Maura Cerioli

May I borrow your logic? (Transporting logical structures along maps)

Abstract It can be very advantageous to borrow key components of a logic for use in another logic. The advantages are both conceptual and practical; due to the existence of software systems supporting mechanized reasoning in a given logic, it may be possible to reuse a system developed for one logic — for example, a theorem-prover — to obtain a new system for another. Translations between logics by appropriate mappings provide a first natural way of reusing tools of one logic in another. This paper generalizes this idea to the case where entire components — for example, the proof theory — of one of the logics involved may be completely missing, so that the appropriate mapping could not even be defined. The idea then is to borrow the missing components (as well as their associated tools if they exist) from a logic that has them in order to create the full-fledged logic and tools that we desire. The relevant structure is transported using maps that only involve a limited aspect of the two logics in question — for example, their model theory. The constructions accomplishing this kind of borrowing of logical structure are very general and simple. They only depend upon a few abstract properties that hold under very general conditions given a pair of categories linked by adjoint functors.

Towards a Rigorous Semantics of UML Supporting Its Multiview Approach

We discuss the nature of the semantics of the UML. Contrary to the case of most languages, this task is far from trivial. Indeed, not only the UML notation is complex and its informal description is incomplete and ambiguous, but we also have the UML multiview aspect to take into account. We propose a general schema of the semantics of the UML, where the different kinds of diagrams within a UML model are given individual semantics and then such semantics are composed to get the semantics on the overall model. Moreover, we fill part of such a schema, by using the algebraic language CASL as a metalanguage to describe the semantics of class diagrams, state machines and the complete UML formal systems.

May I Borrow Your Logic

It can be very advantageous to borrow key components of a logic for use in another logic. The advantages may be not only conceptual; due to the existence of software systems supporting mechanized reasoning in a given logic, it may be possible to reuse a system developed for one logic—for example, a theorem-prover—to obtain a new system for another. Translations between logics by appropriate mappings provide a first way of reusing tools of one logic in another. This paper generalizes this idea to the case where entire components—for example, the proof theory—of one of the logics involved may be completely missing, so that the appropriate mapping could not even be defined. The idea then is to borrow the missing components (as well as their associated tools if they exist) from a logic that has them in order to create the full-fledged logic and tools that we desire. The relevant structure is transported using maps that only involve a limited aspect of the two logics in question—for example, their model theory. The constructions accomplishing this kind of borrowing of logical structure are very general and simple. They only depend upon a few abstract properties that hold under very general conditions given a pair of categories linked by adjoint functors.

Relationships between Logical Frameworks

Adopting the concept of institution to represent logical frames, we have introduced in a previous paper the concept of simulation of an institution by another. Here we first show how simulations can be used to investigate the relationships between frames, distinguishing three levels, corresponding to different kinds of simulations: “set-theoretic”, where the individual models of different frames are related disregarding their categorical and logical interconnection, “categorical”, where the relation is between the categories of models, and “logical”, where the relation is between specifications. Then we propose a concept of translation of inference systems along simulations such that soundness and completeness are preserved.

Permissive Subsorted Partial Logic in CASL

This paper presents a permissive subsorted partial logic used in the CoFI Algebraic Specification Language. In contrast to other ordersorted logics, subsorting is not modeled by set inclusions, but by injective embeddings allowing for more general models in which subtypes can have different data type representations. Furthermore, there are no restrictions like monotonicity, regularity or local filtration on signatures at all. Instead, the use of overloaded functions and predicates in formulae is required to be sufficiently disambiguated, such that all parses have the same semantics. An overload resolution algorithm is sketched.

On the Existence of Initial Models for Partial (Higher-Order) Conditional Specifications

Partial higher-order conditional specifications may not admit initial models, because of the requirement of extensionality, even when the axioms are positive conditional. The main aim of the paper is to investigate in full this phenomenon.

From Total Equational to Partial First-Order Logic

The focus of this chapter is the incremental presentation of partial first-order logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are fully described. Alternatives to partiality, like (variants of) error algebras and order-sortedness, are also discussed, emphasizing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragments are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Finally some more powerful algebraic frameworks are sketched.

Free objects and equational deduction for partial conditional specifications

Abstract Partial conditional specifications consist of conditional axioms, with equalities in the (possibly infinite set of) premises and in the consequence, which are interpreted in partial algebras. Equalities may be existential (= e ) or strong ( = ); t = e t ′ holds in an algebra iff both t and t ′ are defined and equal, while t = t ′ holds in an algebra iff either t = e t ′ holds or both t and t ′ are undefined. Contrary to the well explored case of positive conditional axioms (only existential equalities in the premises), general partial conditional specifications do not always admit free models and the related theory is much more subtle. In this paper we fully investigate and solve the problem of existence of free and initial models, giving necessary and sufficient conditions, first from a model-theoretical and then from a logical deduction viewpoint. In particular we present a deduction system which is complete w.r.t. strong equalities between open terms. Since positive conditional partial specifications and conditional total specifications are special cases of the paradigm investigated here, the presented theory generalizes the related results about free models and the Birkhoff-like deduction theory. The system we exhibit handles also the case of infinitary conjunctions as premises of the axioms; it reduces to a classical one for the positive conditional case by just dropping one rule, and finally it solves the empty-carrier problem without using explicit quantification. The theory presented here also gives the basis for solving, via the usual first-order reduction, the problem of the existence of free and initial models for partial higher-order specifications of term-generated extensional models.

Multiparadigm Specification Languages: A First Attempt at Foundations

This paper is a first attempt at a formal foundation of specification languages allowing their basic modules to be defined in several formalisms. More precisely a rigorous notion of a compositional tool for importing/exporting specifications between two instances of one specification metalanguage on different basic algebraic frameworks is proposed.

Partial higher-order specifications

In this paper we study the classes of extensional models of higher-order partial conditional specifications. After investigating the closure properties of these classes, we show that an inference system for partial higher-order conditional specifications, which is equationally complete w.r.t. the class of all extensional models, can be obtained from any equationally complete inference system for partial conditional specifications. Then, applying some previous results, we propose a deduction system, equationally complete for the class of extensional models of a partial conditional specification.