## Hybrid Logics: Characterization, Interpolation and Complexity (1999)

Venue: | Journal of Symbolic Logic |

Citations: | 101 - 35 self |

### BibTeX

@ARTICLE{Areces99hybridlogics:,

author = {Carlos Areces and Patrick Blackburn and Maarten Marx},

title = {Hybrid Logics: Characterization, Interpolation and Complexity},

journal = {Journal of Symbolic Logic},

year = {1999},

volume = {66},

pages = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Frasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rst-order logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...

### Citations

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Citation Context ...is satisable if and only if ' 0 is, so we can restrict our attention to variable-free H(@) formulas. For denitions of the complexity classes pspace and exptime and other background information, see [P=-=ap94]-=-. 5.1 The language H(@) H(@) is a well-behaved sublanguage of H(#; @). As we have just seen, although H(@) does not enjoy strong interpolation, it does have weak interpolation. Moreover, simple tablea... |

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Citation Context ...ted back-and-forth system linking M and N which contains a partial isomorphism sending m i to n i (0 i k). Note how closely this denition follows the familiar one fromsrst-order logic (cf. e.g., [Ho=-=d93])-=-. In fact, if we think of such a system as describing an Ehrenfeucht-Frasse game, then the sole dierence is that in the \generated back-and-forth game" the universal player must choose his moves ... |

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Citation Context ...s what we needed to show. qed We obtain the corresponding upper bound by translating into first-order logic. It turns out that we end up in (a version of) the guarded fragment with two variables (see =-=[AvBN98]-=-). The satisfaction problem of this fragment is known to be exptime-complete, whence the desired result. In fact we will prove a stronger result. The universal modality E is a modality with the follow... |

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Citation Context ...marx@wins.uva.nl Abstract Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least =-=[Pri67]-=-, but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, ... |

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Citation Context ...careful analysis of weaker versions of the Beth property can be carried out for fragments of H(#; @) as we did with the interpolation property. 5 Complexity H(#; @) is undecidable (this is known from =-=[BS95]-=-, unpublished work by Valentin Goranko, and can be proved directly for the bounded fragment [Woo81]), but the sublanguage H(@) is decidable (this is an easy consequence of results in [PT91, GG93, Bla9... |

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Citation Context ...ll behaved in yet another sense: it has the interpolation property. Interpolation is a much studied notion. Originally considered a property of deductive systems, it was proved forsrst-order logic in =-=[Cra57]-=- using a proof theoretic argument. We shall view interpolation as a property of consequence relations and will prove it using semantic arguments (as is done, for example, in [CK90]). Incidentally, Jer... |

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Citation Context ... at any m 2 M , M;m fl E' iff for some m 0 2 M , M;m 0 fl '. (That is, with the aid of the universal modality we can scan the entire model for information.) The universal modality is very strong (see =-=[GP92]-=- for a detailed discussion) but for present purposes we need merely observe that, given E, we can define @ i ' to be E(i '). In spite of its strength, we have the following upper bound: Theorem 6.3 Th... |

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Citation Context ...s directly from the pspace-hardness of the local K-satisability problem in the ordinary unimodal language [Lad77], while a matching upper bound is easy to obtain by means of model-construction games [=-=ABM9-=-9]. 5.2 The language H t (@) Matters are dierent if we change the underlying modal logic to tense logic. We know from [Spa93b] that the K-satisability problem for the usual language of tense logic is ... |

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Citation Context ...] and [BT99] are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the proof theoretical investigations of =-=[Bla00-=-] and [Tza99], there is [Sel91, Sel97]. For a more detailed guide to theseld, see the hybrid logic home-page (http://turing.wins.uva.nl/~carlos/hybrid). 2 Preliminaries In this section we dene the syn... |

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Citation Context ...ond, some work has been done on very basic hybrid languages (that is, modal or tense languages enriched with nominals, but with no additional mechanisms such as #, @, or 8); early references here are =-=[GG93]-=- and [Bla93]. Third, while [BT98a] and [BT99] are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the pro... |

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Citation Context ...he literature on hybrid languages is relatively small, it is possible to give the reader a swift overview of what is available. First, two early papers on 8-based hybrid languages (namely [Bul70] and =-=[PT91]-=-) deserve to be more widely read: both contain important technical ideas and interesting motivation for the use of hybrid languages. Second, some work has been done on very basic hybrid languages (tha... |

