1. CS 337 Programming Languages Logic Programming I (Logic, Intro to Prolog)

2. week 10CS 337 Concepts of Programming Languages 2 Outline  First-order predicate calculus  Horn clauses  Introduction to Prolog Terms How to “run” a Prolog program Rules

3. week 10CS 337 Concepts of Programming Languages 3 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.

4. week 10CS 337 Concepts of Programming Languages 4 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.

5. week 10CS 337 Concepts of Programming Languages 5 (First order) Predicate Calculus Starts 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.

6. week 10CS 337 Concepts of Programming Languages 6 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.

7. week 10CS 337 Concepts of Programming Languages 7 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).

8. week 10CS 337 Concepts of Programming Languages 8 Horn Clauses Drop 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).

9. week 10CS 337 Concepts of Programming Languages 9 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.

10. week 10CS 337 Concepts of Programming Languages 10 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).

11. week 10CS 337 Concepts of Programming Languages 11 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.

12. week 10CS 337 Concepts of Programming Languages 12 Outline  First-order predicate calculus  Horn clauses  Introduction to Prolog Terms How to “run” a Prolog program Rules What Prolog is good for

13. week 10CS 337 Concepts of Programming Languages 13 Terms  Everything in Prolog is built from terms.  Three kinds of terms: Constants: integers, real numbers, atoms Variables Compound terms

14. week 10CS 337 Concepts of Programming Languages 14 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: []

15. week 10CS 337 Concepts of Programming Languages 15 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"

16. week 10CS 337 Concepts of Programming Languages 16 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

17. week 10CS 337 Concepts of Programming Languages 17 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)

18. week 10CS 337 Concepts of Programming Languages 18 Terms  All Prolog programs and data are built from such terms  | |  | |  ( )  |,

19. week 10CS 337 Concepts of Programming Languages 19 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

20. week 10CS 337 Concepts of Programming Languages 20 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).

21. week 10CS 337 Concepts of Programming Languages 21 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). ?-

22. week 10CS 337 Concepts of Programming Languages 22 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 ?-

23. week 10CS 337 Concepts of Programming Languages 23 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 ?-

24. week 10CS 337 Concepts of Programming Languages 24 Final Period  Queries can take multiple lines  If you forget the final period, Prolog prompts for more input with |

25. week 10CS 337 Concepts of Programming Languages 25 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

26. week 10CS 337 Concepts of Programming Languages 26 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)

27. week 10CS 337 Concepts of Programming Languages 27 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

28. week 10CS 337 Concepts of Programming Languages 28 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

29. week 10CS 337 Concepts of Programming Languages 29 ?- parent(Parent,hacen), parent(Grandparent,Parent). Parent = fatma Grandparent = mohamed ; No ?- parent(mohamed,Child), | parent(Child,Grandchild), | parent(Grandchild,GreatGrandchild). Child = fatma Grandchild = hacen GreatGrandchild = ahmad Yes

30. week 10CS 337 Concepts of Programming Languages 30 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

31. week 10CS 337 Concepts of Programming Languages 31 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).

32. week 10CS 337 Concepts of Programming Languages 32 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

33. week 10CS 337 Concepts of Programming Languages 33 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).

34. week 10CS 337 Concepts of Programming Languages 34 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  | .  :-.  |,

35. week 10CS 337 Concepts of Programming Languages 35 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))

36. week 10CS 337 Concepts of Programming Languages 36 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 = 1 Y = 2*3 Yes ?- 7 = 1+2*3. No

37. week 10CS 337 Concepts of Programming Languages 37 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