Răzvan Diaconescu

Institute of Mathematics of the Romanian Academy | IMAR

3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${3/2}$$\end{document}-Institutions have been introduced as an extension of institution theory that accommodates impli...

The extension of the (ordinary) institution theory of Goguen and Burstall, known as the theory of stratified institutions, is a general axiomatic approach to model theories where the satisfaction is parameterized by states of models. Stratified institutions cover a uniformly wide range of applications from various Kripke semantics to various automa...

We develop an initial study of interpolation for graded consequence relations, which are many-valued consequence relations that arise in connection to many-valued / fuzzy logics. While in general consequence relations represent a foundational structure that supports the mathematical study of reasoning, their graded / many-valued upgrade provides ma...

Institution theory represents the fully axiomatic approach to model theory in which all components of logical systems are treated fully abstractly by reliance on category theory. Here, we survey some developments over the last decade or so concerning the institution theoretic approach to non-classical aspects of model theory. Our focus will be on m...

The theory of L-institutions represents an axiomatic category-based approach to many-valued truth model theory. Technically, L-institutions are many-valued truth extensions of institution theory of Goguen and Burstall. In this paper, in the general framework of L-institutions we study preservation properties of the satisfaction of sentences by filt...

The theory of stratified institutions is a general axiomatic approach to model theories where the satisfaction is parameterized by states of the models. In this paper we further develop this theory by introducing a new technique for representing stratified institutions, which is based on projecting to such simpler structures. On the one hand this c...

We develop a general model theoretic semantics to rewriting beyond the usual confluence and termination assumptions. This is based on preordered algebra which is a model theory that extends many sorted algebra. In this framework we characterise rewriting in arbitrary algebras rather than term algebras (called algebraic rewriting) as a persistent ad...

The theory of stratified institutions is a general axiomatic approach to model theories where the satisfaction is parameterised by states of the models. In this paper we further develop this theory by introducing a new technique for representing stratified institutions which is based on projecting to such simpler structures. On the one hand this ca...

We develop an extension of institution theory that accommodates implicitly the partiality of the signature morphisms and its syntactic and semantic effects. This is driven primarily by applications to conceptual blending, but other application domains are possible (such as software evolution). The particularity of this extension is a reliance on or...

This is a short survey on the development of the formal specification and verification language H with emphasis on the scientific part. H is a modern highly expressive language solidly based upon advanced mathematical theories such as the internalisation of Kripke semantics within institution theory.

This is a short survey on the development of the formal specification and verification language H with emphasis on the scientific part. H is a modern highly expressive language solidly based upon advanced mathematical theories such as the internalisation of Kripke semantics within institution theory.

\frac{3}{2}$-institutions have been introduced as an extension of institution theory that accommodates implicitly partiality of the signature morphisms together with its syntactic and semantic effects. In this paper we show that ordinary institutions that are equipped with an inclusion system for their categories of signatures generate naturally $\...

We develop an extension of institution theory that accommodates implicitly the partiality of the signature morphisms and its syntactic and semantic effects. This is driven primarily by applications to conceptual blending, but other application domains are possible (such as software evolution). The particularity of this extension is a reliance on or...

We provide a precise mathematical definition for first-order views used in parameterised specifications and we also provide a dual semantics for them that captures both their syntactic and model theoretic effects. We show that this semantics is functorial. Parameter instantiation is defined here as a (special kind of) pushout in the category of vie...

A ‘hybridization’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridized institutions’ we mean the result of this process when logics ar...

We propose stratified institutions (a decade old generalized version of the theory of institutions of Goguen and Burstall) as a fully abstract model theoretic approach to modal logic. This allows for a uniform treatment of model theoretic aspects across the great multiplicity of contemporary modal logic systems. Moreover, Kripke semantics (in all i...

Institution theory is a very general mathematical study of formal logical systems—with emphasis on semantics—that is not committed to any particular concrete logical system. This is based upon a mathematical definition for the informal notion of logical system, called institution, which includes both syntax and semantics as well as the relationship...

