Project Abstract: Logic Atlas and Integrator (LATIN)

Codescu, Mihai; Horozal, Fulya; Kohlhase, Michael; Mossakowski, Till; Rabe, Florian

doi:10.1007/978-3-642-22673-1_24

Cited by 35 publications

(24 citation statements)

References 8 publications

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“…We have used this implementation to represent and mechanically verify a number of meta-theorems in DTT. Since the implementation is modular, our approach is well-suited for the integration with the LATIN logic atlas [Codescu et al 2011], which already uses Twelf to formalize the syntax and semantics of a wide variety of formal systems and translation theorems in a systematically modular way. By choosing appropriate DTT-signatures and morphisms, our definition permits uniformly representing both syntactic and semantic logical relations of object logics as DTT logical relations.…”

Section: Resultsmentioning

confidence: 99%

Logical relations for a logical framework

Rabe

Sojakova

2013

ACM Trans. Comput. Logic

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Logical relations are a central concept used to study various higher-order type theories and occur frequently in the proofs of a wide variety of meta-theorems. Besides extending the logical relation principle to more general languages, an important research question has been how to represent and thus verify logical relation arguments in logical frameworks.We formulate a theory of logical relations for Dependent Type Theory (DTT) with β η-equality which guarantees that any valid logical relation satisfies the Basic Lemma. Our definition is syntactic and reflective in the sense that a relation at a type is represented as a DTT type family but also permits expressing certain semantic definitions. We use the Edinburgh Logical Framework (LF) incarnation of DTT and implement our notion of logical relations in the type-checker Twelf. This enables us to formalize and mechanically decide the validity of logical relation arguments. Furthermore, our implementation includes a module system so that logical relations can be built modularly. We validate our approach by formalizing and verifying several syntactic and semantic meta-theorems in Twelf. Moreover, we show how object languages encoded in DTT can inherit a notion of logical relation from the logical framework.

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Section: Resultsmentioning

confidence: 99%

Logical relations for a logical framework

Rabe

Sojakova

2013

ACM Trans. Comput. Logic

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“…flattened theory graph) of the highly modular LATIN [Cod+11] library. flattened theory graph) of the highly modular LATIN [Cod+11] library.…”

Section: Searching the Knowledge Space Of The Latin Logic Atlasmentioning

confidence: 99%

Intelligent Computer Mathematics

Carette

Kaliszyk

Rabe

et al. 2015

Lecture Notes in Computer Science

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Abstract. We summarize math search engines and search interfaces produced by the Document and Pattern Recognition Lab in recent years, and in particular the min math search interface and the Tangent search engine. Source code for both systems are publicly available. "The Masses" refers to our emphasis on creating systems for mathematical non-experts, who may be looking to define unfamiliar notation, or browse documents based on the visual appearance of formulae rather than their mathematical semantics.

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“…The foundation is flexible: for example, we have represented Zermelo-Fraenkel set theory, Mizar's set theory, and higher-order logic HOL as foundations in the applications of our framework Iancu and Rabe 2011;Codescu et al 2011). Since a model may involve any mathematical object, Σ mod must be expressive enough to define arbitrary mathematical objects.…”

Section: Model Theorymentioning

confidence: 99%

“…Moreover, we have implemented our framework and evaluated it in a number of large-scale case studies Iancu and Rabe 2011;Codescu et al 2011). Moreover, we have implemented our framework and evaluated it in a number of large-scale case studies Iancu and Rabe 2011;Codescu et al 2011).…”

Section: Introductionmentioning

confidence: 99%

A logical framework combining model and proof theory

Rabe¹

2013

Math. Struct. Comp. Sci.

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Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics. Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap -for example in applications to software verification -they have fundamentally different perspectives on logic. In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations. † This work was developed over the course of five years, during F. Rabe 948 A similar difference in primacy is found when looking at families of logics. For example, the model-theoretical view takes precedence in description logic (see, for example, Brachman and Schmolze (1985) and Baader et al. (2003)), where the model theory is fixed and proof theory is studied chiefly to provide reasoning tools. This view is arguably paramount as it dominates accounts of (classical) first-order logic and most uses of logic in mathematics. On the other hand, the proof-theoretical view takes precedence in, for example, higher-order logic (see Church (1940) for proof theory and Henkin (1950) for model theory); and some logics such as intuitionistic logics (Brouwer 1907) are explicitly defined by their proof theory.At the level of logical frameworks, proof-theoretical principles have been integrated into model-theoretical frameworks in several ways. Parchments are used in Goguen and Burstall (1986) and Mossakowski et al. (1997) to express the syntax of a logic as a formal language. And, for example, in Meseguer (1989), Fiadeiro andSernadas (1988) and Mossakowski et al. (2005), proof theory is formalised in the abstract style of category theory. These approaches are very elegant, but often fail to exploit a major advantage of proof theory -the constructive reasoning over concrete syntax.Model-theoretical principles have also been integrated into proof-theoretical frameworks, albeit to a lesser extent. Fo...

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Project Abstract: Logic Atlas and Integrator (LATIN)

Cited by 35 publications

References 8 publications

Logical relations for a logical framework

Logical relations for a logical framework

Intelligent Computer Mathematics

A logical framework combining model and proof theory

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