## Kleene algebra with tests and program schematology (2001)

Citations: | 17 - 6 self |

### BibTeX

@TECHREPORT{Angus01kleenealgebra,

author = {Allegra Angus and Dexter Kozen},

title = {Kleene algebra with tests and program schematology},

institution = {},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

The theory of flowchart schemes has a rich history going back to Ianov [6]; see Manna [22] for an elementary exposition. A central question in the theory of program schemes is scheme equivalence. Manna presents several examples of equivalence proofs that work by simplifying the schemes using various combinatorial transformation rules. In this paper we present a purely algebraic approach to this problem using Kleene algebra with tests (KAT). Instead of transforming schemes directly using combinatorial graph manipulation, we regard them as a certain kind of automaton on abstract traces. We prove a generalization of Kleene’s theorem and use it to construct equivalent expressions in the language of KAT. We can then give a purely equational proof of the equivalence of the resulting expressions. We prove soundness of the method and give a detailed example of its use. 1

### Citations

2450 |
The Design and Analysis of Computer Algorithms
- AHO, HOPCROFT, et al.
- 1974
(Show Context)
Citation Context ...aic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs [9, 23], automata and formal languages [20, 21], and the design and analysis of algorithms =-=[1, 7, 10]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [13] and references therein. Kleene algebra with tests (KAT), introduced in [13], combines programs and a... |

374 |
Representation of Events in Nerve Nets and Finite Automata
- Kleene
- 1956
(Show Context)
Citation Context ...lting expressions. We prove soundness of the method and give a detailed example of its use. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions. It was first introduced by Kleene =-=[8]-=-; the name Kleene algebra was coined by Conway [5], who developed much of the algebraic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs [9, 23]... |

277 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...d and give a detailed example of its use. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions. It was first introduced by Kleene [8]; the name Kleene algebra was coined by Conway =-=[5]-=-, who developed much of the algebraic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs [9, 23], automata and formal languages [20, 21], and the ... |

186 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...x, and qp the least solution to q + xp x. Here “least” refers to the natural partial order p q $ p + q = q. The operation + gives the supremum with respect to . This particular axiomatization is from =-=[11]-=-. We normally omit the , writing pq for p q. The precedence of the operators is > > +. Thus p + qr should be parsed p +(q(r )). Typical models include the family of regular sets of strings over a fini... |

131 |
The design and analysis of algorithms
- Kozen
- 1991
(Show Context)
Citation Context ...aic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs [9, 23], automata and formal languages [20, 21], and the design and analysis of algorithms =-=[1, 7, 10]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [13] and references therein. Kleene algebra with tests (KAT), introduced in [13], combines programs and a... |

126 |
Theory of Computation
- Kozen
- 2006
(Show Context)
Citation Context ...ular expression over the alphabet P [ B with the classical interpretation, and construct an equivalent finite automaton M with input alphabet P [ B as in the usual proof of Kleene’s theorem (see e.g. =-=[12]-=-). Conversely, given a finite automaton M with input alphabet P [ B, construct an equivalent regular expression p. Let R(p) denote the regular subset of (P [ B) denoted by p under the classical interp... |

109 | Kleene algebra with tests
- Kozen
- 1997
(Show Context)
Citation Context ...gebra with an embedded Boolean subalgebra. KAT strictly subsumes propositional Hoare Logic (PHL), is of no greater complexity than PHL, and is deductively complete over relational models (PHL is not) =-=[17, 4, 14, 18]-=-. KAT is less expressive than propositional Dynamic Logic (PDL), but also less complex (unless PSPACE = EXPTIME , which 1scomplexity theorists generally regard as unlikely). Moreover, KAT requires not... |

41 | On Hoare logic and Kleene algebra with tests
- Kozen
(Show Context)
Citation Context ...gebra with an embedded Boolean subalgebra. KAT strictly subsumes propositional Hoare Logic (PHL), is of no greater complexity than PHL, and is deductively complete over relational models (PHL is not) =-=[17, 4, 14, 18]-=-. KAT is less expressive than propositional Dynamic Logic (PDL), but also less complex (unless PSPACE = EXPTIME , which 1scomplexity theorists generally regard as unlikely). Moreover, KAT requires not... |

32 | Certification of compiler optimizations using Kleene algebra with tests
- Kozen, Patron
- 2000
(Show Context)
Citation Context ...s or modalities. KAT has been applied successfully in a number of low-level verification tasks involving communication protocols, basic safety analysis, concurrency control, and compiler optimization =-=[2, 3, 16]-=-. A useful feature of KAT in this regard is its ability to accommodate certain basic equational assumptions regarding the interaction of atomic instructions. This feature makes KAT ideal for reasoning... |

20 | On induction vs. *-continuity
- Kozen
- 1981
(Show Context)
Citation Context ...ene [8]; the name Kleene algebra was coined by Conway [5], who developed much of the algebraic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs =-=[9, 23]-=-, automata and formal languages [20, 21], and the design and analysis of algorithms [1, 7, 10]. Many authors have contributed over the years to the development of the algebraic theory; see [13] and re... |

17 | Automata on guarded strings and applications
- Kozen
- 2001
(Show Context)
Citation Context ... given scheme, we will view a scheme as a kind of generalized finite automaton called a schematic automaton. Schematic automata are a generalization of automata on guarded strings (AGS) introduced in =-=[15]-=-. AGS are like ordinary finite automata, except that they take guarded strings as inputs. In turn, schematic automata over are like AGS, except that they take traces of Kripke frames over as inputs. 7... |

16 |
Using Kleene algebra to reason about concurrency control
- Cohen
- 1994
(Show Context)
Citation Context ...s or modalities. KAT has been applied successfully in a number of low-level verification tasks involving communication protocols, basic safety analysis, concurrency control, and compiler optimization =-=[2, 3, 16]-=-. A useful feature of KAT in this regard is its ability to accommodate certain basic equational assumptions regarding the interaction of atomic instructions. This feature makes KAT ideal for reasoning... |

16 |
A semiring on convex polygons and zero-sum cycle problems
- Iwano, Steiglitz
- 1990
(Show Context)
Citation Context ...aic theory. Kleene algebra has appeared in computer science in many guises: semantics and logics of programs [9, 23], automata and formal languages [20, 21], and the design and analysis of algorithms =-=[1, 7, 10]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [13] and references therein. Kleene algebra with tests (KAT), introduced in [13], combines programs and a... |

11 |
The logical schemes of algorithms
- Ianov
- 1960
(Show Context)
Citation Context ...y Allegra Angus Dexter Kozen Department of Computer Science Cornell University Ithaca, NY 14853-7501, USA July 10, 2001 Abstract The theory of flowchart schemes has a rich history going back to Ianov =-=[6]-=-; see Manna [22] for an elementary exposition. A central question in the theory of program schemes is scheme equivalence. Manna presents several examples of equivalence proofs that work by simplifying... |

3 |
Lazy caching. Available as ftp://ftp.telcordia.com/pub/ernie/research/homepage.html
- Cohen
- 1994
(Show Context)
Citation Context ...s or modalities. KAT has been applied successfully in a number of low-level verification tasks involving communication protocols, basic safety analysis, concurrency control, and compiler optimization =-=[2, 3, 16]-=-. A useful feature of KAT in this regard is its ability to accommodate certain basic equational assumptions regarding the interaction of atomic instructions. This feature makes KAT ideal for reasoning... |