Propositional Logic Syntax Acquisition Using Induction and Self-Organization

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Propositional Logic Syntax Acquisition Using
Induction and Self-Organization

• Josefina Sierra Santibáñez
• Departamento de Lenguajes y Sistemas Informáticos
• Universidad Politécnica de Cataluña
• Spain

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• Language Game Guessing (Steels 1999)
• Speaker chooses a formula from a propositional
  language, generates a sentence for expressing the
  formula and communicates the sentence to hearer.
• Hearer tries to interpret the sentence generated
  by the speaker. If it can parse it using its
  lexicon and grammar, it extracts a meaning.
• The speaker communicates the formula it had in
  mind to the hearer. They adjust their grammars to
  become sucessful in future language games.
• Success speakers formula ? hearers
  meaning

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• Goals of the Experiments
• Observe the evolution of
• The communicative success average of sucess-ful
  language games in the last ten language ga-mes
  played by the agents.
• The internal grammars constructed by the
  individual agents.
• The external language used by the population.

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• Definite clause grammarsemantic, score, use
• s(right, 0.25, 20) ? right
• s(light, 0.70, 50) ? light
• s(P,Q, S, 12) ? c1(P,S1,C1), s(Q,S2,C2), S is
  S1?S2?0.01
• c1(not, 0.80, 55) ? not
• s(P,Q,R,S,3) ? c2(P,S1,C1),s(Q,S2,C2),s(R,S3,C3)
  , S is S1?S2?S3?0.01
• c2(and, 0.50, 35) ? and
• Formula Meaning
  Sentence
• right ? light and, right, light
  andrightlight
• light not, light
  notlight

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• Invention
• Generates a sentence E for a meaning M
• If M is atomic, it invents a new word E.
• If M is a list, it tries to construct an
  expression for each of the elements in M using
  the agents grammar.
• If it cannot construct an expression for an
  element using its grammar, it invents a new
  expression.
• It concatenates the expressions associated with
  the elements of M randomly in order to construct
  a sen-tence E for the whole meaning M.
• Adds a new rule to the grammar s(M, 0.01, 0) ?
  E

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• Adoption
• Communication fails because
• The hearer cannot parse the speakers sentence
• The speaker communicates the formula it had in
  mind to the hearer.
• The hearer adopts an association between that
  formula and the sentence used by the speaker.
• s(M, 0.01, 0) ? E
• The hearer can parse the sentence, but its
  interpre-tation is not consistent with the
  speakers meaning.
• Hearer decrase the scores of used associations.
• It may adopt an association between the formula
  and the sentence used by the speaker.

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• Induction
• The agents use some induction mechanisms to
  extract
• generalisations from the grammar rules learnt
  so far.
• The induction rules used in the experiments are
  based
• on the following rules proposed in (Kirby
  2002)
• Simplification
• Chunk
• They are applied whenever the agents invent or
  adopt
• a new association.

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• Simplification scores (Vogt 05) and use counters
  
• r1 ? n(m1, S2, C2) ? e1
• r2 ? left(m1, S1, C1) ? e1
• Rule r2 is replaced with rule r3, of the form
• left(X, S3, 0) ? n(X,S,C), S3 is S?0.01
• or
• letf(X.., S3, 0) ? n(X,S,C), S3 is S?T?0.01
• depending on whether S1 was a constant or a
  variable.
• In the second case, S1 is T?sr1 is the
  arithmetic ex-pression on the right hand side of
  r1.

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• Simplification example 1
• r1 ? s(right, 0.25, 16) ? right
• r2 ? s(and,light,right, 0.10, 6) ?
  andlightright
• Rule r2 is replaced with rule r3
• r3 ? S(and,light,R, S, 0) ? andlight, s(R, S3,
  C3), S is S3?0.01
• r4 ? s(light, 0.70, 25) ? light
• Rule r3 is replaced with rule r5
• r5 ? s(and,Q,R, S, 0) ? and, s(Q, S2, C2), s(R,
  S3, C3), S is S2?S3?0.01

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• Simplification example 2
• r1 ? s(right, 0.25, 16) ? right
• r6 ? s(or,light,right, 0.10, 6) ?
  orlightright
• Rule r6 is replaced with rule r7
• r7 ? S(or,light,R, S, 0) ? orlight, s(R, S3,
  C3), S is S3?0.01
• r4 ? s(light, 0.70, 25) ? light
• Rule r7 is replaced with rule r8
• r8 ? s(or,Q,R, S, 0) ? or, s(Q, S2, C2), s(R,
  S3, C3), S is S2?S3?0.01

