## Higher-order syntax and saturation algorithms for hybrid logic

Venue: | In International Workshop on Hybrid Logic 2006 (HyLo 2006 |

Citations: | 9 - 5 self |

### BibTeX

@INPROCEEDINGS{Hardt_higher-ordersyntax,

author = {Moritz Hardt and Gert Smolka},

title = {Higher-order syntax and saturation algorithms for hybrid logic},

booktitle = {In International Workshop on Hybrid Logic 2006 (HyLo 2006},

year = {}

}

### OpenURL

### Abstract

We present modal logic on the basis of the simply typed lambda calculus with a system of equational deduction. Combining first-order quantification and higher-order syntax, we can maintain modal reasoning in terms of classical logic by remarkably simple means. Such an approach has been broadly uninvestigated, even though it has notable advantages, especially in the case of Hybrid Logic. We develop a tableau-like semi-decision procedure and subsequently a decision procedure for an alternative characterization of

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Citation Context ...ronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcssTaking these points into consideration, we develop a formulation of modal logic based on the simply typed lambda calculus =-=[7]-=- with a system of equational deduction [14]. By combining first-order quantification and higher-order syntax, Kripke semantics is fully internalized in our approach. With a few key definitions, syntax... |

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Citation Context ... the access relation itself is a formula. But then, “Bridge” is just a special case of (C ˙=): {Ruv, v ˙=a} → {. . . , Rua}. In contrast to Blackburn’s system, many modal tableaux as those of Fitting =-=[8,10]-=- and Gabbay [12] maintain external state labels and access information. It turns out that both of these can be naturally represented at the object level. 5 Saturation in Polynomial Space In this secti... |

221 | First-Order Modal Logic
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Citation Context ...dal syntax is preserved on a notational level. This way, standard translations are obsolete. Along the line, our approach remedies the problems with β-conversion as they appear in the work of Fitting =-=[11,9]-=- who gives examples of β/η-equal modal terms denoting differently, which is clearly in conflict with the lambda calculus. In order to avoid phenomena like these, we make names for points of evaluation... |

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Citation Context ...on itself is a formula. But then, “Bridge” is just a special case of (C ˙=): {Ruv, v ˙=a} → {. . . , Rua}. In contrast to Blackburn’s system, many modal tableaux as those of Fitting [8,10] and Gabbay =-=[12]-=- maintain external state labels and access information. It turns out that both of these can be naturally represented at the object level. 5 Saturation in Polynomial Space In this section we describe a... |

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Citation Context ... In a common formulation, often referred to as HL(@, ↓), Hybrid Logic enriches the minimal modal logic K with naming, binding, and referencing constructions, as well as an implicit identity judgement =-=[5,1,4]-=-. We offer an equivalent logic called MFI that extends MF with the identity predicate. Once HL(@) is characterized as the monadic fragment of MFI, we develop a tableau calculus and a tableau-based dec... |

92 | A road-map on complexity for hybrid logics
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Citation Context ... section we describe a decision procedure for the satisfiability problem of MFI1 based on our saturation rules. A central feature of this algorithm is that it matches the PSPACE lower bound for HL(@) =-=[2]-=-. In the case of MF, that is in the absence of identities, a simple standard technique yields an optimal tableau-based decision procedure. We describe this technique as depth-first saturation: When a ... |

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Citation Context ...ed Tzakova’s system by a treatment of the universal modalities and a proof of termination using loop-checking techniques. A semi-decision procedure in form of a tableau calculus is given by Blackburn =-=[3]-=- who discusses the advantages of internalizing labeled deduction in Hybrid Logic. However, none of these results is optimal with respect to computational complexity. Contributions and Overview In Sect... |

29 |
Tableau calculi for hybrid logics
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Citation Context ... predicate. Once HL(@) is characterized as the monadic fragment of MFI, we develop a tableau calculus and a tableau-based decision procedure for this logic. The first such algorithm is due to Tzakova =-=[15]-=-. Recently, Bolander and Braüner [6] extended Tzakova’s system by a treatment of the universal modalities and a proof of termination using loop-checking techniques. A semi-decision procedure in form o... |

21 | Braüner T.: Tableau-based Decision Procedures for Hybrid Logic
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Citation Context ...ed as the monadic fragment of MFI, we develop a tableau calculus and a tableau-based decision procedure for this logic. The first such algorithm is due to Tzakova [15]. Recently, Bolander and Braüner =-=[6]-=- extended Tzakova’s system by a treatment of the universal modalities and a proof of termination using loop-checking techniques. A semi-decision procedure in form of a tableau calculus is given by Bla... |

6 | Hybrid logic is the bounded fragment of first order logic
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Citation Context ... In a common formulation, often referred to as HL(@, ↓), Hybrid Logic enriches the minimal modal logic K with naming, binding, and referencing constructions, as well as an implicit identity judgement =-=[5,1,4]-=-. We offer an equivalent logic called MFI that extends MF with the identity predicate. Once HL(@) is characterized as the monadic fragment of MFI, we develop a tableau calculus and a tableau-based dec... |

1 | Higher-order modal logic – a sketch
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(Show Context)
Citation Context ...dal syntax is preserved on a notational level. This way, standard translations are obsolete. Along the line, our approach remedies the problems with β-conversion as they appear in the work of Fitting =-=[11,9]-=- who gives examples of β/η-equal modal terms denoting differently, which is clearly in conflict with the lambda calculus. In order to avoid phenomena like these, we make names for points of evaluation... |

1 |
Gödel’s god,” Trends
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Citation Context ... the access relation itself is a formula. But then, “Bridge” is just a special case of (C ˙=): {Ruv, v ˙=a} → {. . . , Rua}. In contrast to Blackburn’s system, many modal tableaux as those of Fitting =-=[8,10]-=- and Gabbay [12] maintain external state labels and access information. It turns out that both of these can be naturally represented at the object level. 5 Saturation in Polynomial Space In this secti... |

1 |
Bachelor’s Thesis: Hybrid Logic Revisited
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Citation Context ...ability problem of monadic MFI. We are not aware of any previous tableau-based result that matches the PSPACE lower bound for this problem. Extended discussion of the sections 3 and 4 can be found in =-=[13]-=-.s2 The Logical Base We consider a simply typed lambda calculus where every term has a unique type. Interpretation for terms and types is provided by standard semantics. In our system, logical constan... |

1 | Lecture Notes: Introduction to Computational Logic - Smolka - 2006 |