The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The absence of the desirable interpolation properties from many logics that have been used in refinement, motivates a thorough investigation of methods to expand a specification formalism orthogonally, so that the critical uniform interpolants become available. A potential breakthrough is outlined in this paper. 1. A refinement paradigm Let us consider program development by means of stepwise refinements. One postulates some abstract data typelike specification 1 (ADT), suitable for the problem at hand, which has to be implemented on the available system. The end product consists of (the text of) an abstract program manipulating the postulated ADT, together with a suite of (texts of) modules implementin...

Since the early days of programming and automated reasoning, researchers have developed methods for systematically constructing programs from their specifications. Especially the last decade has seen a flurry of activities including the advent of specialized conferences, such as LOPSTR, covering the synthesis of programs in computational logic. In this paper we analyze and compare three state-of-the-art methods for synthesizing recursive programs in computational logic. The three approaches are constructive/deductive synthesis, schema-guided synthesis, and inductive synthesis. Our comparison is carried out in a systematic way where, for each approach, we describe the key ideas and synthesize a common running example. In doing so, we explore the synergies between the approaches, which we believe are necessary in order to achieve progress over the next decade in this field.

The potential benefits of a uniform version of interpolation are hindered by its absence from many expressive logics. This motivates a thorough investigation of appropriate expansions of known entailments so that an easy-to-derive, uniform presentation of the interpolants is supported. This paper presents the skeleton of a general construction and indicates that a potentially large class of entailments can be extended so that a uniform presentation of the interpolants is available. 1 Introduction There is a well established relation between interpolation [8] and modularity properties of refinements [23, 24, 29, 5, 32, 40, 39, 12, 11] and databases [25]. On the other hand, many logics that have been used in refinement or databases lack the desirable interpolation properties. To compensate for this inadequacy, several groups of researchers have proposed techniques to restrict these logics to fragments that have the desirable modularity properties. Some of these enterprises have focuse...

. Interpolation and Schematic Reasoning are shown to underlie critical and somewhat complementary aspects of designing and (syntactically) manipulating specification modules. In addition, the presence of a Uniform presentation of interpolants facilitates the specification of modules. Also, the ability to encapsulate and manipulate Uniform Schemata may assist us in reasoning with (abstractions of) hidden data. Unfortunately, most formalisms that have been used in fundamental approaches to software engineering lack uniform interpolation and do not directly support schematic reasoning. This paper reveals the critical role of uniform interpolants and uniform schemata from the perspective of modularity, and quotes a general construction indicating that a potentially large class of calculi can be extended conservatively so that a uniform presentation of the critical interpolants becomes available and the manipulation of uniform schemata is supported. 1 Introduction There is a ...