### Citations

1053 |
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
- Nipkow, Paulson, et al.
(Show Context)
Citation Context ...o evaluate all the ATP systems for higher-order reasoning that are integrated in SystemOnTPTP w.r.t. their capabilities for automated reasoning about quantales. In detail, these systems are IsabelleP =-=[24]-=-, LEO-II [5], Satallax [2] and TPS [1]. We have chosen the structure of quantales due to several reasons: • From a mathematical point of view quantales are special cases of the first-order structures ... |

995 |
A Formulation of the Simple Theory of Types
- Church
- 1940
(Show Context)
Citation Context ... aim is again full automation and no user interaction. As a part of TPTP, higher-order logic can now be expressed by the typed higher-order form (THF) which implements Church’s simple theory of types =-=[10]-=-. This theory is based on the simply typed λ-calculus in which functional types are formed from basic types. The decision for simple type theory was made since this theory is already used as a common ... |

428 |
The logic of quantum mechanics.
- Birkhoff, Neumann
- 1936
(Show Context)
Citation Context ...raic reasoning with quantales can be used to verify properties about safety and liveness at an abstract level. Physics: Originally, quantales were derived for modelling phenomena of quantum mechanics =-=[8]-=-. But quantales can also be used to formalise quantum logic — a logic defined for quantum physics [19, 25]. This closes our small list of prospective new applications for quantales where automated rea... |

356 |
Regular Algebra and Finite Machines.
- Conway
- 1971
(Show Context)
Citation Context ...) ⊔ ( · x = yi · x ) . (∗) Moreover 1 is required to be the identity of multiplication, i.e., x · 1 = 1 · x = x. The notion of a quantale is equivalent to Conway’s notion of a standard Kleene algebra =-=[11]-=- and forms a special idempotent semiring. As an immediate consequence of the definition, multiplication is strict, i.e., x · 0 = 0 · x = 0 for all x ∈ S. We now have a closer look at the axiomatisatio... |

250 | A completeness theorem for Kleene algebras and the algebra of regular events.
- Kozen
- 1994
(Show Context)
Citation Context ...his axiom. However THF does not allow arithmetic at the moment. Hence we do not discuss the star operator in this paper though we are aware of equivalent axioms that only need first-order logics (see =-=[20]-=-). 3 THF and Church’s Simple Type Theory In this section we sketch the higher-order approach in the TPTP problem library. We will only mention those points that are necessary for the encoding of quant... |

225 |
The design and implementation of Vampire.
- Riazanov, Voronkov
- 2002
(Show Context)
Citation Context ...has been achieved with ATP, are software verification and mathematics. Full automation (without user interaction) is often only possible by first-order logic. For these tasks ATP systems like Vampire =-=[26]-=- and Prover9 [21] exist and SystemOnTPTP [28] provides a common language and a common interface. First-order ATP systems and the TPTP library have extensively been used by different users in various c... |

106 |
Quantales and (non-commutative) Linear Logic’,
- Yetter
- 1990
(Show Context)
Citation Context ...echanics. Later, this algebraic structure found various fields of application. Examples are classical logic like CTL and CTL ∗ [22], non-classical logic like separation logic or non-commutative logic =-=[12, 25, 32]-=- as well as reasoning within hybrid systems [14]. Further examples can be found in [27]. There are two main contributions of the paper: First we develop the above-mentioned case study. As far as we kn... |

49 |
1975/79]: The Logico-Algebraic Approach to Quantum Mechanics
- HOOKER
(Show Context)
Citation Context ...ct level. Physics: Originally, quantales were derived for modelling phenomena of quantum mechanics [8]. But quantales can also be used to formalise quantum logic — a logic defined for quantum physics =-=[19, 25]-=-. This closes our small list of prospective new applications for quantales where automated reasoning could be applied. However, as mentioned before, tackling these problems is not feasible at the mome... |

34 | Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure.
- Sutcliffe, Benzmuller
- 2010
(Show Context)
Citation Context ...dies covering various areas of sciences (e.g., [9, 15, 17, 18, 31]). Recently, TPTP and SystemOnTPTP were extended to cover not only first-order reasoning but also reasoning within higher-order logic =-=[6, 29]-=-. A general aim is again full automation and no user interaction. As a part of TPTP, higher-order logic can now be expressed by the typed higher-order form (THF) which implements Church’s simple theor... |

34 | Evaluating General Purpose Automated Theorem Proving Systems,
- Sutcliffe, Suttner
- 2001
(Show Context)
Citation Context ...ating system. We set a CPU time limit of 300 s, which is known to be sufficient for the ATP systems to prove almost all the theorems they would be able to prove even with a significantly higher limit =-=[30]-=-. The results of this testing are shown in Table 1; a number indicates that a proof is found in that time, and a “–” indicates that the system reaches the time limit or gives up before reaching this l... |

