## 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... |

600 |
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Citation Context ...show that BC('(x)) j= '(x), from which the result follows by compactness. (In this notation we interpret the x as constants, or equivalently we use the local version ofsrst-order consequence, cf. [End72].) If BC('(x)) is inconsistent we are done. Otherwise, let (M; m) be a model of BC('(x)) and (N; n) be a model of '(x) together with the bounded theory of (M; m). (Such a model can easily be ... |

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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 ...a multimodal language enriched with nominals and @ is pspace-complete. The lower bound follows 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 [ABM99]. 5.2 The language H t (@) Matters are dierent if we change the underlying modal logic to tense logic. We ... |

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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 ...e has a winning strategy for the classic Ehrenfeucht-Frasse game too, contradicting the claim that M 0 j= '[ m] and N 0 j= :'[n]. iii. ) i. A fairly standard diagram-chasing argument (cf. e.g., [vB96]). Let '(x) be as in the hypothesis and BC('(x)) be the bounded consequences of '(x) (that is, the consequences of '(x) that belong to the bounded fragment). We will show that BC('(x)) j= '(x), ... |

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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... |

91 | A road-map on complexity for hybrid logics
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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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(Show Context)
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... |

44 |
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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... |

37 | M.: Hybrid languages and temporal logic
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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... |

24 | A modal logic of relations
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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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(Show Context)
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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(Show Context)
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... |