Affordance-Based Categorization of Road Network Data Using a ...

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

Journal of Geographical Information Science Vol. 00, No. 00, January 2008, 1–19

RESEARCH ARTICLE Affordance-Based Categorization of Road Network Data Using a Grounded Theory of Channel Networks Simon Scheider∗ and Werner Kuhn Institute for Geoinformatics, University of M¨ unster, Germany (v 3.0 released November 2009) We propose a grounded ontological theory of channel networks to categorize features, like junctions, in road network databases. The theory is grounded, because its primitives can be given an unambiguous interpretation into directly observable qualities of physical road networks, like supported movements and their medium, connectedness of such media, and turnoff restrictions. The theory provides a very general approach to automatically annotate and integrate road network data from heterogeneous sources, because it rests on applicationindependent observation principles. We suggest that road network categories like junctions and roads are based on locomotion affordances. Road network databases can be mapped into our channel network theory, so that instances of roads and junctions can be automatically categorized or checked for consistency by what they afford. In this paper, we introduce affordance-based definitions of a road network and a junction, and show that that the latter is satisfied by some of the most common junction types.

Keywords: semantic heterogeneity; semantic annotation; road network categories; symbol grounding problem; affordance-based definition of a junction;

1.

Introduction

Semantic heterogeneity is the main obstacle to use geographic data, like road networks, in service chains (Kuhn 2005). This is because information communities need data from diverse external sources in order to produce information products, and these products are often unintended by the data suppliers. Even though informal standards for road network labeling are available (e.g. the GDF standard, ISO 14825:2004), geometries and category labels in different commercial road network databases are far from denoting equivalent things. Furthermore, the available categories often lack high-level features, like junctions and roads, as well as labels for complex generalization tasks (Scheider and Schulz 2007). The demand for business applications based on road networks largely exceeds their original application in navigation systems (May et al. 2008). Non-standard applications require non-standard (non-available) labels and attributes. User generated geodata, like Open Street Map (OSM), has become a competitor. Here, labels can be freely attached to geometries without any top down constraints. Yet, the many quality issues1 make semantic heterogeneity of such grass root labeling an even greater challenge (Mathes 2004). Formal ontologies are seen as one means to overcome such difficulties (Klien 2008). Today’s practice uses domain ontologies to ∗ Corresponding

author. Email: [email protected]

1 http://wiki.openstreetmap.org/wiki/Category:Quality_Assurance

ISSN: 1365-8816 print/ISSN 1362-3087 online c 2008 Taylor & Francis

DOI: 10.1080/1365881YYxxxxxxxx http://www.informaworld.com

November 13, 2009 tions˙revisedendversion 2

20:22

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

Figure 1. Two subgraphs commonly annotated with ‘road’ and ‘stacked interchange’.

annotate data models and instances with category labels (Mau´e et al. 2008). Yet Hayes (1988) has pointed out that a logical theory aiming to describe the physical world can be at most experimentally complete. This means an ontology is always bound to include unintended models in its universe of interpretations. Thus, one can never know whether a given data set is an intended model of the theory or not. One could handle this problem, Hayes suggested, by tying the meaning of the theory’s tokens to observational systems (Hayes 1988). This symbol grounding problem (Harnad 1990) remains largely unsolved for most ontologies: ultimately, the semantics of the primitive terms in an ontology have to be specified outside symbol systems. Tying domain concepts like road and junction to data about their instances, as was done for the concepts river and lake by Bennett (2008), constrains these in potentially useful ways, but defers the grounding problem to the symbol system of the instance data. Furthermore, the data model is usually incomplete: Essential properties of the theory are not expressed as facts in the data (Scheider and Kuhn 2008). One would prefer a method for grounding ontological primitives in observation procedures in order to support more general ontology and data mappings. If road network categories were definable from directly observable primitives, this would provide a common approach to semantically annotate and translate the contents of heterogeneous road network databases and services. In recent work (Scheider et al. 2009), the authors have introduced a general method for grounding geographic categories in Gibson’s meaningful environment (Gibson 1986). This method uses observable primitives for locations, bodies and their locomotion through media. It allows to individuate bodies and media based on the locomotion they afford. In this article we give a definition of a junction based on locomotion affordances. Navigation oriented road network data models commonly use graphs embedded into the Euclidean plane (Figure 1). A network database expert might annotate the subgraphs of Figure 1 with ‘road’ and ‘stacked interchange’ from left to right (Scheider and Schulz 2007). He can do so because he is skilled in linking the graph components to observable categories, i.e. to what a junction affords. But our purpose is to provide a common semantic ground for such navigation oriented road network databases and their features. In order to ground data in observations, we introduce a grounded formal theory of channel networks in section 2 and show how data models map into it. This section is an advancement and application of previous ideas (Scheider and Kuhn 2008, Scheider et al. 2009). In section 3, we use a graph-theoretical abstraction of this theory to give an affordance-based definition of an n-way junction. This is the paper’s major contribution. In section 4 we show that this definition is satisfied by some of the most common junction types, before we discuss what has been achieved and what remains to be done in the concluding section 5.

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

2.

