CS 337 Programming Languages Logic Programming I (Logic, Intro to Prolog)
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Transcript of CS 337 Programming Languages Logic Programming I (Logic, Intro to Prolog)
CS 337 Programming
Languages
Logic Programming I (Logic, Intro to Prolog)
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Outline First-order predicate calculus Horn clauses Introduction to Prolog
Terms How to “run” a Prolog program Rules
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Logic Programming Uses a set of logical assertions (i.e.
statements that are either true or false), as a program (the facts).
Execution is initiated by a query or goal, which the system attempts to prove true or false, based on the existing set of assertions.
For this reason, logic programming systems are sometimes called deductive databases.
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Examples Computing ancestors:
A parent is an ancestor.If A is an ancestor of B, and B is an ancestor of C, then A is an ancestor of C. A mother is a parent.A father is a parent.mohamed is the father of fatma.fatma is the mother of hacen.ali is the father of hacen.
Computing the factorial function:The factorial of 0 is 1.If m is the factorial of n - 1, then n * m is the factorial of n.
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(First order) Predicate CalculusStarts with a set of axioms ( true assertions),
stated using the following elements: Constants. Usually numbers or names. In the examples,
mohamed and 0 are constants. Predicates. Names for functions that are true or false, like
Boolean functions in a program. Predicates can take a number of arguments. In the examples, ancestor and factorial are predicates.
Functions. First-order predicate calculus distinguishes between functions that are true or false—these are the predicates—and all other functions, which represent non-Boolean values. * is a function in the examples.
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Predicate Calculus (continued) Variables that stand for as yet unspecified quantities. In
the examples, A and m are variables. Connectives. Operations and, or, and not; implication
"" and equivalence " " . Quantifiers. These are operations that introduce
variables: "for all" - the universal quantifier, and "there exists" - the existential quantifier.
Punctuation symbols: left and right parentheses, the comma, and the period.
Note: Arguments to predicates and functions can only be terms: combinations of variables, constants, and functions.
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Examples written in Pred. Calc.: Ancestors:
For all X and Y, parent(X,Y) ancestor(X,Y). For all A, B, and C, ancestor(A,B) and ancestor(B,C) ancestor(A,C). For all X and Y, mother(X,Y) parent(X,Y).For all X and Y, father(X,Y) parent(X,Y).Father(Mohamed,Fatma).Mother(Fatma,Hacen).Father(Ali,Hacen).
Factorials:Factorial(0,1).For all n and m, factorial(n-1,m) factorial(n,n*m).
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Horn ClausesDrop the quantifiers (i.e., assume them implicitly).
Distinguish variables from constants, predicates, and functions by upper/lower case:
parent(X,Y) ancestor(X,Y). ancestor(A,B) and ancestor(B,C) ancestor(A,C). mother(X,Y) parent(X,Y).father(X,Y) parent(X,Y).father(mohamed,fatma).mother(fatma,hacen).father(ali,hacen).
factorial(0,1).factorial(N-1,M) factorial(N,N*M).
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Resolution Resolution is an inference rule for Horn
clauses If we have two Horn clauses, we can
combine left-hand and right-hand sides of both clauses and then cancel those statements that match on both sides.
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Examples of resolution Given
legs(x,2) <- mammal(x), arm(x,2).legs(x,4) <- mammal(x), arms(x,0).mammal(horse).arms(horse,0).
Query:<-legs(horse, 4).
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Unification Unification is the process of matching to
make statement identical. Variables that are set equal to patterns are
said to be instantiated. Example: In the last example, legs(x,4) is
unified with legs(horse,4), so x is instantiated with horse.
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Outline First-order predicate calculus Horn clauses Introduction to Prolog
Terms How to “run” a Prolog program Rules What Prolog is good for
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Terms Everything in Prolog is built from terms.
Three kinds of terms: Constants: integers, real numbers, atoms Variables Compound terms
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Constants Integer constants: 123 Real constants: 1.23 Atoms:
A lowercase letter followed by any number of additional letters, digits or underscores: fred
A sequence of non-alphanumeric characters: *, ., =, @#$
Plus a few special atoms: []
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Atoms Are Not Variables An atom can look like an ML or Java
variable: i, size, length
But an atom is not a variable; it is not bound to anything, never equal to anything else
Think of atoms as being more like string constants: "i", "size", "length"
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Variables Any name beginning with an uppercase
letter or an underscore, followed by any number of additional letters, digits or underscores: X, Child, Fred, _, _123
Most of the variables you write will start with an uppercase letter
Those starting with an underscore get special treatment
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Compound Terms An atom followed by a parenthesized,
comma-separated list of one or more terms: x(y,z), +(1,2), .(1,[]), parent(mohamed,fatma), x(Y,x(Y,Z))
A compound term can look like an ML function call: f(x,y)
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Terms
All Prolog programs and data are built from such terms
<term> <constant> | <variable> | <compound-term><constant> <integer> | <real number> | <atom><compound-term> <atom> ( <termlist> )<termlist> <term> | <term> , <termlist>
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The Prolog Database A Prolog language system maintains a
collection of facts and rules of inference It is like an internal database A Prolog program is just a set of data for
this database The simplest kind of thing in the database
is a fact: a term followed by a period
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Example
A Prolog program of six facts Defining a predicate parent of arity 2 We would naturally interpret these as facts
about families: ali is the parent of othman and so on
parent(ali,othman).parent(fatma,hacen).parent(mohamed,fatma).parent(mohamed,ibrahim).parent(othman,mohamed).parent(hacen,ahmad).
