Safety-Critical Systems: Proposal to Access a Partially Defined Analog Structure

Abstract:
In this paper we discuss phenomena known as partially defined structures in academia, and very complex hybrid systems in real-world automotive and safety critical systems. For a given input, x = (xn-1,xn-2, ...,x0), and output, y = (ym-1,ym-2, ...,y0), of Boolean variables a totally defined Boolean function, y = λ (x), can be represented by several Boolean algebraic expressions hi. A Boolean expression h0 can be transformed into another expression h1 such that h0 and h1 represent the same function. This is the case if and only if for each input assignment the Boolean value of h0 is equal to h1, termed h0 =0 h1. The Boolean output variable y, which is determined by a function λ, is '1', if and only if λ returns '1' by the specific assignment of input variables. Also y = h0 is determined by the specific Boolean expression h0, meaning that all the input assignments which determine the value of h0 to '0' or '1', in consequence determine the value of y to be '0' or '1', respectively. Now, assume that a combinatorial or continuous function li is given partially, such that for some defined input assignments the return value is unknown. It has to clarify what is meant by λ i = hj with λ i being partially defined by reality (physics), while the value of any Boolean expression hj is totally defined for each input assignment by combinatorial realization (structure), and partially defined by timed feed-backed structure, respectively. Such a partially defined λ i may (even partially) specify the behavior of the structure being feed-backed and underlying time regimes. In consequence, they exhibit partially specified states. We show that even a very simple feed-backed structure can exhibit partial behavior. Thus the question raises, whether a Boolean expression hj is capable of representing a partial structure, and what kind of additional information is needed. We make a suggestion on how to overcome the dilemma mentioned above, and provide some examples on how to analyze and specify partially defined hybrid systems, e. g. a safety-critical system.