## Kleene Algebra with Tests: Completeness and Decidability (1996)

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Venue: | Proc. 10th Int. Workshop Computer Science Logic (CSL'96), volume 1258 of Lecture Notes in Computer Science |

Citations: | 22 - 10 self |

### BibTeX

@INPROCEEDINGS{Kozen96kleenealgebra,

author = {Dexter Kozen and Frederick Smith},

title = {Kleene Algebra with Tests: Completeness and Decidability},

booktitle = {Proc. 10th Int. Workshop Computer Science Logic (CSL'96), volume 1258 of Lecture Notes in Computer Science},

year = {1996},

pages = {244--259},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

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### Abstract

. Kleene algebras with tests provide a rigorous framework for equational specification and verification. They have been used successfully in basic safety analysis, source-to-source program transformation, and concurrency control. We prove the completeness of the equational theory of Kleene algebra with tests and *-continuous Kleene algebra with tests over language-theoretic and relational models. We also show decidability. Cohen's reduction of Kleene algebra with hypotheses of the form r = 0 to Kleene algebra without hypotheses is simplified and extended to handle Kleene algebras with tests. 1 Introduction A Kleene algebra with tests is an algebraic structure consisting of a Kleene algebra with an embedded Boolean subalgebra. This formalism provides a rigorous framework for equational specification and verification of programs. It has been applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control [3, 4, 5, 17]. Kleene ...

### Citations

2438 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context .... It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms =-=[1, 11]-=-. See [16] for an introduction and a comprehensive list of citations. Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) [8], in which it is mixed with modal logic to giv... |

376 |
Representation of events in nerve nets and finite automata
- Kleene
- 1951
(Show Context)
Citation Context ...en applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control [3, 4, 5, 17]. Kleene algebra dates back to a 1956 paper of S. C. Kleene =-=[12]-=- and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal ... |

358 |
Propositional dynamic logic of regular programs
- Fischer, Ladner
- 1979
(Show Context)
Citation Context ...and the design and analysis of algorithms [1, 11]. See [16] for an introduction and a comprehensive list of citations. Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) =-=[8]-=-, in which it is mixed with modal logic to give a theoretically appealing and practical system for reasoning about computation at the propositional level. Syntactically, PDL is a two-sorted logic cons... |

302 |
Word Problems Requiring Exponential Time (Preliminary Report
- Stockmeyer, Meyer
- 1973
(Show Context)
Citation Context ...to be exponential time complete [8, 21], thus the equational theory of KAT is decidable in no more than exponential time. It is at least PSPACE-hard, since the equational theory of Kleene algebras is =-=[24]-=-. It can be shown by different methods that the equational theory of KAT is PSPACE-complete [6]. Acknowledgements Ernie Cohen provided valuable comments. The support of the National Science Foundation... |

275 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...rce-to-source program transformation, and concurrency control [3, 4, 5, 17]. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway =-=[7]-=-. It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms ... |

234 | Adherences of languages
- Boasson, Nivat
- 1980
(Show Context)
Citation Context ...ra commutativity conditions of the form pq = qp, even for atomic p and q, is undecidable (see [17]), although with a little extra work this result can be shown to follow from a 1979 result of Berstel =-=[2]-=- (see also [9]). The proof in [17] only needed extra commutativity conditions of the form bp = pb, where b is a test. But as shown in that paper, this equation is equivalent to bpb + bpb = 0. Thus if ... |

211 |
On the caldus of relations
- Tarski
- 1941
(Show Context)
Citation Context ...4, 5, 17]. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra =-=[20, 25]-=-, semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms [1, 11]. See [16] for an introduction and a comprehensive list of citat... |

186 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...ared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms [1, 11]. See =-=[16]-=- for an introduction and a comprehensive list of citations. Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) [8], in which it is mixed with modal logic to give a theore... |

130 | The design and analysis of algorithms - Kozen - 1992 |

108 | Kleene algebra with tests
- Kozen
- 1997
(Show Context)
Citation Context ...d KAT coincide. 2. We show that KAT is complete over relational models. This implies decidability of the equational theory by an essentially trivial reduction to Propositional Dynamic Logic (PDL). In =-=[6]-=-, we show by different methods that the problem is PSPACE-complete, thus of the same complexity as Kleene algebra. 3. We show that the equational theory of Kleene algebra with tests admits free langua... |

84 |
V.: 1979, Models of program logics
- Pratt
(Show Context)
Citation Context ...tructures iff PDL j= !p?c $ !q?c ; where c is a new primitive proposition symbol [8]. By Theorems 6 and 8, this is tantamount to deciding KAT-equivalence. PDL is known to be exponential time complete =-=[8, 21]-=-, thus the equational theory of KAT is decidable in no more than exponential time. It is at least PSPACE-hard, since the equational theory of Kleene algebras is [24]. It can be shown by different meth... |