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• Simplification example 3
• r1 ? s(right, 0.25, 16) ? right
• r9 ? s(or,light,right, 0.10, 6) ?
  lightorright
• Rule r9 is replaced with rule r10
• r10 ? S(or,light,R, S, 0) ? lightor, s(R, S3,
  C3), S is S3?0.01
• r4 ? s(light, 0.70, 25) ? light
• Rule r10 is replaced with rule r11
• r11 ? s(or,Q,R, S, 0) ? s(Q, S2, C2), or, s(R,
  S3, C3), S is S2?S3?0.01

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• Chunk I scores (Vogt 05) and use counters
• r1 ? left(f(m1), S1, C1) ? right(e1)?,
• r2 ? left(f(m2), S2, C2) ? right(e2)?,
• A new category symbol n is created and rules
  added
• n(m1, 0.01, 0) ? e1 n(m2, 0.01,
  0) ? e2
• Rules r1 and r2 are replaced with rule r3, of the
  form
• left(f(X), S3, 0) ? right?(n(X, S, C)), S3 is
  S?0.01
• or
• left(f(X), S3, 0) ? right?(n(X,S,C)), S3 is
  S?T?0.01

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• Chunk I example 1
• r1 ? s(and,Q,R, S, 18) ? and, s(Q, S2, C2),
  s(R, S3, C3), S is S2?S3?0.10
• r2 ? s(or,Q,R, S, 23) ? or, s(Q, S2, C2), s(R,
  S3, C3), S is S2?S3?0.30
• The following new rules are added to grammar
• c2(and, 0.01, 0) ? and c2(or, 0.01,
  0) ? or
• Rules r1 and r2 are replaced with rule r3
• r3 ? s(P,Q,R, S, 0) ? c2(P, S1, C1), s(Q, S2,
  C2), s(R, S3, C3), S is S1?S2?S3?0.01

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• Chunk I example 2
• r1 ? s(and,Q,R, S, 18) ? and, s(Q, S2, C2),
  s(R, S3, C3), S is S2?S3?0.10
• r2 ? s(or,Q,R, S, 23) ? s(Q, S2, C2), or, s(R,
  S3, C3), S is S2?S3?0.30
• Chunk cannot be applied to r1 and r2, because
  they
• place the expressions associated with the
  connectives
• and and or in different positions in the
  sentence.

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• Chunk I example 3
• r1 ? s(and,Q,R, S, 18) ? s(R, S3, C3), and,
  s(Q, S2, C2), S is S2?S3?0.10
• r2 ? s(or,Q,R, S, 23) ? s(Q, S2, C2), or, s(R,
  S3, C3), S is S2?S3?0.30
• Chunk cannot be applied to r1 and r2, because
  they
• place the expressions associated with the
  arguments
• of the binay connective, Q and R, in different
  posi-
• tions in the sentence.
• Rules must agree on the positions of the expres-
• sions associated with the connectives and their
• arguments in the sentence.

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• Chunk II scores (Vogt 05) and use counters
• r1 ? left(f(X), S1, C1) ? right?(n(X, S, C)),
• r2 ? left(f(m1), S2, C2) ? right?(e1),
• Rule r2 is replaced with the following rule
• n(m1, 0.01, 0) ? e1
• right?(X) is the result of removing the
  arithmetic ex-
• pression S3 is S? of the right hand side of
  r1.

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• Chunk II example 1
• r1 ? s(P,Q,R, S, 25) ? c2(P, S1, C1), s(Q, S2,
  C2), s(R, S3, C3), S is S1?S2?S3?0.20
• r2 ? s(iff,Q,R, S, 33) ? iff, s(Q, S2, C2),
  s(R, S3, C3), S is S2?S3?0.50
• Rule r2 is replaced with the following rule
• c2(iff, 0.01, 0) ? iff

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• Chunk II example 2
• r1 ? s(P,Q,R, S, 25) ? c2(P, S1, C1), s(Q, S2,
  C2), s(R, S3, C3), S is S1?S2?S3?0.20
• r2 ? s(iff,Q,R, S, 33) ? s(Q, S2, C2), s(R, S3,
  C3), iff, S is S2?S3?0.50
• Chunk cannot be applied, because rule r1 places
  the
• expresion associated with the connective in first
  po-
• sition in the sentence and rule r2 places the
  expres-
• sion associated with the connective third
  position.

