Artificial Intelligence A Textbook 1st Edition Charu C. Aggarwal - Solutions

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11.3 The procedure depth_first_iterative_deepening/2 in this section may get into an indefinite loop if there is no solution path in the state space. It keeps searching for longer solution paths even when it is obvious that there do not exist any longer paths than those already searched. The same
11.2 Write a depth-first procedure that combines both the cycle-detection and the depth-limiting mechanisms of the procedures in Figures 11.7 and 11.8.
11.1 Write a depth-first search procedure (with cycle detection) depthfirst1(CandidatePath, Solution) to find a solution path Solution as an extension of CandidatePath. Let both paths be represented as lists of nodes in the inverse order, so that the goal node is the head of Solution.
10.4 Trace the execution of the AVL insertion algorithm, starting with the empty tree and successively inserting 5, 8, 9, 3, 1, 6, 7. How is the root item changing during this process?
10.3 Define the relation avl(Tree) to test whether a binary tree Tree is an AVL-tree; that is, all the sibling subtrees may differ in their heights by 1 at the most. Let binary trees be represented by terms of the form t(Left, Root, Right) or nil.
10.2 Modify the program of Figure 10.6 to avoid backtracking (define relations ins2 and combine).
10.1 Define the relation in(Item, Tree) to search for Item in a 2-3 dictionary Tree.
9.10 Our procedure for displaying trees shows a tree in an unusual orientation, so that the root is on the left and the leaves of the tree are on the right. Write a (more difficult) procedure to display a tree in the usual orientation with the root at the top and the leaves at the bottom.
9.9 Modify the procedure in( Item, BinaryDictionary) by adding the third argument, Path, so that Path is the path between the root of the dictionary and Item.
9.8 Define the relation maxelement( D, Item) so that Item is the largest element stored in the binary dictionary D.
9.7 Define the relation linearize(Tree, List) to collect all the nodes in Tree into a list.
9.6 Define the procedure height( Binary Tree, Height) to compute the height of a binary tree. Assume that the height of the empty tree is 0, and that of a one-element tree is 1.
9.5 Define the predicates (a) binarytree(Object) (b) dictionary(Object) to recognize whether Object is a binary tree or a binary dictionary respectively, written in the notation of this section.
9.4 Another good idea for sorting a list is the merge-sort algorithm. It avoids the weakness of quicksort that occurs when lists Small and Big happen to be of very different lengths. Merge-sort is based on dividing the list, then sorting smaller lists, and then merging these sorted smaller lists.
9.3 Our quicksort program performs badly when the list to be sorted is already sorted or almost sorted. Analyse why.
9.2 The difference between the sorting programs of Figures 9.2 and 9.3 is in the representation of lists the latter uses difference lists. Transformation between plain lists and difference lists is straightforward and could be mechanized. Carry out the corresponding changes systematically in the
9.1 Write a procedure to merge two sorted lists producing a third list. For example: ?- merge([2,5,6,6,8], [1,3,5,9], L). L= [1,2,3,5,5,6,6,8,9]
8.7 Implement the updating of the value of an element of an array simulated by functor and arg, using 'holes' for future values along the lines discussed in Section 8.5.4.
8.6 Rewrite program 3 for eight queens (Figure 4.14) using simulated array with arg to represent the sets of free diagonals, as discussed in Section 8.5.4. Measure the improvement in efficiency.
8.5 The following procedure computes the maximum value in a list of numbers: max( [X], X). max( [X | Rest], Max) :- max( Rest, MaxRest), ( MaxRest >= X, !, Max = MaxRest ; Max = X). Transform this into a tail-recursive procedure. Hint: Introduce accumulator argument MaxSoFar.
8.4 Rewrite the collect procedure of Section 8.5.1 using difference pair representation for lists so that the concatenation can be done more efficiently.
8.3 Define the relation reverse( List, ReversedList) where both lists are represented by difference pairs.
8.2 Define the relation add_at_end( List, Item, NewList) to add Item at the end of List producing NewList. Let both lists be represented by difference pairs.
