## Term Equational Systems and Logics (Extended Abstract)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Fiore_termequational,

author = {Marcelo Fiore and Chung-kil Hur},

title = {Term Equational Systems and Logics (Extended Abstract)},

year = {}

}

### OpenURL

### Abstract

We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.

### Citations

393 | Basic concepts of enriched category theory
- Kelly
- 2005
(Show Context)
Citation Context ... isomorphisms C (V ⊗ A, B) ∼ = V [V, C (A, B)] ∼ = C (A, [V, B]) in V for which the former is V -natural in B and the latter V -natural in A, where V [−, =] denotes the internal hom-functor of V (see =-=[13]-=- for details). Further, in the important and quite common case in which V is symmetric monoidal closed, a V -category C with tensors and powers can be equivalently presented as an ordinary category C0... |

261 |
Combinatory Reduction Systems
- Klop
- 1980
(Show Context)
Citation Context ...ameworks for supporting it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories [16] together with their extensions, e.g. to rewriting [18], higher-order =-=[15,19]-=-, and nominal [8,4] settings. The common conceptual core of these frameworks consists of syntactic structure providing a notion of term together with a logical system of equational judgements. Ideally... |

185 |
Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context .... Introduction Formal reasoning is fundamental in computer science, and frameworks for supporting it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories =-=[16]-=- together with their extensions, e.g. to rewriting [18], higher-order [15,19], and nominal [8,4] settings. The common conceptual core of these frameworks consists of syntactic structure providing a no... |

153 | A new approach to abstract syntax involving binders - Gabbay, Pitts - 1999 |

132 |
Higher-order critical pairs
- Nipkow
- 1991
(Show Context)
Citation Context ...ameworks for supporting it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories [16] together with their extensions, e.g. to rewriting [18], higher-order =-=[15,19]-=-, and nominal [8,4] settings. The common conceptual core of these frameworks consists of syntactic structure providing a notion of term together with a logical system of equational judgements. Ideally... |

70 | Rewriting logic: roadmap and bibliography
- Martí-Oliet, Meseguer
- 2002
(Show Context)
Citation Context ...ter science, and frameworks for supporting it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories [16] together with their extensions, e.g. to rewriting =-=[18]-=-, higher-order [15,19], and nominal [8,4] settings. The common conceptual core of these frameworks consists of syntactic structure providing a notion of term together with a logical system of equation... |

65 | Completeness of many-sorted equational logic
- Goguen, Meseguer
- 1985
(Show Context)
Citation Context ...Nominal Sets, Schanuel topos. Introduction Formal reasoning is fundamental in computer science, and frameworks for supporting it abound. Those that are most relevant to this work are equational logic =-=[10]-=- and algebraic theories [16] together with their extensions, e.g. to rewriting [18], higher-order [15,19], and nominal [8,4] settings. The common conceptual core of these frameworks consists of syntac... |

63 |
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
- Kelly, Power
- 1993
(Show Context)
Citation Context ...ription for the definition of signature, but rather consider it as being domain specific. Of course, standard notions of signature (e.g., as they arise in the context of (enriched) algebraic theories =-=[14,20]-=-) may be considered. However, one may also need to go beyond them—see [5]. 2. Select a class of arities (A, C) and give a syntactic description of the generalised terms A � TΣC. This yields a syntacti... |

55 |
Automata and Algebras in Categories
- Adámek, Trnková
- 1990
(Show Context)
Citation Context ...tages as the composition of a left adjoint to UΣ followed by a left adjoint to JS. The inductive construction of a left adjoint to UΣ, or equivalently that of free Σ-algebras, is well-known (see e.g. =-=[1,2]-=-). The free Σ-algebra (T X, τX) on an object X ∈ C and the universal map ηX : X � T X are constructed as follows. The object T X is a colimit of the ω-chain { fn : Xn � Xn+1 }n≥0 inductively defined b... |