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• Need for Coordination The agents must reach
• agreements on how to
• name propositional constants and connectives
• a1 if ? if
  a2 if ? si
• order the expressions associated with the
  different components of non-atomic meanings
  consistently
• a1 not ? un, 2pos not, right ?
  rightnot
• a2 not ? un, 1pos not, right ?
  notright
• a1 if ? bin, 2pos, inv if,right,light ?
  lightifright
• a2 if ? bin, 2pos, noinv if,right,light ?
  rightiflight

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• Self-organization Coordinate agents grammars
• The agents construct a shared external language
  and
• prefer using the rules in that language over the
  rest
• in the rules in their individual grammars.
• The scores of the rules indicate the agents
  preferences
• meaning ? sentence1 highest score
• competing sentences sentence2, ,
  sentenceN
• sentence ? meaning1 highest score
• competing meanings meaning2, ,
  meaningN
• The score of a sentence (or meaning) is computed
  at
• generation (parsing) multiplying the scores of
  the rules
• involved (Vogt 2005).

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• Score of a sentence (meaning) example
• s(right, 0.25, 20) ? right c1(if,
  0.50, 10) ? if
• s(light, 0.70, 50) ? light c2(if,
  0.10, 15) ? si
• s(P,Q,R,S,5) ? c1(P,S1,C1),s(Q,S2,C2),s(R,S3,C3)
  , S is S1?S2?S3?0.10
• s(P,Q,R,S,3) ? c2(P,S1,C1), s(R,S3,C3),s(Q,S2,C2
  ), S is S1?S2?S3?0.01
• Meaning if, right, light ? Generation
• Sentence ifrightlight score
  0.50?0.25?0.70?0.10
• Comp senten silightright score
  0.10?0.25?0.70?0.01

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• Coordination takes place at the third stage of a
  lan-guage game when the speaker communicates the
  meaning it had in mind to the hearer.
• hearers meaning ? speakers meaning
• Speaker increases scores of rules ? sentence
• decreases scores of rules ? competing
  sentences
• Hearer increases scores of rules ? meaning
• decreases scores of rules ? competing
  meanings
• hearers meaning ? speakers meaning
• Speaker and hearer decrease scores of rules they
  used for generating and interpreting the sentence.

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• Reinforcement and Inhibition
• The rules used successfully are reinforced.
• The rules used for generating competing sentences
  or competing meanings are inhibited.
• The rules used for updating scores of grammar
  rules (Steels 1999) replace the original score S
  with
• S1 ? maximum(1, S µ) if the score is
  increased
• S2 ? minimum(1, S ? µ) if the score is
  decreased
• Purging the rules that have been used more than
  30 times and have scores ? 0.01 are removed from
  the agents grammars.

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Experiments Evolution Communicative Success
guessing game 5 agents, 15000 games about
propositional formulas La, b, c, r, l, u
constructed using ?, ?, ?, ?, ?
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Guessing negation ?
a1 s(not,Y, R) ? 2, f, s(Y,Q), R is Q?1
a2 s(X,Y, R) ? 2, c1(X,P), s(Y,Q), R is P?Q?1 c1(not, 1) ? f
a3 s(not,Y, R) ? 2, f, s(Y,Q), R is Q?1
a4 s(not,Y, R) ? 2, f, s(Y,Q), R is Q?1
a5 s(X,Y, R) ? 2, c1(X,P), s(Y,Q), R is P?Q?1 c1(not, 1) ? f
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Guessing conjuction ? (commut)
a1 s(and,Y,Z, T) ? 3, dyp, s(Z,Q), s(Y,R), T is Q?R?1
a2 s(X,Y,Z,T) ? 3, c2(X,P),s(Z,Q),s(Y,R), T is P?Q?R c2(and, 1) ? dyp
a3 s(X,Y,Z,T) ? 3, c1(X,P),s(Y,Q),s(Z,R), T is P?Q?R c1(and, 1) ? dyp
a4 s(X,Y,Z,T) ? 3, c4(X,P),s(Y,Q),s(Z,R), T is P?Q?R c4(and, 1) ? dyp
a5 s(X,Y,Z,T) ? 3, c4(X,P),s(Z,Q),s(Y,R), T is P?Q?R c4(and, 1) ? dyp
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Guessing implication ? (non-com)
a1 s(X,Y,Z,T) ? 1, c1(X,P),s(Y,Q),s(Z,R), T is P?Q?R c1(if, 1) ? bqi
a2 s(X,Y,Z,T) ? 1, c3(X,P),s(Y,Q),s(Z,R), T is P?Q?R c3(if, 1) ? bqi
a3 s(X,Y,Z,T) ? 1, c2(X,P),s(Y,Q),s(Z,R), T is P?Q?R c2(if, 1) ? bqi
a4 s(if,Y,Z, T) ? 1, bqi, s(Y,Q), s(Z,R), T is Q?R?1
a5 s(if,Y,Z, T) ? 1, bqi, s(Y,Q), s(Z,R), T is Q?R?1