### Resumé

To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR.

For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs.

To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR.

For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs.

Sprog | Engelsk |
---|---|

Tidsskrift | Formal Aspects of Computing |

Vol/bind | 29 |

Udgave nummer | 1 |

Sider | 57-95 |

ISSN | 0934-5043 |

DOI | |

Status | Udgivet - 2017 |

### Citer dette

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*Formal Aspects of Computing*, bind 29, nr. 1, s. 57-95. DOI: 10.1007/s00165-016-0396-9

**On proving confluence modulo equivalence for Constraint Handling Rules.** / Christiansen, Henning; Kirkeby, Maja Hanne.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel

TY - JOUR

T1 - On proving confluence modulo equivalence for Constraint Handling Rules

AU - Christiansen,Henning

AU - Kirkeby,Maja Hanne

N1 - Funded by European Union Seventh Framework Programme (FP7/2007- 2013) Grant number 318337, ENTRA - Whole-Systems Energy Transparency and The Danish Council for Independent Research, Natural Sciences, DFF 4181-00442

PY - 2017

Y1 - 2017

N2 - Previous results on proving confluence for Constraint Handling Rules are extended in two ways in order to allow a larger and more realistic class of CHR programs to be considered confluent. Firstly, we introduce the relaxed notion of confluence modulo equivalence into the context of CHR: while confluence for a terminating program means that all alternative derivations for a query lead to the exact same final state, confluence modulo equivalence only requires the final states to be equivalent with respect to an equivalence relation tailored for the given program. Secondly, we allow non-logical built-in predicates such as var/1 and incomplete ones such as is/2, that are ignored in previous work on confluence.To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR.For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs.The Project is supported by The DanishCouncil for IndependentResearch, Natural Sciences, Grant No. DFF4181-00442. The second author’s contribution received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 318337, ENTRA—Whole-Systems Energy Transparency.

AB - Previous results on proving confluence for Constraint Handling Rules are extended in two ways in order to allow a larger and more realistic class of CHR programs to be considered confluent. Firstly, we introduce the relaxed notion of confluence modulo equivalence into the context of CHR: while confluence for a terminating program means that all alternative derivations for a query lead to the exact same final state, confluence modulo equivalence only requires the final states to be equivalent with respect to an equivalence relation tailored for the given program. Secondly, we allow non-logical built-in predicates such as var/1 and incomplete ones such as is/2, that are ignored in previous work on confluence.To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR.For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs.The Project is supported by The DanishCouncil for IndependentResearch, Natural Sciences, Grant No. DFF4181-00442. The second author’s contribution received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 318337, ENTRA—Whole-Systems Energy Transparency.

UR - https://arxiv.org/abs/1611.03628

U2 - 10.1007/s00165-016-0396-9

DO - 10.1007/s00165-016-0396-9

M3 - Journal article

VL - 29

SP - 57

EP - 95

JO - Formal Aspects of Computing

T2 - Formal Aspects of Computing

JF - Formal Aspects of Computing

SN - 0934-5043

IS - 1

ER -