41 |
algebras and automata realizations in the language of categories
- Adámek
- 1974
(Show Context)
Citation Context ...tages as the composition of a left adjoint to UΣ followed by a left adjoint to JS. The inductive construction of a left adjoint to UΣ, or equivalently that of free Σ-algebras, is well-known (see e.g. =-=[1,2]-=-). The free Σ-algebra (T X, τX) on an object X ∈ C and the universal map ηX : X � T X are constructed as follows. The object T X is a colimit of the ω-chain { fn : Xn � Xn+1 }n≥0 inductively defined b... |

30 |
Nominal equational logic
- Clouston, Pitts
(Show Context)
Citation Context ...ng it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories [16] together with their extensions, e.g. to rewriting [18], higher-order [15,19], and nominal =-=[8,4]-=- settings. The common conceptual core of these frameworks consists of syntactic structure providing a notion of term together with a logical system of equational judgements. Ideally, this is further e... |

26 | Variations on algebra: monadicity and generalisations of equational theories
- Robinson
- 2002
(Show Context)
Citation Context ...ription for the definition of signature, but rather consider it as being domain specific. Of course, standard notions of signature (e.g., as they arise in the context of (enriched) algebraic theories =-=[14,20]-=-) may be considered. However, one may also need to go beyond them—see [5]. 2. Select a class of arities (A, C) and give a syntactic description of the generalised terms A � TΣC. This yields a syntacti... |

13 | A formal calculus for informal equality with binding
- Gabbay, Mathijssen
- 2007
(Show Context)
Citation Context ...ng it abound. Those that are most relevant to this work are equational logic [10] and algebraic theories [16] together with their extensions, e.g. to rewriting [18], higher-order [15,19], and nominal =-=[8,4]-=- settings. The common conceptual core of these frameworks consists of syntactic structure providing a notion of term together with a logical system of equational judgements. Ideally, this is further e... |

10 |
Second-order and dependently-sorted abstract syntax
- Fiore
- 2008
(Show Context)
Citation Context ...in specific. Of course, standard notions of signature (e.g., as they arise in the context of (enriched) algebraic theories [14,20]) may be considered. However, one may also need to go beyond them—see =-=[5]-=-. 2. Select a class of arities (A, C) and give a syntactic description of the generalised terms A � TΣC. This yields a syntactic notion of equational theory with an associated model theory arising fro... |

9 | Term rewriting with variable binding: An initial algebra approach, PPDP - Hamana - 2003 |

7 |
Equational systems and free constructions
- Fiore, Hur
- 2007
(Show Context)
Citation Context ...tion is satisfied by all models of a generalised theory iff it is satisfied in a particular free model (see Section 4). 2� Fiore and Hur This, in view of an explicit construction of free models (see =-=[6,7]-=- and Section 1), may be used to synthesise a sound and complete deduction system which, in turn, may suggest how to extend the aforementioned equational logic to make it complete, and allow the use of... |

5 | On the construction of free algebras for equational systems - Fiore, Hur - 2009 |

5 |
A note on actions of a monoidal category. Theory and Applications of Categories
- Janelidze, Kelly
(Show Context)
Citation Context ... presented as an ordinary category C0 equipped with a V -action (−) ⊗ (=) : V × C0 � C0 such that the functors (−) ⊗ C : V � C0 and V ⊗ (−) : C0 � C0 have right adjoints for all C ∈ C0 and V ∈ V (see =-=[12]-=- for details). Remark 2.1 For clarity, and as a notational convention, we will henceforth highlight V -enriched structures by underlining them. Corresponding structures that are not underlined will de... |

1 |
Soliveres Tur, The completeness theorem for monads in categories of sorted sets
- Vidal, B, et al.
(Show Context)
Citation Context ...ogics in universes of Sorted Sets (see Examples 2.6, 2.11, and 3.2) essentially yields the framework of equational theories and logics for monads on categories of Sorted Sets of Climent and Soliveres =-=[3]-=-. 3� � � � � Fiore and Hur In Section 5, as a substantial case study providing evidence for the practicality of the mathematical theory, the above methodology is applied within the universe of Nomina... |