Logic Programming and Prolog

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Logic Programming and Prolog

• Stuart Howard
• CS 550
• Montana State University
• December 5, 2003

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Presentation Outline

• Definition
• History and background
• Logical calculus
• Resolution theorem proving
• Prolog syntax and example program
• Strengths and weaknesses
• Summary and conclusions

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Logic programming

• based on symbolic logic
• write declarations and infer the results
• logic programming languages are declarative
  languages

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What does declarative mean?

• Logic programmers write declarations describing
  relationships between entities
• the what vs. the how
• what does the answer look like?
  (specification)
• vs.
• how do I exactly compute the answer? (implementa
  tion)

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The evolution of programming paradigms
figure from Brookshear
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Lets sort a list nonprocedurally

• permute(oldList, newList) Ù sorted(newList) gt
  sort(oldList, newList)
• j such that 1 j lt n, list(j) list(j 1) gt
  sorted(list)
• example from Sebesta

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Prolog

• Prolog Programmation en Logique
• Early 70s
• Alain Colmerauer and Phillipe Roussel
• Robert Kowalski

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Origins of Prolog

• Q-systems designed for machine natural language
  translation (Colmerauer)
• used to construct automatic English-French
  translations
• used in Canada to translate weather forecasts
  from English to French

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First large Prolog program

• man-machine communication system 1972
• TOUT PSYCHIATRE EST UNE PERSONNE.
• CHAQUE PERSONNE QU'IL ANALYSE, EST MALADE.
• JACQUES EST UN PSYCHIATRE A MARSEILLE.
• EST-CE QUE JACQUES EST UNE PERSONNE?
• OU EST JACQUES?
• EST-CE QUE JACQUES EST MALADE?
• OUI.
• A MARSEILLE.
• JE NE SAIS PAS.

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Prolog example

• Shaun, the bull, and the tractor.

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Logical calculus

• language
• inference rules
• semantics

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Propositional calculus

• the language
• atoms T, F, w w is a string and begins
• with a capital letterH, On_A_B
• connectives Ù, Ú, Ø, ?
• rules for forming sentences (aka. well-formed
  formulas, wffs)

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Rules for wffs

• any atom is a wff
• If w1 and w2 are wffs, so are w1 Ù w2
  conjunction w1 Ú w2 disjunction w1 ? w2
  implication
• Ø w1 negation
• there are no other wffs

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more terms

• literal atom or negated atom (P, Ø P)
• w1 ? w2(antecedent) (consequent)

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Propositional calculus

• the inference rules(6 of them anyway)
• w2 from w1 and w1 ? w2 (modus ponens)
• w1 Ù w2 from w1 and w2 (Ù introduction)
• w2 Ù w1 from w1 Ù w2 (commutativity of Ù)
• w1 from w1 Ù w2 (Ù elimination)
• w1 Ú w2 from w1 or w2 (Ú introduction)
• w1 from Ø(Øw1) (Ø elimination)

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Propositional calculus

• the semantics
• associate atoms with propositions about the
  world
• Shaun_strong with Shaun is strong _____
  ____________ denotation
  __________________________________________
• interpretation

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Propositional calculus

• remember?
• literal an atom or its negation
• a new term!
• clause a set of literals with disjunction
  implied P, Q, Ø R (P Ú Q Ú Ø R)

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Propositional calculus

• a new inference rule resolution
• S1 ? S2 from a? S1 and Øa? S2
• P Ú R from P Ú Q and R Ú ØQ
• from a and Øa

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Resolving (P OR Q) (R OR ? Q)
Brookshear
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Resolving (P OR Q), (R OR ?Q), ?R
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Why not?

• Soundness - OK
• Completeness not OK

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Resolving (P OR Q), (R OR ?Q), ?R, ?P
Brookshear
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limitations

• Propositional calculus limits us to referring to
  hard-coded propositions about the world.
• What if we also want to refer to objects and
  propositions by name?

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Predicate Calculus

• Objects constants (names) or variables
• Functions/relations constants (names)
• weird(MichaelJackson)
• cute(Russell)
• smart(x)
• hate(Michael, Microsoft)
• 
  
  

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Predicate Calculus

• Quantifiers
• Universal
• Existential
• Express properties of entire collections of
  objects

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Predicate Calculus

• Universal quantifiers make statements about every
  object, "x
• A cat is a mammal
• "x Cat(x) Þ Mammal(x)
• Cat(Spot) Þ Mammal(Spot) Ù
• Cat(Rebecca) Þ Mammal(Rebecca) Ù
• Cat(Felix) Þ Mammal(Felix) Ù
• Cat(Richard) Þ Mammal(Richard) Ù
• Cat(John) Þ Mammal(John) Ù
• Comp313A

