## A Note on Relativised Products of Modal Logics (2003)

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Venue: | Advances in Modal Logic |

Citations: | 10 - 6 self |

### BibTeX

@INPROCEEDINGS{Kurucz03anote,

author = {Agi Kurucz and Michael Zakharyaschev},

title = {A Note on Relativised Products of Modal Logics},

booktitle = {Advances in Modal Logic},

year = {2003},

pages = {221--242},

publisher = {King’s College Publications}

}

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### Abstract

this paper. each frame of the class.) For example, K is the logic of all n-ary product frames. It is not hard to see that S5 is the logic of all n-ary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cubic universal product S5 -frames. Note that the `i-reduct' F U 1 U n ; R i of F 1 F n is a union of n disjoint copies of F i . Thus, F and F i validate the same formulas, and so L n L 1 L n : There is a strong interaction between the modal operators of product logics. Every n-ary product frame satis es the following two properties, for each pair i 6= j, i; j = 1; : : : ; n: Commutativity : 8x8y8z xR i y ^ yR j z ! 9u (xR j u ^ uR i z) ^ xR j y ^ yR i z ! 9u (xR i u ^ uR j z) Church{Rosser property : 8x8y8z xR i y ^ xR j z ! 9u (yR j u ^ zR i u) This means that the corresponding modal interaction formulas 2 i 2 j p $ 2 j 2 i p and 3 i 2 j p ! 2 j 3 i p belong to every n-dimensional product logic. The geometrically intuitive many-dimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatio-temporal representation and reasoning (see e.g. [33, 34]) or reasoning about the behaviour of multi-agent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexity|even the product of two NP-complete logics can be non-recursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non- nitely axiomatisable [16]

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Citation Context ...he geometrically intuitive many-dimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatio-temporal representation and reasoning (see e.g. =-=[33, 34]-=-) or reasoning about the behaviour of multi-agent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexity|even the product of tw... |

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Citation Context ...merous examples of applications of fusions in description logic). It is this absence of interaction axioms that ensures the transfer of good algorithmic properties from the components to their fusion =-=[17, 6, 32]-=-. In particular, it is possible to reduce reasoning in the fusion to reasoning in the components. On the semantic level, the fusion of Kripke complete modal logics L 1 ; : : : ; L n can be characteris... |

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Citation Context ...ke complete modal logic L is called a subframe logic if the class of Kripke frames for L is closed under taking (not necessarily generated) subframes. (For a general theory of subframe logics consult =-=[5, 2, 31-=-] and references therein.) Typical examples of subframe logics are modal logics whose classes of Kripke frames are denable by universalsrst-order formulas, such as K, Alt, T, K4, S4, S5, K5, K45, S4.3... |

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Citation Context ...he geometrically intuitive many-dimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatio-temporal representation and reasoning (see e.g. =-=[33, 34]-=-) or reasoning about the behaviour of multi-agent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexity|even the product of tw... |

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Citation Context ...g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexity|even the product of two NP-complete logics can be non-recursively enumerable (see e.g. =-=[29, 27-=-]). In higher dimensions practically all products of `standard' modal logics are undecidable and non-nitely axiomatisable [16]. A natural idea of reducing the strong interaction between modal operator... |

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Citation Context ...lts show that most of these logics are non-nitely axiomatisable and undecidable. Arbitrary relativisations of these extensions of S5-products do result in new, decidable many-dimensional logics; see [=-=24, 30]-=-. Moreover, both the diagonal constants and the substitutions can `detect' some properties of the set of worlds, so it makes sense to consider those frames whose sets of worlds are closed under jumps.... |

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Citation Context ...of two NP-complete logics can be non-recursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non-nitely axiomatisable [=-=16]-=-. A natural idea of reducing the strong interaction between modal operators of product logics in hope to obtain more `user-friendly' but still expressive and useful many-dimensional formalisms is to c... |

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3 |
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2 |
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2 |
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2 |
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Citation Context ...n the ‘domains’ of modal operators is similar to ‘relativisations’ of the quantifiers in first-order logic and algebraic logic, where it indeed results in improving the bad algorithmic behaviour, cf. =-=[24, 20]-=-. As a modification of the product construction, ‘relativisation’ was first suggested in [21]. This idea gives rise to the following ‘product-like’ combinations of logics. First, we choose a class of ... |

2 |
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1 |
Decidable versions of predicate logic and cylindric relativized set algebras
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Citation Context ...n on the `domains' of modal operators is similar to `relativisations' of the quantiers insrst-order logic and algebraic logic, where it indeed results in improving the bad algorithmic behaviour, cf. [=-=24, 20-=-]. As a modication of the product construction, `relativisation' wassrst suggested in [21]. This idea gives rise to the following `product-like' combinations of logics. First, we choose a class of `de... |

1 |
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1 |
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