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Category Theory and Formal Methods Lab
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The lab uses category theory to treat a wide variety of logics in a unified way, supporting the verification of software.
AboutAbout the lab
Understanding how logic works through category theory.
This lab studies universal logic, which treats a wide variety of logical systems in a common framework.
Using category theory as its mathematical language, the lab works on the theory of institutions, which formalizes a logical system as a combination of a signature (a collection of symbols), sentences, models, and a satisfaction relation.
The lab has been involved in laying logical foundations for formal methods, which are used to write software specifications and support their verification, and in designing tools that assist with proofs.
Recently, the lab has been extending its theory to hybrid systems, which combine discrete and continuous behavior, and to quantum computation.
Key terms
- Category theory
- A branch of mathematics that abstractly studies objects and the relationships between them. In this lab, it is used to compare and integrate logical systems.
- Institution
- A formalization, using category theory, of a logical system as a combination of a collection of symbols, sentences, models, and a satisfaction relation.
- Formal methods
- Methods for writing software specifications and carrying out their verification with a mathematical basis.
ResearchResearch themes
The lab studies theories that make it possible to compare and integrate a wide variety of logical systems within a common framework, and has also been involved in laying logical foundations for software specification and verification. Beyond theory, it works on designing theorem-proving tools and extending the theory to hybrid systems and quantum computation.
*Illustrative imageA unified theory of logical systems
The lab studies general frameworks for building and reasoning about a wide variety of logical systems. Through institution-based model theory using category theory, it aims to make it possible to handle comparison, integration, and interconnection of logical systems in a systematic way.
*Illustrative imageLogical foundations of proof scores
The lab was involved in the design of CafeOBJ, an algebraic specification language. Its central goal was to give rigorous logical foundations to proof scores, a formal method that supports both software specification and structured, interactive verification.
*Illustrative imageDeveloping tools that support proofs
The lab designed CITP, a theorem prover that supports inductive reasoning about algebraic specifications. To increase expressive power, it introduced transition algebras, which add features of dynamic logic, such as state transitions and their composition, to many-sorted first-order logic.
*Illustrative imageToward hybrid systems and quantum computation
The lab is extending transition algebras to hybrid systems, which combine discrete and continuous behavior. It has also applied the theory of institutions to quantum computation, presenting a proof system and a Birkhoff-style completeness theorem.
MembersFaculty
- Associate ProfessorDaniel Gaina
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Commutative diagram zelmerszoetrop” Zelmerszoetrop (talk) / CC0 (cropped)
- A unified theory of logical systems: “Mattel Pixter - board - System Logic SL3876D-1731” Raimond Spekking / CC BY-SA 4.0 (cropped)
- Logical foundations of proof scores: “Bhaskaracharya proof of pythagorean Theorem” Mahusha / CC BY-SA 4.0 (cropped)
- Developing tools that support proofs: “20th Century Small Wooden Abacus” Deinodactyla / CC0 (cropped)
- Toward hybrid systems and quantum computation: “Quantum circuit that exhibits Parity Measurement” Vtomole / CC BY-SA 4.0 (cropped)
- Related lab (Partial Differential Equations and Calculus of Variations Lab): “Graphene g-C3N4” Vicente Neto / CC BY 4.0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “Optimization Problem 2” Musical Inquisit / CC BY-SA 4.0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “S asdf rot” Trabandt / CC0 (cropped)
- Related lab (Mathematical Statistics and Statistical Learning Theory Lab): “Dices-probability-def-2” Svjo / CC BY-SA 3.0 (cropped)
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