3

Grounding road networks in observable primitives

Observable primitives for the individuation of road networks and their features can be found in Gibson’s meaningful environment (2.1). We argue that channel networks are special media that afford certain types of locomotion: They are partly bounded by flat support surfaces for continuously movable bodies (2.2). They also have observable turnoff restrictions (indicated e.g. by dotted or full lines on the road surface). We account for these by introducing an affordance primitive for channels. We also show how road network data models can be interpreted in this theory of channels (2.3). We are then able to define a channel network by the movements it affords (subsection 2.4). In the formal part of this section, we use first-order logic and assume for convenience that all quantifers range over the domain of locations of the meaningful environment, and that all free variables are all-quantified. 2.1.

Parts of the environment and the individuation of media

In Scheider et al. (2009), we discussed a directly observable quality space in which geographic categories can be grounded. We summarize our main arguments in the following and introduce primitives for our channel theory. We consider a sensor to be a device to reproducibly transform qualities into symbols. Because technical sensors have to be calibrated by human observers in order to provide reproducible results (Boumans 2005), they rely on the existence of human sensors. In Scheider et al. (2009) we have argued that there exist human sensors for body-related primitives, namely nested locations for virtual bodies and virtual locomotion from one location to another. Supported by evidence from cognitive linguistic, for example (Langacker 2005), we take the view that language semantics is anchored in the individuation of bodies as directly perceivable parts of the environment. Because humans always can agree on names for these parts by pointing at them, they can share a common basic observational language about the world, much in the sense of Quine’s observation sentences (Quine 1960). We use the notion of location for every part of the environment humans can point at, and for arbitrary mereological sums of them1 . We thus suppose a human sensor for locations and their mereological part-whole structure. We also assume that there are granular spherical locations corresponding to the smallest perceivable parts of the environment (defining resolution), and call them loci of attention 2 . Note that these locations do not necessarily coincide with what people normally conceive as places, since they include e.g. arbitrary sums of body parts and free space. The primitive binary parthood relation P denotes that the first location is a part of the second. The binary relations O (overlap) and P P (proper part) are definable from it in the usual way (Casati and Varzi 1999). We assume that locations that are not parts of another location x must have a part that is non-overlapping with x. From this supplementation principle and the usual ground mereological axioms (reflexivity, antisymmetry and transitivity), the following extensionality theorem can be deduced (Casati and Varzi 1999): Theorem 2.1 : ∀z.(P P (z, x) ↔ P P (z, y)) → x = y [extensionality] It states that two locations having the same proper parts are equal. This intro1 In

Scheider et al. (2009), we used the notion ‘place’ instead. But ‘place’ has many other connotations, especially in spatial cognition. 2 We suppose that if humans point to some location, they always focus on some unique locus of attention. We are currently working on a discrete and finite version of the continuous theory in Scheider et al. (2009) with its preliminary notion of granularity.

November 13, 2009 tions˙revisedendversion

20:22

4

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

duces identity into the environment, and is a prerequisite for giving identification criteria to categories, as outlined by Guarino and Welty (2000). A mereological sum + can be easily defined (we assume the same principles for the mereological difference, denoted by the symbol −): Definition 2.2: x + y = ız.(∀w.(O(w, z) → O(w, x) ∨ O(w, y))) Axiom 2.3 : ∃j.φ(j) → (∃z∀y.(O(y, z) ↔ ∃x(φ(x) ∧ O(y, x)))) [existence of arbitrary sums] This is the General Extensional Mereology GEM (Casati and Varzi 1999), in which the fusion axiom schema assures that for any non-empty set of locations, denoted by the predicate wild-card φ, there is a location z in the environment which is the sum of that set. How do humans perceive the properties of locations in the environment? We suggest that fundamental properties are constructed by mental scanning (Langacker 2005), that is imagining a virtual movement of a virtual body (comparable concepts are ‘fictive motion’ (Talmy 1996) and ‘situated simulation’ (Barsalou 2008)). Complex shapes, for example, are perceivable through the length and direction of virtual steps1 . Verticality is also perceivable as a property of steps (Scheider et al. 2009). Without going into detail here, we assume that the environment is wholly covered by steps, and that the domain of a step (consisting of the granular spherical ‘loci of attention’) has a perceivable Euclidean structure (Scheider et al. 2009), similar to the pointless theories of space of Borgo et al. (1996) or Bennett et al. (2000). Note that the formal apparatus of the ‘meaningful environment’, which was described in (Scheider et al. 2009) and is used here in parts, includes and substantially extends the expressiveness of mereotopology, compare Bennett et al. (2000). It allows for example to construct metric concepts like ‘depth’ and ‘length’ of bodies. Our ideas about the things that are accessible by fictive motion were inspired by Gibson (1986). We assume that humans can directly perceive whether locations afford a certain type of action, and are thereby able to individuate bodies, media and their surfaces. Affordances imply perceived virtual actions. A part of a medium, for example, is a location in the environment that affords locomotion through it. The classification of a medium is stable only in a certain locomotion context: water is a medium for fish or divers, but not for pedestrians. In general, we assume that humans have a corresponding sensor for each type of locomotion, leading to a multitude of locomotion affordance primitives. Other simple affordances are smelling, hearing and seeing, giving rise to corresponding media. Complex affordances, like opening a door, are constructed from simple ones, like seeing and moving (Turner 2005). The most complex ones are social affordances, involving interactions with the environment and other actors through conventional signs, like, for example, a building with emergency exits. The central idea of Gibson is that all categories of his meaningful environment can be individuated from such affordances: Substances denote the rigid things in a meaningful environment that do not afford locomotion through them. Surfaces are located exactly where any motion must stop. More complex affordances, like sitting or entering and leaving, give rise to subcategories of substances, like chair and door. In this paper, we re-use the primitive binary affordance relation for air medium connectedness, AirC, introduced in Scheider et al. (2009), which denotes a pair of

