### Abstract

Original language | English |
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Title of host publication | Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science |

Place of Publication | Germany |

Number of pages | 13 |

DOIs | |

Publication status | Published - 30 Nov 2017 |

Event | 42nd International Symposium on Mathematical Foundations of Computer Science - Aalborg, Denmark Duration: 21 Aug 2017 → 25 Aug 2017 http://mfcs2017.cs.aau.dk/ |

### Publication series

Name | Leibniz International Proceedings in Informatics |
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Publisher | Schloss Dagstuhl – Leibniz Center for Informatics |

### Conference

Conference | 42nd International Symposium on Mathematical Foundations of Computer Science |
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Abbreviated title | MFCS 2017 |

Country | Denmark |

City | Aalborg |

Period | 21/08/17 → 25/08/17 |

Internet address |

### Fingerprint

### Keywords

- type theory
- induction recursion
- initial algebra semantics

### Cite this

*Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science*[63] (Leibniz International Proceedings in Informatics ). Germany. https://doi.org/10.4230/LIPIcs.MFCS.2017.63

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*Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science.*, 63, Leibniz International Proceedings in Informatics , Germany, 42nd International Symposium on Mathematical Foundations of Computer Science, Aalborg, Denmark, 21/08/17. https://doi.org/10.4230/LIPIcs.MFCS.2017.63

**Variations on inductive-recursive definitions.** / Ghani, Neil; McBride, Conor; Nordvall Forsberg, Fredrik; Spahn, Stephan.

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

TY - GEN

T1 - Variations on inductive-recursive definitions

AU - Ghani, Neil

AU - McBride, Conor

AU - Nordvall Forsberg, Fredrik

AU - Spahn, Stephan

PY - 2017/11/30

Y1 - 2017/11/30

N2 - Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families — i.e. of families ( U : Set , T : U → D ) such that the inductive definition of U may depend on the recursively defined T — by defining a type DS D E of codes. Each c : DS D E defines a functor J c K : Fam D → Fam E , and ( U , T ) = μ J c K : Fam D is exhibited as the initial algebra of J c K . This paper considers the composition of DS -definable functors: Given F : Fam C → Fam D and G : Fam D → Fam E , is G ◦ F : Fam C → Fam E DS -definable, if F and G are? We show that this is the case if and only if powers of families are DS -definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF ↪ → DS ↪ → PN . Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN -functors exist by adapting Dybjer-Setzer’s proof for DS .

AB - Dybjer and Setzer introduced the definitional principle of inductive-recursively defined families — i.e. of families ( U : Set , T : U → D ) such that the inductive definition of U may depend on the recursively defined T — by defining a type DS D E of codes. Each c : DS D E defines a functor J c K : Fam D → Fam E , and ( U , T ) = μ J c K : Fam D is exhibited as the initial algebra of J c K . This paper considers the composition of DS -definable functors: Given F : Fam C → Fam D and G : Fam D → Fam E , is G ◦ F : Fam C → Fam E DS -definable, if F and G are? We show that this is the case if and only if powers of families are DS -definable, which seems unlikely. To construct composition, we present two new systems UF and PN of codes for inductive-recursive definitions, with UF ↪ → DS ↪ → PN . Both UF and PN are closed under composition. Since PN defines a potentially larger class of functors, we show that there is a model where initial algebras of PN -functors exist by adapting Dybjer-Setzer’s proof for DS .

KW - type theory

KW - induction recursion

KW - initial algebra semantics

UR - http://mfcs2017.cs.aau.dk/

UR - http://www.dagstuhl.de/en/publications/lipics

U2 - 10.4230/LIPIcs.MFCS.2017.63

DO - 10.4230/LIPIcs.MFCS.2017.63

M3 - Conference contribution book

T3 - Leibniz International Proceedings in Informatics

BT - Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science

CY - Germany

ER -