## Expressive Power and Complexity in Algebraic Logic (1997)

Venue: | Journal of Logic and Computation |

Citations: | 20 - 2 self |

### BibTeX

@ARTICLE{Hirsch97expressivepower,

author = {Robin Hirsch},

title = {Expressive Power and Complexity in Algebraic Logic},

journal = {Journal of Logic and Computation},

year = {1997},

volume = {7},

pages = {309--351}

}

### Years of Citing Articles

### OpenURL

### Abstract

Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tied-in with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The model-theoretic notion of interpretation is used to generalise known complexity results to a range of other algebraic logics. In particular a number of relation algebras are shown to have intractable network satisfaction problems. 1 Introduction A basic problem in theoretical computing and applied logic is to select and evaluate the ideal formalism to represent and reason about a given application. Many different formalisms are adopted: classical first-order logic, modal and temporal logics (either...

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Citation Context ... The Allen interval algebra This algebra (I) has thirteen atoms: Id; precedes, meets, overlaps, starts, during, ends together with the converses of the last six. The composition table can be found in =-=[All83]-=-. A natural representation of I is obtained by taking as domain all ordered pairs of rational numbers (p; q) with p ! q. Each of the thirteen atoms is then interpreted in the obvious way, for example ... |

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Citation Context ... relation algebras Here we introduce some well-known relation algebras: the point algebra, the Allen interval algebra, the left-linear algebra, the containment algebra and the metric point algebra of =-=[DMP91]-=- and others. Each of these has had wide application in temporal reasoning, databases and planning (e.g. [AK83b, DM87, Pel88, Hir96, Hir95] etc.). We give the complexity of the network satisfaction pro... |

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Citation Context ...s out that any transitively closed atomic network is satisfiable (hence the NSP has cubic complexity for atomic networks) but for general networks transitive closure does not guarantee satisfiability =-=[All84]-=- and the NSP has been shown to be an NP complete problem [VK86]. The Left Linear Point Algebra The left linear point algebra was first presented in [Com83], where it is referred to as N 1 . A concrete... |

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Citation Context ...ritten r; s) is defined to be f(d; e) : 9x 2 D; (d; x) 2 rs(x; e) 2 sg. Now we move on to the algebraic approach. The algebraic counterpart of a field of sets is a boolean algebra and it is a theorem =-=[Sto36]-=- that every boolean algebra is isomorphic to a field of sets. For proper relation algebras, the algebraic counterpart is called a relation algebra, though the correspondence between relation algebras ... |

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Citation Context ...ower and complexity of certain algebraic logics: cylindric and relation algebras. These algebras were invented primarily to handle algebraically the study of relations of various ranks. Tarski showed =-=[TG87]-=- that relation algebra can act as a vehicle for set theory and hence all of mathematics. Indeed algebraic logic has turned out to have very powerful applications through much of computer science [All8... |

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Citation Context ...of all cylindric set algebras of dimension ff is denoted Cs ff . ffl A cylindric algebra of dimension ff is defined to be a structure C = (C; ; \Gamma; 0; 1; cs; ds) ;!ff obeying the following axioms =-=[HMT71]-=- for every x; y 2 C; ; ;s! ff: 1. (C; ; \Gamma; 0; 1) is a boolean algebra 2. cs0 = 0 3. xscsx 4. cs(xscsy) = csxscsy 5. cscsx = cscsx 6. ds= 1 7. ifs6= ; , then d = cs(dsds) 8. ifs6= , then cs(dsx)sc... |

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Citation Context ...amma b def = \Gamma(b \Gammaa). In a PRA the operation \Gamma is interpreted as complement relative to the top element 1. These axioms are clearly sound over PRA, but they turn out not to be complete =-=[Lyn50]-=- --- there are (even finite) relation algebras which are not isomorphic to any proper relation algebra. Let us define a representation (X; D) of A to be an isomorphism X from A to some proper relation... |

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Citation Context ...1 = 1 which are consequences of axioms 1 to 9. If X(1) = D \Theta D then call (X; D) a square representation. Lyndon's result shows that not every relation algebra is representable. It has been shown =-=[Mon64]-=- that no finite set of axioms can be sound and complete over PRA. Let (X; D); (Y; E) be representations of a relation algebra A. A base-isomorphism h : (X; D) ! (Y; E) is a bijection from D to E prese... |