1 Consider

e.g. Marr’s (Marr and Nishihara 1978) reduction of the shape representation problem to a configuration of cylinders. Each cylinder can be interpreted as a straight path of a certain length, and their configuration by relative directions can reconstruct complex shapes.

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

5

Figure 2. The meaningful environment is wholly covered by virtual paths.

loci of attention x, y connected by a free ‘path’ p. The path p corresponds to a minimal elongated free location, that allows a fictive granular body to be moved from x to y inside of p. This path is self-connected by AirC (see Figure 2 and axiom 8 in Scheider et al. (2009)). Since the fictive body has the shape of a locus of attention — which is basically just a granular sphere — the path must also be strongly connected in a topological sense. This means it is ‘all in one piece’, and there is a surface corresponding to any cut, as if ‘cutting in wood’ (compare the formalization of Borgo et al. (1996)). We furthermore require from this medium primitive to be a partial equivalence relation, i.e. to be symmetric and transitive. Using arbitrary paths of the environment (not only free, air-connected ones that constitute air media), a general notion of strong connectedness SC(x) can be defined (Scheider et al. 2009) by requiring that each pair of granular parts of a location x of the environment is connected by a path which is part of x. In Scheider et al. (2009), we used affordance primitives like AirC to construct Gibson’s concepts of surface, substance and medium, and we called the maximal strongly connected substance parts bodies. Our principle of individuation for media and bodies is based on the idea of unity outlined by Gangemi et al. (2001). A whole is any maximal mereological sum whose parts are self-connected by a certain partial equivalence relation. We slightly divert from Gangemi et al. (2001) in that we require only partial equivalence relations, these being symmetric and transitive but not necessarily reflexive, so that the relation partitions only a subset but not necessarily the whole domain. An air medium is then any location which has all the locations of an equivalence class of AirC as parts: Definition 2.4: Air(x) ↔ W holeAirC (x) The definition of a whole is based on the following first-order axiom schema, where Φ can denote any binary relation, and its sub domain is denoted by the overloaded unary predicate Φ(x) ↔ ∃y.Φ(x, y) ∨ Φ(y, x): Definition 2.5: ΣΦ (x) ↔ (∀y.P (y, x) → (∃z.Φ(z) ∧ P (z, x) ∧ O(z, y))) [x is a sum of entities from the domain of Φ] Definition 2.6: U nityΦ (x) ↔ ΣΦ (x) ∧ (∀z, y.(P (z, x) ∧ P (y, x) ∧ Φ(z) ∧ Φ(y)) → Φ(z, y)) [unities are self-connected by Φ] Definition 2.7: W holeΦ (x) ↔ U nityΦ (x) ∧ (∀y, z.P (y, x) ∧ Φ(z, y) → P (z, x)) [wholes are maximal unities] Note that these definitions imply that an air medium is also strongly connected in a topological sense. Unity implies that every two granular parts of the location are connected by an air-connected path, which is itself — as mentioned above — strongly connected. Since this path must be part of the same air-connected whole by maximality (Definition 2.7), it follows that the whole is strongly connected, too. This is not the case for an air-connected unity.

November 13, 2009 tions˙revisedendversion 6

20:22

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

Figure 3. Illustration of a channel. It is part of an air medium which is located just above a flat smooth surface. Furthermore, it restricts supported movements to enter and leave at separate portals by convention.

2.2.

Roads afford supported locomotion

Applying a primitive affordance relation of the meaningful environment, like AirC, should be interpreted as an observable fact: AirC(a, b) asserts that a certain fictive body can move through air from location a to b. As Gibson argued, this fact is not reducible to other facts, and therefore should not be defined. It is directly observable by humans in the sense that it is an inter-subjective result of their sensorimotor construction, and therefore is at the roots of our method for grounding databases. In this way, we can use more specific affordance primitives based on AirC(a, b) for road network representations. The approach is well suited to capture the multimodality of traffic, since all modes of traffic, e.g. walking, driving, diving and flying, have such observable affordances and corresponding media. Roads are configurations of surfaces, air and substance parts that afford supported locomotion: cars, bicycles and pedestrians do not move through the concrete bodies of roads, nor do they fly arbitrarily around in the air, but they move through a piece of air right above a supporting surface such that they are constantly in touch with it (Kuhn 2007) (Figure 3). More specifically, flat supports afford differentiable movements: A support can be conceived as a regular manifold with a surface whose normal in every point has a vertical component (Scheider and Kuhn 2008). The perception of this affordance therefore presupposes several sensors besides AirC, including ones for flatness and verticality. We call this type of motion friction dominated or supported locomotion (Scheider and Kuhn 2008) and assume that there is a binary affordance primitive SupportC for it. It means that two air-connected locations are also connected by a flat support surface. In this paper, we omit any formal description of the geometric support surface property, because it is not necessary as a premise for the subsequent arguments. One aspect of its intended meaning can however be easily expressed by asserting that the tuple of locations connected by SupportC forms ‘one air-connected piece’, that is, it is a strongly connected unity with respect to AirC: Axiom 2.8 : SupportC(x, y) → U nityAirC (x + y) ∧ SC(x + y)

2.3.

