Logic Programming

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Logic Programming

• Tasanawan Soonklang

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Programming paradigms

• Imperative
• Object-oriented
• Functional
• Logic

Procedural programming
Non-procedural programming or Declarative
programming
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Non-procedural programming

• So far programming has been algorithmic.
• Procedural statements (C, C, FORTRAN)
• Functional expressions (Postscript, ML, LISP)
• Now declarative languages logic programming
• Programs do not state now a result is to be
  computed, but rather the form of the result

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Title
Logic Programming

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• Refers loosely to the use of
• facts and rules to represent information.
• deduction to answer queries.
• algorithm logic control
• We supply logic part, Programming Language
  supplies control part.

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Imperative vs. Logic
fact(0,1) - true.

• int fact(int n)
• int i 1
• for (int j n jgt1 --j)
• i i j
• return I

fact(0,1). fact(N,F) - Ngt0, N1 is N-1,
fact(N1,F1), F is N F1.
?- fact(3,W). W6 Click to see the animation
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Computing

• Deals with relations rather than functions.
• Search facts in order looking for matches.
• Use rules for a logical inferencing process to
  produce results
• Premise
• programming with relations is more flexible than
  with functions.

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Reserve fight

• Information
• fight number
• from city

• to city
• departure time
• arrival time

fight ( fight_number, from_city, to_city,
departure_time, arrival_time ) fight(fight_number,
Bangkok, Los Angeles, departure_time,
arrival_time) fight(fight1,Bangkok, X,
depart1,arrival1), fight(fight2,X,Los
Angeles,depart2,arrival2), depart2 gt
arrive130
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Logic
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Proposition

• A declarative sentence.
• e.g. Woff is a dog.
• e.g. Jim is the husband of Mary.
• A proposition is represented by a logical symbol.
• P Woff is a dog.
• R Jim is the husband of Mary.

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Proposition

• A logical statement that may or may not be true
• Consists of
• objects
• relationships of objects to each other

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Symbolic Logic

• Logic which can be used for the basic needs of
  formal logic
• Express propositions
• Express relationships between propositions
• Describe how new propositions can be inferred
  from other propositions
• Particular form of symbolic logic used for logic
  programming called predicate calculus

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Object Representation

• Objects in propositions are represented by simple
  terms either constants or variables
• Constant - a symbol that represents an object
• Variable - a symbol that can represent different
  objects at different times
• Different from variables in imperative languages

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Compound Terms

• Atomic propositions consist of compound terms
• Compound term one element of a mathematical
  relation, written like a mathematical function
• Mathematical function is a mapping
• Can be written as a table

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Parts of a Compound Term

• Compound term composed of
• Functor - function symbol that names the
  relationship
• Ordered list of parameters (tuple)
• Examples
• student(jon)
• like(seth,OSX)
• like(nick,windows)
• like(jim,linux)

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Forms of a Proposition

• Fact - proposition is assumed to be true
• Query - truth of proposition is to be determined
• Compound proposition
• Have two or more atomic propositions
• Propositions are connected by operators

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Logical Operators
Name Symbol Example Meaning
negation ? ? P not P
conjunction ? P ? Q P and Q
disjunction ? P ? Q P or Q
equivalence ? P ? Q P is equivalent to Q
implication ? ? P ? Q P ? Q P implies Q Q implies P
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Quantifiers
Name Example Meaning
universal ?X.P For all X, P is true
existential ?X.P There exists a value of X such that P is true
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Forms of a Proposition

• Too many ways to state the same thing
• Use a standard form for propositions
• Clausal form
• B1 ? B2 ? ? Bn ? A1 ? A2 ? ? Am
• means if all the As are true, then at least one B
  is true
• Antecedent - right side
• Consequent - left side

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Predicate Calculus and Proving Theorems

• A use of propositions is to discover new theorems
  that can be inferred from known axioms and
  theorems
• Resolution - an inference principle that allows
  inferred propositions to be computed from given
  propositions

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Resolution

• Unification finding values for variables in
  propositions that allows matching process to
  succeed
• Instantiation assigning temporary values to
  variables to allow unification to succeed
• After instantiating a variable with a value, if
  matching fails, may need to backtrack and
  instantiate with a different value

