## A logic for metric and topology (2005)

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Venue: | Journal of Symbolic Logic |

Citations: | 13 - 11 self |

### BibTeX

@ARTICLE{Wolter05alogic,

author = {Frank Wolter and Michael Zakharyaschev},

title = {A logic for metric and topology},

journal = {Journal of Symbolic Logic},

year = {2005},

pages = {828}

}

### OpenURL

### Abstract

Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘ε-definitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘well-behaved ’ common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial

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Citation Context ... problem for M T -terms in M T -models is decidable in exponential time in the length of the term. Proof The proof of both (i) and (ii) is based on the standard elimination method used, e.g., for PDL =-=[20]. Roughl-=-y, it works as follows. Given a term τ, take a suitable closure cl(τ) (the Fischer–Ladner closure in case of PDL) of the set of subterms of τ, form a set Γτ of appropriate types over cl(τ), de... |

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Citation Context ...lete. 7sBefore going into details of the proof, we separate the conceptually interesting part from the ‘folklore’ or trivial observations. Clearly, (3) ⇒ (2) ⇒ (1). By Sahlqvist’s theorem (s=-=ee, e.g., [8]), we-=- also have (4) ⇔ (3). Indeed, AxMT[τ] can be regarded as a normal multi-modal logic with Sahlqvist axioms. So AxMT[τ] is determined by its Kripke frames which clearly coincide with M T [τ]-frames... |

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Citation Context ...s between regions that are usually expressible in the logic S4u of topological spaces (with the interior and closure operators and the quantifiers—the ‘universal modalities’—over all points in=-= space) [27, 15, 19, 7, 29, 1]. Th-=-e intended models are based on—among others and in the decreasing order of abstractness—arbitrary topological spaces, metric spaces with their topologies, and the two-dimensional Euclidean space R... |

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Citation Context ...al in various areas of computer science. For example: • Temporal logics for specification and verification of real-time systems deal with the real line R and the standard Euclidean metric (see, e.g.=-=, [4, 21]).-=- 1s• Spatial representation and reasoning uses various topological relations between regions that are usually expressible in the logic S4u of topological spaces (with the interior and closure operat... |

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Citation Context ...e. On the other hand, it is known that the ‘punctuality’ operators ∃ =a (‘at distance = a’) yield undecidable logics on R (both with and without finite variability constraints on the interpr=-=etations) [3]-=-, but it is an open problem whether the extension of M T with the punctuality operators is decidable over the class of topometric models. The logic of R2 . The situation becomes quite different if we ... |

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Citation Context ...models are based on—among others and in the decreasing order of abstractness—arbitrary topological spaces, metric spaces with their topologies, and the two-dimensional Euclidean space R 2 (see, e.=-=g., [34, 13]).-=- • Similarity measures that are used to classify various sets of objects (e.g., proteins or viruses in bioinformatics) give rise to reasoning about metric spaces not related at all to the standard E... |

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192 |
The computational complexity of provability in systems of modal logic
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Citation Context ...agments of these logics are well known. The ‘pure topological’ fragments (with the only non-Boolean operator ✷) of MT, L(R) and L(R 2 ) coincide with the modal logic S4 [27], which is PSPACE-com=-=plete [25]. The ��-=-�global topological’ fragment (with the operators ✷ and ∃) of MT is S4u, which is also PSPACE-complete [5]. The corresponding fragments of L(R) and L(R 2 ) coincide with the logic of all connect... |

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Complexity of Modal Logics
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Citation Context ...her an M T -term built using only the Booleans and the operators ∀ ≤n , n ∈ N + , belongs to MT. Proof The proofs are by reduction of the global K-consequence relation that is known to be EXPTIM=-=Ehard [31]. -=-We remind the reader that the language LK of modal logic K extends propositional logic (with propositional variables p1, p2,...) by means of one unary operator ✸. LK is interpreted in models of the ... |

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The algebra of topology
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- 1944
(Show Context)
Citation Context ...be regarded as a ‘metric extension’ of Tarski’s programme of the algebraisation of topology (“of creating an algebraic apparatus adequate for the treatment of point-set topology,” to be more=-= precise) [27]; -=-see also [32, 23, 6, 28, 1]. There is another idea underpinning this paper. As is well-known, reasoning about metric and topology is fundamental in various areas of computer science. For example: • ... |

108 |
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Citation Context ...measures that are used to classify various sets of objects (e.g., proteins or viruses in bioinformatics) give rise to reasoning about metric spaces not related at all to the standard Euclidean spaces =-=[12, 17]. -=-Logics for reasoning about similarity have been developed in the field of approximate reasoning [14, 16]. In this respect, we are looking for a logic which can be regarded as a sort of ‘common denom... |

87 | Modal logics for qualitative spatial reasoning
- Bennett
- 1995
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Citation Context ...s between regions that are usually expressible in the logic S4u of topological spaces (with the interior and closure operators and the quantifiers—the ‘universal modalities’—over all points in=-= space) [27, 15, 19, 7, 29, 1]. Th-=-e intended models are based on—among others and in the decreasing order of abstractness—arbitrary topological spaces, metric spaces with their topologies, and the two-dimensional Euclidean space R... |