Turnoff restrictions, channels and their data models

Roads additionally exhibit conventional turnoff restrictions, observable e.g. by road signs or dotted vs. full lines on the side of a road surface. Turnoff restrictions are the main instrument for traffic planning, because they restrict the supported movements to a regulated more or less homogeneous flow into discrete directions. This is a case of physically manifested social affordances, because it presupposes the interpretation of symbols on the road surface. We account for this complex affordance by a further binary locomotion affordance primitive, called LeadsT o, whose domain and range consists of a support-connected

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

7

Figure 4. Channel configurations forming simple road network data elements.

tuple of locations called channels: Axiom 2.9 : LeadsT o(x, y) → ¬O(x, y) [irreflexivity and non-overlap] Axiom 2.10 : LeadsT o(x, y) → SupportC(x, y) [support connectedness] This is the central affordance primitive that allows us to ground road network databases. The idea behind it is that channels force their afforded locomotions to move into only one direction, and to leave into or enter from another channel at portals (see Figure 3). Because they allow for movements into only one direction, channels cannot overlap, and are discrete 1 . The affordance relation must also be irreflexive, because channels always lead to somewhere else, but non-overlap implies irreflexivity. A further important aspect of channels is that their domain is supposed to be finite. One can think of channels as one side of a bidirectional road or a one-way street between two intersections. Because the channel theory is grounded, channels can be directly observed: A channel is a part of a support-connected medium in which movements are restricted to only one direction by convention. This is the reason why in road network data models, channels are usually embedded into the plane by finite sets of line segments whose end nodes indicate some kind of intersection. The intended meaning of channels are sets of discrete locations which occur only in finite amounts. This has important implications for our formalism, since we can have finite models 2 to account for channel semantics. And the channel graph theory we will develop in the next subsection is finite, too. But obviously the data model needs to be more complex than an ordinary graph: In Figure 4 and Figure 5, we have depicted example channel configurations (left) together with their equivalent road network data models (right), denoting e.g. oneway streets and dead ends (Scheider and Schulz 2007). In the data model sub figures, the LeadsT o relation is depicted by dotted arrows, channels are represented as full arrows, and the embedded street segments are represented as a separate undirected graph with lines and nodes. 1 This

strong requirement can be satisfied even in complicated traffic situations, e.g. at signalized crossings or ‘yield’/‘give way’ roads. In these cases, we intend to allow for temporary channels, which cease to exist for that time interval in which a crossing channel happens to exist (Scheider and Kuhn 2008). This, of course, requires a temporal version of the theory, which is not in the scope of this paper. 2 See Ebbinghaus and Flum (2005) for a discussion of the expressiveness of finite first-order theories.

November 13, 2009 tions˙revisedendversion 8

20:22

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

Figure 5. Channel configurations forming simple road network data elements.

Note how the road network data model (right) is supposed to be interpreted in the channel network theory (left): Any adjacent and consecutive pair of sides of embedded street segments which is not affected by turnoff prohibition is equivalent to a tuple of channels of the LeadsT o relation3 (Scheider and Kuhn 2008). The existence of turnoff restrictions has formal consequences: The embedded street segment graph (depicted by the undirected graph on the right hand side of Figure 4 and Figure 5) needs to be supplemented by a second relation on top of it (depicted separately as dotted arrow). This is because the LeadsTo relation would not be captured otherwise: the two branching edges of a diametrical bifurcation (Figure 5.2), for example, are connected at a common node, but do not lead to each other. Especially interesting are pairs of channels that are support connected but neither one leads to the other. A plausible model for this are neighboring channels built on the same surface, like the neighboring sides of a bidirectional road without a median strip (see the bidirectional way in Figure 4.2 or the diametrical bifurcation in Figure 5.2). Hence, a road network can be appropriately represented by a subgraph of the line graph 1 of a street segment graph, i.e. a graph with nodes for channels and edges for only those connected channel pairs where one channel leads to the other. Such a restricted line graph offers also an appropriate way of modeling turning costs in route planning (Winter 2002).

2.4.

Affordance-based definition of channel networks

Channel networks are media that afford mutual reachability between all its parts. The ternary relation ReachableF romIn is recursively definable as the transitive

3 At signalized intersections, this is a triple rather than a tuple, including one temporary channel. Temporary channels exist only during green phases and do not have data equivalents. 1 A line graph L(G) of a graph G is the graph on G-edges in which two G-edges are adjacent as vertices if and only if they are adjacent in G (Diestel 2000).

