## Spatial Reasoning with Propositional Logics (1994)

Venue: | Principles of Knowledge Representation and Reasoning: Proceedings of the 4th International Conference (KR94 |

Citations: | 98 - 15 self |

### BibTeX

@INPROCEEDINGS{Bennett94spatialreasoning,

author = {Brandon Bennett},

title = {Spatial Reasoning with Propositional Logics},

booktitle = {Principles of Knowledge Representation and Reasoning: Proceedings of the 4th International Conference (KR94},

year = {1994},

pages = {51--62},

publisher = {Morgan Kaufmann}

}

### OpenURL

### Abstract

I present a method for reasoning about spatial relationships on the basis of entailments in propositional logic. Formalisms for representing topological and other spatial information (e.g. [2] [10] [11]) have generally employed the 1st-order predicate calculus. Whilst this language is much more expressive than 0-order (propositional) calculi it is correspondingly harder to reason with. Hence, by encoding spatial relationships in a propositional representation automated reasoning becomes more effective. I specify representations in both classical and intuitionistic propositional logic, which --- together with well-defined meta-level reasoning algorithms --- provide for efficient reasoning about a large class of spatial relations. 1 INTRODUCTION This work has developed out of research done by Randell, Cui and Cohn (henceforth RCC) on formalising spatial and temporal concepts used in describing physical situations [11]. A set of classical 1st-order logic axioms has been formulated in whi...

### Citations

2281 | Maintaining knowledge about temporal intervals. C o m m u n i c a t i o n s o f t h e ACM26(1 l
- Allen
- 1983
(Show Context)
Citation Context ...e locus of composition is the same as what in [12] was referred to as the `transitive closure' of two base relations. This terminology derives from Allen's `transitivity table' for temporal intervals =-=[1]-=-. However,`transitive closure' already has a meaning different from what is intended here, so a new term is required to avoid potential confusion. In describing the more general problem of determining... |

564 | A Spatial Logic Based on Regions and Connection
- Randell, Cui, et al.
- 1992
(Show Context)
Citation Context ...stract I present a method for reasoning about spatial relationships on the basis of entailments in propositional logic. Formalisms for representing topological and other spatial information (e.g. [2] =-=[10]-=- [11]) have generally employed the 1st-order predicate calculus. Whilst this language is much more expressive than 0-order (propositional) calculi it is correspondingly harder to reason with. Hence, b... |

60 |
2.0 Users Guide
- McCune, OTTER
- 1990
(Show Context)
Citation Context ...ble for the 8 base relations shown in Figure 1 in under 244 seconds. This is a substantial improvement over the method described in [12]. In generating the table given there, the theorem prover Otter =-=[7]-=- was used, working with the 1st-order axiomatisation of the RCC theory. Otterstook a total of 2460 seconds to prove the required theorems but some proofs required human assistance (addition of hand ch... |

50 | Modelling topological and metrical properties of physical processes - RANDELL, CORN - 1989 |

49 |
Computing transitivity tables: a challenge for automated theorem provers. Lecture notes in computer science
- Randell, Cohn, et al.
- 1992
(Show Context)
Citation Context ...been formulated in which a large number of spatial and temporal relations can be defined [10]. One problem with this formalism is that computing inferences in the theory is far from easy --- see e.g. =-=[12]-=-. Of course one can use any general purpose 1st-order theorem prover, but the complexity of the theory means that for many significant problems this approach is impractical. In this paper I present an... |

37 |
Complexity, convexity and combinations of theories
- Oppen
- 1980
(Show Context)
Citation Context ... set equation is not equivalent to applying the complement operation to its set term). However, it can be established that in the domain of sets, entailments of this kind are convex 3 in the sense of =-=[9]-=-. A class of entailments is convex iff whenever \Gamma j= OE 1 : : :sOE n then \Gamma j= OE i , for some i in 1 : : : n. Consider the entailment associated with the impossibility of \Theta. Suppose no... |

30 |
An O(n log n)-space decision procedure for intuitionistic propositional logic
- Hudelmaier
- 1993
(Show Context)
Citation Context ...Clearly, to use I + 0 as a representation language for effective spatial reasoning we need to be able to reason efficiently in I 0 . Theorem proving in I 0 is decidable but potentially very hard (see =-=[5]-=-). A proof-theory for the language can be specified in terms of a simple cut-free Gentzen sequent calculus which is only a slight modification of the corresponding classical system. The formalisation ... |

19 |
Qualitative spatial reasoning and representation
- Cohn, Randell, et al.
- 1993
(Show Context)
Citation Context ...nt of the conv axioms in a Prolog implemented I + 0 reasoning program. It was produced in 3h 31mins on a Sparc10 workstation. Such a table has hitherto never been computed by a proof oriented method. =-=[3]-=- contains a similar table constructed using a model building approach but it has subsequently been found that the table given there is too strict in that it rules out certain configurations, which are... |

17 | 1972]: Introduction to set theory and topology - Kuratowski |

13 |
What is elementary geometry?, in: "The Axiomatic Method, with Special Reference to Geometry and
- Tarski
- 1959
(Show Context)
Citation Context ...ow that they are faithful to the interpretation in terms of the betweenness relation, which has a straightforward algebraic definition in a model which is a Cartesian space over the real numbers (see =-=[14]-=-). For example [10] gives the following definitions: ffl INSIDE(X; Y ) j def DR(X;Y )sP(X; conv(Y )) ffl P-INSIDE(X; Y ) j def DR(X;Y )sPO(X; conv(Y )) ffl OUTSIDE(X;Y ) j def DR(X; conv(Y )) 10 More ... |

12 |
Elements of Intuitionism. Oxford Logic Guides
- Dummett
(Show Context)
Citation Context ...fied in terms of a simple cut-free Gentzen sequent calculus which is only a slight modification of the corresponding classical system. The formalisation I use is essentially the same as that given in =-=[4]-=-. Theorem proving in the I 0 sequent calculus is more complex than that of C 0 : in C 0 all connectives can be eliminated deterministically because the rules produce Boolean combinations of sequents w... |

7 | Sentential calculus and topology - Tarski - 1956 |

6 |
Thirty Years of Foundational Studies
- MOSTOWSKI
- 1966
(Show Context)
Citation Context ...any situation hU ; \Sigma; fi. 2 TOPOLOGICAL INTERPRETATION OF PROPOSITIONAL LOGIC There is a close connection between classical propositional calculus, which I shall refer to as C 0 , and set theory =-=[8, p14]-=-. The simplest semantics for C 0 is to take propositions as denoting truth values and to correlate the connectives with truth functions. However, if we interpret propositional letters as denoting arbi... |

4 |
A Calculus of Individuals based on â€˜Connection
- Clark
- 1981
(Show Context)
Citation Context ...k Abstract I present a method for reasoning about spatial relationships on the basis of entailments in propositional logic. Formalisms for representing topological and other spatial information (e.g. =-=[2]-=- [10] [11]) have generally employed the 1st-order predicate calculus. Whilst this language is much more expressive than 0-order (propositional) calculi it is correspondingly harder to reason with. Hen... |