8.1 Procedures sub1, sub2 and sub3, shown below, all implement the sublist relation. subl is a more procedural definition whereas sub2 and sub3 are written in a more declarative style. Study the behaviour, with reference to efficiency, of these three procedures on some sample lists. Two of them
7.8 Generalize the eight-queens CLP(FD) program to an N queens program. For large N, a good labelling strategy for N queens is 'middle-out', which starts in the middle of the domain and then continues with values further and further away from the middle. Implement this labelling strategy and
7.7 Measure the time needed by the program in Figure 7.5 to solve the cryptarithmetic puzzle. Then replace the labelling goal by: labeling( [ff], Vars) The labelling option 'ff' stands for 'first fail'. That is, the variable with currently the smallest domain will be assigned a value first. Having
7.6 Modify the program of Figure 7.4 for the case of three-dimensional blocks and boxes. The structure of the program and the main principles will remain the same, so that relatively straightforward modifications are needed: (1) dimensions and positions become three-dimensional, like dim(5.0, 3.0,
7.5 The fit/5 predicate in our program only works for four blocks. All the blocks and pairs of blocks are explicitly enumerated to check for the no-overlap requirement. Extending the program in this style for a larger number of blocks (beyond 10, say) would become too cumbersome. Re-define the fit
7.4 Run times of our program for just four blocks are trivial. However, how about larger numbers of blocks? If the program of Figure 7.4 was extended to handle any number of blocks, its complexity would grow rapidly with the number of blocks. Fortunately, our program usually recognizes many
7.3 Demonstrate precisely how in the example query above '?- fib(N,4).' the constraint solver may realize that the accumulated constraints are not satisfiable.
7.2 Execute the arc consistency algorithm on the final state of the trace in Figure 7.2, after Tf is assigned value 9.
7.1 Try to execute the arc consistency algorithm with different orders of the arcs. What happens?
6.15 Write the procedure search(KeyWord, Sentence) that will, each time it is called, find a sentence in the current input file that contains the given KeyWord. Sentence should be in its original form, represented as a sequence of characters or as an atom (procedure getsentence of this section can
6.14 Define the procedure plural that will convert nouns into their plural form. For example: ?- plural( table, X). X = tables
6.13 Define the relation starts( Atom, Character) to check whether Atom starts with Character.
6.12 Generalize the squeeze procedure to handle commas as well. All blanks immediately preceding a comma are to be removed, and we want to have one blank after each comma.
6.11 Let f be a file of terms. Write a procedure findallterms(Term) that displays on the monitor all the terms in f that match Term. Make sure that Term is not instantiated in the process (which could prevent its match with terms that occur later in the file).
6.10 Let f be a file of terms. Define a procedure findterm(Term) that displays on the monitor the first term in f that matches Term.
6.9 Use bagof to define the relation copy_term( Term, Copy) such that Copy is Term with all its variables renamed.
6.8 Use bagof to define the relation powerset( Set, Subsets) to compute the set of all subsets of a given set (all sets represented as lists).
6.7 Define the relation copy_term(Term, Copy) which will produce a copy of Term so that Copy is Term with all its variables renamed. This can be easily programmed by using asserta and retract. In some Prologs copy_term is provided as a built-in predicate.
6.6 (a) Write a Prolog question to remove the whole product table from the database. (b) Modify the question so that it only removes those entries where the product is 0.
6.5 Define the relation subsumes(Term1, Term2) so that Term1 is more general than Term2. For example: ?-subsumes( X, c). yes ?- subsumes(g(x), g(t(Y))). yes ?- subsumes( f(X,X), f(a,b)). no
6.4 The substitute procedure of this section only produces the 'outer-most' substitution when there are alternatives. Modify the procedure so that all possible alternative substitutions are produced through backtracking. For example:?-substitute(a+b, f(A+B), new, NewTerm). A=a B = b NewTerm =
