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## Automated reasoning in Kleene algebra (2007)

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Venue: | CADE 2007, LNCS 4603 |

Citations: | 25 - 11 self |

### Citations

1758 |
The Temporal Logic of Reactive and Concurrent Systems Specification
- Manna, Pnueli
- 1992
(Show Context)
Citation Context ... ]p, where x stands for an arbitrary action. We can also define the initial state by initx = [x|0; it is the set of states with no xpredecessors. A set of axioms has been proposed by Manna and Pnueli =-=[17]-=- and further been adapted by von Karger [25] to a setting of second-order quantales. However, von Karger’s axioms can easily be translated to the first-order setting of modal Kleene algebras. |(px) ∗ ... |

992 | Dynamic Logic
- Harel
- 1984
(Show Context)
Citation Context ...amic Logics Modal Kleene algebras are very similar to propositional dynamic logics. More precisely, they are strongly related to variants of dynamic algebras developed by Kozen, Parikh and Pratt (cf. =-=[12]-=-). The axioms of dynamics algebras look like those of Kleene modules, but the induction axiom of Kleene modules is replaced by Segerberg’s induction axiom |x ∗ 〉p − p ≤ |x ∗ 〉(|x〉p − p), (5) with p − ... |

613 |
Modal Logic
- BLACKBURN, RIJKE, et al.
- 2002
(Show Context)
Citation Context ...n example from modal logics to further demonstrate the balance between expressive and computational power of modal Kleene algebras. We will automate a modal correspondence proof of Löb’s formula (cf. =-=[5]-=-), which in modal logic expresses well-foundedness of a transitive relation. In its usual form it is written as ✷(✷p → p) → p. To represent it algebraically, we first replace ✷ by |x] and then dualise... |

353 |
Regular algebra and finite machines
- Conway
- 1971
(Show Context)
Citation Context .... Generation of Kleene algebras with Mace4, however, requires isomorphism checking and therefore storing models (7 elements need > 2GB RAM). Interestingly, Conway’s classical book on regular algebras =-=[7]-=- lists 21 Kleene algebras with four elements. But his examples (5.) and (7.) are flawed and another one is missing. According to the Mace4 manual, the results of the isomorphism checker should be take... |

242 | A completeness theorem for kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...ication: domain-specific algebras for standard provers instead of domain-specific provers for standard algebras. More concretely, we investigate the potential of automated reasoning in Kleene algebra =-=[15]-=- with the resolutionand paramodulation-based Prover9 and the counterexample generator Mace4 [2]. Over the last few years, variants of Kleene algebras emerged as fundamental structures in computing. Th... |

102 |
Term Rewriting Systems
- Terese
- 2003
(Show Context)
Citation Context ...em for abstract reduction systems. The Church-Rosser theorem is usually proved by induction over the number of y ∗ x ∗ -peaks that arise from (x + y) ∗ , i.e., with an external induction measure (cf. =-=[24]-=-). However, equational proofs with the internal induction provided by Kleene algebra can also be given [22]. We can automatically prove the reduction law in about 3min from scratch; we can also automa... |

64 |
Commutation, transformation, and termination
- Bachmair, Dershowitz
- 1986
(Show Context)
Citation Context ...considerable part of diagrammatic reasoning [10] which can therefore be verified by using a theorem prover in the background. Our second example is a separation theorem due to Bachmair and Dershowitz =-=[4]-=-. It states that, in the presence of a suitable commutation condition, concurrent processes terminate iff individual processes do. The theorem can be specified and proved by hand in omega algebra [22]... |

56 | On Hoare logic and Kleene algebra with tests
- Kozen
- 2000
(Show Context)
Citation Context ...tests from a given set of tests. This can be checked with Prover9. Idempotent semirings with tests are expressive enough for (indirectly) encoding Hoare logic without the assignment and the loop-rule =-=[16]-=-. Validity of a 3sHoare triple {p}x{q} is captured by px¬q = 0: no action x transforms a precondition p into a postcondition ¬q. We will discuss and automation of Hoare logic and the associated weakes... |

54 | Resolution-based methods for modal logics
- Nivelle, Schmidt, et al.
(Show Context)
Citation Context ... Kleene algebras. This treatment of modal logic is completely axiomatic whereas previous approaches usually translate the Kripke semantics for modal logics more indirectly into first-order logic (cf. =-=[20, 8]-=-). These translational approaches therefore reason in one particular model whereas ours, beyond relations, also covers models based on traces, paths and languages. Finally, an extension to first-order... |

53 | Kleene algebra with domain
- Desharnais, Möller, et al.
- 2003
(Show Context)
Citation Context ... possible to prove theorems of considerable complexity and practical relevance. Our main contributions are as follows: First, we specify Kleene algebras, omega algebras [6] and their modal extensions =-=[9, 18]-=- in Prover9 and Mace4. We chose this particular tool primarily because it integrates automated deduction with counterexample search. Any other paramodulation-based theorem prover should lead us to sim... |

48 |
Separation and reduction
- Cohen
- 2000
(Show Context)
Citation Context ...rem proving technology makes it possible to prove theorems of considerable complexity and practical relevance. Our main contributions are as follows: First, we specify Kleene algebras, omega algebras =-=[6]-=- and their modal extensions [9, 18] in Prover9 and Mace4. We chose this particular tool primarily because it integrates automated deduction with counterexample search. Any other paramodulation-based t... |