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Proof by Contradiction

• Hypotheses a set of pertinent propositions
• Goal negation of theorem stated as a proposition
• Theorem is proved by finding an inconsistency

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Theorem Proving

• Basis for logic programming
• When propositions used for resolution, only
  restricted form can be used
• Horn clause - can have only two forms
• Headed single atomic proposition on left side
• Headless empty left side (used to state facts)
• Most propositions can be stated as Horn clauses

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First Order Logic (FOL)

• Allows the following to be modeled
• Objects
• properties of objects
• relations among the objects
• Like propositional logic, FOL has sentences
• Additionally it has terms which allow the
  representation of objects

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Terms

• A term is a logical expression which refers to an
  object.
• Elements of a term
• Constant Symbols e.g. A, B, John
• Predicate Symbols - refer to a relation
• Function Symbols - refer to a relation which is
  a function

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Sentences

• Atomic Sentences - state a fact
• e.g. company(gordon).
• e.g. location(gordon,usa).
• Complex Sentences - formed from
• atomic sentences
• logical connectives
• quantifiers

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Quantifier

• Universal Quantifier (?)
• All cats are mammals.
• ?x (Cat(x) ? Mammal(x))
• Extensile Quantifier (?)
• Spot has a sister who is a cat.
• ?x ( Sister(x,Spot) ? Cat(x))

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Prolog

• Programming Logic
• Prolog is based upon First Order Logic (FOL)
• A Prolog program consists of a Knowledge Base
  composed of
• facts
• rules
• All facts and rules must be expressed as Horn
  Clauses

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Applications

• Relational DBMS
• Artificial Intelligence
• Expert systems
• Natural language processing
• Automatic theorem proving

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The Origins of Prolog

• University of Aix-Marseille
• Natural language processing
• University of Edinburgh
• Automated theorem proving

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Syntax rules

• Predicates (functors) must start with lower-case
  letter.
• Constants begin with a lower-case letter or
  number.
• Variables begin with an upper-case letter or an
  (_).

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Syntax rules

• All clauses have a head and a body.
• head - body.
• The symbol - is read if
• All sentences (clauses) must end with a period.

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Edinburgh syntax

• Term a constant, variable, or structure
• Constant an atom or an integer
• Atom symbolic value of Prolog
• Atom consists of either
• a string of letters, digits, and underscores
  beginning with a lowercase letter
• a string of printable ASCII characters delimited
  by apostrophes

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Edinburgh syntax

• Variable any string of letters, digits, and
  underscores beginning with an uppercase letter
• Instantiation binding of a variable to a value
• Lasts only as long as it takes to satisfy one
  complete goal
• Structure represents atomic proposition
• functor(parameter list)

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Fact statements

• Used for the hypotheses
• Headless Horn clauses
• female(shelley).
• male(bill).
• father(bill,jake).

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Rule statements

• Used for the hypotheses
• Headed Horn clause
• Right side body (if part)
• May be single term or conjunction
• Left side Head (then part)
• Must be single term
• Conjunction multiple terms separated by logical
  AND operations (implied)

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Rule statements

• ancestor(mary,shelley)- mother(mary,shelley).
• Can use variables (universal objects) to
  generalize meaning
• parent(X,Y)- mother(X,Y).
• parent(X,Y)- father(X,Y).
• grandparent(X,Z)-
• parent(X,Y), parent(Y,Z).
• sibling(X,Y)- mother(M,X), mother(M,Y),
• father(F,X), father(F,Y).

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Goal statements

• For theorem proving, theorem is in form of
  proposition that we want system to prove or
  disprove
• Same format as headless Horn
• man(fred)
• Conjunctive propositions and propositions with
  variables also legal goals
• father(X,mike)

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Simple Prolog facts

• A database of facts
• inclass(john, cmsc330).
• inclass(mary, cmsc330).
• inclass(george, cmsc330).
• inclass(jennifer, cmsc330).
• inclass(john, cmsc311).
• inclass(george, cmsc420).
• inclass(susan, cmsc420).
• Queries Prolog can confirm these facts
• ?-inclass(john, cmsc330). ? yes
• ?- inclass(susan, cmsc420). ? yes
• ?- inclass(susan, cmsc330). ? no

