is an
  SOL-deduction D from (P ?T R i) s.t.
• The top clause is ?Q(X)?ANS(X).
• The production field P is ltANS , nonegt.
• D generates a clause ANS(X) s1?... ?ANS(X)sk
  which subsumes ANS(X)?1?... ?ANS(X)?n .
• Note The uncertain literals are not
  considered here.

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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)?1?... ?Q(X)?n 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)?1?... ?ANS(X)?n s.t.
  A1??Am belongs to P

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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 and TCS-freeness constraints.

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Constraint Skip-Regularity
Any complementary literals of skipped literals
can be forbidden to appear in an SOL tableau,
without losing the completeness.
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Irrational Answers Violating Skip-Regularity
The tableau violates the skip-regularity wrt f(a).
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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
? a clausal set as an axiom theory
?
L1
L2
Ln
a tableau clause C
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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).
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Rational Answers Satisfying Skip-Regularity and
TCS-Freeness
f(a) ? f(c) ? m(l,a,b,c)
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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)?1?...
  ?Q(X)?n is a member of Th(P ?TRi ) and A1??Am
  belongs to lt(?U), nonegt, then there is an
  SOL-deduction D from P s.t.
• The top clause is ?Q(X)?ANS(X).
• The production field P is lt (TRi)- ?ANS
  ?(?U), nonegt.
• D generates a CA-clause
• B1??Bs?C1??Ct ?ANS(X) s1?... ?ANS(X)sk
• B1??Bs belongs to lt (TRi)-, nonegt.
• C1??Ct belongs to lt (?U), nonegt.
• C1??Ct ?ANS(X)s1?... ?ANS(X)sk subsumes
  A1??Am?ANS(X)?1?... ?ANS(X)?n .

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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
? G-subumption rule
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SOL-S(G) calculus SOL Skip-Preference
G-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.
• G-subsumption check Check whether a selected
  subgoal is subsumed by a tentative answer or not.
  G-subsumption check only partially simulates
  TCS-subsumption, but is enough for speculative
  computation.

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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
)
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Survived Rational Tableaux in SOL with
Skip-Preference and G-subsumption
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3rd step Speculative Computation in SOL with
Skip-Preference and G-subsumption

• Theorem Suppose that P ?TRi is consistent.
  Let ? Q(X) be a query. If A1??Am?Q(X)?1?...
  ?Q(X)?n is a member of Th(P ?TRi ) and A1??Am
  belongs to lt(?U), nonegt, then there is an
  SOL-S(G) deduction D from P s.t.
• The top clause is ?Q(X)?ANS(X) . 2. G is (TRi)-
  .
• The production field P is lt(TRi)- ?ANS ?(?U),
  nonegt.
• D generates a CA-clause
• B1??Bs?C1??Ct ?ANS(X) s1?... ?ANS(X)sk
• B1??Bs belongs to lt (TRi)-, nonegt.
• C1??Ct belongs to lt (?U), nonegt.
• C1??Ct ?ANS(X)s1?... ?ANS(X)sk subsumes
  A1??Am?ANS(X)?1?... ?ANS(X)?n .

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Computation Process

• Pri ltRi , TRi , Si , Hi gt
• 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 ltf, f, f, f gt
• Pro1 ltf, D, S1, H1 gt

42
Updating Computation Processes (1/2)

• Input Pri ltRi , TRi , Si , Hi gt
• Rnew new replies from slave agents
• Output Pri1 ltRi1, TRi1 , Si1 , Hi1 gt
• Step1 Rold ?L? Ri L? Rnew
• Ri1 Rnew ? (Ri \
  Rold)
• Step2 Told Rold ? ?L? TRi L? Rnew
• TRi1 Rnew? (TRi \ Told)
• Step3 If TRi1 TRi , then Si1 Si and
  Hi1 Hi

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Updating Computation Processes (2/2)

• Step4 Check if there is a CA-clause Ajk ?
  Ojk (j?i) in Hi such that TRi1 does not
  contradict Ajk
• if exists, then Hi1 Hi and collect all
  such pairs (Aik, Oik) as Si1
• else recompute SOL-deductions to obtain new
  CA-clauses, which is added to Hi1. Si1 is the
  set of all pairs (A, O) for such new A ? O.

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Process Example (1/2)

• Pro0 lt f, f, f, fgt
• Pro1 lt f, f(b),f(c) , S1, H1 gt
• 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 lt f(b), f(b),f(c), S1, H1 gt

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Process Example (2/2)

• Agent B changes the answer into ?free(b)
• Pro3 lt?f(b) , ?f(b), f(c), S3, H3 gt
• 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 ltf(a),f(b), f(a),f(b),f(c), S4, H3 gt
• where S4 (f(a),f(b),f(c), ans(l,a,b,c)).

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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 G-subsumption prevents
  generating irrational consequences.
• The history set is used for updating computation
  processes without recomputing the same goals.

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Future Work

• Efficient implementation of SOL and SOL-S(G)
• 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