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

9

closure 2 of LeadsT o. A channel y is reachable from channel x in a larger support f if both are parts of f and either x directly leads to y or y is reachable from x via z that is also part of f: Definition 2.11: (y)ReachableF rom(x)In(f ) ↔ P (x, f ) ∧ P (y, f ) ∧ (LeadsT o(x, y) ∨ (∃z.(z)ReachableF rom(x)In(f ) ∧ (y)ReachableF rom(z)In(f ))) Note that the relation ReachableF romIn is neither symmetric nor reflexive with respect to x and y. There is an even stronger notion of reachability, which we call 2-reachability: Definition 2.12: (y)2-ReachableF rom(x)In(f ) ↔ (y)ReachableF rom(x)In(f ) ∧ (∀z.(z)ReachableF rom(x)In(f ) ∧ (y)ReachableF rom(z)In(f ) ∧ z 6= x ∧ z 6= y → (x)ReachableF rom(y)In(f − z)) We consider a channel y 2-reachable from x iff one can reach y from x and still reach x from y after removing any intermediate channel z from x to y. The idea behind it is that a support f affords mutually independent to and from paths between remote locations. This is a plausible assumption for road networks, because one never uses the same road into the same direction when driving back. We can now define a channel network as a whole with respect to 2-reachability: Definition 2.13: ChannelN etwork(x) ↔ W holeλa,b.((a)2-ReachableF rom(b)In(x)) (x) Note that wholeness requires for a channel network that reachability amongst its channels be reflexive (all channels of the network can be reached by navigation from themselves in the network) and symmetric (it is always possible to return to the origin). It also requires that the network exclusively consists of channels. Furthermore, as was discussed by Scheider and Kuhn (2008) in greater detail, every channel network must consist of at least two mutually reachable channels (minimal model): There must be at least one channel by definition of a whole (Definition 2.7), but because this channel has to be reachable from itself inside the network (Definition 2.13), and leadsTo is irreflexive by Axiom 2.9, there has to be another non-overlapping one. A channel network also does not contain any graveyards and factories, channels without the possibility to leave or enter. This follows directly from Definition 2.6 and the minimal model. Our theory of channel networks partly satisfies a certain graph theory. We call this theory the channel digraph. Channel digraphs will be used in the following text and figures in order to describe our formalism on another abstraction level, and in order to make it more amenable to data structures and algorithmic standards that are based on graphs. If we take the set of channels of a channel network as the set of vertices V , and the relation LeadsTo as the arc relation A (directed edges), then the resulting graph D(V, A) is a directed graph without loops (arcs connecting one vertex to itself, due to Axiom 2.9) and multiple arcs (arcs with the same incident vertices). For an arc e = (x, y) we call the incident vertex x the initial, and y the terminal vertex of e. The following graph theoretical notions will be used to further describe channel digraphs (Berge 1991): A chain is a sequence of vertices and arcs c = (x0 , e1 , x1 , e2 , x2 , ..., xq−1 , eq , xq ), such that either ek = (xk , xk−1 ) or ek = (xk−1 , xk ). Chains are called elementary if each inner vertex x1 ...xq−1 appears only once. A chain that does not contain the same arc twice is called simple. A 2 Even

though the transitive closure is not in general first-order definable, this restriction does not apply here because the domain of channels is finite. The semantics of this recursive formula can be given by a least fixpoint in a finite model, compare Ebbinghaus and Flum (2005), page 220.

November 13, 2009 tions˙revisedendversion 10

20:22

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

Figure 6. a) Not strongly connected block b) strongly connected, but not a block c) strongly connected block, but not 2-reachable whole d) 2-reachable whole

walk is a forward traversable chain such that ek = (xk−1 , xk ). A path is a simple walk. A chain is called closed if x0 = xq . A cycle is a simple closed chain. A circuit is a closed path. The recursively defined reachability relation Definition 2.11 between x and y on z is equivalent to the existence of a walk between vertices x and y on a subgraph z of the channel digraph D. What graph-theoretical properties can be asserted for channel digraphs? It can be shown that 2-reachability (Definition 2.12) implies strong connectedness of D as well as 2-connectedness, so that D is a strong block (every arc is part of a circuit and an elementary cycle; compare standard definitions, e.g. in Berge (1991) or Diestel (2000)). As Figure 6) shows, 2-reachability is even more restrictive, and requires that every elementary walk in a channel digraph is part of an elementary circuit.

3.

A junction affords navigational choices

In section 2 we have introduced a grounded theory of channel networks. The theory can be used to check and categorize road network data with respect to what their ’observable world’ equivalents afford. In this section, we will set the stage for categorization of the most prominent road network features, namely junctions.

3.1.

