Chapter 8: The Logical Paradigm Lecturer: Xinming (Simon) Ou CIS 505: Programming Languages Fall 2010 Kansas State University 1

What we have covered so far: Imperative paradigm – Computes effects Functional paradigm – Computes values – Used ML as an example Logical paradigm – Computes relations – Use Prolog as an example 2

Example relations Parent relation: – parent(A, B) means A is B’s parent – e.g. parent(bill, mary). parent(mary, john). Ancestor relation – Can be defined inductively: A is B’s ancestor if A is B’s parent. A is B’s ancestor if A is C’s ancestor, and C is B’s ancestor – The resulting relation is the smallest one that satisfies the above two rules. 3

Some definitions for Prolog Atoms: – Any sequence of alpha-numeric characters that starts with a lower- case letter, or a single-quoted string, or a number e.g. mary, john01, ‘Mary Doe’, 100 Variables: – Any sequence of alpha-numeric characters that starts with an upper- case letter, or an underscore “_” – e.g. Mary, _mary, _ Literal – predicate(t 1, …, t k ), where t i is either an atom, a variable, or a data- structure (function applied to parameters). – e.g. parent(mary, john). parent(mother(john), john). ancestor(mother(father(john)), john). 4

Horn Clauses A Horn clause is a logical clause with a single positive literal: L 0 ∨ L 1 ∨ … L n This is equivalent to L 1 ∧ … ∧ L n => L 0 In Prolog, we use “,” to mean logical and, and write implication “backward”. Each clause is concluded with a “.” L 0 :- L 1, …, L n. – Example: ancestor(A, B) :- parent(A, B). ancestor(A, B) :- parent(A, C), ancestor(C, B). We call the left-hand side of the clause its head, and the right-hand side of the clause its body. – A clause may have an empty body. e.g. parent(mother(X), X). 5

Variables in Clauses All variables are implicitly universally bound at the beginning of the clause – e.g. ancestor(A, B) :- parent(A, B). Logically it is equivalent to: Forall A, B. parent(A,B) => ancestor(A, B) Thus A and B can be instantiated with any term. An underscore “_” is a wild card and can match anything. – e.g. isParent(A) :- ancestor(A, _). 6

Query in Prolog A query is in the form of a literal. The answer to the query is all the instantiations of the variables that make the literal true. – e.g. ? - ancestor(X,Y). X = bill Y = mary; X = mary Y = john; X = bill Y = john; no – Logically it is equivalent to “exists X, Y. ancestor(X,Y)?” 7

Execution Semantics of Prolog When a query is issued, it is “compared” against the head of all the clauses one by one. – If a “match” is found, the body of the clause becomes the new goals – This process will iterate and may either succeed or fail. – In either case the execution will backtrack to the first “choice point”, and try another match. This is called “SLD resolution” 8

Z2=john X=mary Y=john X=bill Y=mary Example SLD resolution ancestor(X,Y) :- parent(X,Y). ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y). parent(bill,mary). parent(mary,john). ?- parent(X,Y). ?- Success ?- Success ?- parent(X,Z), ancestor(Z,Y). ?- ancestor(X, Y). X=bill Z=mary ?- ancestor(mary,Y). ?- parent(mary,Y). ?- Success ?- parent(mary,Z2), ancestor(Z2,Y). … Failure … Failure ?- ancestor(john,Y). X=mary Z=john ?- ancestor(john,Y). 9

Logic deduction as a program The advantage of Prolog is that it has both a logic meaning, and an execution semantics – Ideally you do not need to think about the SLD resolution process when writing Prolog code – A Prolog program is simply a collection of logical statements. A query is simply asking whether a fact can be derived as a logical consequence of the statements. However… – When the result does not match your expectation, knowing the SLD resolution process will help in debugging. – Moreover, Prolog is not always declarative, which we will see in the next lecture. 10

Practice XSB We will be using the XSB Prolog system – Is installed on all the departmental Linux machines – Can be downloaded from: Installation is relatively hassle-free. However, if you need to compile XSB under Mac OS X Snow Leopard, please let me know and there will be special instructions. 11

The first simple Prolog program Put the following Prolog statements in a text file named “ancestor.P” parent(bill, mary). parent(mary, john). ancestor(A, B) :- parent(A, B). ancestor(A, B) :- parent(A, C), ancestor(C, B). Load the file in XSB: bash-3.2$ xsb [xsb_configuration loaded] [sysinitrc loaded] XSB Version 3.2 (Kopi Lewak) of March 15, 2009 [i386-apple-darwin10.4.0; mode: optimal; engine: slg-wam; scheduling: local; word size: 64] | ?- [ancestor]. 12

Experiment with it Issue various queries: e.g. ?- ancestor(X,Y). ?- ancestor(bill, X). ?- ancestor(john, X). … Change the order of the clauses and see what will happen: parent(bill, mary). parent(mary, john). ancestor(A, B) :- parent(A, C), ancestor(C, B). ancestor(A, B) :- parent(A, B). 13