A new approach to the semantics of model diagrams

Sometimes, a diagram can say more than a thousand lines of code. But, sadly, most of the time, software engineers give up on diagrams after the design phase, and all real work is done in code. The supremacy of code over diagrams would be leveled if diagrams were code. This paper suggests that model and instance diagrams, or, which amounts to the same, class and object diagrams, become first level entities in a suitably expressive programming language, viz., type theory. The proposed semantics of diagrams is compositional and self-describing, i.e., reflexive, or metacircular. Moreover, it is well suited for metamodelling and model driven engineering, as it is possible to prove model transformations correct in type theory. The encoding into type theory has the additional benefit of making diagrams immediately useful, given an implementation of type theory

A Functional Programming Language with Patterns and Copatterns

Since the emergence of coinductive data types in functional programming languages, various languages such as Haskell and Coq tried different ways in dealing with them. Yet, none of them dealt with coinductive data types properly. In lazy languages such as Haskell, both inductive data types and coinductive data types are gathered and mixed in one list. Moreover, some languages such as Coq used the same constructors that are used for inductive data types as a tool to tackle coinductive data types, and while other languages such as Haskell did use destructors, they did not use them properly. Coinductive data types behave differently than inductive data types and therefore, it is more appropriate to deal with them differently. In this thesis, we propose a new functional programming language where coinductive data types are dealt with in a dual approach in which coinductive data types are defined by observation and inductive data types are defined by constructors. This approach is more appropriate in dealing with coinductive data types whose importance comes from their role in creating a safer and more sophisticated software

Algorithmic Theories of Problems. A Constructive and a Non-Constructive Approach

In this paper we examine two approaches to the formal treatment of the notion of problem in the paradigm of algorithmic semantics. Namely, we will explore an approach based on Martin-Löf’s Constructive Type Theory (CTT), which can be seen as a direct continuation of Kolmogorov’s original calculus of problems, and an approach utilizing Tichý’s Transparent Intensional Logic (TIL), which can be viewed as a non-constructive attempt of interpreting Kolmogorov’s logic of problems. In the last section we propose Kolmogorov and CTT-inspired modifications to TIL-based approach. The focus will be on non-empirical (i.e., mathematical and logical) problems only

Multimodal Dependent Type Theory

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations

An Investigation into Intuitionistic Logic with Identity

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective

Constructive Type Theory and the Dialogical Approach to Meaning

In its origins Dialogical logic constituted one part of a new movement called the Erlangen School or Erlangen Constructivism. Its goal was to provide a new start to a general theory of language and of science. According to the Erlangen-School, language is not just a fact that we discover, but a human cultural accomplishment whose construction reason can and should control. The resulting project of intentionally constructing a scientific language was called the Orthosprache-project. Unfortunately, the Orthosprache-project was not further developed and seemed to fade away. It is possible that one of the reasons for this fading away is that the link between dialogical logic and Orthosprache was not sufficiently developed - in particular, the new theory of meaning to be found in dialogical logic seemed to be cut off from both the project of establishing the basis for scientific language and also from a general theory of meaning. We would like to contribute to clarifying one possible way in which a general dialogical theory of meaning could be linked to dialogical logic. The idea behind the proposal is to make use of constructive type theory in which logical inferences are preceded by the description of a fully interpreted language. The latter, we think, provides the means for a new start not only for the project of Orthosprache, but also for a general dialogical theory of meaning

Constructive Type Theory, an appetizer

Recent debates in metaphysics have highlighted the significance of type theories, such as Simple Type Theory (STT), for our philosophical analysis. In this chapter, I present the salient features of a constructive type theory in the style of Martin-Löf, termed CTT. My principal aim is to convey the flavour of this rich, flexible and sophisticated theory and compare it with STT. I especially focus on the forms of quantification which are available in CTT. A further aim is to argue that a comparison between a plurality of theories is beneficial to the philosophical analysis. We may, for example, discover helpful features of one theory that we may want to import into another context, thus enriching our repertoire of formal tools. Or, through comparison with a less well-known theory, we may gain a better understanding of the characteristics of the theories we are familiar with. As argued in this chapter, CTT has much to offer in all of these respects