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Citation Context ...f MT, L(R) and L(R 2 ) coincide with the modal logic S4 [27], which is PSPACE-complete [25]. The ‘global topological’ fragment (with the operators ✷ and ∃) of MT is S4u, which is also PSPACE-c=-=omplete [5]. Th-=-e corresponding fragments of L(R) and L(R 2 ) coincide with the logic of all connected topological spaces (induced by metric spaces); it can be obtained from S4u by adding the ‘connectivity axiom’... |

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Citation Context ...s not difficult to see that r ∈ (τ0 ∧∀(τ0 → ϕ ♯ 2 ))K but r �∈ (ϕ ♯ 1 )K . Therefore, (τ0 ∧∀(τ0 → ϕ ♯ 2 )) → ϕ♯ 1 �∈ MT. (ii) Call an LK-formula ϕ valid if ϕN=-= = V , for every model N. It is proved in [11] that, for any ϕ1,ϕ2 ∈ LK, �-=-�2 ⊢ ϕ1 iff (ϕ2 ∧ ✷ϕ2 ∧ ··· ∧ ✷ 2l(ϕ1 )+l(ϕ2 ) ϕ2) → ϕ1 is valid, where l(ϕi) is the number of subformulas of ϕi. Now, it follows immediately from the reduction (i) that ϕ2 ... |

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Citation Context ...taining no topological operators ✷ and ✸) in topometric model over subspaces of R × R or over R × R itself is undecidable. Proof The proof is by reduction to the undecidable Z × Z tiling proble=-=m (see [9]-=- and references therein), which is formulated as follows. Given a finite set T = {T0,...,Tl} of tile types (i.e., squares Ti with colours left(Ti), right(Ti), up(Ti), and down(Ti) on their edges), dec... |

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Citation Context ...a ‘metric extension’ of Tarski’s programme of the algebraisation of topology (“of creating an algebraic apparatus adequate for the treatment of point-set topology,” to be more precise) [27];=-= see also [32, 23, 6, 28, 1]. -=-There is another idea underpinning this paper. As is well-known, reasoning about metric and topology is fundamental in various areas of computer science. For example: • Temporal logics for specifica... |

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Diskrete Räume
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Citation Context ... → y ∈ X)}. Such spaces are known as Aleksandrov spaces. Alternatively they can be defined as topological spaces where arbitrary (not only finite) intersections of open sets are open; for details =-=see [2, 10]. The -=-constraints on the relations in M T [M]-frames reflect the connection between metric and topology: (qoR) corresponds to the S4-axioms for ✷, and (rsD)–(D < R) to axioms (3)–(13). In fact, it is ... |

29 |
General Topology Part 1
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- 1966
(Show Context)
Citation Context ... → y ∈ X)}. Such spaces are known as Aleksandrov spaces. Alternatively they can be defined as topological spaces where arbitrary (not only finite) intersections of open sets are open; for details see =-=[2, 11]-=-. The constraints on the relations in MT [M]-frames reflect the connection between metric and topology: (qoR) corresponds to the S4-axioms for ✷, and (rsD)–(D < R) to axioms (3)–(13). In fact, it is a... |

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- 2003
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Citation Context ...ches to devising languages that are capable of speaking about topometric models. The first natural choice would be the appropriate (fragment of) two-sorted first-order logic. However, as was shown in =-=[24], ev-=-en the two-variable fragment of monadic predicate logic with extra binary predicates d(x,y) < a, a ∈ N, is undecidable. It was also proved in [24] that propositional operators ‘somewhere in the op... |

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- 1997
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Citation Context ...ive rise to reasoning about metric spaces not related at all to the standard Euclidean spaces [12, 17]. Logics for reasoning about similarity have been developed in the field of approximate reasoning =-=[14, 16]. In-=- this respect, we are looking for a logic which can be regarded as a sort of ‘common denominator’ of the formalisms constructed in these fields and which reveals most important expressivity and co... |

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Aussagenkalkül und die Topologie, Fund
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- 1938
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Citation Context ...a ‘metric extension’ of Tarski’s programme of the algebraisation of topology (“of creating an algebraic apparatus adequate for the treatment of point-set topology,” to be more precise) [27];=-= see also [32, 23, 6, 28, 1]. -=-There is another idea underpinning this paper. As is well-known, reasoning about metric and topology is fundamental in various areas of computer science. For example: • Temporal logics for specifica... |

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Citation Context ...a ‘metric extension’ of Tarski’s programme of the algebraisation of topology (“of creating an algebraic apparatus adequate for the treatment of point-set topology,” to be more precise) [27];=-= see also [32, 23, 6, 28, 1]. -=-There is another idea underpinning this paper. As is well-known, reasoning about metric and topology is fundamental in various areas of computer science. For example: • Temporal logics for specifica... |

22 |
Computational Molecular Biology
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- 2000
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Citation Context ...measures that are used to classify various sets of objects (e.g., proteins or viruses in bioinformatics) give rise to reasoning about metric spaces not related at all to the standard Euclidean spaces =-=[12, 17]. -=-Logics for reasoning about similarity have been developed in the field of approximate reasoning [14, 16]. In this respect, we are looking for a logic which can be regarded as a sort of ‘common denom... |

