## Multimodal and Intuitionistic Logics in Simple Type Theory

Citations: | 9 - 9 self |

### BibTeX

@MISC{Benzmüller_multimodaland,

author = {C. Benzmüller and L. C. Paulson},

title = {Multimodal and Intuitionistic Logics in Simple Type Theory },

year = {}

}

### OpenURL

### Abstract

We study straightforward embeddings of propositional normal multimodal logic and propositional intuitionistic logic in simple type theory. The correctness of these embeddings is easily shown. We give examples to demonstrate that these embeddings provide an effective framework for computational investigations of various non-classical logics. We report some experiments using the higher-order automated theorem prover LEO-II.

### Citations

847 |
A formulation of the simple theory of types
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- 1940
(Show Context)
Citation Context ... in 〈W, (Rr)r∈S, |=〉 for all possible |=. If s is valid for all possible Kripke frames 〈W, (Rr)r∈S〉 then s is called valid and we write |= K s. Classical higher-order logic or simple type theory ST T =-=[4, 17]-=- is a formalism built on top of the simply typed λ-calculus. The set T of simple types is usually freely generated from a set of basic types {o, ι} (where o denotes the type of Booleans) using the rig... |

304 |
An Introduction To Mathematical Logic and Type Theory: To Truth Through Proof
- Andrews
- 1986
(Show Context)
Citation Context ... in 〈W, (Rr)r∈S, |=〉 for all possible |=. If s is valid for all possible Kripke frames 〈W, (Rr)r∈S〉 then s is called valid and we write |= K s. Classical higher-order logic or simple type theory ST T =-=[4, 17]-=- is a formalism built on top of the simply typed λ-calculus. The set T of simple types is usually freely generated from a set of basic types {o, ι} (where o denotes the type of Booleans) using the rig... |

181 |
Completeness in the Theory of Types
- Henkin
- 1950
(Show Context)
Citation Context ... converted to t using both β and η. For each s ∈ ST T there is a unique β-normal form and a unique βη-normal form. The semantics of ST T is well understood and thoroughly documented in the literature =-=[1, 2, 12, 24]-=-; our summary below is adapted from Andrews [5]. A frame is a collection {Dα}α∈T of nonempty domains (sets) Dα, such that Do = {T, F } (where T represents truth and F represents falsehood). The Dα→β a... |

148 |
Type-Logical Semantics
- Carpenter
- 1997
(Show Context)
Citation Context ... first. He presents an embedding of modal logic into a 2-sorted type theory. This idea is picked up by Gamut [20] and a related embedding has recently been studied by Hardt and Smolka [22]. Carpenter =-=[16]-=- proposes to use lifted connectives, an idea that also underlies the embeddings presented by Merz [27], Brown [15], Harrison [23, Chap. 20], and Kaminski and Smolka [25]. In this article we pick up an... |

125 | E – A Brainiac Theorem Prover
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(Show Context)
Citation Context ...P/D070511/1] and by the German Research Foundation (DFG) [grant number BE 2501/6-1].within our higher-order theorem prover LEO-II [14] via cooperation with the first-order automated theorem prover E =-=[31]-=-. We also discuss some challenges to higher-order reasoning implied by this application direction. New in this article with regard to previous (non-reviewed) work [13, 14] are the soundness and comple... |

106 | Resolution in Type Theory
- Andrews
- 1971
(Show Context)
Citation Context ...e is a unique β-normal form and a unique βη-normal form. The semantics of ST T is well understood and thoroughly documented in the literature [1, 2, 12, 24]; our summary below is adapted from Andrews =-=[5]-=-. A frame is a collection {Dα}α∈T of nonempty domains (sets) Dα, such that Do = {T, F } (where T represents truth and F represents falsehood). The Dα→β are collections of functions mapping Dα into Dβ.... |

85 | A resolution calculus for modal logics
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(Show Context)
Citation Context ...utomate reasoning in modal logics: the direct approach and the translational approach. The direct approach [9, 10, 18, 28] develops specific calculi and tools for the task; the translational approach =-=[29, 30]-=- transforms modal logic formulas into first-order logic and applies standard first-order tools. Embeddings of modal logics into higher-order logic, however, have not yet been widely studied, although ... |

77 |
Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse
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(Show Context)
Citation Context ...pleteness proofs for the embedding. So far this has only been proven for the monomodal logics S4 and T [11]. In the second part of this article we then combine our results with Gödel’s interpretation =-=[21]-=- of propositional intuitionistic logic in modal logic S4 to obtain a sound and complete embedding of propositional intuitionistic logic in simple type theory. 2 Preliminaries We assume familiarity wit... |

71 | Proofs in higher-order logic
- Miller
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(Show Context)
Citation Context ...ation cannot be synthesized in LEO-II by higher-order pre-unification. In fact, LEO-II needs to guess an appropriate instantiation by applying primitive substitutions (also called set instantiations) =-=[3]-=- and what LEO-II essentially proposes based on primitive substitutions is to consider the universal relation as a candidate relation (a predecessor version of LEO-II suggested the equality relation [1... |

