J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....performance of a task plays a cru cial role. The most difficult tasks in estimating the timing behavior of a program are to determine the number of iterations of a certain loop and to handle problems originating from the use of recursion. A solution to the first problem has been given in [1], the second one will be treated in this paper. If recursive procedures are to be used in implementing real time applications, several problems occur: 1. It is not clear, whether a recursive procedure completes or not. 2. If it completes, it must be guaranteed that its result is delivered ....

.... source code of a recursive procedure is consi dered to consist of simple segments of linear code, the performance of which is known a priori, if statements, loops with known upper bounds of the number of iterations which can be derived at compile time, e.g. for loops or discrete loops (cf. [1]) and recursive calls to the procedure itself. In terms of a context free grammar this is stated as follows code(f) iff 0 then nonrecursive(f) else recursive(f) end if recursive(f) seq(f) seq(f) statement(f) statement(f) statement(f) simple(f) compound(f) ....

J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193-212, 1994.

....cases where it is possible to statically decide the minimum and maximum number of iterations from the syntax of the loop construct. There are even unterminating loop constructs which we can identify. This is similar to what Patterson is doing in [Pat95] Blieberger is studying this problem in [Bli94] where he defines a new kind of loop, discrete loops, which are more general than for loops, but still analyzable. Of course, syntactic analysis cannot handle all cases, so we have to rely on semantic analysis in the general and more complex cases. Syntactic analysis should be regarded as an ....

J. Blieberger. Discrete loops and worst case performance. Computing Languages, 20(3):193--212, 1994.

....transforming the general loop into a forloop with an additional exit statement, or they directly require a constant time bound within which the loop has to complete (e. g [6] Other researchers attempt to do static and dynamic program path analysis using regular expressions (e.g. 5] In [1] the concept of discrete loops is introduced. Like a for loopa discrete loop uses a loop variable that is incremented decremented on every iteration of the loop until it does no longer fall into a given range. With a common for loop a fixed value (usually one) is added to the loop variable on ....

....1 1 y24S T 1 V B , Definition 4.6 (Complete, Partial and Bounded History) Depending on C vK various forms of MSDL can be distinguished. If C )v (e.g. Example 3 Factorial Numbers) we have a (plain, single staged) discrete loop. This case as been studied extensively in [1], and will not be treated in this paper. If 9 vK 9 (e.g. Example 2 Fibonacci Numbers) we call a MSDL with bounded history. Otherwise C )v . If C S 1 V (e.g. Example 1 Catalan Numbers) than we say uses the complete history. Otherwise (e.g. Example 4) we speak of partial ....

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J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.

....performance of a task plays a crucial role. The most difficult tasks in estimating the timing behavior of a program are to determine the number of iterations of a certain loop and to handle problems originating from the use of recursion. A solution to the first problem has been given in [1], the second one will be treated in this paper. If recursive procedures are to be used in implementing real time applications, several problems occur: 1. It is not clear, whether a recursive procedure completes or not. 2. If it completes, it must be guaranteed that its result is delivered ....

.... code of a recursive procedure is considered to consist of ffl simple segments of linear code, the performance of which is known a priori, ffl if statements, ffl loops with known upper bounds of the number of iterations which can be derived at compile time, e.g. for loops or discrete loops (cf. [1]) and ffl recursive calls to the procedure itself. In terms of a context free grammar this is stated as follows code(f) if f 2 F 0 then nonrecursive(f) else recursive(f) end if recursive(f) seq(f) seq(f) statement(f) fstatement(f)g statement(f) simple(f) j compound(f) j ....

J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.

....time bounded loops and simple for loops. An algorithm for calculating an upper bound of the WCET of Real Time Euclid programs is described in (Stoyenko et al. 1991) Although in the meantime it has been proved that more general loop statements can be used in real time programming languages (cf. (Blieberger, 1994)) and that recursion can be employed without harm in real time systems (cf. Blieberger and Lieger, 1996; Blieberger, 2000) the concept of schedulability analysis, which is also introduced in (Halang and Stoyenko, 1991) is still very important for real time applications. One of the minor ....

....relations defined in Definition 14 is equal to that of recursive functions with the operator (cf. Rogers, 1992) solving such systems of recurrence relations is undecidable; in fact it is equivalent to the halting problem. However, for simple loops, such as for loops or discrete loops (cf. (Blieberger, 1994)) which generalize for loops, the corresponding systems of recurrence relations can be solved. The undecidability of the above problem shows that we cannot get rid of the oracle completely, but certainly today s real time programming languages and their designers place too heavy a burden on the ....