We provide a set of sufficient conditions for the existence of translations of structured specifications across specification formalisms. The most basic condition is the existence of a translation between the logical systems underlying the specification formalisms, which corresponds to the unstructured situation. Our approach is based upon institut...

We give clear algebraic sense to logical opposition by providing negation-free lattice theoretic lattice-theoretic definitions for the concepts around the notori-ous square of opposition. These include contradiction, contrariety, subcontrariety, the square and the hexagon of opposition, etc. This constitutes a platform for an analysis of the mathem...

This book constitutes the thoroughly refereed post-conference proceedings of the 22nd International Workshop on Algebraic Development Techniques, WADT 2014, held in September 2014 in Sinaia, Romania. The 8 revised papers presented were carefully reviewed and selected from 13 presentations and focus together with one invited paper on foundations of...

In this essay we analyse and elucidate the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model the- oretic developments carried out within institution theory. The power of the proposed general method is illustrated with the examples of (Craig) i...

Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic.

We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multi-signature...

We develop foundations for structuring behavioural specifications based on the logic tradition of hidden algebra. This includes an analysis of a number of important technical compositional properties for behavioural signatures, such as pushouts, inclusions and unions, as well as an investigation of algebraic rules for behavioural module composition...

We survey two important distinctive features of CafeOBJ, namely behavioural specification based upon coherent hidden algebra and heterogeneous specification based upon Grothendieck institutions. Both of them represent seminal contributions to formal specification culture that go much beyond the realm of CafeOBJ. Our presentation includes rather det...

We define and develop the concept of quasi-variety for models of hybrid logics and we apply this for determining initial semantics for classes of hybrid logics theories. The hybrid logic is considered here in a very general sense, internal to abstract institutions (in the sense of the so-called institution theory of Goguen and Burstall). This means...

We develop many-valued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important many-valued logic model theories, such as traditional first-order many-valued logic and fuzzy multi-algebras, may be conservatively embedded into our abstract framework. Our development is technically based upon the so-calle...

We present a generic method for establishing interpolation properties by 'borrowing' across logical sys-tems. The framework used is that of the so-caled 'institution theory' which is a categorical abstract model theory providing a formal definition for the informal concept of 'logical system' and a mathematical con-cept of 'homomorphism' between lo...

In this paper we develop an axiomatic approach to structured specifications in which both the underlying logical system and corresponding institution of the structured specifications are treated as abstract institutions, which means two levels of institution independence. This abstract axiomatic approach provides a uniform framework for the study o...

We give a logic-independent semantics for predefined (data) types within the categorical abstract model theoretic framework of the theory of institutions. We develop a generic interpolation result for this semantics, which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study o...

Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logic-independent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothendieck flattenings to inclusion systems. An importa...

We develop a general logic-independent structural induction proof method at the level of abstract institutions. This provides a solid and uniform mathematical foundation to induction proof methodologies for a wide variety of actual logic-based formal specification frameworks. Our development is based technically upon an axiomatic approach to substi...

Modal logics are successfully used as specification logics for reactive systems. However, they are not expressive enough to refer to individual states and reason about the local behaviour of such systems. This limitation is overcome in hybrid logics which introduce special symbols for naming states in models. Actually, hybrid logics have recently r...

We develop module algebra for structured specifications with model oriented denotations. Our work extends the existing theory with specification building operators for non-protecting importation modes and with new algebraic rules (most notably for initial semantics) and upgrades the pushout-style semantics of parameterized modules to capture the (p...

We extend the concept of quasi-variety of first-order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi-varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi-varieties of MVL models we derive the existence of initial models...

We develop a combination, called hidden preordered algebra, between preordered algebra, which is an algebraic framework supporting specification and reasoning about transitions, and hidden algebra, which is the algebraic framework for behavioural specification. This combination arises naturally within the heterogeneous framework of the modern forma...

Saturated models constitute one of the powerful methods of conventional model theory, with many applications. Here we develop a categorical abstract model theoretic approach to saturated models within the theory of institutions. The most important consequence is that the method of saturated models becomes thus available to a multitude of logical sy...