69 | TPS: A Theorem Proving System for Classical Type Theory
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- 1996
(Show Context)
Citation Context ... are encouraging [13]. For instance, Example 2 can be proven automatically in less than 0.1 seconds on a standard notebook computer and similar performance results can be achieved with the prover TPS =-=[6, 7]-=-. We can even use higher-order theorem provers to investigate meta-theoretic properties of various modal logics automatically. This issue has been studied in our previous work [13]; here we give two e... |

52 |
Some theorems about the sentential calculi of Lewis and Heyting
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- 1948
(Show Context)
Citation Context ... tι→o where tι→o is the IPL ST T term for t according to Definition 5. Alternatively we may translate t into t ′′ : t ′′ := (refl r) ∧ (trans r) ⊃ valid tι→o Combining soundness [21] and completeness =-=[26]-=- of Gödel’s embedding with our soundness and completeness theorems 1 and 2 we obtain the following corollary: Corollary 1 (Soundness and Completeness of Embedding IPL ST T ). Let t ∈ IPL and let t ′ ∈... |

34 |
General models and extensionality
- Andrews
- 1972
(Show Context)
Citation Context ... converted to t using both β and η. For each s ∈ ST T there is a unique β-normal form and a unique βη-normal form. The semantics of ST T is well understood and thoroughly documented in the literature =-=[1, 2, 12, 24]-=-; our summary below is adapted from Andrews [5]. A frame is a collection {Dα}α∈T of nonempty domains (sets) Dα, such that Do = {T, F } (where T represents truth and F represents falsehood). The Dα→β a... |

34 | LEO-II - a cooperative automatic theorem prover for higher-order logic
- Benzmüller, Paulson, et al.
- 2008
(Show Context)
Citation Context ...he Engineering and Physical Sciences Research Council [grant number EP/D070511/1] and by the German Research Foundation (DFG) [grant number BE 2501/6-1].within our higher-order theorem prover LEO-II =-=[14]-=- via cooperation with the first-order automated theorem prover E [31]. We also discuss some challenges to higher-order reasoning implied by this application direction. New in this article with regard ... |

33 |
General Models, Descriptions, and Choice in Type Theory
- Andrews
- 1972
(Show Context)
Citation Context ... converted to t using both β and η. For each s ∈ ST T there is a unique β-normal form and a unique βη-normal form. The semantics of ST T is well understood and thoroughly documented in the literature =-=[1, 2, 12, 24]-=-; our summary below is adapted from Andrews [5]. A frame is a collection {Dα}α∈T of nonempty domains (sets) Dα, such that Do = {T, F } (where T represents truth and F represents falsehood). The Dα→β a... |

24 |
TPS: A hybrid automatic-interactive system for developing proofs
- Andrews, Brown
(Show Context)
Citation Context ... are encouraging [13]. For instance, Example 2 can be proven automatically in less than 0.1 seconds on a standard notebook computer and similar performance results can be achieved with the prover TPS =-=[6, 7]-=-. We can even use higher-order theorem provers to investigate meta-theoretic properties of various modal logics automatically. This issue has been studied in our previous work [13]; here we give two e... |

18 | A framework for modal logic programming
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- 1996
(Show Context)
Citation Context ...tomated theorem prover LEO-II. 1 Introduction There are two well investigated approaches to automate reasoning in modal logics: the direct approach and the translational approach. The direct approach =-=[9, 10, 18, 28]-=- develops specific calculi and tools for the task; the translational approach [29, 30] transforms modal logic formulas into first-order logic and applies standard first-order tools. Embeddings of moda... |

18 |
Higher Order Semantics and Extensionality
- Benzmüller, Brown, et al.
(Show Context)
Citation Context |

14 |
How to use modalities and sorts in Prolog
- Nonnengart
(Show Context)
Citation Context ...utomate reasoning in modal logics: the direct approach and the translational approach. The direct approach [9, 10, 18, 28] develops specific calculi and tools for the task; the translational approach =-=[29, 30]-=- transforms modal logic formulas into first-order logic and applies standard first-order tools. Embeddings of modal logics into higher-order logic, however, have not yet been widely studied, although ... |

11 | Automating access control logics in simple type theory with LEO-II’, in Emerging Challenges for Security
- Benzmüller
- 2009
(Show Context)
Citation Context ...ew in this article with regard to previous (non-reviewed) work [13, 14] are the soundness and completeness proofs for the embedding. So far this has only been proven for the monomodal logics S4 and T =-=[11]-=-. In the second part of this article we then combine our results with Gödel’s interpretation [21] of propositional intuitionistic logic in modal logic S4 to obtain a sound and complete embedding of pr... |