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Blieberger, J.: 1994, `Discrete Loops and Worst Case Performance'. Computer Languages 20(3), 193--212.

....real time and safety related systems and proving such systems correct. 2 Discrete Loops In this section we give an informal introduction to discrete loops, before we perform a theoretical treatment, i.e. an exact definition and some mathematical results. Further results can be found in [Bli94]. 2.1 Introduction to Discrete Loops In contrast to for loops, discrete loops allow for more complex dependency between two successive values of the loop variable. In fact an arbitrary functional dependency between two successive values of the loop variable is admissible, but this dependency ....

....respectively. Theorem 4.1. If for all 1 i e f i (1) 1 and f i (x 1) Gamma f i (x) 1 for all x 2 N , then 1. the corresponding discrete loop completes, 2. the length of (k (max) is equal to l, and 3. the length of (k (min) is equal to u. A proof of Theorem 4. 1 can be found in [Bli94]. If f min (x) min i ff i (x)g and f max (x) max i ff i (x)g can be determined independently of x, Theorem 4.1 enables us to restrict our interest to two single functions in estimating lower and upper bounds of the number of iterations of a discrete loop. 4.2 Some Results on Special ....

J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.

....Another approach is to let the user specify a time bound within the loop has to complete. In any case the user, i.e. the programmer, has to react to such exceptional cases. Our approach is different in that we define a new kind of loops, so called discrete loops, which are described in detail in [Bli94]. In contrast to for loops, discrete loops allow for a more complex dependency between two successive values of the loop variable. Like for loops discrete loops have a loop variable and an integer range associated with them. The major difference to for loops is that the loop variable need not be ....

....node pointer : node pointer.left j node pointer.right with h : height new h = h 1 loop loop body end loop; Fig. 2. Template for Traversing Binary Trees In any case the number of iterations of discrete loops can be determined at compile time if the iteration functions are monotonic (cf. [Bli94]) 3 Real Time Recursions In view of the problems that arise when recursions are to be used in real time applications, most designers of real time programming languages decide to forbid them in their languages, e.g. RT Euclid, PEARL, Real Time Concurrent C, and the MARS approach. Other ....

Johann Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.

....27 09 1999; 12:56; p.1 2 J. Blieberger, T. Fahringer, B. Scholz The focus of this paper is on accurate cache behavior analysis. Note that modelling caches is only one performance aspect that must be considered in order to determine execution times. There are many other performance characteristics (Blieberger, 1994; Blieberger and Lieger, 1996; Blieberger, 1997; Fahringer, 1996; Park, 1993; Healy et al. 1995) to be analyzed which however are beyond the scope of this paper. In this paper we introduce a novel approach for deriving a highly accurate analytical function of the precise number of cache hits 2 ....

Blieberger, J.: 1994, `Discrete Loops and Worst Case Performance'. Computer Languages 20(3), 193--212.

....transforming the general loop into a forloop with an additional exit statement, or they directly require a constant time bound within which the loop has to complete (e. g [6] Other researchers attempt to do static and dynamic program path analysis using regular expressions (e.g. 5] In [1] the concept of discrete loops is introduced. Like a for loopa discrete loop uses a loop variable that is incremented decremented on every iteration of the loop until it does no longer fall into a given range. With a common for loop a fixed value (usually one) is added to the loop variable on ....

....k ] f (k) a k ] f (k) b k ] Definition 4.6 (Complete, Partial and Bounded History) Depending on D(L) various forms of MSDL can be distinguished. If D(L) 1 (e.g. Example 3 Factorial Numbers) we have a (plain, single staged) discrete loop. This case as been studied extensively in [1], and will not be treated in this paper. If 1 D(L) 1 (e.g. Example 2 Fibonacci Numbers) we call L a MSDL with bounded history. Otherwise D(L) 1. If D(f (k) k for all k (e.g. Example 1 Catalan Numbers) than we say L uses the complete history. Otherwise (e.g. Example 4) we speak of ....

[Article contains additional citation context not shown here]

J. Blieberger. Discrete loops and worst case performance. Computer Languages, 20(3):193--212, 1994.

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