42 | More Church-Rosser proofs (in Isabelle/HOL - Nipkow - 1996 |

25 |
D.: Encoding two-valued nonclassical logics in classical logic
- Ohlbach, Nonnengart, et al.
- 2001
(Show Context)
Citation Context ... Kleene algebras. This treatment of modal logic is completely axiomatic whereas previous approaches usually translate the Kripke semantics for modal logics more indirectly into first-order logic (cf. =-=[20, 8]-=-). These translational approaches therefore reason in one particular model whereas ours, beyond relations, also covers models based on traces, paths and languages. Finally, an extension to first-order... |

20 | B.: Temporal algebra
- Karger
- 1998
(Show Context)
Citation Context ... We can also define the initial state by initx = [x|0; it is the set of states with no xpredecessors. A set of axioms has been proposed by Manna and Pnueli [17] and further been adapted by von Karger =-=[25]-=- to a setting of second-order quantales. However, von Karger’s axioms can easily be translated to the first-order setting of modal Kleene algebras. |(px) ∗ 〉q = q + p|x〉|(px) ∗ 〉q, 〈(xp) ∗ |q = q + p〈... |

18 |
Algebras of modal operators and partial correctness, Theor
- Möller, Struth
(Show Context)
Citation Context ... possible to prove theorems of considerable complexity and practical relevance. Our main contributions are as follows: First, we specify Kleene algebras, omega algebras [6] and their modal extensions =-=[9, 18]-=- in Prover9 and Mace4. We chose this particular tool primarily because it integrates automated deduction with counterexample search. Any other paramodulation-based theorem prover should lead us to sim... |

12 | D.: KAT-ML: An interactive theorem prover for Kleene algebra with tests
- Aboul-Hosn, Kozen
(Show Context)
Citation Context ...at yields particularly short and abstract proofs, and they are supported by powerful automata-based decision procedures. Kleene algebras have already been integrated into higher-order theorem provers =-=[21, 14, 3]-=- and their applicability as a formal method has successfully been demonstrated in that setting. But their potential for automated deduction has not yet been explored. At first sight, feeding an automa... |

9 | Calculating Church-Rosser proofs in Kleene algebra
- Struth
- 2002
(Show Context)
Citation Context ...at yields particularly short and abstract proofs, and they are supported by powerful automata-based decision procedures. Kleene algebras have already been integrated into higher-order theorem provers =-=[21, 14, 3]-=- and their applicability as a formal method has successfully been demonstrated in that setting. But their potential for automated deduction has not yet been explored. At first sight, feeding an automa... |

9 |
G.: Can refinement be automated
- Höfner, Struth
- 2007
(Show Context)
Citation Context ...bvious axioms. Using this alternative approach, we could automatically verify some key refinement laws for concurrent systems, which are far more sophisticated than the examples treated in this paper =-=[13]-=-. The equational coding failed on most of these examples. Second, an integration of domain-specific solvers and decision procedures promises a full automation of partial correctness analysis of progra... |

7 |
G.: Kleene modules
- Ehm, Möller, et al.
- 2004
(Show Context)
Citation Context ...They can be automatically verified; only the last implication requires a simple interaction. The relevance of this alternative approach to modalities over Kleene algebras has further been explored in =-=[11]-=-. Essentially, the above laws define a Kleene module, a two-sorted structure over a Kleene algebra and a Boolean algebra in which the diamond operator acts as a scalar product. By using Kleene modules... |

6 | Calculational relation-algebraic proofs in Isabelle/Isar
- Kahl
- 2004
(Show Context)
Citation Context ...at yields particularly short and abstract proofs, and they are supported by powerful automata-based decision procedures. Kleene algebras have already been integrated into higher-order theorem provers =-=[21, 14, 3]-=- and their applicability as a formal method has successfully been demonstrated in that setting. But their potential for automated deduction has not yet been explored. At first sight, feeding an automa... |

6 | Superposition Theorem Proving for Commutative Algebraic Theories - Stuber - 1999 |

5 |
G.: Diagram chase in relational system development
- Ebert, Struth
- 2005
(Show Context)
Citation Context ... level that is suitable for automation. Reasoning about abstract reduction systems is traditionally diagrammatic. Kleene algebra provides a semantics for a considerable part of diagrammatic reasoning =-=[10]-=- which can therefore be verified by using a theorem prover in the background. Our second example is a separation theorem due to Bachmair and Dershowitz [4]. It states that, in the presence of a suitab... |

3 | Cancellative Abelian Monoids in Refutational Theorem Proving. Dissertation, Universitat des Saarlandes
- Waldmann
- 1997
(Show Context)
Citation Context ...wever, this task seems still a challenge. Over the last decades, considerable effort has been put into the development of special purpose calculi for automated deduction with algebraic theories, e.g. =-=[26, 23]-=-, but the practical impact on formal methods has been rather limited. Nevertheless, the specific balance of expressive and computational power and the user-friendliness of automated deduction could co... |

2 |
Abstract abstract reduction
- Struth
- 2006
(Show Context)
Citation Context ...x ∗ (yx ∗ ) ∗ connect concurrency with interleaving while separation laws infer global system properties from those of the particular processes. Kleene algebras are very useful for deriving such laws =-=[6, 22]-=-. We present two examples that show how such derivations can be automated. Our first example is the reduction law y ∗ x ∗ ≤ x ∗ y ∗ ⇒ (x + y) ∗ ≤ x ∗ y ∗ which says that repeated concurrent executions... |