Lists, Operators, Arithmetic

1. Lists, Operators, Arithmetic Notes for Ch.3 of Bratko For CSCE 580 Sp03 Marco Valtorta

2. Representation of Lists • Prolog provides a convenient representation of lists • [ ann,tennis,tom,skiing] • ann is the head of the list above • [ tennis,tom,skiing] is its tail • The head of a list is its first element • The tail of a list is the list without its first element

3. Lists as Terms • The empty list: [ ] (a constant) • Any other list is a structure whose principal functor is the list constructor, which is represented by the dot (.) in Prolog • Any non-empty list matches .( Head,Tail) • [ ann, tennis, tom, skiing] is a “cosmetic improvement” (“syntactic sugar”) for .( ann, .( tennis, .(tom, .(skiing, [ ])))) • See Fig. 3.1 for a tree representation of this term

4. More List Notation • Prolog converts the internal representation of a list to its “neater” form: • ?- L = .( ann, .( tennis, .( tom, .(skiing, []) ) ) ). L = [ann, tennis, tom, skiing] • 3 ?- L = .( .( ann, []), .( .( tom, []), [])). L = [[ann], [tom]] . . . . [ ] [ ] ann tom [ ]

5. More List Notation • L = [a | Tail] • [ a,b,c] = [ a | [ b,c]] = [ a,b | [c]] = [ a,b,c | [ ]]

6. Some Operations on Lists • Membership: checking whether some object is an element of a list • Concatenation of two lists, obtaining a third list, which is the juxtaposition of the first two • Adding an element to a list • Deleting an element from a list • Checking whether a list is a sublist of another • Generating permutations of a list • There are built-in versions of some of these predicates in SWI: see section 4.29 (p.114) • pr3_1.pl

7. pr3_1.pl • Note the non-determinism and reversibility of many of these operations • member is built-in; we call ours memb • Note that member can be used to generate lists • Alternate member: memb1( X,L) :- conc( L1,[ X | L2], L).

8. Operator Notation • 2*a + b*c • Infix notation is a convenience for external presentation • The principal functor is (one of the) highest precedence operator(s) • Fig.3.8 is a table of predefined operators; it is a subset of the predefined operators in SWI (Section E.4, p.255)

9. Operator Precedence • Xfx vs. xfy vs. yfx (a – b) – c or a – (b – c)? • Since – is defined as yfx, a – (b – c) • Is not not p correct? Yes, because not is defined as fy

10. Arithmetic • Try: ?- X = 1 + 2. Surprise! • Try ?- X is 1 + 2. is is not reversible! • Notation: -Number is +Expr • Moreover, Expr must have a value and Number should be unbound. Do not use is to test for equality • See Section 4.26 (p.109) and Appendix E3 of manual

11. Equality • = for unification: tries to match terms by finding a unifier • =:= for equality: forces arithmetic evaluation, but no substitution is made • Try ?- 1+2 = 2+1 (fails) ?- 1+2 =:= 2+1 (succeeds) ?- 1+A = B+2 (succeeds) ?- 1+A =:= B+2 (fails) 5 ?- 1+A =:= B+2. ERROR: Arguments are not sufficiently instantiated

12. Euclid’s Algorithm • euclid.pl • Another non-reversible program 2 ?- gcd( 10, 55, D). D = 5 3 ?- gcd( 10, Y, 5). ERROR: Arguments are not sufficiently instantiated

13. Just for Fun… Correctness • gcd( X,X,X). Obvious • Clause 2. Induction. Assume X < Y. • case 1: Y = nX. Obvious • case 2: Y = nX + r. Any divisor D of Y and X is also a divisor of r. Since r < X, the last subgoal in the body of the clause is (eventually) gcd(X,r,D). We only need to show that, if D is a GCD of X and r, it is also a GCD of X and Y. Any divisor of X and r is also a divisor of r. So, the set of divisors of Y and X is the same as the set of divisors of X and r. So, the Greatest CD of X and Y is the same as the GCD of X and r. But r < X, so the last subgoal in the body of the clause is (eventually) set up with X, r, D. • Clause 3. Induction. Assume Y < X. Analogous to clause 2 case. • See Knuth, vol.1, Section 1.1.

14. Length • A case study in evaluated vs. non-evaluated terms: length.pl • length is built-in. It is reversible. Warning: dirty secrets of SWI-Prolog ahead! length(List, Length) :- $length(List, Length), !. % written in C length(List, Length) :- var(Length), length2(List, Length). length2([], 0). length2([_|List], N) :- length2(List, M), succ(M, N).

15. Length II • len1 is interpreted. It is reversible, but it will loop infinitely if asked for multiple lists of some length! • len2 fixes this (at the cost of using a cut) • len3 returns a not interpreted term • len4 does the same with a shorter second clause • len5 uses len4 and then interprets the result • len6 is to len5 what len2 is to len1

16. Axiomatizing Arithmetic • Unary (“tally”) arithmetic: Giuseppe Peano’s axioms, also known as “Peano’s Postulates.” • See Russell and Norvig, Section 8.3 for an informal presentation of the axioms. • See http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html for a biographical sketch of Giuseppe Peano) • Binary arithmetic: UseDefK7.ppt • Note: examples are in Cilog, a dialect of Prolog, based on an exercise in Poole et al.’s AI Textbook

17. Peano’s Postulates (P1) 0 is a natural number (P2) If x is natural number, there is another natural number denoted by x’ (and called the successor of x) (P3) 0 != x’ for any natural number x (P4) If x’ = y’, then x = y (P5) If Q is a property that may or may not hold of the natural numbers, and if (I) 0 has the property Q, and (II) whenever a natural number x has the property Q, then x’ has the property Q, then all numbers have the property Q (Principle of Induction) (Mendelson,E. Introduction to Mathematical Logic. Princeton, NJ: Van Nostrand, 1965.)