Chapter Three: Lists, Operators, Arithmetic 1

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Important things about lists  List elements are enclosed in square brackets  The length of a list is the number of elements it has  There is a special list: the empty list [ ]

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail  A non-empty list can be thought of as consisting of two parts  The head  The tail  The head is the first item in the list  The tail is everything else  The tail is the list that remains when we take the first element away  The tail of a list is always a list

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 1  [mia, vincent, jules, yolanda] Head: Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 1  [mia, vincent, jules, yolanda] Head: mia Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 1  [mia, vincent, jules, yolanda] Head: mia Tail: [vincent, jules, yolanda]

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 2  [[ ], dead(z), [2, [b,c]], [ ], Z, [2, [b,c]]] Head: Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 2  [[ ], dead(z), [2, [b,c]], [ ], Z, [2, [b,c]]] Head: [ ] Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 2  [[ ], dead(z), [2, [b,c]], [ ], Z, [2, [b,c]]] Head: [ ] Tail: [dead(z), [2, [b,c]], [ ], Z, [2, [b,c]]]

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 3  [dead(z)] Head: Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 3  [dead(z)] Head: dead(z) Tail:

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Head and Tail example 3  [dead(z)] Head: dead(z) Tail: [ ]

© Patrick Blackburn, Johan Bos & Kristina Striegnitz The built-in operator |  Prolog has a special built-in operator | which can be used to decompose a list into its head and tail  The | operator is a key tool for writing Prolog list manipulation predicates

© Patrick Blackburn, Johan Bos & Kristina Striegnitz The built-in operator | ?- [Head|Tail] = [mia, vincent, jules, yolanda]. Head = mia Tail = [vincent,jules,yolanda] yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz The built-in operator | ?- [X|Y] = [mia, vincent, jules, yolanda]. X = mia Y = [vincent,jules,yolanda] yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz The built-in operator | ?- [X,Y|Tail] = [[ ], dead(z), [2, [b,c]], [], Z, [2, [b,c]]]. X = [ ] Y = dead(z) Tail = [[2, [b,c]], [ ], Z, [2, [b,c]]] yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Anonymous variable  Suppose we are interested in the second and fourth element of a list ?- [X1,X2,X3,X4|Tail] = [mia, vincent, marsellus, jody, yolanda]. X1 = mia X2 = vincent X3 = marsellus X4 = jody Tail = [yolanda] yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz Anonymous variables  There is a simpler way of obtaining only the information we want: ?- [ _,X2, _,X4|_ ] = [mia, vincent, marsellus, jody, yolanda]. X2 = vincent X4 = jody yes ?- The underscore is the anonymous variable

3.2 Some operations on lists Membership 2. Concatenation 3. Deleting an item 4. Permutations

3.2 Some operations on lists Member member(X,L) where X is an object and L is a list member(X,L) is true if X occurs in L e.g. member(b, [a,b,c]) member(b, [a,[b,c]]) member([b,c], [a,[b,c]]) Definition of member: X is a member of L if either: (1) X is the head of L, or (2) X is a member of the tail of L. 

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(yolanda,[yolanda,trudy,vincent,jules]).

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(yolanda,[yolanda,trudy,vincent,jules]). yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(vincent,[yolanda,trudy,vincent,jules]).

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(vincent,[yolanda,trudy,vincent,jules]). yes ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(zed,[yolanda,trudy,vincent,jules]).

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(zed,[yolanda,trudy,vincent,jules]). no ?-

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(X,[yolanda,trudy,vincent,jules]).

© Patrick Blackburn, Johan Bos & Kristina Striegnitz member(X,[X|T]). member(X,[H|T]):- member(X,T). ?- member(X,[yolanda,trudy,vincent,jules]). X = yolanda; X = trudy; X = vincent; X = jules; no

3.2 Some operations on lists Concatenation conc(L1,L2,L3) here L1 and L2 are two lists and L3 is their concatenation. conc([a,b],[c,d],[a,b,c,d]) conc([a,b],[c,d],[a,b,a,c,d]) Definition of conc: (1) If the first argument is empty, then the second and third arguments must be the same list conc([],L,L). the result of concatenation of L1 and L2 is the list L3 where L3 is the concatenation of L1 and L2 

3.2 Some operations on lists Concatenation Examples ?- conc([a,b,c],[1,2,3],L) L=[a,b,c,1,2,3] ?-conc([a,[b,c],d],[a,[],b],L) L=[a,[b,c],d,a,[],b]

3.2 Some operations on lists Concatenation Examples ?- conc(L1,L2,[a,b,c]) (decompose) L1=[] L2=[a,b,c]; L1=[a] L2=[b,c]; L1=[a,b] L2=[c]; L1=[a,b,c] L2=[]; no

3.2 Some operations on lists Concatenation Examples How can we find the months that precede and the months that follow a given month? ?-conc(Before, [may|After], [jan,feb,mar,apr,may,jun,jul,aug,sep,oct,nov,dec]). Before = [jan, feb, mar, apr] After = [jun, jul, aug, sep, oct, nov, dec]

3.2 Some operations on lists Deleting an item Deleting item X from a list L can be programmed as a relation: del(X,L,L1) where L1 is equal to the list L with the item X removed. Definition of del: It can be defined similarly to the membership relation, again we have 2 cases: 1. If X is the head. 2. If X is in the tail. del(X,[X|Tail],Tail). del(X,[Y|Tail],[Y,Tail1]):- del(X,Tail,Tail1).

3.2 Some operations on lists Deleting an item Example: ?- del(a,[a,b,a,a],L). L = [b,a,a]; L = [a,b,a]; no

3.2 Some operations on lists Deleting an item Example: If we want to insert ‘a’ in the list [1,2,3] then we can do that by asking the question: What is L such that after deleting ‘a’ from L we obtain [1,2,3]? ?- del(a,L,[1,2,3]). L = [a,1,2,3]; L = [1,a,2,3]; L = [1,2,a,3]; L = [1,2, 3,a]; no

3.2 Some operations on lists Permutations Permutation relation has two arguments(lists), one is a permutation of the other. e.g. ?- permutation([a,b,c],P). P=[a,b,c]; P=[a,c,b]; P=[b,a,c]; …

3.2 Some operations on lists Permutations 1. If the first list is empty, then the second must be also empty. permutation([],[]).