Representing model theory in a type-theoretical logical framework

Horozal, Fulya; Rabe, Florian

doi:10.1016/j.tcs.2011.03.022

Cited by 13 publications

(21 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The representation of ZFC we used in Horozal and Rabe (2011) required several thousand declarations just to build up the basic notions of ZFC set theory to a degree that supports simple examples. The representation of ZFC we used in Horozal and Rabe (2011) required several thousand declarations just to build up the basic notions of ZFC set theory to a degree that supports simple examples.…”

Section: Foundationsmentioning

confidence: 99%

“…For example, in Horozal and Rabe (2011), we use a simply typed higher-order logic for F 0 and ZFC set theory as F, and P interprets types as sets and terms as elements of these sets. For example, standard constructions and results like the real numbers are usually assumed without giving or relying on a specific definition.…”

Section: Foundationsmentioning

confidence: 99%

“…Counter-examples arise when F 0 is too weak and cannot prove all theorems of F. Horozal and Rabe (2011) showed that it is a necessary condition in the special case F 0 = F = ZFC, but the general case is open. The statement follows immediately because σ is a model-theoretical theory morphism if and only if Mod M P (σ) maps Θ -models to Θ-models.…”

Section: Example 432 (Modal Logic Continued)mentioning

confidence: 99%

“…The most extensive case studies were presented in Horozal and Rabe (2011) and Iancu and Rabe (2011). In Horozal and Rabe (2011), we gave a comprehensive representation of first-order logic, which includes a formalisation of ZFC set theory to formalise the model theory of FOL and a formalised soundness proof of FOL in the sense of Theorem 4.25.…”

Section: F Rabe 988mentioning

confidence: 99%

See 3 more Smart Citations

A logical framework combining model and proof theory

Rabe¹

2013

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Foundationsmentioning

confidence: 99%

Section: Foundationsmentioning

confidence: 99%

Section: Example 432 (Modal Logic Continued)mentioning

confidence: 99%

Section: F Rabe 988mentioning

confidence: 99%

See 2 more Smart Citations

A logical framework combining model and proof theory

Rabe¹

2013

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…between HOL Light and Isabelle in the Flyspeck project [18], between HOL Light and Coq [12,25,19] or between HOL and Nuprl [15]. To avoid the quadratic blowup in the number of translators to develop, proof formats are emerging either for proofs in a specific logic such as the OpenTheory format [17] for ITPs in the HOL family or relying on logical frameworks [20,24,14] such as λ-prolog and Twelf to represent proofs in several logics. We propose to combine proofs coming from different theorem provers relying on the Dedukti logical framework [23], a typed λ-calculus with dependent types and rewriting, as a common formalism in which proofs can be translated and combined.…”

Section: Introductionmentioning

confidence: 99%

FoCaLiZe and Dedukti to the Rescue for Proof Interoperability

Cauderlier

Dubois

2017

Interactive Theorem Proving

View full text Add to dashboard Cite

Abstract. Numerous contributions have been made for some years to allow users to exchange formal proofs between different provers. The main propositions consist in ad hoc pointwise translations, e.g. between HOL Light and Isabelle in the Flyspeck project or uses of more or less complete certificates. We propose in this paper a methodology to combine proofs coming from different theorem provers. This methodology relies on the Dedukti logical framework as a common formalism in which proofs can be translated and combined. To relate the independently developed mathematical libraries used in proof assistants, we rely on the structuring features offered by FoCaLiZe, in particular parameterized modules and inheritance to build a formal library of transfer theorems called MathTransfer. We finally illustrate this methodology on the Sieve of Eratosthenes, which we prove correct using HOL and Coq in combination.

show abstract

Towards MKM in the Large: Modular Representation and Scalable Software Architecture

Kohlhase

Rabe

Zholudev

2010

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

MKM has been defined as the quest for technologies to manage mathematical knowledge. MKM "in the small" is well-studied, so the real problem is to scale up to large, highly interconnected corpora: "MKM in the large". We contend that advances in two areas are needed to reach this goal. We need representation languages that support incremental processing of all primitive MKM operations, and we need software architectures and implementations that implement these operations scalably on large knowledge bases. We present instances of both in this paper: the MMT framework for modular theory-graphs that integrates meta-logical foundations, which forms the base of the next OMDOC version; and TNTBase, a versioned storage system for XMLbased document formats. TNTBase becomes an MMT database by instantiating it with special MKM operations for MMT.

show abstract

Representing model theory in a type-theoretical logical framework

Cited by 13 publications

References 19 publications

A logical framework combining model and proof theory

A logical framework combining model and proof theory

FoCaLiZe and Dedukti to the Rescue for Proof Interoperability

Towards MKM in the Large: Modular Representation and Scalable Software Architecture

Contact Info

Product

Resources

About