31 |
C.E.: TPS: A hybrid automatic-interactive system for developing proofs.
- Andrews, Brown
- 2006
(Show Context)
Citation Context ...er-order reasoning that are integrated in SystemOnTPTP w.r.t. their capabilities for automated reasoning about quantales. In detail, these systems are IsabelleP [24], LEO-II [5], Satallax [2] and TPS =-=[1]-=-. We have chosen the structure of quantales due to several reasons: • From a mathematical point of view quantales are special cases of the first-order structures of semirings and Kleene algebras. Sinc... |

29 | Automated reasoning in Kleene algebra.
- Hofner, Struth
- 2007
(Show Context)
Citation Context ... a common language and a common interface. First-order ATP systems and the TPTP library have extensively been used by different users in various case studies covering various areas of sciences (e.g., =-=[9, 15, 17, 18, 31]-=-). Recently, TPTP and SystemOnTPTP were extended to cover not only first-order reasoning but also reasoning within higher-order logic [6, 29]. A general aim is again full automation and no user intera... |

25 | C.E.: Analytic Tableaux for Higher-Order Logic with Choice.
- Backes, Brown
- 2010
(Show Context)
Citation Context ...ems for higher-order reasoning that are integrated in SystemOnTPTP w.r.t. their capabilities for automated reasoning about quantales. In detail, these systems are IsabelleP [24], LEO-II [5], Satallax =-=[2]-=- and TPS [1]. We have chosen the structure of quantales due to several reasons: • From a mathematical point of view quantales are special cases of the first-order structures of semirings and Kleene al... |

19 | Algebra and sequent calculus for epistemic actions.
- Baltag, Coecke, et al.
- 2005
(Show Context)
Citation Context ...itive) linear logic is more suitable — its connection to quantales is discussed in [32]. Last but not least we want to mention that quantales are also used for reasoning about dynamic epistemic logic =-=[3, 4]-=-. This logic is used to model multi-agent systems and phenomena in philosophy. Computer science: Besides reasoning in logic, quantales have further applications. For this paper we only mention hybrid ... |

16 | Combined Reasoning by Automated Cooperation
- Benzmuller, Sorge, et al.
- 2007
(Show Context)
Citation Context ...xiomatisation for quantales we have given is quite natural. Moreover, it has been stated that set-theoretic theorems are solved more efficiently in higher-order logic than using first-order encodings =-=[7]-=-. 5 Automating Basic Properties In this section we use the encoding of Section 4 to automatically verify basic properties of quantales. We proved around 50 theorems, in particular theorems that involv... |

15 | On automating the calculus of relations.
- Hofner, Struth
- 2008
(Show Context)
Citation Context ...om a mathematical point of view quantales are special cases of the first-order structures of semirings and Kleene algebras. Since the latter structures are particular suitable for automated reasoning =-=[15, 16, 17]-=-, the hope is that quantales are also suitable. 40 R.A.Schmidt, S.Schulz, B.Konev (eds.); PAAR-2010; Volume 123, pp. 40–51Automated Higher-order Reasoning about Quantales Dang and Höfner • Following ... |

14 | THF0—The core of the TPTP language for higher-order logic
- Benzmüller, Rabe, et al.
- 2008
(Show Context)
Citation Context ...dies covering various areas of sciences (e.g., [9, 15, 17, 18, 31]). Recently, TPTP and SystemOnTPTP were extended to cover not only first-order reasoning but also reasoning within higher-order logic =-=[6, 29]-=-. A general aim is again full automation and no user interaction. As a part of TPTP, higher-order logic can now be expressed by the typed higher-order form (THF) which implements Church’s simple theor... |

12 | Algebraic Separation Logic.
- Dang, Hofner, et al.
- 2011
(Show Context)
Citation Context ...echanics. Later, this algebraic structure found various fields of application. Examples are classical logic like CTL and CTL ∗ [22], non-classical logic like separation logic or non-commutative logic =-=[12, 25, 32]-=- as well as reasoning within hybrid systems [14]. Further examples can be found in [27]. There are two main contributions of the paper: First we develop the above-mentioned case study. As far as we kn... |

12 | System description: SystemOnTPTP
- Sutcliffe
- 2000
(Show Context)
Citation Context ...fication and mathematics. Full automation (without user interaction) is often only possible by first-order logic. For these tasks ATP systems like Vampire [26] and Prover9 [21] exist and SystemOnTPTP =-=[28]-=- provides a common language and a common interface. First-order ATP systems and the TPTP library have extensively been used by different users in various case studies covering various areas of science... |

