THE THEORETICAL ASPECTS OF THE OPTIMAL FltiEDPOlNTm bY

In this paper we define a new type of fixedpoint of recursive definitions and investigate some of its properties. This opt imal fixedpoint (which always uniquely exists) contains, in some sense, the maximal amount of “interesting” information which can be extracted from the recursive definition, and it may be strictly more defined than the program’s least fixedpoint. This fixedpoint can be the basis for assigning a new semantics to recursive programs. Reproduced in he U.S.A. Available from the National Tech&al Information Service, SpringtTeld, Virginia 22151. * INTRODUCTION Recursive definitions are usually considered from two different points of view, namely: (i) As an algorithm for computing a function by repeated substitutions of the function definition for its name. (ii) As a functional equation, expressing the required relations between values of the defined function for various arguments. A function that satisfies these relations (a solution of the equation) is called a fixedpoint. The functional equation represented by a recursive definition may have many fixedpoints, all of which satisfy the relations dictated by the definition. There is no a priori preferred solution and therefore, if the definition has more than one fixedpoint, one of them must be chosen. A number of works describing a least (defined) fixedpoint approach towards the semantics of recursive definitions have been published recently (e.g., Scott [8]). Researchers in the field have chosen the least fixedpoint as the "best solution" for three reasons: (i) It uniquely exists for a wide class of practically applicable recursive definitions. (ii) The classical stack implementation technique computes this fixedpoint for any recursive definition. (iii) There is a powerful method (computational induction) for proving properties of this fixedpoint. However, as a mathematical model for extracting information from an implicit functional equation, the selection of the least defined solution seems a poor _. choice; for many recursive definitions, the least fixedpoint does not reveal all the useful information embedded in the definition. In general, the more defined the solution, the more valuable it is. On the other hand, this argument 1 should be applied with caution, as there are inherently underdefined recursive definitions. Consider the extreme example F(x) <= F(x), for which any partial function is a solution. A randomly chosen total function is by no means superior to the totally undefined least fixedpoint in this case. The optimal fixedpoint, defined in this paper, tries to remedy this situation. It is intended to supply the maximally defined solution relevant to the given recursive definition. Consider, for example, the following recursive definition for solving the discrete form of the Laplace equation, where F(x,y) maps pairs of integers in [-lOO,lOO]x[-lOO,lOO] into reals: F(x,y) <= if x<-100 V x>lOO V y<-100 V y>lOO then x2+y2 else $[F(x-l,y)+F(xtl,y)+F(x,y-l)+F(x,y+l)]. This concise organization of knowledge is defined enough to have a unique total fixedpoint (which is our optimal fixedpoint), but its least fixedpoint is totally undefined inside the square [-lOO,lOO]x[-lOO,lOO]. While the notion of the optimal fixedpoint is theoretically well-defined, its computation aspects contain many pitfalls, since the optimal fixedpoints of certain recursive definitions are non-computable partial functions. We do not pursue in this paper the practical aspects of the optimal fixedpoint approach; in Manna and Shamir[4,5], and in more detail in Shamir[8], we suggest several techniques directed toward the computation of the optimal fixedpoint. In Part I of this paper, a few structural properties of the set of all fixedpoints of recursive definitions are proven. The otpimal fixedpoint is then introduced 'in Part 1I)as the formalization of our intuitive notion of the "best solution" 3f recursive definitions. The existence of a unique optimal fixedpoint for any