6.3 Define the predicate ground( Term) so that it is true if Term does not contain any uninstantiated variables.
6.2 Define the procedure: add_to_tail( Item, List) to store a new element into a list. Assume that all of the elements that can be stored are non-variables. List contains all the stored elements followed by a tail that is not instantiated and can thus accommodate new elements. For example, let the
6.1 Write a procedure simplify to symbolically simplify summation expressions with numbers and symbols (lower-case letters). Let the procedure rearrange the expressions so that all the symbols precede numbers. These are examples of its use:?- simplify(1+1+a, E). E = a +2 ?- simplify(1+a+4+2+b+c,
5.6 Define the predicate unifiable(List1, Term, List2) where List2 is the list of all the members of List1 that match Term, but are not instantiated by this matching. For example: ?- unifiable([X,b, t(Y)], t(a), List). List = [X, t(Y)] Note that X and Y have to remain uninstantiated although the
5.5 Define the set subtraction relation set_difference(Set1, Set2, SetDifference) where all the three sets are represented as lists. For example: set_difference([a,b,c,d], [b,d,e,f], [a,c])
5.4 Given two lists, Candidates and Ruled Out, write a sequence of goals (using member and not) that will through backtracking find all the items in Candidates that are not in Ruled Out.
5.3 Define the procedure split( Numbers, Positives, Negatives) which splits a list of numbers into two lists: positive ones (including zero) and negative ones. For example: split([3, -1, 0, 5, -2], [3, 0, 5], [-1, -2]) Propose two versions: one with a cut and one without.
5.2 The following relation classifies numbers into three classes: positive, zero and negative: class( Number, positive) :- Number > 0. class(0, zero). class( Number, negative) :- Number < 0. Define this procedure in a more efficient way using cuts.
5.1 Let a program be: P( 1). P(2) - !. P(3). Write all Prolog's answers to the following questions: (a) ?-p(X). (b) ?-p(X), p(Y). (c) ?-p(X), !, P(Y).
4.6 Let the squares of the chessboard be represented by pairs of their coordinates of the form X/Y, where both X and Y are between 1 and 8.(a) Define the relation jump( Square1, Square2) according to the knight jump on the chessboard. Assume that Squarel is always instantiated to a square while
4.5 When searching for a solution, the program of Figure 4.10 explores alternative values for the Y-coordinates of the queens. At which place in the program is the order of alternatives defined? How can we easily modify the program to change the order in which the alternative values for
4.4 Another drawback of our program is that it only solves the particular puzzle DONALD + GERALD = ROBERT. But to solve another famous puzzle SEND + MORE MONEY, we would have to write another program. Of course, we can just modify the variables list and the arithmetic condition in the program. It
4.3 Consider the following two questions that are logically equivalent: (a) ? schedule(riva at Start, riva at End, Schedule), member( arrive( malcesine at), Schedule), before( 17:00, Start). (b) ? sail( riva at Start, _), before( 17:00, Start), schedule( riva at Start, riva at End, Schedule),
4.2 Explain why was it useful to represent the location Pos of the rubbish on the floor by the term floor(Pos) and not just Pos (e.g. floor(middle) and not just middle). Hint: Omit the functor floor in the program and correspondingly simplify the clause for action pickup. Then study what happens
4.1 Experiment with the planning program of Figure 4.4. For example, use the program to solve the task with multiple goals: rubbish in basket, and basket in corner 1, and robot at door. What happens when using the procedure plan on its own, and combination of procedures conc and plan?
3.22 Define the operators 'if', 'then', 'else' and ':' so that the following becomes a legal term: if X > Y then Z := X else Z := Y Choose the precedences so that 'if' will be the principal functor. Then define the predicate 'if' as a small interpreter for a kind of 'if-then-else' statement of the
3.21 Define the procedure between( N1, N2, X) which, for two given integers N1 and N2, generates through backtracking all the integers X that satisfy the constraint N1 < x < N2.
3.20 Define the predicate subsum(Set, Sum, SubSet) so that Set is a list of numbers, SubSet is a subset of these numbers, and the sum of the numbers in SubSet is Sum. For example: ?- subsum( [1,2,5,3,2], 5, Sub). Sub = [1,2,2]; Sub [2,3]; Sub = [5];
3.19 Define the predicate ordered(List) which is true if List is an ordered list of numbers. For example, ordered([1,5,6,6,9,12]).
