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 an interpreter where the search tree is pruned to guarantee termination for certain kinds of predicate definition; then we introduce finally clauses, to provide a default value for all those cases where unification with a coinductive hypothesis is not correct. Finally, we propose a finer grain semantics where the user can specify only a subset of the arguments that have to be considered when coinductive hypotheses are unified. The semantics defined by these vanilla meta-interpreters are an interesting starting point for a more mature design and implementation of coinductive Prolog.

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.

It is well known that big-step operational semantics are not suitable for proving soundness of type systems, because of their inability to distinguish stuck from non-terminating computations. We show how this problem can be solved by interpreting coinductively the rules for the standard big-step operational semantics of a Java-like language, thus making the claim of soundness more intuitive: whenever a program is well-typed, its coinductive operational semantics returns a value. Indeed, coinduction allows non-terminating computations to return values; this is proved by showing that the set of proof trees defining the semantic judgment forms a complete metric space when equipped with a proper distance function. In this way, we are able to prove soundness of a nominal type system w.r.t. the coinductive semantics. Since the coinductive semantics is sound w.r.t. the usual small-step operational semantics, the standard claim of soundness can be easily deduced.

In previous work we have shown that more precise type analysis can be achieved by exploiting union types and static single assignment (SSA) intermediate representation (IR) of code. In this paper we exploit static single information (SSI), an extension of SSA proposed in literature and adopted by some compilers, to allow assignments of more precise types to variables in conditional branches. In particular, SSI can be exploited rather easily and effectively to infer more precise types in dynamic object-oriented languages, where explicit runtime typechecking is frequently used. We show how the use of SSI form can be smoothly integrated with abstract compilation, our approach to static type analysis. In particular, we define abstract compilation based on union and nominal types for a simple dynamic object-oriented language in SSI form with a runtime typechecking operator, to show how precise type inference can be.

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.

This paper further investigates the potential and practical applicability of abstract compilation in two different directions. First, we formally define an abstract compilation scheme for precise prediction of uncaught exceptions for a simple Java-like language; besides the usual user declared checked exceptions, the analysis covers the runtime ClassCastException. Second, we present a general implementation schema for abstract compilation based on coinductive CLP with variance annotation of user-defined predicates, and propose an implementation based on a Prolog prototype meta-interpreter, parametric in the solver for the subtyping constraints.

In recent work we have proposed a novel approach to define idealized type systems for object-oriented languages, based on abstract compilation of programs into Horn formulas which are interpreted w.r.t. the coinductive (that is, the greatest) Herbrand model. In this paper we investigate how this approach can be applied also in the presence of imperative features. This is made possible by con- sidering a natural translation of Static Single Assignment intermediate form programs into Horn formulas, where phi functions correspond to union types.

We define a coinductive semantics for a simple Java-like language by simply interpreting coinductively the rules of a standard big-step operational semantics. We prove that such a semantics is sound w.r.t. the usual small-step operational semantics, and then prove soundness of a conventional nominal type system w.r.t. the coinductive semantics. From these two results, soundness of the type system w.r.t. the small-step semantics can be easily deduced. This new proposed approach not only opens up new possibilities for proving type soundness, but also provides useful insights on the connection between coinductive big-step operational semantics and type systems.

Coinductive abstract compilation is a novel technique, which has been recently introduced for defining precise type systems for object- oriented languages. In this approach, type inference consists in translating the program to be analyzed into a Horn formula f, and in resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f. Type systems defined in this way are idealized, since types and, con- sequently, goal derivations, are not finitely representable. Hence, sound implementable approximations have to rely on the notions of regular types and derivations, and of subtyping and subsumption between types and atoms, respectively. In this paper we address the problem of defining a sound and complete axiomatization of a subtyping relation between coinductive object and union types, defined as set inclusion between type interpretations. Besides being an important theoretical result, completeness is useful for reasoning about possible implementations of the subtyping relation, when restricted to regular types.

This paper further investigates the potential and practical applicability of abstract compilation in two different directions. First, we formally define an abstract compilation scheme for precise prediction of uncaught exceptions for a simple Java-like language; besides the usual user declared checked exceptions, the analysis covers the runtime ClassCastException. Second, we present a general implementation schema for abstract compilation based on coinductive CLP with variance annotation of user-defined predicates, and propose an implementation based on a Prolog prototype meta-interpreter, parametric in the solver for the subtyping constraints.

