Embedding RCC8D in the Collective Spatial Logic CSLCS

Abstract

Discrete mereotopology is a logical theory for the specification of qualitative spatial functions and relations defined over a discrete space, intended as a set of basic elements, the pixels, with an adjacency relation defined over it. The notions of interest are that of region, intended as an arbitrary aggregate of pixels, and of specific relations between regions. The mereotopological theory RCC8D extends the mereological theory RCC5D—a theory of region parthood for discrete spaces—with the topological notion of connection and the remaining relations (disconnection, external connection, tangential and nontangential proper parthood and their inverses). In this paper, we propose an encoding of RCC8D into CSLCS, the collective extension of the Spatial Logic of Closure Spaces SLCS. We show how topochecker, a model-checker for CSLCS, can be used for effectively checking the existence of a RCC8D relation between two given regions of a discrete space.

A Proof of Proposition 1

Proof

We prove that, for all models $\mathcal {M}= ((X,\mathcal {C}),\mathcal{V})$ and points $x \in X$, the following holds:

$$ \mathcal {M}, x \not \models \varPhi _2 \, \vee \, (\varPhi _1 \, \mathcal {S}\, \varPhi _2) \text{ iff } \mathcal {M}, x \not \models A(\varPhi _1 \, \mathcal {W}\, \varPhi _2). $$

For the direct implication, we proceed as follows:

For the one but last step of the above derivation, note that: (i) $\mathcal {M}, \pi (\ell ) \not \models \varPhi _1$ implies that $\mathcal {M}, \pi (i) \models \varPhi _1$ for all i does not hold; and (ii) $\mathcal {M}, \pi (j) \not \models \varPhi _2, \text{ for } \text{ all } j \text{ s.t. } 0 \le j \le \ell$ implies that, if there exists k s.t. $\mathcal {M}, \pi (k) \models \varPhi _2$, then, it necessarily must be $k > \ell$; but then $\mathcal {M}, \pi \models \varPhi _1 \, \mathcal {U}\, \varPhi _2$ cannot hold because this would not allow $\mathcal {M},\pi (\ell ) \not \models \varPhi _1$, with $\ell < k$.

The derivation for the reverse implication is given below:

Take the minimal $\ell$ as above. If $\ell = 0$, then clearly $\mathcal {M},x \not \models \varPhi _1 \mathcal {S}\varPhi _2$, by definition of $\mathcal {S}$, and since we also have $\mathcal {M},x \not \models \varPhi _2$, we get $\mathcal {M},x \not \models \varPhi _2 \, \vee \,(\varPhi _1 \mathcal {S}\varPhi _2)$, i.e. the assert. If instead $\ell > 0$, then clearly $\mathcal {M}, \pi (j) \models \varPhi _1$ for $0 \le j < \ell$, by minimality of $\ell$, and since we also have $\mathcal {M},\pi \not \models \varPhi _1 \, \mathcal {U}\, \varPhi _2$, we get $\mathcal {M}, \pi (j) \not \models \varPhi _2$ for $0 \le j \le \ell$. So, there exist $\pi ,\ell$ s.t. $\pi (0)=x$, $\mathcal {M}, \pi (\ell )\models \lnot \varPhi _1$ and for all j, $0 < j \le \ell$, $\mathcal {M},\pi (j) \not \models \varPhi _2$, which, by definition of $\mathcal {S}$, is equivalent to $\mathcal {M},x \not \models \varPhi _1 \, \mathcal {S}\varPhi _2$. Moreover, since we also know that $\mathcal {M},\pi (0)\not \models \varPhi _2$, we get $\mathcal {M},x \not \models \varPhi _2 \, \vee \, (\varPhi _1 \mathcal {S}\varPhi _2)$, i.e. the assert.