We develop a general study of the algebraic specification practice, originating from the OBJ tradition, which encodes atomic sentences in logical specification languages as Boolean terms. This practice originally motivated by operational aspects, but also leading to significant increase in expressivity power, has recently become important within th...

We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to o...

We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading...

Formal specification practice involve specifications which are finite. We study the concept of finiteness for specifications within the general framework of the so-called π-institutions, which constitute an abstract categorical proof oriented formalization of the informal concept of logical system. In this paper we argue that finite specifications...

This book develops model theory independently of any concrete logical system or structure, within the abstract category-theoretic framework of the so called ‘institution theory’. The concept of institution provides a model theory oriented mathematical formulation for the intuitive concept of logical system and had arisen within computer science as...

CafeOBJ is an executable industrial-strength algebraic specification language; it is a modern successor of OBJ and incorporates several new algebraic specification paradigms. It was developed in Japan with large-scale support from the Japanese government. Its definition is given in [12], a presentation of its logical foundations can be found in [14...

Interpolation is one of the most important topics of logic and model theory. Below is a very simple example. Consider the following semantic deduction in PL (propositional logic): $$ p_1 \wedge q \vDash p_2 \vee q $$ where p1, p2, q are propositional symbols (i.e., relation symbols of zero arity). The simplest justification for this deduction is by...

The already familiar semantic consequence relation E ⊨ E′ between sets of sentences constitutes the semantic way to establish truth because it involves the models and the satisfaction relation between models and sentences. The syntactic approach to truth consists of establishing consequence relations, called proofs, between sets of sentences involv...

In many institutions the satisfaction relation between models and sentences is defined by induction on the structure of the sentences. Usually sentences are formed from ‘atomic’ sentences, which constitute the starting building blocks, by applying iteratively constructs such as quantifiers and connectives. The connectives may be Boolean or potentia...

The ultraproduct construction on models is one of the most important devices used by ‘first order model theory’, which is that part of model theory relying upon ‘first order’ quantifiers (handled by representable signature morphisms) and finiteness at various syntactic levels such as arities of symbols, atoms, quantification, and logical connective...

A lot of deep results in model theory can be reached by the method of saturated models. Two of the most useful properties of saturated models are their existence and their uniqueness. The existence means that each model can be elementarily extended to a saturated model, while uniqueness holds when the model is ‘sufficiently‘ small. The main topic o...

The logic programming paradigm in its purely logical form can be defined in arbitrary institutions. This liberation of the logic programming ideal from its conventional framework (based upon the Horn sub-institution of REL1, the single sorted variant of relational logic) gives the opportunity of developing the logic programming paradigm over variou...

Institution-independentmodel theory as a categorical abstract model theory relies heavily on category theory. This preliminary chapter gives a brief overviewof the categorical concepts and results used by this book. The reader without enough familiarity with category theory is advised to use one of the textbooks on category theory available in the...

Grothendieck institutions generalize the flattening Grothendieck construction from (indexed) categories, (see Sect. 2.5), to (indexed) institutions. Regarded from a fibration theoretic angle, Grothendieck institutions are just institutions for which their category of signatures is fibred. For example, the actual institutions with many-sorted signat...

Definability theory provides answers to the question of to what extent implicit definitions can be made explicit. The inverse function of groups is a simple example.

Axiomatizability results express a rather subtle relationship between semantics and syntax. They give complete characterizations of certain classes of theories in purely semantic terms, formulated as closure properties of classes of models under some categorical operators. Perhaps the most famous example is the Birkhoff Variety theorem of equationa...

In this chapter we first give a model theoretic presentation of classical first order logic with equality and show the invariance of the satisfaction relation between models and sentences with respect to the change of notation. This is our first example of an institution. We then introduce the abstract concept of institution and illustrate it by a...

This chapter is devoted to some applications of institution-independent model theory to specification theory, thus it digresses slightly from the main topic of the book.

We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called ‘institutions’. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ul...