Channel network features as induced subgraphs

We now turn our focus to the properties of subgraphs of this channel digraph. An induced subgraph F (U, O) has a subset of vertices U ⊆ V of the channel digraph, and the set of arcs with elements e ∈ A connecting any pair {x1, x2} ⊂ U , is exactly its set of arcs, e ∈ O. A channel network feature, like a road or junction, is an induced subgraph of D, because the selection of channels does not influence the affordance relation between them. The essential affordance properties of such induced subgraphs now seem to arise from the connection properties of some of their vertices, called entries, exits and gates (Figure 7). Consider all arcs from A \ O that are incident with vertices in U (connecting F with its complement in D), and call them in-/out bridges of F . We call a vertex v ∈ U an entry of F , if and only if it is a terminal vertex of an in-bridge, and an exit if and only if it is an initial vertex of an out-bridge. The set of bridge-incident vertices that are either entries or exits (but not both), is called gates of F . Gates allow traffic to either enter or leave the subgraph, but not both. Note that a gate forces a moving object traversing it to move on a walk of length ≥ 1 inside the subgraph F , because it cannot leave F immediately. Definition 3.1: Entry(en, x, y) ↔ ChannelN etwork(y) ∧ P P (x, y) ∧ P (en, x) ∧ (∃z.¬P (z, x) ∧ P (z, y) ∧ LeadsT o(z, en))

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

11

Figure 7. An induced subgraph of a channel network and its entries and exit vertices (indicated by dotted arrows, gates are narrowly dotted).

Definition 3.2: Exit(ex, x, y) ↔ ChannelN etwork(y) ∧ P P (x, y) ∧ P (ex, x) ∧ (∃z.¬P (z, x) ∧ P (z, y) ∧ LeadsT o(ex, z)) Definition 3.3: Gate(e, x, y) ↔ ((Entry(e, x, y)∧¬Exit(e, x, y))∨(Exit(e, x, y)∧ ¬Entry(e, x, y))) Because a channel digraph D is 2-reachable (Definition 2.13), it follows that for any proper subgraph F with more than one arc, there always exist at least one entry and one distinct exit. To see this, remember that each pair of vertices of the 2-reachable channel digraph lies on an elementary circuit, and so must any pair v, v’ with v ∈ U and v 0 ∈ V \ U . Therefore the paths p : v → v 0 and p0 : v 0 → v must contain an entry on p0 and an exit on p that are either distinct or equal to v. If we take v to be the initial vertex of the arc in F, there is always a path p : v → v 0 that contains this arc, and therefore the corresponding exit on p must be distinct from v. For the same reason, there must be an F -internal walk from each entry to an exit and to each exit from an entry (F is no sink or source of traffic; Figure 7): Theorem 3.4 : ∀x, y.ChannelN etwork(y) ∧ P P (x, y) ∧ ΣLeadsT o (x)) → (∃en, ex.Entry(en, x, y) ∧ Exit(ex, x, y) ∧ en 6= ex) ∧ (∀en.Entry(en, x, y) → (∃ex.Exit(ex, x, y) ∧ (ex)ReachableF rom(en)In(x))) ∧ (∀ex.Exit(ex, x, y) → (∃en.Entry(en, x, y) ∧ (ex)ReachableF rom(en)In(x)))

3.2.

Affordance based definition of n-way junctions

Now we are in a position to define a junction based on the actions it affords as part of a channel network. We can do this because our graph theory is grounded in locomotion affordances. An n-way junction is an induced subgraph of a channel digraph, which (1) affords n − 1 (n ≥ 3) navigational choices for n entries, as there is a walk from each of the n entries to each of n exits except into the opposite direction (total reachability), (2) affords movements to enter and leave through distinct entry and exit channels (discreteness of navigational action), (3) does not contain a smaller n-way junction (minimality I) and (4) has entries and exits with a minimal vertex degree of two. Entries have either more than one internal successor or an internal predecessor. Analogously, exits have either more than one internal predecessor or an internal successor (minimality II). We discuss and motivate these properties in the rest of this section by analyzing a median u-turn junction (see Figure 8). At a median u-turn intersection, the main

November 13, 2009 tions˙revisedendversion 12

20:22

Journal

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

Figure 8. A median u-turn (Michigan left) intersection data model.

Figure 9. Illustration of junction properties by a median u-turn channel subgraph (a). If one leaves out the grey parts in b, c and d, this violates some of the required properties for junctions. For details see text.

road, a dual carriageway, intersects the minor road, which is a bidirectional road. It is prohibited to turn off to the left from the major road or onto the major road at the intersection point. These left turns are only possible via u-turn channels (see data model in Figure 8). Channels are indicated by arrows alongside their embedded street segment lines. Note that the bidirectional street segments have two channels, one for each direction, while the dual carriageway consists of one-way street segments. The LeadsTo relation among channels is indicated by grey curved arrows that lead from a channel to its successor. Exits and entries of the subgraph are numbered anticlockwise. Feature external channels are in grey. We will refer to the correspondent channel digraph in Figure 9 a, in which channels are depicted as vertices and the LeadsTo relation as arcs. The exits and entries in these figures have equivalent numbers. The first and most obvious property is that junctions afford paths from each entry to each exit (except the one in the opposite direction) (total reachability). Thus they enforce a navigational choice. If a driver has entered a junction, he is afterwards forced to take a directional decision by taking one of a set of n − 1 paths inside of it. Note that this property also allows drivers coming from different directions to

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

13

Figure 10. Illustration of the minimality properties by a median u-turn channel subgraph. a) It is always possible to add roads (left) and superfluous channels (right) and still meet requirements one and two for junctions. b) This subgraph even meets the first minimality requirement. For details see text.

