## Calculating Church-Rosser Proofs in Kleene Algebra (2002)

Venue: | Relational Methods in Computer Science, 6th International Conference, volume 2561 of LNCS |

Citations: | 10 - 4 self |

### BibTeX

@INPROCEEDINGS{Struth02calculatingchurch-rosser,

author = {Georg Struth},

title = {Calculating Church-Rosser Proofs in Kleene Algebra},

booktitle = {Relational Methods in Computer Science, 6th International Conference, volume 2561 of LNCS},

year = {2002},

pages = {276--290},

publisher = {Springer}

}

### OpenURL

### Abstract

We prove Church-Rosser theorems for non-symmetric transitive relations, quasiorderings and equations in Kleene algebra. Proofs are simple, rigorous and general, using solely algebraic properties of the regular operations. They are fixed point-based, induction-free and often amenable to automata. They are mere calculations as opposed to deduction and in particular suited to automation. In the Church-Rosser proofs for the -calculus, the term and algebra part are cleanly separated. In all our considerations, Kleene algebra is an excellent means of abstraction.

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Citation Context ...regular events. Besides formal languages and automata, Kleene algebras also arise, for instance, in the context of relation algebra (c.f. [20]) and logics, analysis and construction of programs (c.f. =-=[11-=-]). We follow Kozen's denition [13]. These structures are called Kozen semirings in [3]. For concise proofs of all statements in this text see [24]. Some formal proofs with the Isabelle proof checker ... |

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Citation Context ...f checkers by many researchers (c.f. [21,17,18] and the references given there.). Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait-MartinL of (c.f. [=-=2]-=-) or Takahashi [26]. They use induction and therefore higher-order logic. Our proofs aresrst-order and oftensnite combinatorics. We algebraically reconstruct an alternative proof of Barendregt [2], wh... |

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Citation Context ...uages and automata, Kleene algebras also arise, for instance, in the context of relation algebra (c.f. [20]) and logics, analysis and construction of programs (c.f. [11]). We follow Kozen's denition [=-=13]-=-. These structures are called Kozen semirings in [3]. For concise proofs of all statements in this text see [24]. Some formal proofs with the Isabelle proof checker can be found in [25]. A semiring is... |

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Citation Context ... hypotheses are not only relevant to Church-Rosser theorems. They can be used to express independence or precedence in execution sequences or Mazurkiewicz traces in imperative and concurrent programs =-=[14,4,7]-=-. Kleene algebra is related to certain allegories (c.f. [10]). Allegories can be understood either as categories with relations as arrows or as typed relation algebras. In particular, semicommuting re... |

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Citation Context ...f general interest for formal methods, sincesrst-order proof search methods and even decision procedures can be used then. Relational specications and proofs are at the heart of methods like Z and B [=-=22,1]-=-. We have tried to include the most important proofs in the paper. A full formal treatment can be found in an extended version [24]. The remainder is organized as follows. Section 2 and section 3 intr... |

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Citation Context ...0 a i ) Q n i=1 a i . (ii) a j a i a i a j for all 0 is n implies ( P n i=0 a i ) Q n i=1 a i . Proposition 2 (i) is related to Church-Rosser theorems modulo and equivalence relation [12]. Proposition 2 (ii) is an algebraic completeness proof of bubble sort. At the end of this section we calculate some Church-Rosser statements for the Kleene plus. Proposition 3. Let A be a Kleene alge... |

57 | Some lambda calculus and type theory formalized
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Citation Context ...perative and concurrent programs works in a similar way [4,14,15]. The Church-Rosser theorem in the -calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [=-=21,17,18-=-] and the references given there.). Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait-MartinL of (c.f. [2]) or Takahashi [26]. They use induction and ... |

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Citation Context ...f general interest for formal methods, sincesrst-order proof search methods and even decision procedures can be used then. Relational specications and proofs are at the heart of methods like Z and B [=-=22,1]-=-. We have tried to include the most important proofs in the paper. A full formal treatment can be found in an extended version [24]. The remainder is organized as follows. Section 2 and section 3 intr... |

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Citation Context ...perative and concurrent programs works in a similar way [4,14,15]. The Church-Rosser theorem in the -calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [=-=21,17,18-=-] and the references given there.). Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait-MartinL of (c.f. [2]) or Takahashi [26]. They use induction and ... |

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Citation Context ...perative and concurrent programs works in a similar way [4,14,15]. The Church-Rosser theorem in the -calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [=-=21,17,18-=-] and the references given there.). Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait-MartinL of (c.f. [2]) or Takahashi [26]. They use induction and ... |

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Citation Context ...means of abstraction. Some simple lemmata must be proven on the term structure. They can then be lifted to the algebra level. Reasoning about imperative and concurrent programs works in a similar way =-=[4,14,15-=-]. The Church-Rosser theorem in the -calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [21,17,18] and the references given there.). Almost all previous ... |