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SWI-Prolog
Prompting for a query with ?- Normally interactive: get query, print
result, repeat
Welcome to SWI-Prolog (Version 3.4.2)Copyright (c) 1990-2000 University of Amsterdam.Copy policy: GPL-2 (see www.gnu.org)
For help, use ?- help(Topic). or ?- apropos(Word).
?-
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The consult Predicate
Predefined predicate to read a program from a file into the database
File relations (or relations.pl) contains our parent facts
?- consult(relations).% relations compiled 0.00 sec, 0 bytes
Yes?-
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Simple Queries
A query asks the language system to prove something
The answer will be Yes or No (Some queries, like consult, are executed
only for their side-effects)
?- parent(mohamed,ibrahim).
Yes?- parent(ahmad,mohamed).
No?-
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Final Period
Queries can take multiple lines If you forget the final period, Prolog
prompts for more input with |
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Queries With Variables
Any term can appear as a query, including a term with variables
The Prolog system shows the bindings necessary to prove the query
?- parent(P,hacen).
P = ali
Yes?- parent(P,ali).
No
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Flexibility Normally, variables can appear in any or
all positions in a query: parent(Parent,mohamed) parent(fatma,Child) parent(Parent,Child) parent(Person,Person)
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Conjunctions
A conjunctive query has a list of query terms separated by commas
The Prolog system tries to prove them all (using a single set of bindings)
?- parent(mohamed,X), parent(X,hacen).
X = fatma
Yes
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Multiple Solutions
There might be more than one way to prove the query
By typing ; rather than Enter, you ask the Prolog system to find more
?- parent(mohamed,Child).
Child = ibrahim ;
Child = fatma ;
No
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?- parent(Parent,hacen), parent(Grandparent,Parent).
Parent = fatmaGrandparent = mohamed ;
No?- parent(mohamed,Child),| parent(Child,Grandchild),| parent(Grandchild,GreatGrandchild).
Child = fatmaGrandchild = hacenGreatGrandchild = ahmad
Yes
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A Rule
A rule says how to prove something: to prove the head, prove the conditions
To prove greatgrandparent(GGP,GGC), find some GP and P for which you can prove parent(GGP,GP), then parent(GP,P) and then finally parent(P,GGC)
greatgrandparent(GGP,GGC) :- parent(GGP,GP), parent(GP,P), parent(P,GGC).
conditions
head
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A Program With The Rule
A program consists of a list of clauses A clause is either a fact or a rule, and ends
with a period
parent(ali,othman).parent(fatma,hacen).parent(mohamed,fatma).parent(mohamed,ibrahim).parent(othman,mohamed).parent(hacen,ahmad). greatgrandparent(GGP,GGC) :- parent(GGP,GP), parent(GP,P), parent(P,GGC).
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Example
This shows the initial query and final result Internally, there are intermediate goals:
The first goal is the initial query The next is what remains to be proved after
transforming the first goal using one of the clauses (in this case, the greatgrandparent rule)
And so on, until nothing remains to be proved
?- greatgrandparent(mohamed,GreatGrandchild).
GreatGrandchild = ahmed
Yes
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Rules Using Other Rules
Same relation, defined indirectly Note that both clauses use a variable P The scope of the definition of a variable is
the clause that contains it
grandparent(GP,GC) :- parent(GP,P), parent(P,GC).
greatgrandparent(GGP,GGC) :- grandparent(GGP,P), parent(P,GGC).
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Core Syntax Of Prolog You have seen the complete core syntax:
There is not much more syntax for Prolog than this: it is a very simple language
<clause> <fact> | <rule><fact> <term> .<rule> <term> :- <termlist> .<termlist> <term> | <term> , <termlist>
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Arithmetic Operators Predicates +, -, * and / are operators too,
with the usual precedence and associativity
?- X = +(1,*(2,3)).
X = 1+2*3
Yes?- X = 1+2*3.
X = 1+2*3
Yes
Prolog lets you use operator notation, and prints it out that way, but the underlying term is still +(1,*(2,3))
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Not Evaluated
The term is still +(1,*(2,3)) It is not evaluated To make Prolog evaluate such terms,
use the is built-in predicate.
?- +(X,Y) = 1+2*3.
X = 1Y = 2*3
Yes?- 7 = 1+2*3.
No
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Declarative Languages Each piece of the program corresponds to
a simple mathematical abstraction Prolog clauses – formulas in first-order logic ML fun definitions – functions
Many people use declarative as the opposite of imperative, including both logic languages and functional languages