### Abstract

Original language | English |
---|---|

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 |

Publisher | Springer |

Pages | 176-191 |

Number of pages | 15 |

ISBN (Print) | 9783642229435 |

Publication status | Published - 2011 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Publisher | Springer |

Volume | 6859 |

ISSN (Print) | 0302-9743 |

### Fingerprint

### Keywords

- automata theory
- reversible processes
- duality
- computational logic
- coalgebraic semantics
- coalgebraic logics
- categories

### Cite this

*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.

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*Algebra and coalgebra in computer science : Proceedings of the 4th international conference, CALCO 2011, Winchester, UK, August 30 - September 2, 2011.*Lecture Notes in Computer Science, vol. 6859 , Springer, pp. 176-191.

**Indexed induction and coinduction, fibrationally.** / Ghani, Neil; Johann, Patricia; Fumex, Clement.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

TY - GEN

T1 - Indexed induction and coinduction, fibrationally.

AU - Ghani, Neil

AU - Johann, Patricia

AU - Fumex, Clement

PY - 2011

Y1 - 2011

N2 - 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).

AB - 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).

KW - automata theory

KW - reversible processes

KW - duality

KW - computational logic

KW - coalgebraic semantics

KW - coalgebraic logics

KW - categories

UR - http://www.springer.com/computer/theoretical+computer+science/book/978-3-642-22943-5

M3 - Conference contribution book

SN - 9783642229435

T3 - Lecture Notes in Computer Science

SP - 176

EP - 191

BT - Algebra and coalgebra in computer science

A2 - Corradini, Andrea

A2 - Klin, Bartek

A2 - Cirstea, Corina

PB - Springer

ER -