We present two extensions of the LF constructive type theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for plugging-in and gluing together, different metalanguages and proof development environments. Oracles can be invoked either to check that a constraint holds or to provide a witness. The systems are presented in the canonical style developed by the ‘CMU School.’ The first system, CLLF𝒫, is the canonical version of the system LLF𝒫, presented earlier by the authors. The second system, CLLF𝒫?, features the possibility of invoking the oracle to obtain also a witness satisfying a given constraint. In order to illustrate the advantages of our new frameworks, we show how to encode logical systems featuring rules that deeply constrain the shape of proofs. The locks mechanisms of CLLF𝒫 and CLLF𝒫? permit to factor out naturally the complexities arising from enforcing these ‘side conditions,’ which severely obscure standard LF encodings. We discuss Girard's Elementary Affine Logic, Fitch–Prawitz set theory, call-by-value λ-calculi and functions, both total and even partial.

The Theory of Contexts is a type-theoretic axiomatization aiming to give a metalogical account of the fundamental notions of variable and context as they appear in Higher Order Abstract Syntax. In this paper, we prove that this theory is consistent by building a model based on functor categories. By means of a suitable notion of forcing, we prove that this model validates Classical Higher Order Logic, the Theory of Contexts, and also (parametrised) structural induction and recursion principles over contexts. Our approach, which we present in full detail, should also be useful for reasoning on other models based on functor categories. Moreover, the construction could also be adopted, and possibly generalized, for validating other theories of names and binders.

Taking the view that infinite plays are draws, we study Conwaynon-terminating games and non-losing strategies. These admit asharp coalgebraic presentation, where non-terminating games are seen as afinal coalgebra and game contructors, such as disjunctivesum, as final morphisms. We have shown, in a previous paper,that Conway’s theory of terminating games can be rephrased naturally in terms of game(pre)congruences. Namely, various conceptually independent notions ofequivalence can be defined and shown to coincide on Conway’sterminating games. These are the equivalence induced by the ordering on surrealnumbers, the contextual equivalence determined by observingwhat player has a winning strategy, Joyal’s categoricalequivalence, and, for impartial games, the denotationalequivalence induced by Grundy semantics. In this paper, wediscuss generalizations of such equivalences to non-terminating games andnon-losing strategies. The scenario is even more rich and intriguing inthis case. In particular, we investigate efficient characterizations of the contextualequivalence, and we introduce a category of fair strategies and acategory of fair pairs of strategies, both generalizing Joyal’s categoryof Conway games and winning strategies. Interestingly, the category of fair pairs capturesthe equivalence defined by Berlekamp, Conway, Guy on loopy games.

We extend the coalgebraic account of specification and refinement of objects and classes in object-oriented programming given by Reichel and Jacobs to (generalised) binary methods. These are methods that take more than one parameter of a class type. Class types include products, sums and powerset type constructors. To allow for class constructors, we model classes as bialgebras. We study and compare two solutions for modelling generalised binary methods, which use purely covariant functors.

In the first solution, which applies when we already have a class implementation, we reduce the behaviour of a generalised binary method to that of a bunch of unary methods. These are obtained by freezing the types of the extra class parameters to constant types. If all parameter types are finitary, the bisimilarity equivalence induced on objects by this model yields the greatest congruence with respect to method application.

In the second solution, we treat binary methods as graphs instead of functions, thus turning contravariant occurrences in the functor into covariant ones.

We show the existence of final coalgebras in both cases.

We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on λ-terms and, more generally, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss, in particular, what we call the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor ([Pscr ](- × -))[oplus ]([Pscr ](- × -)). In this paper we present two general methods for showing the soundness of this principle. The first applies to approximable applicative structures – many CPO λ-models in the literature and the corresponding quotient models of indexed terms turn out to be approximable applicative structures. The second method is based on Howe's congruence candidates, which applies to many observational equivalences. Structures satisfying cartesian coinduction are precisely those applicative structures that can be modelled using the standard set-theoretic application in non-wellfounded theories of sets, where the Foundation Axiom is replaced by the Axiom X1 of Forti and Honsell.

In this paper we give a big-step Structured Operational Semantics (SOS), in the style of Plotkin, Kahn and Milner, of a significant fragment of the functional programming language Scheme, including quote, eval, quasiquote and unquote. The SOS formalism allows us to discuss incrementally the various features of the language and to keep a low mathematical overhead, thus producing a rigorous account of the semantics of a ‘real’ programming language, which nonetheless has a pedagogical value. More specifically, we formalize four strictly increasing fragments of Scheme, using a number of formal systems which express the evaluation of expressions, the display of output results, and the handling of errors.

We propose a general theory of partial n-place operations based solely on the primitive notion of the application of a (possibly partial) operation to n objects. This theory is strongly selfdescriptive in that the fundamental manipulations of operations, that is, application, composition, abstraction, union, intersection and so on, are themselves internal operations. We give several applications of this theory, including implementations of partial n-ary λ-calculus, and other operation description languages. We investigate the issue of extensionality and give weakly extensional models of the theory.

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.

The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:

CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that

(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)

The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.

In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.