Three decades of institution theory

institution to a system of institutions, i.e. a diagram of institutions, more precisely. However the thinking along these lines has led to a rather different and much more efficient solution, namely that of the ‘flattening’ of the respective diagram of institutions to a single institution by extending a corresponding construction from category theory [49] to institutions. The resulting concept of Grothendieck institution [22, 62] is emerging as the fundamental mathematical structure for the multi-logic heterogeneous specification paradigm. Apart of CafeOBJ, the heterogeneous specification framework with CASL extensions [64] is also based upon the theory of Grothendieck institutions. Quite surprisingly, Grothendieck institutions have been applied to pure model theory, such as for obtaining interpolation results [29]. Doing model theory without concrete structure. The development of model theory at the very general level of abstract institutions is based upon the observation that the most important model theory methods are independent of the conventional first order logic context in which they have originally been developed. This means that all these methods can be formulated and developed at a much more abstract level independent of any particular logical structure. The breakthrough was given by the institution-independent method of ultraproducts [23], which was followed by a rather drastic reformulation in [24] of the institution-independent method of diagrams of [80, 81]. The development of institution-independent saturated model theory [33, 29] came a bit later. These have been used for developing general results about compactness [23], axiomatizability [80, 81, 29], elementary chains [51], interpolation [25, 52], definability [72], completeness [20, 71], generating a big array of novel concrete results in actual unconventional, or even in conventional well studied logics. Moreover, the institution-independent approach to model theory makes the access to highly difficult model theoretic results considerably easier, an example being the Keisler-Shelah isomorphism theorem [33, 29]. Illuminating model theoretic phenomena. The institution-independent approach has lead to the redesign of important fundamental logic concepts and to the clarification of some causality relationships between model theoretic phenomena including the demounting of some deep theoretical preconceptions. One such example is that of interpolation which has been extended to sets of sentences instead of single sentences and to arbitrary commutative squares of signature morphisms instead of the traditional intersection-union squares of signatures. The first extension corrects a traditional misunderstanding about the lack of interpolation properties of logics such Horn clause logic or equational logic. It is the merit of [76] to have proved a Craig interpolation property for sets of sentences in equational logic based upon its Birkhoff-style axiomatizability property, thus revealing a previously unknown cause for interpolation. This idea has been generalized to abstract institutions in [25], thus leading to a myriad of new concrete interpolation results (for fragments of first order logic see also [73]). The second extension of the interpolation concept comes from the practice of algebraic specification which requires interpolation for arbitrary pushout squares of signature morphisms. When interpolation is considered in this way a significant difference between the single and the many sorted logics shows up. The interpolation problem for many sorted first order logic, which stayed for several years as a conjecture, had received a rather elegant solution in [52] as a particular concrete case of a general institution-independent interpolation result. The institution theoretic study of interpolation has also revealed that the Craig-Robinson form of interpolation [78], which stregthens the Craig formulation by adding to the set of the premises a set of ‘secondary’ premises from the second signature, is actually more appropriate than the traditional Craig formulation. This conclusion is motivated by applications such as definability [72, 29], translation of interpolation [27, 29], modularisation of formal specifications [32, 83, 37], completeness of structured specifications proof calculi [12, 29]. A somehow similar situation happens with (Beth) definability, it can also be extended to arbitrary signature morphisms and formulated more properly in terms of sets of sentences [66, 72], and it can also be obtained as a consequence of Birkhoff-style axiomatizability properties [72]. Another example is given by completeness, which was discovered to have a ‘layered’ structure as explained below. Both Birkhoff and Gödel-Henkin forms of completeness have been developed at the generic level of abstract institutions in [20] and [71, 50], respectively, by a Three decades of institution theory 7 technique common to both of them, originally developed by [12], and which consists of separating the proof rules and the completeness phenomenon on several layers. In this approach the base layer consists of an institution with a given sound and complete proof system. Since this base layer refers usually to the ‘atomic’ sentences, its completeness is rather easy to establish in each particular case. The other layers are built on top of the base layer succesively by considering more complex sentences and consequently adding new proof rules and meta-rules. This layered construction is done fully abstractly and the respective completeness results are proved fully generally relative to the completeness of the predecessor level, thus leading especially in the Birkhoff case to a multitude of concrete complete proof calculi for various logics, some of them rather unconventional. Many of these complete proof calculi are new, and quite surprising in that they appear rather remote from the original Birkhoff completeness. Stratified institutions. This is a recent refinement of the concept of institution which captures uniformly the concept of open formulæ and the concept of models with states (such as possible worlds semantics for modal logics) in a fully abstract setting. Stratified institutions have been developed in [9, 2], however a precursor can be found in [34]. They have already been used to develop a very general version of Tarski’s elementary chain theorem applicable to both classical and non-classical (i.e. modal) logics. Stratified institutions also represent a big promise for logic combination, which is one of the great challenges in contemporary logic. Proof theoretic developments. Although institution theory is primarily model theoretic approach, there have been a proof theory development within institution theory [59, 66, 26, 29, 74] motivated primarily by the foundations for formal verifications. The main goal of the recent approach to proof theory of [66, 26, 29] is to liberate it from the Curry-Howard isomorphism dogma in order to achieve greater simplicity, generality, and harmony with the model theory. Another recent approach to extend institutions with proofs is proposed by [74], its most interesting feature being the conceptual symmetry between the model and the proof theory. Technically speaking, the proof theory of [66, 26, 29], as well as that of [74], follows the proofs-as-arrows idea of categorical logic [55], but it has a much broader range of applications than the latter. Moreover it treats concepts such as implication or quantifiers in a more realistic manner than in categorical logic (for example in categorical logic implication presuposes conjunctions). Categorical abstract algebraic logic. Although algebraic logic is not a model theoretic approach, we should also mention here the new trend called ‘categorical abstract algebraic logic’ which develops algebraic logic at the generic abstract level of the π-institutions. The paper [84] is one from a long series of papers on this topic. The UNILOG connection. Institution theory appears naturally as a major actor in the current universal logic trend, known as UNILOG. Starting with 2005 the UNILOG community is organising world congresses at a rate of each 2-3 years. In each of these congresses it is a custom to organise a competition of papers answering a specific question. In the first congress, held in Switzerland, the institution theory paper [66] failed short to win the first prize for the question ‘what is the identity of a logic’, but a follow-up paper [65] by the same authors won it at the next congress, held in China, for the question ‘what is a logic translation’. 5. Looking to the future Future is hard to predict, especially in the current climate of scientific research in which theories are developing and trends are changing at an increased speed. Institution theory is already established as the most fundamental mathematical structure for logic based specification theory, and in this sense it will continue to play its foundational role. Moreover institution theoretic ideas will continue to spread in other areas of computing science, however it is difficult to see exactly in which of these and how. In the next period I think the interest for developing model theory at the very general level of abstract institutions, as part of the universal logic trend, will continue to grow. A related area of great interest consists of applying institution-independent model theory to provide a model