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Citation Context ...ebras have been studied carefully, for example in [J ' 82] and an overview of the theory of relation algebras can be found in [Mad91a, J ' 91]. For a good history of the study of relation algebra try =-=[Mad91b]-=-. 2.2 Cylindric Algebra For higher order relations we use cylindric algebra. Let ff be any ordinal --- mostly finite, here. Corresponding to a proper relation algebra we define a cylindric set algebra... |

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Citation Context ...gebra can be decomposed as a subdirect product of simple relation algebras (called the components) and the relation algebra is representable if and only if all the simple components are representable =-=[JT48]-=-. For a simple relation algebra, every representation is a disjoint union of square representations and thus a simple, representable relation algebra always has a square representation. In this paper,... |

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Citation Context ... 6.3 Complexity of NSP for Relation Algebras and CA 3 There is a correspondence between relation algebras and those three dimensional cylindric algebras that are generated by two dimensional elements =-=[Mad91a]-=-. We have seen that the following relation algebras have a network satisfaction problem with polynomial-time complexity: A 2 ; A 3 ; P and L. On the other hand M and I have NP-complete network satisfa... |

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(Show Context)
Citation Context ... necessary, but not always sufficient, condition for the satisfiability of N (see the pentagonal algebra, section 3.2 for a case where not all transitively closed atomic networks are satisfiable. See =-=[HH97a]-=- for an analysis of what happens in the infinite case). ffl Let k be a natural number. A network N is called k-consistent if for each subnetwork M of N with less than k nodes there is a transitively c... |

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(Show Context)
Citation Context ...ry of interpretations, we will be able to use this result to show that most cylindric algebras of dimension four or more have an NP-hard NSP. The results obtained here are similar to results found in =-=[Hir94b]-=- where a certain type of relation algebra (called a pair algebra) is shown to have NP-hard NSP. Recall that, for C 2 CA 4 , a C-network N is a set of nodes, N 1 , and a map N 2 : (N 1 ) 4 ! C. March 1... |

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14 |
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(Show Context)
Citation Context ...dric algebra always has a cubic representation. 2.3 Languages For both relation and finite-dimensional cylindric algebra there is a very natural, first-order language corresponding to an algebra (see =-=[McK66]-=-). Let A be any relation algebra, and let L = L(A) be the first-order language with one binary predicate symbol for each element of A. We use the same symbol for an element of A as for the correspondi... |

13 |
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(Show Context)
Citation Context ...oblem [VK86]. The Left Linear Point Algebra The left linear point algebra was first presented in [Com83], where it is referred to as N 1 . A concrete representation of it appeared in [D 91], see also =-=[AGN94]-=-, page 642. A left-linear structure (L; !) is a partial order such that if s; t; u 2 L and s; t ! u then either s ! t; t ! s or s = t. The algebraic counterpart to this type of structure is the relati... |

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11 | A symbolic approach to interval constraint problems - Ladkin, Reinefeld - 1993 |

10 |
A remark on chromatic polygroups
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(Show Context)
Citation Context ...ve closure does not guarantee satisfiability [All84] and the NSP has been shown to be an NP complete problem [VK86]. The Left Linear Point Algebra The left linear point algebra was first presented in =-=[Com83]-=-, where it is referred to as N 1 . A concrete representation of it appeared in [D 91], see also [AGN94], page 642. A left-linear structure (L; !) is a partial order such that if s; t; u 2 L and s; t !... |

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6 | Representations for small relation algebras
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(Show Context)
Citation Context ...entation, but we have no better estimate of the complexity of the network satisfaction problem over PA except to say that it lies in NP. For more about relation algebras with at most three atoms read =-=[AM94]-=- where the interested reader can find the smallest representations. The Allen interval algebra This algebra (I) has thirteen atoms: Id; precedes, meets, overlaps, starts, during, ends together with th... |

4 |
Model Theory, Encylopedia of Mathematics and its Applications 42
- Hodges
- 1993
(Show Context)
Citation Context ...king a known complexity result for a fixed relation algebra and generalising the result to several other relation algebras. The reader who wishes to find out more about interpretations is referred to =-=[Hod95]-=-. Many of the definitions and some of the results in this section are taken from Hodges' book and made appropriate here for algebraic logic. Interpretations have been used very successfully before wit... |

2 | Intractability in the Allen and Koomen planner - Hirsch - 1995 |

1 | From points to intervals - Hirsch - 1994 |