Symbolic Programming First of two assignments Solution

Recall that a non-negative integer n can be encoded (in unary) as n succ’s applied to 0




numeral(0).




numeral(succ(X)) :- numeral(X).




In general, Prolog terms constructed from a constant null and k + 1 unary functors f0; : : : ; fk amount to strings over an alphabet fa0; : : : ; akg , with null encoding the empty string, and fi(null) encoding the string ai. For strings of length 1, it will prove useful (for some purposes such as the arithmetic encoding below) to encode the string in reverse, representing, for example, a2a3 as f3(f2(null)), rather than f2(f3(null)).




To simplify notation, let us work with the alphabet f0; 1g (with k = 1; a0 = 0; a1 = 1) and unary functors f0, f1. Let \pterm" abbreviate \Prolog term built from null, f0, f1" as described by the clauses




pterm(null).




pterm(f0(X)) :- pterm(X).




pterm(f1(X)) :- pterm(X).




Putting numbers in binary form,




(y) 0 becomes the bitstring 0 and pterm f0(null),




1 becomes the bitstring 1 and pterm f1(null),




2 becomes the bitstring 10 and pterm f0(f1(null)),




3 becomes the bitstring 11 and pterm f1(f1(null)),




...




Note that pterms such as null and f1(f1(f0(f0(null)))) are excluded from (y), even though we can associate non-negative integers (0 and 3) with them.




N.B. In solving the problems below, you are banned from using built-in arithmetic predicates or lists in Prolog.







1Due Oct 16 (Tuesday): submit to Blackboard and, if possible, demonstrate during lab (Mon 4-5, Tues 2-3). For any extensions beyond Oct 16, email your demonstrator, David Woods (dwoods@tcd.ie). E-mail submissions to Tim Fernando will receive an F3.




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Problem 1. De ne a predicate incr(P1,P2) over pterms P1 and P2 such that under (y), P2 is the successor of P1. For example, as 3 is 2+1,




?- incr(f0(f1(null)),X). X = f1(f1(null)) ;



no




You are free to de ne incr such that incr(null,X) holds for no X or else only for X=f1(null). Likewise for incr(P,X) where P is some other pterm not in (y).




Problem 2. De ne a predicate legal(P) true exactly of pterms P men-tioned by (y). Hence,




?- legal(X). X = f0(null) ; X = f1(null) ;



X = f0(f1(null)) ; X = f1(f1(null)) ;




X = f0(f0(f1(null))) ; X = f1(f0(f1(null))) ;




...




Using legal, revise your predicate incr to incrR such that




?- incrR(X,Y).




X = f0(null), Y = f1(null) ;




X = f1(null), Y = f0(f1(null)) ;




X = f0(f1(null)), Y = f1(f1(null)) ;




X = f1(f1(null)), Y = f0(f0(f1(null))) ;




X = f0(f0(f1(null))), Y = f1(f0(f1(null))) ;




X = f1(f0(f1(null))), Y = f0(f1(f1(null))) ;




...




Problem 3. De ne a predicate add(P1,P2,P3) over pterms P1,P2 and P3 such that under (y), P3 is P1 plus P2. For example, as 3 is 1+2,




?- add(f1(null),f0(f1(null)),X). X = f1(f1(null)) ;



no













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Problem 4. De ne a predicate mult(P1,P2,P3) over pterms P1,P2 and P3 such that under (y), P3 is P1 times P2. For example, as 2 is 1 2,




?- mult(f1(null),f0(f1(null)),X). X = f0(f1(null)) ;



no




Problem 5. De ne a predicate revers(P, RevP) that takes a pterm P and reverses it to RevP so that, for example,




?- revers(f0(f1(null)),X). X = f1(f0(null)) ;



no




Problem 6. De ne a predicate normalize(P, Pn) true of pterms P and Pn such that legal(Pn) and P and Pn encode the same number, enc(P) = enc(Pn), where




enc(null) := 0




enc(f0(X)) := 2 enc(X)




enc(f1(X)) := 2 enc(X) + 1:




For example,




?- normalize(null, X). X = f0(null) ;



no




?- normalize(f1(f0(f0(null))), X). X = f1(null) ;



no




Feel free to use Prolog’s built-in binary predicate n= for inequality (e.g. null n= f0(null)).




Final note. To help your demonstrator (and yourself) test your solutions, please add the following clauses to your Prolog code.







% test add inputting numbers N1 and N2




testAdd(N1,N2,T1,T2,Sum,SumT) :- numb2pterm(N1,T1), numb2pterm(N2,T2), add(T1,T2,SumT), pterm2numb(SumT,Sum).







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% test mult inputting numbers N1 and N2




testMult(N1,N2,T1,T2,N1N2,T1T2) :- numb2pterm(N1,T1), numb2pterm(N2,T2),




mult(T1,T2,T1T2), pterm2numb(T1T2,N1N2).




% test revers inputting list L




testRev(L,Lr,T,Tr) :- ptermlist(T,L), revers(T,Tr), ptermlist(Tr,Lr).




% test normalize inputting list L




testNorm(L,T,Tn,Ln) :- ptermlist(T,L), normalize(T,Tn), ptermlist(Tn,Ln).




% make a pterm T from a number N numb2term(+N,?T)




numb2pterm(0,f0(null)).




numb2pterm(N,T) :- N0, M is N-1, numb2pterm(M,Temp), incr(Temp,T).




make a number N from a pterm T pterm2numb(+T,?N) pterm2numb(null,0).



pterm2numb(f0(X),N) :- pterm2numb(X,M), N is 2*M.




pterm2numb(f1(X),N) :- pterm2numb(X,M), N is 2*M +1.




reversible ptermlist(T,L)



ptermlist(null,[]).




ptermlist(f0(X),[0|L]) :- ptermlist(X,L).




ptermlist(f1(X),[1|L]) :- ptermlist(X,L).







Apart from these clauses, your program should make no use of Prolog’s built-in arithmetic and list predicates.























































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