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Summer 2011 REGS Report

The focus of this project is on the development of a unifying, categorical framework for complete deduction for coalgebra. From an algebraic perspective, deduction systems have been studied extensively by mathematicians and computer scientists. However, their investigations from a coalgebraic point of view have been more modest. Nonetheless, work by Roşu and others using a category theoretic approach indicates a great potential for dualization, and thus, for applications to coalgebra. Combining notions and techniques from category theory, state-based computation, and logic [5], coalgebra aims to be the mathematics of computational dynamics [4]. Coalgebras consist of a state space, say X, and a transition map of the form X : X −→ F (X ), where F is an endofunctor of a category C (usually the category of sets). Coalgebras, which are the duals of algebras, turn out to be suitable models of general dynamical systems and of infinite data types, and satisfy the powerful proof principle of coinduction, first introduced by Aczel in his seminal work on non-well-founded sets [1]. Even though coalgebra is still in its infancy, it has already led to interesting connections between seemingly unrelated fields such as differential equations and automata theory [9], as well as provided a new perspective on the study of classical mathematical structures such as metric spaces and partial orders [10]. Our research aims to expand our understanding of coalgebras as they relate to deduction systems. Mathematicians have been interested in the definitional power of equations for a long time. One of the earliest results characterizing a class of algebras as one containing exactly all algebras which satisfy a given set of equations was obtained by Birkhoff in 1935 [3]. Birkhoff’s variety theorem states that given a signature Σ, a class K of Σ-algebras is closed under homomorphic images, subalgebras, and products—such a class is called a variety—if and only if K is the collection of all Σ-algebras satisfying some set of equations. The classical notion of Σ-algebra easily generalizes to the