programming techniques

cs774 (Prasad) L17ProgTech 1

Programming Techniques

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Generalization/Abstraction

Analogy:

[a,b,c] [f(a),f(b),f(c)]

maplist(_,[],[]).

maplist(P,[X|T],[NX|NT]) :-

G =.. [P,X,NX],

call(G),

maplist(P,T,NT).

(G p(N,NX))


Application

transpose([],[]).

transpose([[]|_],[]) :- !.

transpose([R|Rs],[C|Cs]) :-

maplist(first,[R|Rs],C),

maplist(rest,[R|Rs],RC),

transpose(RC,Cs).

first([H|T],H).

rest([H|T],T).

/* Built-in maplist exists*/


Enhancing Efficiency

• Interpreted vs Compiled code (order of magnitude improvement observed)

• Improving data structures and algorithm • 8-Queens problem, Heuristic Search, Quicksort, etc

• Tail-recursive optimization

• Memoization • storing partial results / caching intermediate results

• Difference lists• DCGs


(cont’d)

• Prolog implementations that index on the first argument of a predicate improve determinism.

• Cuts and other meta-programming primitives can be used to program in new search strategies for controlled backtracking.


Optimizing Fibonacci Number Computation

fib(0,0) :- !.

fib(1,1) :- !.

fib(N,F) :-

N1 is N - 1, N2 is N1 -1, fib(N1,F1), fib(N2,F2),

F is F1 + F2.

?-fib(5,F).

Complexity: Exponential time algorithm


Fibonacci Call Tree with Parameter Value


(cont’d)

f(0,F,_,F).

f(1,_,F,F).

f(N,Fpp,Fp,F) :- N >= 2,

N1 is N – 1, F0 is Fp + Fpp,

f(N1,Fp,F0,F).

fib(N,F) :- f(N,0,1,F).

?-fib(5,F).

Complexity: Linear time algorithm

(tail-recursive version)


Last call optimization

• Activation record normally stores a continuation and a backtrack point, to be used when the goal succeeds or fails respectively.

p :- q, r.

p :- s.– LCO avoids allocating a new activation record

for s, but rather reuses one for p.


Caching intermediate results

• Instead of explicitly modifying the code to improve performance, XSB uses tabling to store intermediate results and avoids recomputing earlier goals.

• Ironically, double-recursive (exponential-time) Fibonacci Number definition serves as a benchmark for testing efficiency of implementation of recursion!


Different Lists : Motivation


STACK push pop

Array-Impl. O(1) O(1)

Linked-list Impl. O(1) O(1)

QUEUE enqueue dequeue

Circular Buffer or Linked List Impl.

(front& rear pointer)

O(1) O(1)

Linked-List Impl.(front but no rear

pointer)

O(1) O(n)

(cont’d)• In Prolog, pointers implementing list structures are not

available for inspection/manipulation. Hence, complexity of enqueue (resp. dequeue) is O(1) and that of dequeue (resp. enqueue) is O(n).

enqueue(Q,E,[E|Q]).

dequeue([E],E).

dequeue([_|F|T],E) :- dequeue([F|T],E).

• Difference list is a techqniue to get O(1) complexity for both the operations.


Difference Lists : Details• Represent list L as a difference of two lists

L1 and L2– E.g., consider L = [a,b,c] and various L1-L2

combinations given below.


L1 L2

[a,b,c] []

[a,b,c,d,e] [d,e]

[a,b,c|T] T

[a,b,c,d|T] [d|T]

BenefitL = L1 – L2

• Both enqueue and dequeue are O(1) operations obtained by cons-ing an element to L1 and L2 respectively.

enqueue(L1-L2, E, [E|L1] – L2).

dequeue(L1-L2, E, L1 – [E|L2]).

E.g.,

enqueue([a]-[], b, [b,a] – []).

dequeue([a]-[], a, [a]–[a]).cs774 (Prasad) L17ProgTech 14

Append using Difference Lists

append(X-Y, Y-Z, X-Z).• Ordinary append complexity = O(length of first list)

• Difference list append complexity = O(1)


X

Y

Z

X-Y

Y-Z

Y

Z

Z

X-Z

Restrictionappend(X-Y, Y-Z, X-Z).

?-append([a,b,c|[d]]-[d], [1,2]-[], N).

• Fails because the second lists must be a variable. Incomplete data structure is a necessity.


Interpreter-based Semantics vs Declarative Semantics

• IS is an over-specification but may provide an efficient implementation.

• DS specifies correctness criteria and may permit further optimization.

• Overall research goal: Characterize classes of programs for which the declarative and the procedural semantics coincide.


Relational Algebra (Operations on Relations)

• Select, Project, Join, Union, Intersection, difference– Transitive closure cannot be expressed in terms

of these operations.

• A query language is relationally complete if it can perform the above operations.


Deductive Databases : Datalog (Function-free/Finite Domain Prolog)

• Datalog + Negation is relationally complete.

• What effects query evaluation efficiency?– Characteristics of data (cyclic vs acyclic)– Ordering of rules and body literals– Search strategy (top-down vs bottom-up)

• Tuple-at-a-time vs Set-at-a-timecs774 (Prasad) L17ProgTech 20

Middle Ground:Top-down vs Bottom-up

• Improve efficiency by caching. (cf. tabling)

• Remove Incompleteness by loop detection.

• Focused search.• Propagate bindings in

the query. (cf. Magic sets)


In general, the efficiency of query evaluation can be improved by sequencing goals on the basis of their bindings and dependencies among rule literals.

Heuristics for rearranging rules and body literals for efficiency

• Order body literals by decreasing values of failure probability

• Order rules by decreasing values of success probability

• Order body literals to maximize dependencies among adjacent literals.

• Metric for comparison – e.g., extent of base relation graphs inspected


Backtracking

• Chronological

• Dependency directed– focus on the reason for backtracking

ans(X,Y) :- p(X), q(Y), r(X).

p(1). p(2). p(3).

q(1). q(2). q(3).

r(3).


Data Dependency Graph


ans(X,Y) :-

p(X), r(X),

q(Y),

If r(X) fails, then backtrack to p(X)rather than q(Y).

Indexing• Prolog indexes on

– predicate symbol and arity– principal functor of first argument (cf. constant -> hash)

• Randomly accessed rule groupsp(a) :- …

p(22) :- …

p(f(X)) :- …

p([]) :- …, p([a]) :- …, …


Robert Kowalski

• Algorithm = Logic + Control

Niklaus Wirth

• Programs = Data Structures + Algorithms


programming techniques

Documents

nxcs774 prasadl17progtech

list structures

lists l1

linked list impl

detailsrepresent list

o1 complexity

complexity of enqueue

el1 l2