For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parame- terized. For instance, in the first-order logic, the satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further by possible worlds. This paper proposes a uniform treatment of...

We develop possible worlds (Kripke) semantics at the categorical abstract model theoretic level provided by the so-called 'institutions'. Our general abstract modal logic framework provides a method for systematic Kripke semantics extensions of logical systems from computing science and logic. We also extend the institution-independent method of ul...

This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. We generalise the concept of definability to arbitrary logics, formalised as institutions, and we develop three general definability results. One generalises the cla...

Institutions with proof-theoretic structure, here called ‘institutions with proofs’, provide a complete formal notion for the intuitive notion of logic, including both the model and the proof theoretic sides. This paper introduces a concept of proof rules for institutions and argues that the proof systems of the actual institutions with proofs are...

This paper is dedicated to Joseph Goguen, my beloved teacher and friend, on the ocassion of his 65th anniversary. It is a survey of institution-independent model theory as it stands today, the true form of abstract model theory which is based on the concept of institution. Institution theory was co-fathered by Joseph Coguen and Rod Burstall in late...

Behavioural specification based on hidden (sorted) algebra constitutes one of the most promising recently developed formal specification and verification paradigms for system development. Here we formally introduce a novel concept of behavioural object within the hidden algebra framework. We formally define several object composition operators on...

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representin...

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representin...

We generalize the method of diagrams from conventional model theory to a simple institution-independent (i.e. independent of the details of the actual logic formalized as an institution) framework based on a novel categorical concept of elementary diagram of a model. We illustrate the power of our institution-independent method of elementary diagra...

We formulate a general institution-independent (i.e. independent of the details of the actual logic formalised as institution) version of the Craig Interpolation Theorem and prove it in dependence of Birkhoff-style axiomatizability properties of the actual logic. We formalise Birkhoff-style axiomatizability within the general abstract model theoret...

The basic logic programming semantic concepts, query, solutions, solution forms, and the fundamental results such as Herbrand theorems, are developed over any logical system, formalised as institution, by employing ‘institution-independent’ concepts of variable, substitution, quantifier, and atomic formulae. This sets semantical foundations for a u...

It is well known that interpolation properties of logics underlying specification formalisms play an important rule in the study of structured specifications, they have also many other useful logical consequences.In this paper, we solve the interpolation problem for Grothendieck institutions which have recently emerged as an important mathematical...

We define abstract modal semantics using institutions. Modalities can then be generated over a wide variety of logics. Using tools from institution-independent model theory we state a preservation result for the modal satisfaction.

We generalise the ultraproducts method from conventional model theory to an institution-independent (i.e. independent of the details of the actual logic formalised as an institution) framework based on a novel very general treatment of the semantics of some important concepts in logic, such as quantification, logical connectives, and ground atomic...

CafeOBJ is an executable industrial strength multi-logic algebraic specification language which is a, modern successor of OBJ and incorporates several new algebraic specification paradigms. In this paper we survey its logical foundations and present some of its methodologies.

We generalise the ultraproducts method from conventional model theory to an institution-independent (i.e. independent of the details of the actual logic formalised as an institution) framework based on a novel very general treatment of the semantics of some important concepts in logic, such as quantification, logical connectives, and ground atomic...

This paper surveys the logical and mathematical foundations of CafeOBJ, which is a successor of the famous algebraic specification language OBJ but adds to it several new primitive paradigms such as behavioural concurrent specification and rewriting logic.We first give a concise overview of CafeOBJ. Then we focus on the actual logical foundations o...

We extend indexed categories, fibred categories, and Grothendieck constructions to institutions. We show that the 2-category of institutions admits Grothendieck constructions (in a general 2-categorical sense) and that any split fibred institution is equivalent to a Grothendieck institution of an indexed institution. We use Grothendieck institution...

The research reported in this paper exploits the view of constraint programming as computation in a logical system, namely constraint logic. The basic ingredients of constraint logic are: constraint models for the semantics (they form a comma-category over a fixed model of ‘built-ins’); generalized polynomials in the role of basic syntactic ingredi...

