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This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE).
|Title of host publication||Algebra and coalgebra in computer science|
|Subtitle of host publication||Proceedings of the 4th international conference, CALCO 2011, Winchester, UK, August 30 - September 2, 2011|
|Editors||Andrea Corradini, Bartek Klin, Corina Cirstea|
|Number of pages||15|
|Publication status||Published - 2011|
|Name||Lecture Notes in Computer Science|
- automata theory
- reversible processes
- computational logic
- coalgebraic semantics
- coalgebraic logics
Ghani, N., Johann, P., & Fumex, C. (2011). Indexed induction and coinduction, fibrationally. In A. Corradini, B. Klin, & C. Cirstea (Eds.), Algebra and coalgebra in computer science : Proceedings of the 4th international conference, CALCO 2011, Winchester, UK, August 30 - September 2, 2011 (pp. 176-191). (Lecture Notes in Computer Science; Vol. 6859 ). Springer.