#### DMCA

## On the coalgebraic theory of Kleene algebra with tests (2008)

Citations: | 25 - 2 self |

### Citations

464 |
Representation of events in nerve nets and finite automata,” in Automata Studies
- Kleene
- 1956
(Show Context)
Citation Context ...proofs can be generated automatically in PSPACE. This matches Worthington’s bound [16] for equational proofs. 2 KA and KAT 2.1 Kleene Algebra Kleene algebra (KA) is the algebra of regular expressions =-=[6, 8]-=-. The axiomatization used here is from [9]. A Kleene algebra is a structure (K, +, ·, ∗ , 0, 1) such that K is an idempotent semiring under +, ·, 0, and 1 and satisfies the axioms 1 + pp ∗ ≤ p ∗ 1 + p... |

384 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ...tomatic Proof Generation in PSPACE The results of Sections 6.2 and 6.3 give rise to a nondeterministic linear-space algorithm for deciding the equivalence of two given KAT terms. By Savitch’s theorem =-=[15]-=-, there is a deterministic quadratic-space algorithm. The deterministic algorithm can be used to create bisimulation proofs of equivalence or inequivalence automatically. To obtain the linear space bo... |

353 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...roofs can be generated automatically in PSPACE . This matches Worthington’s bound [16] for equational proofs. 2 KA and KAT 2.1 Kleene Algebra Kleene algebra (KA) is the algebra of regular expressions =-=[6, 8]-=-. The axiomatization used here is from [9]. A Kleene algebra is a structure (K, +, ·, ∗, 0, 1) such that K is an idempotent semiring under +, ·, 0, and 1 and satisfies the axioms 1 + pp∗ ≤ p∗ q + pr ≤... |

295 | Derivatives of regular expressions
- Brzozowski
- 1964
(Show Context)
Citation Context ... Σ and T also by KCT. 4.1 The Brzozowski Derivative There is a natural KCT over Σ and T defined in terms of the Brzozowski derivative on sets of guarded strings. The traditional Brzozowski derivative =-=[3]-=- is a kind of residuation operator on sets of ordinary strings. The current form is quite similar, except that we extend the definition to accommodate tests. We define two maps where for R ⊆ GS, D : A... |

242 | A completeness theorem for kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...CE. This matches Worthington’s bound [16] for equational proofs. 2 KA and KAT 2.1 Kleene Algebra Kleene algebra (KA) is the algebra of regular expressions [6, 8]. The axiomatization used here is from =-=[9]-=-. A Kleene algebra is a structure (K, +, ·, ∗ , 0, 1) such that K is an idempotent semiring under +, ·, 0, and 1 and satisfies the axioms 1 + pp ∗ ≤ p ∗ 1 + p ∗ p ≤ p ∗ q + pr ≤ r ⇒ p ∗ q ≤ r q + rp ≤... |

144 | Kleene Algebra with tests
- Kozen
- 1997
(Show Context)
Citation Context ...is known to require PSPACE and to produce exponential-size proofs in the worst case. This worst-case bound is unlikely to be significantly improved, as the equational theory of KAT is PSPACE-complete =-=[5]-=-. Chen and Pucella’s treatment has a few technical shortcomings, as they themselves point out. In their words: The “path independence” of a mixed automaton gives any mixed automaton a certain form of ... |

85 | Automata and coinduction (an exercise in coalgebra
- Rutten
- 1998
(Show Context)
Citation Context ... tasks. Traditionally, KAT is axiomatized equationally [10]. In [4], Chen and Pucella develop a coalgebraic theory of KAT inspired by Rutten’s coalgebraic theory of KA based on deterministic automata =-=[14]-=-. Remarking that “the known automatatheoretic presentation of KAT [12] does not lend itself to a coalgebraic theory,” and that “the notion of derivative, essential to the coinduction proof principle i... |