3.18 Define the predicate sumlist( List, Sum) so that Sum is the sum of a given list of numbers List.
3.17 Define the predicate maxlist( List, Max) so that Max is the greatest number in the list of numbers List.
3.16 Define the relation max(X, Y, Max) so that Max is the greater of two numbers X and Y.
3.15 In the previous section, relations involving lists were written as: member( Element, List), conc( List1, List2, List3), del( Element, List, NewList),... Suppose that we would prefer to write these relations as: Element in List, concatenating List1 and List2 gives List3, deleting Element from
3.14 Consider the program: t(0+1, 1+0). t(X+0+1, X+1+0). t(X+1+1, Z) :- t(X+I, X1), t(X1+1, Z).How will this program answer the following questions if '+' is an infix operator of type yfx (as usual): (a) ?-t(0+1, A). (b) ?-t(0+1+1, B). (c) t(1+0+1+1+1, C). (d) ?-t(D, 1+1+1+0).
3.13 Suggest an appropriate definition of operators ('was', 'of', 'the') to be able to write clauses like diana was the secretary of the department. and then ask Prolog: ?- Who was the secretary of the department. Who = diana ?- diana was What. What the secretary of the department
3.12 Assume the operator definitions :- op(300, xfx, plays). :- op( 200, xfy, and). Then the following two terms are syntactically legal objects: Term1 = jimmy plays football and squash Term2 = susan plays tennis and basketball and volleyball How are these terms understood by Prolog? What are their
3.11 Define the relation flatten( List, FlatList) where List can be a list of lists, and FlatList is List 'flattened' so that the elements of List's sublists (or sub-sublists) are reorganized as one plain list. For example: ?- flatten( [a,b,[c,d],[],[[[e]]],f], L). L = [a,b,c,d,e,f]
3.10 Define the predicate equal length(L1, L2) which is true if lists L1 and L2 have equal number of elements.
3.9 Define the relation dividelist( List, List1, List2) so that the elements of List are partitioned between List1 and List2, and List1 and List2 are of approximately the same length. For example: dividelist( [a,b,c,d,e], [a,c,e], [b,d]).
3.8 Define the relation subset(Set, Subset) where Set and Subset are two lists representing two sets. We would like to be able to use this relation not only to check for the subset relation, but also to generate all possible subsets of a given set. For example: ?- subset( [a,b,c], S). S = [a,b,c];
3.7 Define the relation translate(List1, List2) to translate a list of numbers between 0 and 9 to a list of the corresponding words. For example: translate([3,5,1,3], [three, five, one, three]) Use the following as an auxiliary relation: means(0, zero). means( 1, one). means(2, two).
3.6 Define the relation shift( List1, List2) so that List2 is List1 'shifted rotationally' by one element to the left. For example, ? shift([1,2,3,4,5], L1), shift(L1, L2). produces: L1 = [2,3,4,5,1] L2 = [3,4,5,1,2]
3.5 Define the predicate palindrome( List). A list is a palindrome if it reads the same in the forward and in the backward direction. For example, [m,a,d,a,m].
3.4 Define the relation reverse( List, ReversedList) that reverses lists. For example, reverse( [a,b,c,d], [d,c,b,a]).
3.3 Define two predicates evenlength(List) and oddlength( List) so that they are true if their argument is a list of even or odd length respectively. For example, the list [a,b,c,d] is 'evenlength' and [a,b,c] is 'oddlength'.
3.2 Define the relation last(Item, List) so that Item is the last element of a list List. Write two versions: (a) using the conc relation, (b) without conc.
3.1 (a) Write a goal, using conc, to delete the last three elements from a list L producing another list Ll. Hint: L is the concatenation of Ll and a three-element list. (b) Write a goal to delete the first three elements and the last three elements from a list L producing list L2.