take the same exit. Each entry is used by people driving into different directions, and each exit is used by people coming from different directions. Together with the minimality assumption, this property also implies that a junction is a connected subgraph. If one leaves out the dotted arc in Figure 9 b, then entries one, two and four are affected and lose their paths to exits three and four, so that this subgraph is not a junction anymore. The second property, discreteness of navigational action, seems to be common to every road network feature, thus also to a road. If people are speaking of ‘staying on or leaving a road/junction’, we suggest that they mean discrete actions: The process of entering, staying on, or leaving a road/junction is unambiguous for a moving object inside of a channel. So roads and junctions are assumed to have mutually exclusive entry and exit channels (gates). An obvious reason for this is that navigational actions are kept simpler and more transparent for other road users. The subgraph in Figure 9 c (in black) is a part of the median u-turn that also satisfies the total reachability assumption. But because neighboring exits and entries collapse, this subgraph violates the discreteness property. Even if one adds the two entries (En1, En3) and exits (Ex4, Ex2) like in Figure 9 d, one will not yet have identified a proper junction for the same reasons. Only if one adds the two channels at the centre of the bidirectional road (Figure 9 a), this assumption is met. We furthermore need minimality assumptions for junctions. This is because one can usually supplement a junction with further channels such that it satisfies the first two properties. In Figure 10 a, we have extended the median u-turn at the left end by completing the dual carriageway road. At the right end, we have added two channels. Junctions are required to be minimal in the following two senses: In a first sense, a junction is minimal because it never contains a smaller junction. We imply a criterion of individuation based on minimality, because junctions keep the first two properties if one adds an extra road to them (Figure 10). In a second sense, we require a minimal vertex degree for entries and exits: This is because entries and exits should afford either a navigational choice or a path from another entry, or to another exit. In the first case, the entry has more than one successor, and the exit has more than one predecessor. In the second case, the entry has an internal predecessor, and the exit has an internal successor. Why is this a legitimate property of junctions? Take for example the entry En2’ with cardinality one in Figure 10 a. All paths from this entry must cross the one edge incident to En2’. And because no other path (from another entry) uses this edge, the subgraph can safely be shortened by this edge while retaining its total reachability property. The same applies for exits.

November 13, 2009 tions˙revisedendversion

20:22

Journal

14

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

(a)

(b)

(c)

Figure 11. (a) A 4-way intersection (source: public domain), (b) its data model and (c) the corresponding channel digraph.

The second minimality criterion is necessary because it is generally not implied by the first. To see this, one has to slightly increase the complexity of the median u-turn junction (see Figure 10 b) by including additional vertices into the arcs that allow right turns from and to the vertical bidirectional street. Properties one and two are satisfied by the subgraph with full black arcs, because all entries and exits are gates and mutually reachable from each other. This subgraph is also minimal in the first sense, because it does not contain a smaller junction satisfying the properties one and two. But the minimal vertex degree of these new gates is one, so that this condition is able to identify the false positive example. We use an abbreviation for the fact that in a pair of channels, one channel leads to the other: Definition 3.5: LConn(x, y) → LeadsT o(x, y) ∨ LeadsT o(y, x) This leads to the following formal definition in our grounded theory of channel networks: Definition 3.6: Junctionn−way (x) ↔ ∃y. (∀en.Entry(en, x, y) → ¬Exit(en, x, y))∧ (∀ex.Exit(ex, x, y) → ¬Entry(ex, x, y))∧ [discreteness of navigational action] (∃!n ex.Exit(ex, x, y) ∧ ∃!n en.Entry(en, x, y))∧ [there exist n entries and exits] (∀en, ex.Entry(en, x, y) ∧ Exit(ex, x, y) → (ex)ReachableF rom(en)In(x))∧ [total reachability] (¬∃z.Junctionn−way (z) ∧ P P (z, x))∧ [minimality I] (∀e.(Entry(e, x, y)∨Exit(e, x, y)) → (∃c, d.c 6= d∧P (c, x)∧P (d, x)∧LConn(e, c)∧ LConn(e, d))) [minimality II]

4.

Application to common junction models

Will will now show that our definition is satisfied by instances of the most common junction types. The first example we examine is a simple 4-way intersection (Figure 11). The subgraph consists of four entry and four exit gates without internal channels, and these are directly connected by LeadsTo edges. As the subgraph is the minimal model for a 4-way junction, it is minimal in the first sense, and because each gate is a vertex with three successors (resp. predecessors), it is also minimal in the second sense. A junction type similar to the median u-turn is a diamond interchange (Figure 12). Here two major roads with different speed limits intersect. The long ramps indicate the faster road. The channels of the faster road are unnecessary for the individuation of a 4-way junction, therefore they are marked as feature external

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

Science

Junc-

International Journal of Geographical Information Science

(a)

(b)

15

(c)

Figure 12. (a) A diamond interchange aerial photo (source: public domain (USGS)), (b) its data model and (c) channel digraph.

(a)

(b)

(c)

Figure 13. (a) A jughandle intersection (source: public domain (USGS)), (b) data model and (c) channel digraph .

