First Order Predicate Logic to English Translation

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First Order Predicate Logic to English Translation

• Donald O. Davis
• Department of Computer Science
• Old Dominion University
• Norfolk VA 23529
• dod_at_cs.odu.edu
• Adviser Professor Shunichi Toida
• April 29, 2005

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Overview

• Natural Language Generation
• First Order Predicate Logic to English Translation

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Natural Language Generation

• Content Determination
• Document Structuring
• Sentence Aggregation
• Lexicalisation
• Referring Expression Generation
• Syntactic and Morphological Realization
• Orthographic Realization

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Content Determination

• The process of deciding what to say
• Different communicative goals may require
  different information to be expressed
• Content required may depend on characteristics of
  the reader
• Constraints upon the output
• Questions of what information should be included
  are application dependent

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Document Structuring

• Problem of imposing ordering and structure over
  the information
• A text is not just a random collection of
  sentences
• Texts have an underlying structure in which the
  parts are related together
• Readers have an expectation of the structure of
  text
• Two related issues
• conceptual grouping
• rhetorical relationships

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Sentence Aggregation

• Mapping of linguistic structures and elements
  such as sentences and paragraphs
• A one-to-one mapping from messages to sentences
  results in disfluent text
• Messages need to be combined to produce larger
  and more complex sentences
• The result is a sentence specification or
  SENTENCE PLAN
• Natural languages afford many different ways of
  expressing the same thing

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Lexicalisation

• Lexicalisation determines the particular words to
  be used to express domain concepts and relations
• We can express information about ownership or
  possession
• The car owned by Mary
• Marys car
• We need to provide some mechanism for choosing
  among alternatives
• There was rain on every day for eight days from
  the the 11th
• There was rain on every day from the 11th to the
  18th

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Referring Expression Generation

• Referring expression generation is concerned with
  how we describe domain entities in such a way
  that the hearer will know what we are talking
  about
• The same entity may be referred to in different
  ways
• Initial Reference
• Subsequent Reference
• Major issue is avoiding ambiguity
• March 1995
• March
• Last Month
• Takes into account previous communications with
  the user

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Syntactic and Morphological Realization

• Every natural language has grammatical rules that
  govern how words and sentences are constructed
• Morphology rules of word formation
• Syntax rules of sentence formation
• English rules for forming negative sentence are
  not straightforward
• I do not
• I am not bored

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Orthographic Realization

• Orthographic realization is concerned with
  matters like casing and punctuation
• This also extends into typographic issues font
  size, column width
• May also include capitalization of section
  headings or final sentence punctuation in the
  case of exclamations or questions

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First Order Predicate Logic to English Translation

• System Description
• Outline of Method
• Universal Quantifiers
• Existential Quantifiers
• Example
• Translation of Single Quantifiers
• Translation of Single Quantifiers using IMPLY
• Translation of Double Quantifiers
• Translation of Double Quantifiers using IMPLY
• CatchAll Function

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System Description

• Used in conjunction with a undergraduate
  web-based discrete logic course
• Used to translate well-form predicate logic
  formulas to English sentences
• Assisting students in identifying their mistakes
• The system can handle AND, OR and NOT connectives
  and is mainly intended for the class of formulas
  with one and two variables

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Outline of Method

• A stack structure contains the well-formed
  formula
• A determination is made of the occurrence of an
  IMPLY operator
• If no IMPLY operator is not used, the stack is
  evaluated in reverse order
• If IMPLY operator is used, pattern-matching
  determines output
• \E y J(x) -gt Q(x,y) would be \E y (x) -gt
  (x,y)
• Remaining cases covered by CatchAll Function
• The universe of discourse is the set of people

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Universal Quantifiers

• For every person x,
• All persons
• Everyone
• For every person x, if
• Every person
• Every person who

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Existential Quantifiers

• There exists a person x such that
• There exists a person who
• Someone
• There exists a person x such that if
• There exists a person who
• Some

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Example

• \A x J(x) V K(x)
• J(x) x studies hard
• K(x) x loves sports
• Stack result )x(K V )x(J x A\
• If we pop the stack from left to right, when we
  get to K we could write
• x loves sports

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Example

• When we arrive at V we could append OR with the
  previous result
• or x loves sports
• At the letter J we could append x studies hard
  with the previous result
• x studies hard OR x loves sports
• Once we finally arrive at \A we could append For
  every person x with the previous result for the
  answer
• For every person x, x studies hard or x loves
  sports

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Translation of Single Quantifiers

• J(x) x studies hard
• K(x) x loves sports
• \A x J(x) V K(x)
• For every person x, x studies hard or loves
  sports
• All persons study hard or love sports
• Everyone studies hard or loves sports

