Recursive programs

Electronic digital computers—Programming. 2. Recursive programming. 3. Computer programs. I. Title. II. Series. 

The program is again put into the system using consult(user). This very simple recursive program is quite powerful. Obviously we can ask whether b is a member of the list [a,b,c] by typing in... 

This is a recursion formula for the exact case. We would like to be able to apply this to any number n of CSTRs in series and find an analytical and then quantitative result for comparison to the exact PFR result. To do this we need recursive programming. There are three programming styles in Mathematica Rule-Based, Functional, and Procedural. We will attack this problem in recursion with Rule-Based, Functional, and Procedural programming. We can begin by looking at the rule-based recursion codes for Ca and Cb in any n CSTRs. 

Now we will return to the recursive programming part of this problem because it is a prototypical type problem we can expect to encounter often in chemical engineering analysis... 

When Stephen used his recursive program to perform an exhaustive search for longest paths in Worm World 1, he found a range of path lengths as follows (R=right, L=left, U=up, D=down) ... 

FIGURE 27.2. Coral-like form generated from recursive program. 

The factorial funciton can be accomplished with a simple recursive program ... 

Figure 8.4 Program to find an element in a list of elements by means of list processing language (LISP) and a recursive programming in logic (PROLOG) procedure.

There is an obvious interest in proofs-by-induction, because these allow the synthesis of recursive programs. Note that there often is a similar proof content in transformational synthesis and proofs-as-programs synthesis, and it seems that the same proof construction techniques should be applicable to both. This may suggest that these approaches are probably two facets of the same process. For instance, the work of [Neugebauer 93] shows how the specification forms of these approaches may be reconciled. 

Z. Manna and R. Waldinger. The automatic synthesis of systems of recursive programs. In Proc. of IJCAI ll, pp. 405-411. 

The sum in Eq. (43) can be obtained by a recursion algorithm used commonly in dynamic programming [62]. 

Software to predict the properties of formulated products is made more powerful by a recursive procedure which can use formulas stored in files as raw materials. Particular care must be taken with program flow control and data structures for the recursion to be effective. This paper illustrates these issues using an example derived from a working formulation system for coatings development. 

Recursion, when used in the context of computer programs, refers to a procedure that calls itself as a subprocedure. The classic examples cited in programming texts Q, ate computation of the factorial function and... 

By contrast, in a recursive modelling procedure the program designer must distinguish between variables used in different invocations of the procedure for a single concept, such as a characteristic value of a formula. For instance, the formula entered by the user has a PBR, and so will any intermediates used in the formula. Thus, the recursive call to calculate characteristic values of the intermediate will return a PBR in a variable which must have a different name than the variable used for the current formula s PBR. 

Data is passed between levels of recursive procedures in variables cited in the CALL statement for the procedure. As indicated in the section on programming issues, care must be taken in assigning names to the data variables so that values returning from a recursive CALL do not overwrite the values in variables within the calling procedure. Note the difference between the variable names in the SUBROUTINE statement of... 

Recursive techniques have been in use since the early days of digital computers. However, we have seen little use of this technique in formula calculations. Implementation requires the ability to calculate and return the same type of data for a formula as is normally retrieved from a raw material database. We have outlined some of the difficulties in using recursion. None of them present insurmountable difficulties if the programming environment being used supports recursion. The benefit of implementation is in the power of the software to simplify experimentation and formula development. 

The procedure illustrated here, besides containing only trivial technical calculations, lacks important features that are required in production programs. Extensive error checking and recovery must be performed. The procedure must detect the occurrence of a self-referential system of formulas, which would result in attempting endless recursive calls. Access to multiple raw material and formula databases adds power to the program, but must be implemented by complex code to allow flexible control of that access. The structural and input/output statements to support these features may greatly exceed the number of statements that perform modelling calculations. 

More specifically, the basic notions of a Turing Machine, of computable functions and of undecidable properties are needed for Chapter VI (Decision Problems) the definitions of recursive, primitive recursive and partial recursive functions are helpful for Section F of Chapter IV and two of the proofs in Chapter VI. The basic facts regarding regular sets, context-free languages and pushdown store automata are helpful in Chapter VIII (Monadic Recursion Schemes) and in the proof of Theorem 3.14. For Chapter V (Correctness and Program Verification) it is useful to know the basic notation and ideas of the first order predicate calculus a highly abbreviated version of this material appears as Appendix A. 

The definition of program scheme is "recursive" or "computable" in the sense that we can decide whether a given flow diagram (finite of course) meets this definition. Conditions 1 and 3a and b are clearly verifiable by inspection of the diagram. Since accessibility is decidable for finite state graphs, we can determine whether a given statement lies on a path from START and so verify condition 2. 

We shall return to the concept of value languages later in our study of monadic recursion schemes this concept is a powerful tool in studying the relative power of monadic program schemes and monadic recursion schemes. For the present we need only the following simple fact. 

The transformation t we saw at the end of the last section, which changes liberal schemes into free schemes, is such a canonical transformation. The corresponding canonical class of schemes is the class of schemes such that tests are applied initially on the input variables and are applied after assignment statements on the program variables involved, and at no other time. This transformation t is clearly recursive and equivalence preserving. The class of free schemes is not a canonical form class, since, as we saw, there are schemes not strongly equivalent to any free scheme. 

F. WHILE PROGRAMS, STEP PROGRAMS, AND PRIMITIVE RECURSIVE FUNCTIONS... 

We can use some elementary facts from recursive function theory to show that every partially computable or partial recursive function from nonnegative integers to nonnegative integers can be computed by a WHILE program of a particularly simple form. This result can be extended to functions computable over other domains by standard encoding techniques. 

A particular subset of the total recursive functions, the primitive recursive functions, can be computed by a special type of WHILE program, a STEP program or LOOP program. We define a STEP construction ... 

Notice that f can be defined from h and g by primitive recursion as the output of the following program ... 

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