40 | On Kleene algebras and closed semirings
- Kozen
- 1990
(Show Context)
Citation Context ...he natural order . In the presence of the other axioms, the *-continuity condition (7) implies (3--6), and is strictly stronger in the sense that there exist Kleene algebras that are not *-continuous =-=[14]-=-. The main result of [16] says that all true identities between regular expressions, interpreted as regular sets of strings, are derivable from the axioms of Kleene algebra [16], and only such identit... |

33 | Dynamic algebras as a well-behaved fragment of relation algebras
- Pratt
(Show Context)
Citation Context ...956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs =-=[13, 23]-=-, automata and formal language theory [18], and the design and analysis of algorithms [1, 11]. See [16] for an introduction and a comprehensive list of citations. Kleene algebra forms an essential com... |

27 | On folk theorems
- Harel
- 1980
(Show Context)
Citation Context ...at paper, the utility of KAT was illustrated by giving a purely equational proof of the following classical result: every while program can be simulated by a while program with at most one while loop =-=[10, 19]-=-. E. Cohen has taken a slightly different approach in which tests are defined to be elements b satisfying the condition bs1. He has given several practical examples of the use of Kleene algebra with c... |

27 |
Relation algebras with transitive closure
- Ng
- 1984
(Show Context)
Citation Context ...4, 5, 17]. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra =-=[20, 25]-=-, semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms [1, 11]. See [16] for an introduction and a comprehensive list of citat... |

26 |
Representation of events in nerve nets and nite automata, in: C.E
- Kleene
- 1956
(Show Context)
Citation Context ...een applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control [3, 4, 5,17]. Kleene algebra dates back to a 1956 paper of S. C. Kleene =-=[12]-=- and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal ... |

20 | On induction vs. *-continuity
- Kozen
- 1981
(Show Context)
Citation Context ...956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs =-=[13, 23]-=-, automata and formal language theory [18], and the design and analysis of algorithms [1, 11]. See [16] for an introduction and a comprehensive list of citations. Kleene algebra forms an essential com... |

16 |
Using Kleene algebra to reason about concurrency control
- Cohen
- 1994
(Show Context)
Citation Context ...work for equational specification and verification of programs. It has been applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control =-=[3, 4, 5, 17]-=-. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25],... |

16 |
A semiring on convex polygons and zero-sum cycle problems
- Iwano, Steiglitz
- 1990
(Show Context)
Citation Context .... It has appeared in one form or another in relational algebra [20, 25], semantics and logics of programs [13, 23], automata and formal language theory [18], and the design and analysis of algorithms =-=[1, 11]-=-. See [16] for an introduction and a comprehensive list of citations. Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) [8], in which it is mixed with modal logic to giv... |

12 |
Dynamic algebras and the nature of induction
- Pratt
- 1980
(Show Context)
Citation Context ...ional Models In this section we establish completeness over relational models. It will suffice to construct a relational model isomorphic to G. This construction is similar to a construction of Pratt =-=[22]-=- for regular sets. For A any set of guarded strings, define h(A) def = f(x; x \Pi y) j x 2 GS; y 2 Ag : Lemma 5. The language-theoretic model P and its submodel G are isomorphic to relational models. ... |

10 | Kleene algebra with tests and commutativity conditions - Kozen - 1996 |

6 |
On the decidability of some problems about rational subsets of free partially commutative monoids
- Gibbons, Rytter
- 1986
(Show Context)
Citation Context ...ty conditions of the form pq = qp, even for atomic p and q, is undecidable (see [17]), although with a little extra work this result can be shown to follow from a 1979 result of Berstel [2] (see also =-=[9]-=-). The proof in [17] only needed extra commutativity conditions of the form bp = pb, where b is a test. But as shown in that paper, this equation is equivalent to bpb + bpb = 0. Thus if Cohen's reduct... |

3 |
Algorithmic logic and its applications
- Mirkowska
- 1972
(Show Context)
Citation Context ...at paper, the utility of KAT was illustrated by giving a purely equational proof of the following classical result: every while program can be simulated by a while program with at most one while loop =-=[10, 19]-=-. E. Cohen has taken a slightly different approach in which tests are defined to be elements b satisfying the condition bs1. He has given several practical examples of the use of Kleene algebra with c... |

1 |
Hypotheses in Kleene algebra. ftp://ftp.bellcore.com/pub/ernie/research/homepage.html
- Cohen
- 1994
(Show Context)
Citation Context ...work for equational specification and verification of programs. It has been applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control =-=[3, 4, 5, 17]-=-. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25],... |

1 |
Lazy caching. ftp://ftp.bellcore.com/pub/ernie/research/homepage.html
- Cohen
- 1994
(Show Context)
Citation Context ...work for equational specification and verification of programs. It has been applied successfully to problems in basic safety analysis, source-to-source program transformation, and concurrency control =-=[3, 4, 5, 17]-=-. Kleene algebra dates back to a 1956 paper of S. C. Kleene [12] and was developed extensively in a 1971 monograph of Conway [7]. It has appeared in one form or another in relational algebra [20, 25],... |