55 | The computational complexity of hybrid temporal logics
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Citation Context ... exptime. And in fact, we can: Theorem 5.3 The satisability problem for nominal Propositional Dynamic Logic with converse programs and the universal modality is solvable in exptime. Proof. Proved in [=-=ABM00-=-] using a spypoint argument. qed Corollary 5.4 The local K-satisability problem for H t (@) plus the universal modality is exptime-complete. Because of the availability of the Kleene star in PDL, we c... |

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Citation Context ...rst-order logic in [Cra57] using a proof theoretic argument. We shall view interpolation as a property of consequence relations and will prove it using semantic arguments (as is done, for example, in =-=[CK90]-=-). Incidentally, Jerry Seligman has recently announced a proof-theoretic proof of interpolation for H(#; @). Before plunging into the details, note that in modal logic we can distinguish between stron... |

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Citation Context ...ork has been done on very basic hybrid languages (that is, modal or tense languages enriched with nominals, but with no additional mechanisms such as #, @, or 8); early references here are [GG93] and =-=[Bla93]-=-. Third, while [BT98a] and [BT99] are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the proof theoretic... |

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Citation Context ... or 8); early references here are [GG93] and [Bla93]. Third, while [BT98a] and [BT99] are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in =-=[Gor96]-=-. Finally, in addition to the proof theoretical investigations of [Bla00] and [Tza99], there is [Sel91, Sel97]. For a more detailed guide to theseld, see the hybrid logic home-page (http://turing.wins... |

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Citation Context ...ask: Open Question 5.11 Is there any decidable hybrid language extending H(@) that enjoys arrow interpolation? To close this section, some remarks on Beth Definability. The Beth Definability Property =-=[Bet53]-=- is commonly studied together with interpolation (and in many cases interpolation is considered just a step in the proof of the Beth Property). Loosely speaking, a logic has the Beth property if any i... |

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Citation Context ...brid languages (that is, modal or tense languages enriched with nominals, but with no additional mechanisms such as #, @, or 8); early references here are [GG93] and [Bla93]. Third, while [BT98a] and =-=[BT99]-=- are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the proof theoretical investigations of [Bla00] and ... |

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Citation Context ...ds as the modal depth of �� and the information on the playing board is polynomial in the length of ��. Using the close correspondence between Alternating Turing Machines (ATM's) and two playe=-=r games [Chl86], it-=- is straightforward to implement the problem of whether the existential player has a winning strategy in the ��-game on a ptime ATM. Because any ptime ATM algorithm can be turned into a pspace Tur... |

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Citation Context ... are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the proof theoretical investigations of [Bla00] and =-=[Tza99-=-], there is [Sel91, Sel97]. For a more detailed guide to theseld, see the hybrid logic home-page (http://turing.wins.uva.nl/~carlos/hybrid). 2 Preliminaries In this section we dene the syntax and sema... |

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Citation Context ...@) as we did with the interpolation property. 5 Complexity H(#; @) is undecidable (this is known from [BS95], unpublished work by Valentin Goranko, and can be proved directly for the bounded fragment =-=[Woo81]-=-), but the sublanguage H(@) is decidable (this is an easy consequence of results in [PT91, GG93, Bla93]). In this section we examine the computational complexity of H(@) and related systems, and sharp... |

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Citation Context ...ause for the universal modality: ST x (E') = 9x:ST x ('). The translation only uses 2 variables. Moreover, inspection shows that the range of the translation belongs to the packed fragment defined in =-=[Mar97]-=-. (It does not fall in the guarded fragment as defined in [AvBN98] because the translation of E is not guarded, though a translation into the guarded fragment could be obtained by relativizing with a ... |

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Citation Context ... note that the meaning of TIP depends on the way we dene the consequence relation ' j=s. In Section 2 we introduced two consequence relations: local consequence and global consequence (see [vB83] or [=-=MV9-=-7] for a discussion of their relative merits). So there are two plausible denitions of the turnstile interpolation property, TIP loc and TIP glo . In modal logic these dierent notions of interpolation... |