21 |
Everywhere” and “Here
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- 1999
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Citation Context ...e corresponding fragments of L(R) and L(R 2 ) coincide with the logic of all connected topological spaces (induced by metric spaces); it can be obtained from S4u by adding the ‘connectivity axiom’=-= of [30] ∃✷P1 ⊓ ∃✷P2 ⊓ ∀-=-(✷P1 ⊔ ✷P2) ⊑ ∃(✷P1 ⊓ ✷P2). (2) Language II. In the language M T we can formalise metric relations, but we cannot compare two distances without specifying their absolute values. To ena... |

18 |
A modal account of similarity-based reasoning
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- 1997
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Citation Context ...ive rise to reasoning about metric spaces not related at all to the standard Euclidean spaces [12, 17]. Logics for reasoning about similarity have been developed in the field of approximate reasoning =-=[14, 16]. In-=- this respect, we are looking for a logic which can be regarded as a sort of ‘common denominator’ of the formalisms constructed in these fields and which reveals most important expressivity and co... |

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- 2003
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Citation Context ...a ‘common denominator’ of various ‘qualitative’ temporal, description, spatial, epistemic, dynamic, etc. logics.) The logic we construct here is a natural combination of a logic of metric spac=-=es from [33] (equi-=-pped with operators ‘somewhere in the sphere of radius a,’ a ∈ Q + , including or excluding the boundary) and the well established logic of topological spaces S4u mentioned above. Besides the in... |

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Citation Context ...s that are used to classify various sets of objects (e.g., proteins or viruses in bio-informatics) give rise to reasoning about metric spaces not related at all to the standard Euclidean spaces [13]. =-=[18, 19]-=- suggest a combination of topological relations between regions in spaces with similarity measures for classifying and identifying objects. Logics for reasoning about similarity have been developed in... |

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- 1999
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Citation Context ...al in various areas of computer science. For example: • Temporal logics for specification and verification of real-time systems deal with the real line R and the standard Euclidean metric (see, e.g.=-=, [4, 21]).-=- 1s• Spatial representation and reasoning uses various topological relations between regions that are usually expressible in the logic S4u of topological spaces (with the interior and closure operat... |

16 | A tableau algorithm for reasoning about concepts and similarity
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- 2003
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Citation Context ...nominals (atomic terms interpreted as singleton sets) would make the language powerful enough for reasoning about concepts, similarities, and prototypes in combinations with, e.g., description logics =-=[17, 26]. It i-=-s an interesting open problem whether the resulting language is still decidable. Note that the nominals would require new axioms like where n is a nominal. Acknowledgements ✸{n} = {n}, ✸∃ <a {n}... |

16 | A completeness proof for propositional S4 in cantor space, in: E. Orlowska (Ed.), Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa - Mints - 1998 |

16 |
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Citation Context ...ome topometric model. The reader familiar with algebraic semantics of modal logics and duality theory should not have problems with reformulating these results in the algebraic manner (consult, e.g., =-=[20]-=-). Remark 6. It is to be noted that the axiomatic system AxMT[τ] is not complete if we do not include in it the axioms for all parameters from the closure of N(τ) under both (+) and (−). For example, ... |

13 |
On the complexity of qualitative spatial reasoning
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- 1999
(Show Context)
Citation Context ...s between regions that are usually expressible in the logic S4u of topological spaces (with the interior and closure operators and the quantifiers—the ‘universal modalities’—over all points in=-= space) [27, 15, 19, 7, 29, 1]. Th-=-e intended models are based on—among others and in the decreasing order of abstractness—arbitrary topological spaces, metric spaces with their topologies, and the two-dimensional Euclidean space R... |

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- 2002
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Citation Context ...fficiently large parameters. 3.2. Representation theorem. Now we show how to prove the implication (1) ⇒ (5). The proof actually extends a representation theorem of McKinsey and Tarski [29] (see also =-=[32, 8, 1]-=- for more recent proofs) which they use to show the implication (1) ⇒ (5) for the topological fragment ML of our language. To start with, we formulate their representation theorem for the metric space... |

5 |
Mathematics of Modality. Number 43
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- 1993
(Show Context)
Citation Context ...ome topometric model. The reader familiar with algebraic semantics of modal logics and duality theory should not have problems with reformulating these results in the algebraic manner (consult, e.g., =-=[18]). R-=-emark 6. It is to be noted that the axiomatic system AxMT[τ] is not complete if we do not include in it the axioms for all parameters from the closure of N(τ) under both (+) and (−). For example, ... |

5 |
Axiomatizing the next-interior fragment of dynamic topological logic
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- 1997
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(Show Context)
Citation Context ...ly if there is no point y > x until which ‘not X’ and there is no point y < x since which ‘not X.’ • Spatial representation and reasoning uses various topological and metric relations between regions =-=[29, 16, 21, 7, 31, 1]-=-. The intended models are based on—among others and in the decreasing order of abstractness— arbitrary topological spaces, metric spaces with their topologies, and the two-dimensional Euclidean space ... |