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Florian Rabe

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3 papers

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TCS Journal 2017 Journal Article

Morphism axioms

Florian Rabe

We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle problems where it is difficult to give a theory that specifies the intended model category, or where seemingly isomorphic theories actually have non-isomorphic model categories. Our morphism axioms remedy this by providing new syntax for axiomatizing and reasoning about the properties of model morphisms. Additionally, our system resolves a subtle incompatibility between theory morphisms and model morphisms: the semantics that maps theories to model categories is functorial. While this result is standard in principle, previous formulations had to restrict the allowed theory morphisms or the allowed model morphisms. Our system allows establishing the result in full generality.

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I&C Journal 2013 Journal Article

A scalable module system

Florian Rabe
Michael Kohlhase

Symbolic and logic computation systems ranging from computer algebra systems to theorem provers are finding their way into science, technology, mathematics and engineering. But such systems rely on explicitly or implicitly represented mathematical knowledge that needs to be managed to use such systems effectively. While mathematical knowledge management (MKM) “in the small” is well-studied, scaling up to large, highly interconnected corpora remains difficult. We hold that in order to realize MKM “in the large”, we need representation languages and software architectures that are designed systematically with large-scale processing in mind. Therefore, we have designed and implemented the Mmt language – a module system for mathematical theories. Mmt is designed as the simplest possible language that combines a module system, a foundationally uncommitted formal semantics, and web-scalable implementations. Due to a careful choice of representational primitives, Mmt allows us to integrate existing representation languages for formal mathematical knowledge in a simple, scalable formalism. In particular, Mmt abstracts from the underlying mathematical and logical foundations so that it can serve as a standardized representation format for a formal digital library. Moreover, Mmt systematically separates logic-dependent and logic-independent concerns so that it can serve as an interface layer between computation systems and MKM systems.

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TCS Journal 2011 Journal Article

Representing model theory in a type-theoretical logical framework

Fulya Horozal
Florian Rabe

In a broad sense, logic is the field of formal languages for knowledge and truth that have a formal semantics. It tends to be difficult to give a narrower definition because very different kinds of logics exist. One of the most fundamental contrasts is between the different methods of assigning semantics. Here two classes can be distinguished: model theoretical semantics based on a foundation of mathematics such as set theory, and proof theoretical semantics based on an inference system possibly formulated within a type theory. Logical frameworks have been developed to cope with the variety of available logics unifying the underlying ontological notions and providing a meta-theory to reason abstractly about logics. While these have been very successful, they have so far focused on either model or proof theoretical semantics. We contribute to a unified framework by showing how the type/proof theoretical Edinburgh Logical Framework (LF) can be applied to the representation of model theoretical logics. We give a comprehensive formal representation of first-order logic, covering both its proof and its model theoretical semantics as well as its soundness in LF. For the model theory, we have to represent the mathematical foundation itself in LF, and we provide two solutions for that. Firstly, we give a meta-language that is strong enough to represent the model theory while being simple enough to be treated as a fragment of untyped set theory. Secondly, we represent Zermelo–Fraenkel set theory and show how it subsumes our meta-language. Specific models are represented as LF morphisms. All representations are given in and mechanically verified by the Twelf implementation of LF. Moreover, we use the Twelf module system to treat all connectives and quantifiers independently. Thus, individual connectives are available for reuse when representing other logics, and we obtain the first version of a feature library from which logics can be pieced together. Our results and methods are not restricted to first-order logic and scale to a wide variety of logical systems, thus demonstrating the feasibility of comprehensively formalizing large scale representation theorems in a logical framework.

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