56 | On Hoare logic and Kleene algebra with tests
- Kozen
- 2000
(Show Context)
Citation Context ..., and KAT are all PSPACE-complete. For KAT expressions e1, e2, we write e1 ≤ e2 if this inequality holds in the free KAT on generators Σ, T ; that is, if it is a consequence of the axioms of KAT. See =-=[9, 10, 11]-=- for a more detailed introduction. 3 Automata on Guarded Strings Automata on guarded strings (AGS), also known as automata with tests, were introduced in [12]. They are a generalization of ordinary fi... |

34 | F.: Kleene algebra with tests: Completeness and decidability
- Kozen, Smith
- 1996
(Show Context)
Citation Context ...expressions e, GS(e) = L(e). Thus the set accepted by the automaton (Exp, D, E, e) is GS(e). Proof. We wish to show that for all x ∈ GS, x ∈ GS(e) iff L(e)(x) = 1. By the completeness theorem for KAT =-=[13]-=-, we have x ∈ GS(e) iff x ≤ e, so it suffices to show that x ≤ e iff L(e)(x) = 1. We proceed by induction on the length of x. The basis for x an atom α is immediate from the definition of Eα. For x = ... |

28 | Automata on guarded strings and applications
- Kozen
- 2001
(Show Context)
Citation Context ...hen and Pucella develop a coalgebraic theory of KAT inspired by Rutten’s coalgebraic theory of KA based on deterministic automata [14]. Remarking that “the known automatatheoretic presentation of KAT =-=[12]-=- does not lend itself to a coalgebraic theory,” and that “the notion of derivative, essential to the coinduction proof principle in this context, is not readily definable for KAT expressions as define... |

18 | A Kleene theorem for polynomial coalgebras
- Bonsangue, Rutten, et al.
- 2009
(Show Context)
Citation Context ...dependence, fixed variable ordering, and pseudo-bisimulation do not arise in this setting. This treatment places KCT within the general coalgebraic framework described by Bonsangue, Rutten, and Silva =-=[1, 2]-=-. We also give a complexity analysis of the coinductive proof principle. We show that an efficient implementation is tantamount to the construction of nondeterministic automata from the given expressi... |

12 |
Regular Expressions and the Equivalence of Programs
- Kaplan
- 1969
(Show Context)
Citation Context ...dinary finite-state automata to include tests. An ordinary automaton with null transitions is an AGS over the two-element Boolean algebra. 3.1 Guarded Strings Guarded strings were first introduced in =-=[7]-=-. A guarded string over Σ, T is an alternating sequence α0p0α1p1 · · · αn−1pn−1αn, where pi ∈ Σ and the αi are atoms (minimal nonzero elements) of the free Boolean algebra B generated by T . The set o... |

7 | R.: A coalgebraic approach to Kleene algebra with tests. Theoretical Computer Science 327(1-2
- Chen, Pucella
(Show Context)
Citation Context ...a number of areas, including the verification of compiler 1optimizations and communication protocols and various other program analysis tasks. Traditionally, KAT is axiomatized equationally [10]. In =-=[4]-=-, Chen and Pucella develop a coalgebraic theory of KAT inspired by Rutten’s coalgebraic theory of KA based on deterministic automata [14]. Remarking that “the known automatatheoretic presentation of K... |

7 | Automatic Proof Generation in Kleene Algebra
- Worthington
- 2008
(Show Context)
Citation Context ...ofs can be generated purely mechanically via repeated application of the Brzozowski derivative, whereas classical equational logic “requires creativity” [4]. This is not strictly true, as Worthington =-=[16]-=- has shown that equational proofs can also be generated automatically. However, it is fair to say that the coinductive approach does provide a more natural mechanism. Still unresolved is the issue of ... |

2 | Regular expressions for polynomial coalgebras
- Bonsangue, Rutten, et al.
- 2007
(Show Context)
Citation Context ...dependence, fixed variable ordering, and pseudo-bisimulation do not arise in this setting. This treatment places KCT within the general coalgebraic framework described by Bonsangue, Rutten, and Silva =-=[1, 2]-=-. We also give a complexity analysis of the coinductive proof principle. We show that an efficient implementation is tantamount to the construction of nondeterministic automata from the given expressi... |