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Translation of Single Quantifiers

• J(x) x studies hard
• K(x) x loves sports
• \E x J(x) K(x)
• There exists a person x such that x studies hard
  and loves sports
• There exists a person who studies hard and love
  sports
• Someone studies hard and loves sports

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Translation of Single Quantifiers

• J(x) x studies hard
• K(x) x loves sports
• \E x J(x) V K(x)
• There exists a person x such that x studies hard
  or does not love sports
• There exists a person who studies hard or does
  not love sports
• Someone studies hard or does not love sports

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Translation of Single Quantifiers using IMPLY

• J(x) x studies hard
• K(x) x loves sports
• \A x J(x) -gt K(x)
• For every person x, if x studies hard, x loves
  sports
• Every person loves sports if he/she studies hard

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Translation of Single Quantifiers using IMPLY

• J(x) x studies hard
• K(x) x loves sports
• \E x J(x) -gt K(x)
• There exists a person x such that if x studies
  hard, x loves sports
• There exists a person who loves sports if he/she
  studies hard

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Translation of Single Quantifiers using IMPLY

• J(x) x studies hard
• K(x) x loves sports
• \E x J(x) -gt K(x)
• There exists a person x such that if x studies
  hard, x does not love sports
• There exists a person who does not love sports if
  he/she studies hard

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Translation of Double Quantifiers

• P(x,y) x runs faster than y
• Q(x,y) x is older than y
• \A x P(x,y) V Q(x,y)
• For every person x the following holds x runs
  faster than y or x is older than y

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Translation of Double Quantifiers

• P(x,y) x runs faster than y
• Q(x,y) x is older than y
• \E x P(x,y) Q(x,y)
• There exist a person x which satisfies the
  following x runs faster than y and x is older
  than y

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Translation of Double Quantifiers

• P(x,y) x runs faster than y
• Q(x,y) x is older than y
• \E x P(x,y) V Q(x,y)
• There exist a person x which satisfies the
  following x run faster than y or x is not older
  than y

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Translation of Double Quantifiers using IMPLY

• R(x,y) x swims faster than y
• J(y) y studies hard
• K(x) x loves sports
• \A x \E y K(x) J(y) -gt R(x,y)
• For every person x and for some person y the
  following holds If x loves sports and y studies
  hard then x swims faster than y

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Translation of Double Quantifiers using IMPLY

• R(x,y) x swims faster than y
• L(x) x is a student
• \E x \A y R(x,y) -gt L(x)
• There exist a person x for every person y which
  satisfies the following If x swims faster than y
  then x is a student

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Translation of Double Quantifiers using IMPLY

• S(x,y) x is taller than y
• \E y \E x S(y,x)
• There exist a person y which satisfies the
  following some person y is not taller than x

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CatchAll Function

• P(x,y) x runs faster than y
• S(x,y) x is taller than y
• \A x P(x,y) -gt S(x,y)
• For every person x the following holds x runs
  faster than y implies x is taller than y

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CatchAll Function

• P(x,y) x runs faster than y
• Q(x,y) x is older than y
• T(x,y) x talks louder than y
• \E x P(x,y) Q(x,y) -gt T(x,y)
• There exist a person x which satisfies the
  following x runs faster than y and x is older
  than y implies x talks louder than y

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CatchAll Function

• J(x) x studies hard
• N(x) x likes baseball
• R(x,y) x swims faster than y
• \A x J(x) N(x) -gt R(x,y)
• For every person x the following holds x study
  hard and x like baseball implies x does not swim
  faster than y

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Conclusion

• The system attempts to take the last in first out
  nature of the stack data type to combine
  multi-sentences separated by AND or OR in a
  meaningful way
• Other cases make use of pre-determined patterns
  to produce relevant text output in a general way
• Predetermined patterns can not cover the infinite
  number of WWFs available to the user
• The system contains one catchall method design to
  produce one legible sentence from a WWF not
  already covered

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Conclusion

• While this sentence is grammatically correct, it
  is far from the style of the average English
  speaking person
• Improvement of this catchall function is the
  single most necessary component to cover a larger
  number of cases while communicating in a style
  common to most people

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References

• Cole et al. Survey of the State of the Art in
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• Geoffrey Hunter. Metalogic An Introduction to
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  1971.
• Ehud Reiter and Robert Dale. Building applied
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• Rolf Schwitter and Norbert E. Fuchs. Attempto
  controlled english (ACE) a seemingly informal
  bridgehead in formal territory (poster abstract).
  In Joint International Conference and Symposium
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• L. Wanner and E. Hovy. The healthdoc sentence
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