Bernard Elspas

SELECT—a formal system for testing and debugging programs by symbolic execution

SELECT is an experimental system for assisting in the formal systematic debugging of programs. It is intended to be a compromise between an automated program proving system and the current ad hoc debugging practice, and is similar to a system being developed by King et al. of IBM. SELECT systematically handles the paths of programs written in a LISP subset that includes arrays. For each execution path SELECT returns simplified conditions on input variables that cause the path to be executed, and simplified symbolic values for program variables at the path output. For conditions which form a system of linear equalities and inequalities SELECT will return input variable values that can serve as sample test data. The user can insert constraint conditions, at any point in the program including the output, in the form of symbolically executable assertions. These conditions can induce the system to select test data in user-specified regions. SELECT can also determine if the path is correct with respect to an output assertion. We present four examples demonstrating the various modes of system operation and their effectiveness in finding bugs. In some examples, SELECT was successful in automatically finding useful test data. In others, user interaction was required in the form of output assertions. SELECT appears to be a useful tool for rapidly revealing program errors, but for the future there is a need to expand its expressive and deductive power.

Graphs with circulant adjacency matrices

Abstract Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied. Examples are given showing that the connection set determined by the first row of such a matrix need not be multiplicatively related to the connection set of an isomorphic graph. Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related. The first criterion is that the graphs have a prime number of vertices. The second criterion is that the adjacency matrices have non-repeated eigenvalues. The final section gives a partial characterization of graphs with n vertices whose automorphism group is the cyclic group Cn.

SELECT---a formal system for testing and debugging programs by symbolic execution

SELECT is an experimental system for assisting in the formal systematic debugging of programs. It is intended to be a compromise between an automated program proving system and the current ad hoc debugging practice, and is similar to a system being developed by King et al. of IBM. SELECT systematically handles the paths of programs written in a LISP subset that includes arrays. For each execution path SELECT returns simplified conditions on input variables that cause the path to be executed, and simplified symbolic values for program variables at the path output. For conditions which form a system of linear equalities and inequalities SELECT will return input variable values that can serve as sample test data. The user can insert constraint conditions, at any point in the program including the output, in the form of symbolically executable assertions. These conditions can induce the system to select test data in user-specified regions. SELECT can also determine if the path is correct with respect to an output assertion. We present four examples demonstrating the various modes of system operation and their effectiveness in finding bugs. In some examples, SELECT was successful in automatically finding useful test data. In others, user interaction was required in the form of output assertions. SELECT appears to be a useful tool for rapidly revealing program errors, but for the future there is a need to expand its expressive and deductive power.

A note on optimum burst-error-correcting codes

A detailed study has been made of a certain class of systematic binary error-correcting codes that will correct the error bursts typical of some digital channels. These codes--generalizations of codes discovered by Abramson and Melas--are cyclic codes designed to correct any single burst of errors per n -digit word provided that the width of the burst (regarded cyclically) does not exceed a certain number of digits, b . Moreover, these codes are optimum in the sense that they employ the minimum number of redundant digits theoretically possible for a cyclic code with given values of n and b . A cyclic code is completely characterized by its generator polynomial g(x) , hence, the properties of the code can be determined by analysis of the corresponding g(x) . Necessary and sufficient conditions on g(x) have been formulated for the corresponding cyclic code to be an optimum burst- b correcting code. These conditions have been formulated into a series of tests that can be carried out (in principle) on any g(x) . All optimum burst- b cyclic codes with n and b have been found in this way and their generators are tabulated in the paper. In all, 98 codes are listed (not counting reciprocals) for b = 3 and b = 4 ; it was shown that no optimum codes exist for b = 5 within the limits stated. Practical codes for b \geq 6 will probably be nonoptimum codes because of the extreme word lengths required for optimum ones.

A Bound on the Run Measure of Switching Functions

The run measure of a switching function has arisen in several contexts as an indication of the complexity, or cost, of a realization of the function. The run measure of a function can be defined in terms of its conventional truth-table representation. The output column of the truth table is an ordered sequence of zeros and ones that are disposed in runs; i.e., groups of like digits, of various lengths. The run measure of the function is simply the number of runs in this output sequence. It is often convenient to consider two functions to be equivalent if one can be obtained from the other by some permutation or complementation of the input variables. In this context, the cost of a function can be taken as the minimum value of the run measure over the equivalence class that contains the function. This paper derives a firm upper bound on the run measure for arbitrary n-variable switching functions when arbitrary permutations and complementations of the input variables are permitted. It is also shown that this bound is attained only in the case of the parity functions and that, hence in this sense, all other functions are less complex.

Single-error-correcting codes for constant-weight data words

A family of single-error-correcting codes is described for the protection of binary data words of fixed length k , each of which has the same number w of 1 s. The code family is shown to be valid for all integers k and w (where 0 . Some other related codes, based upon conventional Hamming codes, Latin squares, and block designs are also developed, and some of these are more efficient for certain values of w and k .