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Simple Prolog facts rules

• A database of facts
• dog(woff).
• barks(woff).
• barks(spot).
• wags_tail(woff).
• A database of rules
• dog(X) - barks(X), wags_tail(X).
• Queries
• ?- dog(woff) gt yes
• ?- dog(spot) gt no
• ?- dog(Y) gt Y woff

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Inferencing Process

• Facts and rules are Knowledge base (KB)
• Prolog uses a goal directed search of the KB
• Depth first search is used
• Query clauses are used as goals and searched left
  to right
• KB clauses are searched in the order they occur
  in the KB
• Goals are matched to the head of clauses
• Terms must unify based upon variable substitution
  before they match

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Inferencing Process

• Queries are called goals
• If a goal is a compound proposition, each of the
  facts is a subgoal
• To prove a goal is true, must find a chain of
  inference rules and/or facts. For goal Q
• B - A
• C - B
• Q - P
• Process of proving a subgoal called matching,
  satisfying, or resolution

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Approaches

• Bottom-up resolution, forward chaining
• Begin with facts and rules of database and
  attempt to find sequence that leads to goal
• Works well with a large set of possibly correct
  answers
• Top-down resolution, backward chaining
• Begin with goal and attempt to find sequence that
  leads to set of facts in database
• Works well with a small set of possibly correct
  answers
• Prolog implementations use backward chaining

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Subgoal Strategies

• When goal has more than one subgoal, can use
  either
• Depth-first search find a complete proof for
  the first subgoal before working on others
• Breadth-first search work on all subgoals in
  parallel
• Prolog uses depth-first search
• Can be done with fewer computer resources

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Backtracking

• With a goal with multiple subgoals, if fail to
  show truth of one of subgoals, reconsider
  previous subgoal to find an alternative solution
  backtracking
• Begin search where previous search left off
• Can take lots of time and space because may find
  all possible proofs to every subgoal

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Unification

• Can use a form of substitution called unification
  to derive other relationships.
• inclass(susan, X).
• Prolog searches database and responds
  Xcmsc420.
• Hitting Enter key, Prolog says No since no
  other fact.
• inclass(john, Y).
• Prolog has following responses
• Ycmsc330.
• Ycmsc311.
• no.

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Unification

• Can define more complex queries
• takingboth(X)-
• inclass(X, cmsc330),
• inclass(X, cmsc311).
• ?-takingboth(john)
• yes
• ?-takingboth(Y)
• Yjohn
• no

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Simple Arithmetic

• Prolog supports integer variables and integer
  arithmetic
• is operator takes an arithmetic expression as
  right operand and variable as left operand
• A is B / 17 C
• Not the same as an assignment statement!

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Data

• Data
• Integers 1, 2, 3, 4
• Reals 1.2, 3.4, 6.7
• Strings 'abc', '123'
• Facts lower case names
• Variables Upper case names
• Lists a, b, c, d

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Example

• speed(ford,100).
• speed(chevy,105).
• speed(dodge,95).
• speed(volvo,80).
• time(ford,20).
• time(chevy,21).
• time(dodge,24).
• time(volvo,24).
• distance(X,Y) - speed(X,Speed),
• time(X,Time),
• Y is Speed Time.

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List Structures

• basic data structure
• List is a sequence of any number of elements
• Elements can be atoms, atomic propositions, or
  other terms (including other lists)
• apple, prune, grape, kumquat
• (empty list)
• X Y (head X and tail Y)

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Append Example

• append(, List, List).
• append(Head List_1, List_2, Head List_3)
  -
• append (List_1, List_2, List_3).

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Reverse Example

• reverse(, ).
• reverse(Head Tail, List) -
• reverse (Tail, Result),
• append (Result, Head, List).

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Reverse Example

• reverse(, ).
• reverse(Head Tail, List) -
• reverse (Tail, Result),
• append (Result, Head, List).

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Summary

• So Prolog is
• 1. A database of facts.
• 2. A database of queries.
• 3. A sequential execution model
• Search facts in order looking for matches.
• If failure, back up to last match and try next
  entry in database.
• Because of this last item, Prolog is not truly
  declarative. It does have an algorithmic
  execution model buried in structure of database.

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The End.