Corecursion

Corecursion is the ability of defining a function that produces some infinite data in terms of the function and the data itself, as supported by lazy evaluation. However, in languages such as Haskell strict operations fail to terminate even on infinite regular data, that is, cyclic data. Regular corecursion is naturally supported by coinductive Prolog, an extension where predicates can be interpreted either inductively or coinductively, that has proved to be useful for formal verification, static analysis and symbolic evaluation of programs. In this paper we use the meta-programming facilities offered by Prolog to propose extensions to coinductive Prolog aiming to make regular corecursion more expressive and easier to program with. First, we propose a new interpreter to solve the problem of non-terminating failure as experienced with the standard semantics of coinduction (as supported, for instance, in SWI-Prolog). Another problem with the standard semantics is that predicates expressed in terms of existential quantification over a regular term cannot directly defined by coinduction; to this aim, we introduce finally clauses, to allow more flexibility in coinductive definitions. Then we investigate the possibility of annotating arguments of coinductive predicates, to restrict coinductive definitions to a subset of the arguments; this allows more efficient definitions, and further enhance the expressive power of coinductive Prolog. We investigate the effectiveness of such features by showing different example programs manipulating several kinds of cyclic values, ranging from automata and context free grammars to graphs and repeating decimals; the examples show how computations on cyclic values can be expressed with concise and relatively simple programs. The semantics defined by these vanilla meta-interpreters are an interesting starting point for a more mature design and implementation of coinductive Prolog.

In previous work we have presented coFJ, an extension to Featherweight Java that promotes coinductive programming, a sub-paradigm expressly devised to ease high-level programming and reasoning with cyclic data structures. The coFJ language supports cyclic objects and regularly corecursive methods, that is, methods whose invocation terminates not only when the corresponding call trace is finite (as happens with ordinary recursion), but also when such a trace is infinite but cyclic, that is, can be specified by a regular term, or, equivalently, by a finite set of recursive syntactic equations. In coFJ it is not easy to ensure that the invocation of a corecursive method will return a well-defined value, since the recursive equations corresponding to the regular trace of the recursive calls may not admit a (unique) solution; in such cases we say that the value returned by the method call is undetermined. In this paper we propose two new contributions. First, we design a simpler construct for defining corecursive methods and, correspondingly, provide a more intuitive operational semantics. For this coFJ variant, we are able to define a type system that allows the user to specify that certain corecursive methods cannot return an undetermined value; in this way, it is possible to prevent unsafe use of such a value. The operational semantics and the type system of coFJ are fully formalized, and the soundness of the type system is proved.

Co-recursion is the ability of defining a function that produces some infinite data in terms of the function and the data itself, and is typically supported by languages with lazy evaluation. However, in languages as Haskell strict operations fail to terminate even on infinite regular data. Regular co-recursion is naturally supported by co-inductive Prolog, an extension where predicates can be interpreted either inductively or co-inductively, that has proved to be useful for formal verification, static analysis and symbolic evaluation of programs. In this paper we propose two main alternative vanilla meta-interpreters to support regular co-recursion in Prolog as an interesting programming style in its own right, able to elegantly solve problems that would require more complex code if conventional recursion were used. In particular, the second meta-interpreters avoids non termination in several cases, by restricting the set of possible answers. The semantics defined by these vanilla meta-interpreters are an interesting starting point to study new semantics able to support regular co-recursion for non logical languages.

Corecursive FeatherWeight Java (coFJ) is a recently proposed extension of the calculus FeatherWeight Java (FJ), supporting cyclic objects and regular recursion, and explicitly designed to promote a novel programming paradigm inspired by coinductive Logic Programming (coLP), based on coinductive, rather than inductive, interpretation of recursive function definitions. We present a slightly modified version of coFJ where the application of a coinductive hypothesis can trigger the evaluation of a specific expression at declaration, rather than at use site. Following an approach inspired by abstract compilation, we then show how coFJ can be directly translated into coLP, when coinductive SLD is extended with a similar feature for explicitly solving a goal when a coinductive hypothesis is applied. Such a translation is quite compact and, besides showing the direct relation between coFJ and coinductive Prolog, provides a first prototypical but simple and effective implementation of coFJ.

Despite cyclic data structures occur often in many application domains, object-oriented programming languages provide poor abstraction mechanisms for dealing with cyclic objects. Such a deficiency is reflected also in the research on theoretical foundation of object-oriented languages; for instance, Featherweigh Java (FJ), which is one of the most widespread object-oriented calculi, does not allow creation and manipulation of cyclic objects. We propose an extension to Featherweight Java, called coFJ, where it is possible to define cyclic objects, {abstractly corresponding to regular terms}, and where an abstraction mechanism, called regular corecursion, is provided for supporting implementation of coinductive operations on cyclic objects. We formally define the operational semantics of coFJ, and provide a handful of examples showing the expressive power of regular corecursion; such a mechanism promotes a novel programming style particularly well-suited for implementing cyclic data structures, and for supporting coinductive reasoning.