11 |
Quantales and their applications, volume 234
- Rosenthal
- 1990
(Show Context)
Citation Context ...ssical logic like CTL and CTL ∗ [22], non-classical logic like separation logic or non-commutative logic [12, 25, 32] as well as reasoning within hybrid systems [14]. Further examples can be found in =-=[27]-=-. There are two main contributions of the paper: First we develop the above-mentioned case study. As far as we know it is one of the largest proof experiments of THF. As a consequence, we show that fu... |

9 |
Formal design verification of digital systems
- Wojcik
(Show Context)
Citation Context ... a common language and a common interface. First-order ATP systems and the TPTP library have extensively been used by different users in various case studies covering various areas of sciences (e.g., =-=[9, 15, 17, 18, 31]-=-). Recently, TPTP and SystemOnTPTP were extended to cover not only first-order reasoning but also reasoning within higher-order logic [6, 29]. A general aim is again full automation and no user intera... |

8 | Automated verification of refinement laws
- Höfner, Struth, et al.
(Show Context)
Citation Context ... a common language and a common interface. First-order ATP systems and the TPTP library have extensively been used by different users in various case studies covering various areas of sciences (e.g., =-=[9, 15, 17, 18, 31]-=-). Recently, TPTP and SystemOnTPTP were extended to cover not only first-order reasoning but also reasoning within higher-order logic [6, 29]. A general aim is again full automation and no user intera... |

8 | Quantales and temporal logics
- Möller, Höfner, et al.
- 2006
(Show Context)
Citation Context ...Originally, quantales were introduced to formalise phenomena of quantum mechanics. Later, this algebraic structure found various fields of application. Examples are classical logic like CTL and CTL ∗ =-=[22]-=-, non-classical logic like separation logic or non-commutative logic [12, 25, 32] as well as reasoning within hybrid systems [14]. Further examples can be found in [27]. There are two main contributio... |

7 | The LEO-II project
- Benzmüller, Paulson, et al.
- 2007
(Show Context)
Citation Context ...l the ATP systems for higher-order reasoning that are integrated in SystemOnTPTP w.r.t. their capabilities for automated reasoning about quantales. In detail, these systems are IsabelleP [24], LEO-II =-=[5]-=-, Satallax [2] and TPS [1]. We have chosen the structure of quantales due to several reasons: • From a mathematical point of view quantales are special cases of the first-order structures of semirings... |

6 |
Database for automated proofs of Kleene algebra. http://www.kleenealgebra.de (accessed
- Höfner
- 2011
(Show Context)
Citation Context ...α → $o and x has type α. In the remainder we give an extract of the complete input file to demonstrate the encoding. A full encoding of the Axioms (1)–(6) is given in the Appendix 1 and at a web site =-=[13]-=-. We only consider the supremum definition here since it forms the most interesting part of the encoding. 14 thf(sup,type,( 15 sup: ( ( $i > $o ) > $i ) )). The formula defines the type of the supremu... |

5 | Bommel. Automated Theorem Proving for Quality-checking Medical Guidelines
- Hommersom, Lucas, et al.
- 2005
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4 |
Applied Theorem Proving - Natural Language Testsuite. http://www.coli.unisb.de/ bos/atp
- Bos
- 2000
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4 |
Algebraic Calculi for Hybrid Systems
- Höfner
- 2009
(Show Context)
Citation Context ...fields of application. Examples are classical logic like CTL and CTL ∗ [22], non-classical logic like separation logic or non-commutative logic [12, 25, 32] as well as reasoning within hybrid systems =-=[14]-=-. Further examples can be found in [27]. There are two main contributions of the paper: First we develop the above-mentioned case study. As far as we know it is one of the largest proof experiments of... |

3 | Reasoning about dynamic epistemic logic
- Baltag, Coecke, et al.
- 2004
(Show Context)
Citation Context ...itive) linear logic is more suitable — its connection to quantales is discussed in [32]. Last but not least we want to mention that quantales are also used for reasoning about dynamic epistemic logic =-=[3, 4]-=-. This logic is used to model multi-agent systems and phenomena in philosophy. Computer science: Besides reasoning in logic, quantales have further applications. For this paper we only mention hybrid ... |

2 |
Automated Higher-order Reasoning about Quantales Dang and Höfner
- Quantales
- 2000
(Show Context)
Citation Context ...echanics. Later, this algebraic structure found various fields of application. Examples are classical logic like CTL and CTL ∗ [22], non-classical logic like separation logic or non-commutative logic =-=[12, 25, 32]-=- as well as reasoning within hybrid systems [14]. Further examples can be found in [27]. There are two main contributions of the paper: First we develop the above-mentioned case study. As far as we kn... |