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Citation Context ...lgebra. For all a; b; c 2 A, 1 = 1 ; (12) 1 a ; (13) a a = a ; (14) a a ; (15) a = a ; (16) ac cb ) a c cb ; (17) cb ac ) cb a c; (18) (a + b) = a (ba ) : (19) In particular, (ab) a = a(ba) and a a = aa are simple consequences of (17). Lemma 4. Let A be a Kleene algebra. For all a; b; c 2 A, a + a + a + ; (20) a a + ; (21) a + a ; (22) aa + ... |

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Citation Context ...volve well-foundedness. 9 Discussion The Church-Rosser theorems in section 5 are interesting for the foundations of rewriting. Theorem 4 is the Church-Rosser theorem for rewriting with quasiorderings =-=[16]-=-, proposition 3 that for rewriting with non-symmetric transitive relations [23]. The commutation and semicommutation relations in the hypotheses are not only relevant to Church-Rosser theorems. They c... |

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Citation Context ...tabulation techniques for a Kleene algebra extended by an ! -operation, to model innite iteration and well-foundedness. A proof of Newman's lemma in a dierent extension of Kleene algebra appears in [8=-=-=-]. We believe that Newman's lemma is an excellent test example for extensions of Kleene algebra to innite behavior. Finally, Church-Rosser proofs in Kleene algebra are also well-suited for automation.... |

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Citation Context ...nductive proofs of Church-Rosser statements with more than two generators. Theorem 4 and theorem 5 are the basis for non-symmetric rewriting with quasiorderings and non-symmetric transitive relations [23]. The inequalities b a a b are examples of semi-commutation properties (as opposed to commutation properties like ab = ba). As we will see in the next section, their interpretations in the r... |

10 |
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(Show Context)
Citation Context ...means of abstraction. Some simple lemmata must be proven on the term structure. They can then be lifted to the algebra level. Reasoning about imperative and concurrent programs works in a similar way =-=[4,14,15-=-]. The Church-Rosser theorem in the -calculus has been considered interesting for interactive proof checkers by many researchers (c.f. [21,17,18] and the references given there.). Almost all previous ... |

9 |
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Citation Context ...eir equational counterparts. 8 6 Conversion and Functions To obtain precisely an algebraic variant of the Church-Rosser theorem (theorem 1), we add a converse operator. A Kleene algebra with converse [6] is a tuple (A; +; ; ; ; 0; 1), where (A; +; ; ; 0; 1) is a Kleene algebra and the unary operation is a contravariant involution that distributes with + and : a = a; (36) (a + b) = a + b ; (37... |

6 |
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Citation Context ...elations. 3 3 Kleene Algebra A Kleene algebra is a structure for modeling sequential composition, non-deterministic choice and iteration orsxed point computation. Theseld has been pioneered by Conway =-=[5]-=- in the context of the algebra of regular events. Besides formal languages and automata, Kleene algebras also arise, for instance, in the context of relation algebra (c.f. [20]) and logics, analysis a... |

1 |
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Citation Context ... hypotheses are not only relevant to Church-Rosser theorems. They can be used to express independence or precedence in execution sequences or Mazurkiewicz traces in imperative and concurrent programs =-=[14,4,7]-=-. Kleene algebra is related to certain allegories (c.f. [10]). Allegories can be understood either as categories with relations as arrows or as typed relation algebras. In particular, semicommuting re... |

1 |
Church-Rosser proofs in Kleene algebra and allegories
- Struth
- 2001
(Show Context)
Citation Context ... specications and proofs are at the heart of methods like Z and B [22,1]. We have tried to include the most important proofs in the paper. A full formal treatment can be found in an extended version [=-=24]-=-. The remainder is organized as follows. Section 2 and section 3 introduce some Church-Rosser and Kleene algebra basics. Section 4 recalls further properties of Kleene algebra for the Church-Rosser pr... |

1 |
Isabelle-speci and proofs of Church-Rosser theorems
- Struth
- 2001
(Show Context)
Citation Context ... believe that Newman's lemma is an excellent test example for extensions of Kleene algebra to innite behavior. Finally, Church-Rosser proofs in Kleene algebra are also well-suited for automation. In [=-=2-=-5] we have specied Kleene algebra in the Isabelle proof-checker and formalized all proofs up to theorem 4. Both the specication and the proofs are very short and simple. The degree of automation is qu... |

1 |
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(Show Context)
Citation Context ... researchers (c.f. [21,17,18] and the references given there.). Almost all previous attempts formally reconstruct a proof via the nowadays standard methods of Tait-MartinL of (c.f. [2]) or Takahashi [=-=26]-=-. They use induction and therefore higher-order logic. Our proofs aresrst-order and oftensnite combinatorics. We algebraically reconstruct an alternative proof of Barendregt [2], which puts more empha... |