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Predicate Calculus

• Existential Quantifiers make statements about
  some objects, x
• Spot has a sister who is a cat
• x Sister(x, Spot) Ù Cat(x)
• (Sister(Spot, Spot) Ù Cat(Spot)) Ú
• (Sister(Rebecca, Spot) Ù Cat(Rebecca)) Ú
• (Sister(Felix, Spot) Ù Cat(Felix)) Ú
• (Sister(Richard, Spot) Ù Cat(Richard)) Ú
• (Sister(John, Spot) Ù Cat(John)) Ú
• Comp313A

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Predicate Calculus

• " is a conjunction over the universe of objects
• is a disjunction over the universe of objects

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Predicate Calculus Example

• The George W. Bush Family
• mother(Barbara, George Jr.)
• father(George Jr., Jenna)
• father(George Jr., Barbara II)

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Predicate Calculus Example

• Q Barbara is the grandmother of whom?
• " x " y "z mother(x, y) Ù parent(y, z) gt
  grandmother(x, z)
• " x "y father(x,y) gt parent(x, y)
• " x "y mother(x,y) gt parent(x, y)

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Predicate Calculus Example

• mother(Barbara, George Jr.)
• father(George Jr., Jenna)
• father(George Jr., Barbara II)
• " x " y "z mother(x, y) Ù parent(y, z) gt
  grandmother(x, z)
• " x "y father(x,y) gt parent(x, y)
• " x "y mother(x,y) gt parent(x, y)

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Resolution in Predicate Calculus

• " x " y "z mother(x, y) Ù parent(y, z) gt
  grandmother(x, z)
• mother(x, y) Ù parent(y, z) gt grandmother(x, z)
• Ø mother(x, y) Ù parent(y, z) Ú grandmother(x,
  z)
• Ø mother(x, y) Ú Ø parent(y, z) Ú grandmother(x,
  z)

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Resolution in Predicate Calculus

• " x "y father(x,y) gt parent(x, y)
• father(a, b) gt parent(a, b)
• Ø father(a, b) Ú parent(a, b)

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Resolution in Predicate Calculus

• " x "y mother(x,y) gt parent(x, y)
• mother(c, d) gt parent(c, d)
• Ø mother(c, d) Ú parent(c, d)

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• mother(Barbara, George Jr.)
• father(George Jr., Jenna)
• father(George Jr., Barbara II)
• Ø mother(x, y) Ú Ø parent(y, z) Ú grandmother(x,
  z)
• Ø father(a, b) Ú parent(a, b)
• Ø mother(c, d) Ú parent(c, d)
• Ø grandmother(Barbara, Jenna)
• Ø mother(Barbara, e) Ú Ø parent(e, Jenna) 7, 4
• Ø mother(Barbara, f) Ú Ø father(f, Jenna) 8, 5
• Ø father(George Jr., Jenna) 9, 1

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Prolog Syntax

• Rules
• Facts
• Goals
• They are all variants of an implication!

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Rule

• General
• a gt b
• a Ù b Ù c gt d

Prolog b - a. d - a, b, c.
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Fact

• General
• gt b

Prolog b.
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Goal

• General
• b gt

Prolog b.
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Horn what?

• Ø a Ú Ø b Ú Ø c Ú d
• (Ø a Ú Ø b Ú Ø c) Ú d
• Ø (a Ù b Ù c) Ú d
• a Ù b Ù c gt d

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Horn what?

• b
• Ø Ú b
• gt b

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Horn what?

• Ø a Ú Ø b Ú Ø c
• Ø (a Ù b Ù c) Ú
• a Ù b Ù c gt

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More Prolog
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1
2
4
5
3
6
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1
2
4
3
5
6
7
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Uses of logic programming

• relational database management systems
• expert systems
• natural language processing

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Weaknesses of Prolog

• Resolution order control
• Closed world assumption

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Summary and Conclusions

• Proponets say
• Logical language gt logically organizedgt fewer
  errors, less maintenance
• Programs concise gt less development time gt good
  prototyping language
• Opponents say
• BUNK!
• http//www.mozart-oz.org/features.html

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Presentation Review

• Definition
• History and background
• Logical calculus
• Resolution theorem proving
• Prolog syntax and example program
• Strengths and weaknesses
• Summary and conclusions

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Thanks for staying awake!
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References

• Concepts of Programming Languages (Fourth
  Edition) by Robert Sebesta (An excellent
  overview of Prolog... very helpful!) Notes from
  CS 436 (Artificial Intelligence) Dr. John Paxton
  (My first introduction to predicate calculus and
  Prolog) Artificial Intelligence A New
  Synthesis by Nils Nilsson (Excellent
  presentation of propositional and predicate
  calculus) Computer Science An Overview by
  Glenn Brookshear (Nice short overview of logic
  programming and Prolog) Compiler Design by
  Reinhard Wilhelm and Dieter Maurer (Obtuse,
  highly mathemetized, and hard to read.... i.e.
  it's Greek to me) http//www.cs.waikato.ac.nz/Te
  aching/COMP313A/lecture_notes.html (6 excellent
  PowerPoint lectures on logic programming) Alain
  Colmerauer's web site (a cofounder of Prolog)
  History of Prolog Prolog compiler that ESUS
  uses