By Ryszard Kubiak; Roman Rudzinski; Stefan Sokolowski
A function of contemporary complicated computing is the sensible method of programming. during this ebook, the authors current an advent to the maths which underline sensible programming, emphasizing the certainty of definition and specification--a prerequisite of fine programming and challenge fixing with a working laptop or computer. The e-book is self-contained, requiring a low point of mathematical sophistication and should be used as an creation to the maths of programming.
Provides an creation to the useful method of programming**Emphasizes the matter to be solved, no longer the programming language**Takes the view that every one laptop courses are a definition of a function**Includes workouts for every chapter**Can be used as a pre-programming language creation to the maths of computing.
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Extra info for An introduction to programming with specifications : a mathematical approach
X ) + /(j/) = 5 6. D o t h e following f u n c t i o n s m e e t t h e c o r r e s p o n d i n g specifications? a. function: c : Int —> Int 2 c(i) = i specification: h : Int —> Int Vz : Int. 5. b. Exercises 29 function: specification: q : Int χ Int —> Nat 2 2 χ Να£ ρ : Int χ Int —• Afa£ χ iVa£ g(i,j)^(z ,j ) c. p(l,2)=p(-2,-l) function: g : A t o χ Real —• £ 0 0 / specification: / : Afai χ iteaZ —> £ooZ g(n, x) = ( ^ / x > 0) Vx > 0. / ( 2 , x ) 7. R e a l i z e t h e following specifications a n d prove t h e r e a l i z a t i o n s correct.
2). The usefulness of local definitions will become apparent later when re cursive function definitions are discussed. 4 Exercises 1. What are the types of the following expressions: a. LET a = 5y IN (a, false) for y : Real b. 2 2 LET (a, b) = (x + y, χ - y) IN a + b for x, y : Int 2. Evaluate the following expressions: a. LET m = IF χ > y THEN χ ELSE y IN LET η = IF χ < y THEN χ ELSE y IN (ra,n) for χ = 1, y = — 1 b. IF abs(x) > abs(y) THEN LET α ΞΞ χ - y IN xa ELSE LET a = χ + y IN xa for χ = —3, y = — 4 c.
0)[0/x] = (3x η > 1) 2 : Real, x = 0) Note that there are no free xs in the last formula. A warning regarding the operation of substitution is due. This is a syntactic operation and the value of the resulting expression may depend not only on the value of the expression in which the substitution is performed, but also on its syntactic shape. In other words, even if e\ and e 2 are equal for any valuations the values of e\\ejx\ and e2[e/x] are not necessarily equal (see Exercise 3 to this chapter).
An introduction to programming with specifications : a mathematical approach by Ryszard Kubiak; Roman Rudzinski; Stefan Sokolowski