9 | Higher-order syntax and saturation algorithms for hybrid logic
- Hardt, Smolka
(Show Context)
Citation Context ...mention the idea first. He presents an embedding of modal logic into a 2-sorted type theory. This idea is picked up by Gamut [20] and a related embedding has recently been studied by Hardt and Smolka =-=[22]-=-. Carpenter [16] proposes to use lifted connectives, an idea that also underlies the embeddings presented by Merz [27], Brown [15], Harrison [23, Chap. 20], and Kaminski and Smolka [25]. In this artic... |

9 | HOL Light Tutorial (for version 2.20
- Harrison
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Citation Context ...id.3 Propositional Normal Multimodal Logics in Simple Type Theory Simple type theory is an expressive logic and it is thus no surprise that modal logic can be encoded in several ways in it. Harrison =-=[23]-=-, for instance, presents a ‘deep embedding’ of modal logics by formalizing standard Kripke semantics and a ‘shallow embedding’ of the temporal logic LTL. The latter encoding more naturally exploits th... |

9 | Terminating tableaux for hybrid logic with the difference modality and converse
- Kaminski, Smolka
- 2008
(Show Context)
Citation Context ...rdt and Smolka [22]. Carpenter [16] proposes to use lifted connectives, an idea that also underlies the embeddings presented by Merz [27], Brown [15], Harrison [23, Chap. 20], and Kaminski and Smolka =-=[25]-=-. In this article we pick up and extend the embedding of multimodal logics in simple type theory as proposed by Brown [15]. The starting point is a characterization of multimodal logic formulas as par... |

6 |
Intensional and Higher-Order Modal Logic, volume 19 of North-Holland Mathematics Studies
- Gallin
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Citation Context ...der tools. Embeddings of modal logics into higher-order logic, however, have not yet been widely studied, although multimodal logic can be regarded as a natural fragment of simple type theory. Gallin =-=[19]-=- appears to mention the idea first. He presents an embedding of modal logic into a 2-sorted type theory. This idea is picked up by Gamut [20] and a related embedding has recently been studied by Hardt... |

6 | Progress in the development of automated theorem proving for higher-order logic
- Sutcliffe, Benzmüller, et al.
- 2009
(Show Context)
Citation Context ...nary experiments with our approach have been promising [13]. A small corpus of related example problems has meanwhile been entered into the new TPTP library for automated higher-order theorem proving =-=[32]-=- in order to stimulate further experiments with our approach and to foster the improvement of existing higher-order automated theorem provers for the task. Future work includes the study of further em... |

4 |
Festschrift in Honor of Peter B. Andrews on His 70th Birthday, chapter Exploring
- Benzmüller, Paulson
- 2008
(Show Context)
Citation Context ...order automated theorem prover E [31]. We also discuss some challenges to higher-order reasoning implied by this application direction. New in this article with regard to previous (non-reviewed) work =-=[13, 14]-=- are the soundness and completeness proofs for the embedding. So far this has only been proven for the monomodal logics S4 and T [11]. In the second part of this article we then combine our results wi... |

4 |
Fariñas del Cerro. MOLOG: A system that extends PROLOG with modal logic
- Luis
- 1986
(Show Context)
Citation Context ...tomated theorem prover LEO-II. 1 Introduction There are two well investigated approaches to automate reasoning in modal logics: the direct approach and the translational approach. The direct approach =-=[9, 10, 18, 28]-=- develops specific calculi and tools for the task; the translational approach [29, 30] transforms modal logic formulas into first-order logic and applies standard first-order tools. Embeddings of moda... |

3 |
Encoding hybrid logic in higher-order logic. Unpublished slides from an invited talk presented at Loria
- Brown
- 2005
(Show Context)
Citation Context ...and a related embedding has recently been studied by Hardt and Smolka [22]. Carpenter [16] proposes to use lifted connectives, an idea that also underlies the embeddings presented by Merz [27], Brown =-=[15]-=-, Harrison [23, Chap. 20], and Kaminski and Smolka [25]. In this article we pick up and extend the embedding of multimodal logics in simple type theory as proposed by Brown [15]. The starting point is... |

1 | 12-15, 2008, Proceedings, volume 5195 of LNCS - Armando, Baumgartner, et al. |

1 |
Yet another encoding of TLA in Isabelle. Available on the Internet: http://www.loria.fr/~merz/projects/isabelle-tla/doc/design.ps.gz, 1999. Linh Anh Nguyen. A fixpoint semantics and an SLD-resolution calculus for modal logic programs
- Merz
(Show Context)
Citation Context ... Gamut [20] and a related embedding has recently been studied by Hardt and Smolka [22]. Carpenter [16] proposes to use lifted connectives, an idea that also underlies the embeddings presented by Merz =-=[27]-=-, Brown [15], Harrison [23, Chap. 20], and Kaminski and Smolka [25]. In this article we pick up and extend the embedding of multimodal logics in simple type theory as proposed by Brown [15]. The start... |