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COMP313A Programming Languages. Logic Programming (1). Lecture Outline. Conceptual foundations of Logic Programming The Basics of Logic Programming Predicate Calculus A little bit of logic programming Prolog. Conceptual Foundations. What versus how specification versus implementation

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Comp313a programming languages l.jpg

COMP313A Programming Languages

Logic Programming (1)


Lecture outline l.jpg
Lecture Outline

  • Conceptual foundations of Logic Programming

  • The Basics of Logic Programming

    • Predicate Calculus

  • A little bit of logic programming

    • Prolog


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Conceptual Foundations

  • What versus how

    • specification versus implementation

  • Declarative Programming

    • Programmer declares the logical properties that describe the property to be solved

    • From this a solution is inferred

    • Inference engine


An example l.jpg
An example

Searching for an element in a list

Predicate is_in(x,L) true whenever element x is in the list L.

For all elements x and lists L: is_in(x,L) IFF

L = [x]

or

L = L1 . L2 and

(is_in (x,L1) or is_in(x, L2))


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Example continuedImplementation

  • Need to know how to split a list into right and left sublists

  • How to order the elements stored in the list


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A solution in C++

int binary_search(const int val, const int size, const int array[]){

int high, low, mid;

if size <= 0{

return (-1);

}

high = size;

low = 0;

for(;;) {

mid = (high + low) / 2;

if (mid = low){

return (val != array[low]) ?-1:mid;

}

if (val < array[mid]) {

high = mid;

}

else if (val > array[mid]) {

low = mid;

}

else return mid

}}}


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A Declarative Solution

  • Given an element x and a list L, to prove that x is in L,

  • proceed as follows:

  • Prove that L is [x]

  • Otherwise split L into L1 . L2 and prove one of the following

    • (2.1) x is in L1, or

    • (2.2) x is in L2


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A sorting example

A predicate sort(X,Y)

Sort(X,Y) is true if the nonempty list Y is the sorted version of X

Use two auxiliary predicates: permutation(X,Y) and is_sorted(Y)

For all integer lists X,Y: sort(X,Y) iff

permutation(X,Y) and sorted(Y)


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Logic and Logic Programming

  • First-order predicate calculus

    • Logic statements

      Examples

      John is a man. man(John).

      John is a human. human(John).

      Tristan is the son of Margaret. son(Margaret,Tristan).

      A horse is a mammal. loathes(Margaret, Heavy_Metal).

      0 is a natural number . natural(0).

      Mammals have four legs and no arms or two legs and two arms.

      For all X, mammal (x) -> legs(x,4) and arms(x,0) or legs(x,2) and arms(x,2).

      Humans have two legs and two arms.

      For all X, human(x) -> legs(x,2) and arms(x,2).

      If x is a natural number then so is the successor of x.

      For all x, natural(x) -> natural(successor(x)).


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First-Order Predicate Calculus

  • Constants

  • Predicates

  • Functions

  • Variables that stand for as yet unamed quantities

  • Atomic sentences

  • Connectives construct more complex sentences

  • Quantifiers

  • Punctuation

  • Arguments to predicates can only be terms – variables, constants and functions


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First-Order Predicate Calculus cont…

  • Quanitifiers

    • Universal, existential

  • Express properties of entire collections of objects

  • Universal quantifiers make statements about every object, "x

    A cat is a mammal

    "x Cat(x) ÞMammal(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


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First-Order Predicate Calculus cont…

  • 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


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First-Order Predicate Calculus cont…

  • Connections between $and "

  • Negation

    Everyone dislikes rugbyº Noone likes rugby

    "x ØLikes (x, rugby) ºØ$x Likes(x, rugby)

    Everyone likes icecream º Noone dislikes icecream

    "x Likes (x, icecream) ºØ$x ØLikes(x, icecream)


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First-Order Predicate Calculus cont…

  • " is a conjunction over the universe of objects

  • $Is a disjunction over the universe of objects

    "x ØP ºØ$x P

    Ø "x P º$x ØP

    "x P ºØ$x ØP

    Ø "x ØP º$x P


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De Morgan’s Laws

ØPÙØQ º (PÚQ)

Ø(PÙQ) ºØPÚØQ

PÙQ ºØ(ØPÚØQ)

Ø(ØPÙØQ) º PÚ Q


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Using First-Order Predicate Calculus

  • Marcus was a man

  • Marcus was a Pompeian

  • All Pompeians were Romans

  • Caesar was a ruler

  • All Romans were either loyal to Caesar or hated him


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  • Everyone is loyal to someone

  • People only try to assassinate rulers they are not loyal to

  • Marcus tried to assassinate Caesar

    Was Marcus loyal to Caesar?

    Prove Ø loyalto(Marcus, Caesar)


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  • Turn the following sentences into formulae in first order predicate logic

  • John likes all kinds of food

  • Apples are food

  • Chicken is food

  • Anything anyone eats and isn’t killed by is food

  • Bill eats peanuts and is still alive

  • Sue eats everything Bill eats

  • Prove that John likes peanuts using backward chaining


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A little bit of Prolog predicate logic

  • Objects and relations between objects

  • Facts and rules

    parent(pam, bob). parent(tom,bob).

    parent(tom, liz). parent(bob, ann).

    parent(bob, pat). parent(pat, jim).

    ? parent(bob, pat).

    ? parent(bob, liz).

    ? parent(bob, ben).

    ? parent(bob, X).

    ? parent(X, Y).


Prolog l.jpg
Prolog predicate logic

grandparent (X,Y) :- parent(X, Z), parent(Z, Y).

For all X and Y

X is the grandparent of Y if

X is a parent of Z and

Z is a parent of Y

sister (X,Y) :- parent(Z, X), parent(Z, Y), female(X)

For all X and Y

X is the sister of Y if

Z is the parent of both X and Y and

X is a female


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