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First Order Predicate Logic to English Translation

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Title: First Order Predicate Logic to English Translation


1
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

2
Overview
  • Natural Language Generation
  • First Order Predicate Logic to English Translation

3
Natural Language Generation
  • Content Determination
  • Document Structuring
  • Sentence Aggregation
  • Lexicalisation
  • Referring Expression Generation
  • Syntactic and Morphological Realization
  • Orthographic Realization

4
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

5
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

6
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

7
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

8
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

9
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

10
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

11
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

12
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

13
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

14
Universal Quantifiers
  • For every person x,
  • All persons
  • Everyone
  • For every person x, if
  • Every person
  • Every person who

15
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

16
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

17
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

18
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

19
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

20
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

21
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

22
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

23
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

24
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

25
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

26
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

27
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

28
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

29
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

30
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

31
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

32
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

33
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

34
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

35
References
  • Cole et al. Survey of the State of the Art in
    Human Language Technology. Cambridge University
    Press, Cambridge, UK, 1998.
  • Geoffrey Hunter. Metalogic An Introduction to
    the Metatheory of Standard First Order Logic.
    University of California Press, California, USA,
    1971.
  • Ehud Reiter and Robert Dale. Building applied
    natural language generation systems. Journal of
    Natural Language Engineering, 3(1)5787, 1997.
  • 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
    on Logic Programming, page 536, 1996.
  • L. Wanner and E. Hovy. The healthdoc sentence
    planner. In INLG96, pages 110, Herstmonceux
    Castle, Sussex, 1996.
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