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Citation Context ...matching upper bound is easy to obtain by means of model-construction games [ABM99]. 5.2 The language H t (@) Matters are dierent if we change the underlying modal logic to tense logic. We know from [=-=Spa93b-=-] that the K-satisability problem for the usual language of tense logic is pspacecomplete. However expanding such a language with even a single nominal (or free variable) results in an exptime-hard sa... |

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Citation Context ...n instruction to move to the world labeled by the variable x and evaluate ' there. Previous work on H(#; @) has concentrated on relating it to other hybrid languages [BS98], studying it axiomatically =-=[BT98a]-=-, and developing analytic proof techniques [Bla00, Tza99]. Taken together, this work suggests that H(#; @) and certain of its sublanguages (notably H(@)) are important systems. The purpose of the pres... |

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Citation Context ...class frames(8x:'(x)) enjoys strong interpolation (AIP). This result stands in sharp contrast to the scarcity of general interpolation results obtained for the basic modal language; see for example [M=-=ak91-=-]. Indeed, it can be viewed as delineating the syntactic form of interpolants in modal logic as follows. Let L be the modal logic in the basic modal language of asrst-order denable class frames of fra... |

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Citation Context ...g # and @ we have in effect enriched the modal language with an explicit substitution operator . Such operators are used in the study of cylindric algebras, and were added to cylindric modal logic in =-=[Ven94]-=-. The link between #and explicit substitution can be made even more clear if we expand the first-order language with an explicit substitution operator (like s i j in the theory of cylindric algebras) ... |

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Citation Context ...). Conversely, if is pure nominal-free sentence, then denes frames(8x:ST ((x))), and K[H(#; @)] + is complete with respect to frames(8x:ST ((x))). Proof. The converse condition was proved in [BT98b=-=]-=-, so let's examine the other direction. For item i., wesrst remark that as '(x) belongs to the frame language, it contains no unary predicate symbols, hence HT ('(x)) is a pure formula; that # x:HT ('... |

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Citation Context ...rsrst-order logic these notions are equivalent, but in modal logic this is not the case (as we see below, equivalence depends on both compactness and the availability of a deduction theorem; cf. also =-=[Cze82-=-]). Further, note that the meaning of TIP depends on the way we dene the consequence relation ' j=s. In Section 2 we introduced two consequence relations: local consequence and global consequence (see... |

3 |
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Citation Context ...g # and @ we have in eect enriched the modal 7 language with an explicit substitution operator. Such operators are used in the study of cylindric algebras, and were added to cylindric modal logic in [=-=Ven94]-=-. The link between # and explicit substitution can be made even more clear if we expand thesrst-order language with an explicit substitution operator (like s i j in the theory of cylindric algebras) a... |

3 |
Logical matrices and the amalgamation property
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Citation Context ...first-order logic these notions are equivalent, but in modal logic this is not the case (as we see below, equivalence depends on both compactness and the availability of a deduction theorem; cf. also =-=[Cze82]-=-). Further, note that the meaning of TIP depends on the way we define the consequence relation ' j= /. In Section 2 we introduced two consequence relations: local consequence and global consequence (s... |

3 |
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Citation Context ... are the basic references for H(#; @), an interesting discussion of # as part of a stronger system can be found in [Gor96]. Finally, in addition to the proof theoretical investigations of [Bla98] and =-=[Tza98]-=-, there is also [Sel91, Sel97]. Seligman's work deals with stronger (8-based) systems, but many of the key ideas underlying hybrid deduction (in particular, the deductive significance of @) were first... |

2 |
On Padoa's method in the theory of de
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(Show Context)
Citation Context ...sk: 23 Open Question 4.10 Is there any decidable hybrid language extending H(@) that enjoys arrow interpolation? To close this section, some remarks on Beth Denability. The Beth Denability Property [B=-=et53]-=- is commonly studied together with interpolation (indeed, sometimes interpolation is considered to be just a step in the proof of the Beth Property). Loosely speaking, a logic has the Beth property if... |

2 |
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Citation Context ...lities in the common language, then interpolation does not follows immediately, especially if the modalities interact (for example, if the theory contains axioms involving more than one modality; see =-=[Mar99]-=- for examples of this type). Nonetheless, we advance the following conjecture: Conjecture 5.6 Multi-modal H(#; @) has arrow interpolation even if the interpolant's modalities are restricted to the com... |