2.11 What happens if we ask Prolog: ?- X = f(x). Should this request for matching succeed or fail? According to the definition of unification in logic this should fail, but what happens according to our definition of matching in Section 2.2? Try to explain why many Prolog implementations answer the
2.10 Prolog implementations usually provide debugging facilities that enable the interactive tracing of the execution of goals. Such tracing gives similar insights as our diagrammatic traces in Figures 1.9 and 2.13-15. A typical use of the Prolog trace facility is as follows. Assume that you have
2.9 Consider the program in Figure 2.10 and simulate, in the style of Figure 2.10, Prolog's execution of the question: ?- big(X), dark(X). Compare your execution trace with that of Figure 2.10 when the question was essentially the same, but with the goals in the order: ?- dark(X), big(X). In which
2.8 Rewrite the following program without using the semicolon notation. translate( Number, Word) :- Number 1, Word = one; Number 2, Word = two; = Number 3, Word = three. =
2.7 The following program says that two people are relatives if (a) one is an ancestor of the other, or (b) they have a common ancestor, or (c) they have a common successor: relatives(X, Y) :- ancestor(X, Y). relatives(X, Y) :- ancestor( Y, X). relatives(X, Y) :- % X and Y have a common ancestor
2.6 Consider the following program: f( 1, one). f(s(1), two). f(s(s(1)), three). f(s(s(s(X))), N) :- f( X, N). How will Prolog answer the following questions? Whenever several answers are possible, give at least two. (a) ?- f(s(1), A). (b) ?-f(s(s(1)), two). (c) ?-f(s(s(s(s(s(s(1)))))), C). (d) ?-
2.5 Assume that a rectangle is represented by the term rectangle( P1, P2, P3, P4) where the P's are the vertices of the rectangle positively ordered. Define the relation: regular(R) which is true if R is a rectangle whose sides are vertical and horizontal.
2.4 Using the representation for line segments as described in this section, write a term that represents any vertical line segment at x = 5.
2.3 Will the following matching operations succeed or fail? If they succeed, what are the resulting instantiations of variables? (a) point(A, B) = point(1, 2) (b) point(A, B) = point(X, Y, Z) (c) plus(2, 2) = 4 (d) +(2, D)+(E, 2) (e) triangle( point(-1,0), P2, P3) = triangle( P1, point(1,0),
2.2 Suggest a representation for rectangles, squares and circles as structured Prolog objects. Use an approach similar to that in Figure 2.4. For example, a rectangle can be represented by four points (or maybe three points only). Write some example terms that represent some concrete objects of
2.1 Which of the following are syntactically correct Prolog objects? What kinds of object are they (atom, number, variable, structure)? (a) Diana (b) diana (c) 'Diana' (d) _diana (e) 'Diana goes south' (f) goes( diana, south) (g) 45 (h) 5(X, Y) (i) +(north, west) (j) three( Black(Cats))
1.7 Try to understand how Prolog derives answers to the following questions, using the program of Figure 1.8. Try to draw the corresponding derivation diagrams in the style of Figure 1.9. Will any backtracking occur at particular questions? (a) ?-parent(pam, bob). (b) ?-mother( pam, bob). (c)
1.6 Consider the following alternative definition of the ancestor relation: ancestor(X, Z) :- parent(X, Z). ancestor( X, Z) :- parent( Y, Z), ancestor(X, Y). Does this also seem to be a correct definition of ancestors? Can you modify the diagram of Figure 1.7 so that it would correspond to this new
1.5 Define the relation aunt( X, Y) in terms of the relations parent and sister. As an aid you can first draw a diagram in the style of Figure 1.3 for the aunt relation.
1.4 Define the relation grandchild using the parent relation. Hint: It will be similar to the grandparent relation (see Figure 1.3).
1.3 Translate the following statements into Prolog rules: (a) Everybody who has a child is happy (introduce a one-argument relation happy). (b) For all X, if X has a child who has a sister then X has two children (introduce new relation hastwochildren).
1.2 Formulate in Prolog the following questions about the parent relation: (a) Who is Pat's parent? (b) Does Liz have a child? (c) Who is Pat's grandparent?
1.1 Assuming the parent relation as defined in this section (see Figure 1.1), what will be Prolog's answers to the following questions? (a) ? parent( jim, X). (b) ?-parent( X, jim). (c) ?-parent(pam, X), parent( X, pat). (d) ?- parent(pam, X), parent(X, Y), parent(Y, jim).
How does the bias and variance of a bagged classifier compare to that of an individual classifier from the bagged set of classifiers?

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Artificial Intelligence A Textbook 1st Edition Charu C. Aggarwal - Solutions

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