This chapter gives an overview of the main features and methodologies, and, logical and mathematical foundations of CafeOBJ. CafeOBJ is an executable industrial strength algebraic specification language, which is a modern successor of OBJ and incorporating several new algebraic specification paradigms. CafeOBJ also permits equational specification...

We extend the classical hidden algebra formalism by a re-arrangement of the basic concepts. This re-arrangement of the hidden algebra formalism permits an extension to novel concepts which bring new practical strength to the specification and verification methodologies. The main novel concept, which constitutes the core of this work, is that of beh...

. We present a formal method for component-based system specification and verification which is based on the new algebraic specification language CafeOBJ, which is a modern successor of OBJ incorporating several new developments in algebraic specification theory and practice. We first give an overview of the main features of CafeOBJ, including its...

A new method is introduced to concurrently compose an object from already verified objects. The most important new feature of our method is that the verification of the composed object can be done by re-using the verifications of component objects. That is, the verification of composed object is also composable. This is not always true. We can show...

We extend behavioural specification based on hidden sorts to rewriting logic by constructing a hybrid between the two underlying logics. This is achieved by defining a concept of behavioural satisfaction for rewriting logic. Our approach is semantic in that it is based on a general construction on models, called behaviour image, which uses final mo...

this paper we are going to propose how it can be applied to component based software constructions. We call our new specification style component based algebraic specification

The article contains an overview presentation of the CafeOBJ language design and of its current implementation. CafeOBJ is a modern successor of the OBJ language incorporating several new major developments in algebraic specification theory and practice. It is aimed to be an industrial strength language, suitable both for researchers and for practi...

We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphisms map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of logical multi-paradigm languages, especially structu...

: We introduce the concept of semantic paramodulation as a "semantic " definition of paramodulation (in the sense that it applies to any model, not only to the term algebra) within the framework of category-based equational logic (introduced by [8, 9]). This not only generalises the traditional syntactic approaches to paramodulation, but also provi...

We extend behavioural specification based on hidden sorts to rewriting logic by constructing a hybrid between the two underlying logics. This is achieved by defining a concept of behavioural satisfaction for rewriting logic. In the case of concurrent distributed systems, this provides behavioural abstraction for rewriting logic not only by extendin...

リサーチレポート（北陸先端科学技術大学院大学情報科学研究科）

Although modularisation is basic to modern computing, it has been little studied for logic-based programming. We treat modularisation for equational logic programming using the institution of categorybased equational logic in three different ways: (1) to provide a generic satisfaction condition for equational logics; (2) to give a category-based se...

This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying ce...

. This paper exploits the point of view of constraint programming as computation in a logical system, namely constraint logic. We define the basic ingredients of constraint logic, such as constraint models and generalised polynomials. We show that constraint logic is an institution, and we internalise the study of constraint logic to the framework...

Equational deduction is generalised within a category-based abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. Applications include many and order sort...

Theories with hidden sorts provide a setting to study the idea of behaviour and behavioural equivalence of elements. But there are variants on the notion of theory: many sorted algebras, order sorted algebras and so on; we would like to use the theory of institutions to develop ideas of some generality. We formulate the notion of behavioural equiva...

: This paper surveys our current state of knowledge (and ignorance) on the use of hidden sorted algebra as a foundation for the object paradigm. Our main goal is to support equational reasoning about properties of concurrent systems of objects, because of its simple and efficient mechanisation. We show how equational specifications can describe obj...

: Modularisation is important for managing the complex structures that arise in large theorem proving problems, and in large software and/or hardware development projects. This paper studies some properties of logical systems that support the definition, combination, parameterisation and reuse of modules. Our results show some new connections among...

This paper surveys several different variants of order sorted algebra (abbreviated OSA ), comparing some of the main approaches (overloaded OSA, universe OSA, unified algebra, term declaration algebra, etc .), emphasising motivation and intuitions, and pointing out features that distinguish the original ‘overloaded’ OSA approach from some later dev...