Coinductive abstract compilation is a novel technique, which has been recently introduced, for defining precise type systems for object-oriented languages. In this approach, type inference consists in translating the program to be analyzed into a Horn formula f, and in resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f. Type systems defined in this way are idealized, since types and, consequently, goal derivations, are not finitely representable. Hence, sound implementable approximations have to rely on the notions of regular types and derivations, and of subtyping and subsumption between types and atoms, respectively. In this paper we address the problem of defining a complete subtyping relation <= between types built on object and union type constructors: we interpret types as sets of values, and investigate on a definition of subtyping such that t_1 <= t_2 is derivable whenever the interpretation of t_1 is contained in the interpretation of t_2. Besides being an important theoretical result, completeness is useful for reasoning about possible implementations of the subtyping relation, when restricted to regular types.

In recent work we have shown how it is possible to define very precise type systems for object-oriented languages by abstractly compiling a program into a Horn formula f. Then type inference amounts to resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of f. Type systems defined in this way are idealized, since in the most interesting instantiations both the terms of the coinductive Herbrand universe and goal derivations cannot be finitely represented. However, sound and quite expressive approximations can be implemented by considering only regular terms and derivations. In doing so, it is essential to introduce a proper subtyping relation formalizing the notion of approximation between types. In this paper we study a subtyping relation on coinductive terms built on union and object type constructors. We define an interpretation of types as set of values induced by a quite intuitive relation of membership of values to types, and prove that the definition of subtyping is sound w.r.t. subset inclusion between type interpretations. The proof of soundness has allowed us to simplify the notion of contractive derivation and to discover that the previously given definition of subtyping did not cover all possible representations of the empty type.

We propose a novel general approach for defining expressive type systems for object-oriented languages, based on abstract compilation of programs into coinductive constraint logic programs defined on a specific constraint domain X called type domain. In this way, type checking and type inference amount to resolving a certain goal w.r.t. the coinductive (that is, the greatest) Herbrand model of a logic program (that is, a Horn formula) with constraints over a fixed type domain X. In particular, we show an interesting instantiation where the constraint predicates of X are syntactic equality and subtyping over coinductive object and union types. The corresponding type system is so expressive to allow verification of simple properties like data structure invariants. Finally, we show a prototype implementation, written in Prolog, of the inference engine for coinductive CLP(X), which is parametric in the solver for the type domain X.

We propose a novel approach to constraint-based type inference based on coinductive logic programming. That is, constraint generation corresponds to translation into a conjunction of Horn clauses P, and constraint satisfaction is defined in terms of the maximal coinductive Herbrand model of P. We illustrate the approach by formally defining this translation for a small object-oriented language similar to Featherweight Java, where type annotations in field and method declarations can be omitted. In this way, we obtain a very precise type inference and provide new insights into the challenging problem of type inference for object-oriented programs. Since the approach is deliberately declarative, we define in fact a formal specification for a general class of algorithms, which can be a useful road maps to researchers. Moreover, despite we consider here a particular language, the methodology could be used in general for providing abstract specifications of type inference for different kinds of programming languages.

We propose a novel approach based on coinductive logic to specify type systems of programming languages. The approach consists in encoding programs in Horn formulas which are interpreted w.r.t. their coinductive Herbrand model. We illustrate the approach by first specifying a standard type system for a small object-oriented language similar to Featherweight Java. Then we define an idealized type system for a variant of the language where type annotations can be omitted. The type system involves infinite terms and proof trees not representable in a finite way, thus providing a theoretical limit to type inference of object-oriented programs, since only sound approximations of the system can be implemented. Approximation is naturally captured by the notions of subtyping and subsumption; indeed, rather than increasing the expressive power of the system, as it usually happens, here subtyping is needed for approximating infinite non regular types and proof trees with regular ones.

Although coinductive logic programming (Co-LP) has proved to have several applications, including verification of infinitary properties, model checking and bisimilarity proofs, type inference via Co-LP has not been properly investigated yet. In this paper we show a novel approach to solve the problem of type inference in the context of Java-like languages, that is, object-oriented languages based on nominal subtyping. The proposed approach follows a generic scheme: first, the program P to be analyzed is translated into an approximating logic program P' and a goal G; then, the solution of the type inference problem corresponds to find an instantiation of the goal G which belongs to the greatest model of P', that is, the coinductive model of P'. Operationally, this corresponds to find a co-SLD derivation of G in P', according to the operational semantics of Co-LP recently defined by Simon et al. [ICLP06,ICALP07]. A complete formalization of an instantiation of this scheme is considered for a simple object-oriented language and a corresponding type soundness theorem is stated. A prototype implementation based on a meta-interpreter of co-SLD has been implemented. Finally, we address scalability issues of the approach, by sketching an instantiation able to deal with a much more expressive object-oriented language.