(grey). As one can see all entries and exits are totally reachable gates with vertex degree two. The model is also minimal in the first sense: Although there is an elementary circuit due to the u-turns in the middle of the minor road, satisfying properties one, two, and four in section 3.2, this circuit does not have gates. Any other totally 4-reachable subgraph does not comply with property four. Another well known junction type is a jughandle intersection (Figure 13). It prevents left-turns and right-turns from the major road at the intersection point by providing only appropriate ramps. Properties one, two, and four of section 3.2 are satisfied, and the model is also minimal because all its induced subgraphs are either not totally reachable or introduce non-gates. A ‘prototype’ junction is the cloverleaf interchange (Figure 14), mainly used for highways and high speed roads to prevent left turns by so called loop roads. The junction is totally reachable (it is possible to drive on an elementary circuit consisting of exactly those loops), and obviously has gates of degree two (properties one, two and four of section 3.2). It is also minimal: although the inner circuit is totally reachable in four directions and all its vertices have a degree of two, they are not gates. All other subgraphs contradict one of the junction properties. In addition to these models, our definition of a junction (Definition 3.6) is also satisfied by the following 4-way types: quadrant roadway intersection, stacked interchange, continuous flow intersection, and the following 3-way types: trumpet interchange and semi-directional T-interchange1 . The considered junction types

1 See

Rodegerdts et al. (2004) for an extensive list of signalized junction types with description.

November 13, 2009 tions˙revisedendversion

20:22

Journal

16

of

Geographical

Information

Science

Junc-

Simon Scheider and Werner Kuhn

(a)

(b)

(c)

Figure 14. (a) A cloverleaf interchange (by kind permission of Rijkswaterstaat, Adviesdienst Geoinformatie en ICT, Beeldbank VenW, Netherlands), (b) its data model and (c) channel digraph.

are deemed most common also by Wikipedia2 . The question of whether our first-order logic definition is a correct and complete classifier, thus a solution to an inductive prediction learning problem covering all and only the positive examples (Wrobel 1996), turns out to be problematic from a methodological point of view. First, because the concept of a junction and other higher order network features is not unambiguously established in existing network data models, it is unclear where to get an unbiased sample from, and who applies the ‘right’ concept. Second, road network categories—like all human categories— have a graded structure (Lakoff 1990), so there are many more or less prototypical versions of a definition. Our junction definition is also applicable to roundabouts that have only gates as entries, though for a general definition of roundabouts (which can include non-gate entries), the discreteness property of Definition 3.6 would have to be dropped. So, at least in our understanding, roundabouts are less prototypical types of junctions needing special treatment.

5.

Conclusion and future work

We presented a formal theory of channel networks which is grounded in observable qualities of the environment. Following Gibson, media, bodies and what they afford are facts directly accessible to human observation and measurement. An example is the medium connectedness relation AirC, which denotes an afforded motion of a body from one location to another. Media, e.g. Air, can be individuated by using such a primitive relation as unity criterion. We introduced affordance primitives for flat supports and channels as subcategories of AirC in order to account for the observable facts that cars require a medium with a smooth support surface and that potential movements on a road are further restricted by observable turnoff restrictions. We gave a definition of a channel network as a medium being a 2reachable whole of channels. We also gave a standard interpretation of our theory into common road network data models and provided a graph theoretical view on it. Using this grounded theory, it is now possible to categorize navigation oriented road network graphs, and to check their formal properties against the theory’s

2 http://en.wikipedia.org/wiki/Junction_(traffic)

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

REFERENCES

Science

Junc17

theorems or other observations (in the latter case using the theory as a semantic link). We demonstrated this by giving an affordance-based definition of a junction which is satisfied by the most common junction models. The theory could be used in order to develop explanation-based classifiers (in the spirit of Mitchell et al. (1986)) and semantic consistency checkers for road network categories, such as those of OSM. The theory is not bound to be interpreted into a certain data model because it comes with a domain independent observation procedure. It therefore constitutes a link between the worlds of data, observation and domain ontologies. We are aware that there are other useful perspectives on road network databases, for example as a representation of constructed infrastructure, or as a reference frame for address data. Still, the affordance based approach and the proposed formalism, which also encompasses complex metric concepts (compare the example ‘waterdepth’ in Scheider et al. (2009)), are useful as a common semantic ground for these perspectives. As a next step, we intend to develop appropriate graph based classification and search algorithms to demonstrate the usefulness of the theory. We furthermore intend to account for the graded structure (Lakoff 1990) of road network categories. In addition, the underlying grounded theory of the meaningful environment is still under development, in particular with respect to the issues of finite granularity, uncertainty and temporality.

Acknowledgements

We would like to thank the reviewers for their insightful comments. This work is funded by the Semantic Reference Systems II project granted by the German Research Foundation (DFG KU 1368/4-2).

References

Barsalou, L.W., 2008. Grounded cognition. Annual review of psychology, 59, 617– 645. Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M., 2000. A foundation for regionbased qualitative geometry. In: W. Horn ed. Proc. 14th European conference on artificial intelligence. Amsterdam: IOS Press, 204–208. Bennett, B., Mallenby, D., Third, A., 2008. An ontology for grounding vague geographic terms. In: C. Eschenbach, M. Gruninger, eds. Proc. 5th intern. conf. on formal ontology in information systems. Saarbr¨ ucken: IOS Press. Berge, C., 1991. Graphs. 3rd rev. ed. North-Holland mathematical library, Vol. 6. Pt. 1. Amsterdam: North-Holland. Borgo, S., Guarino, N., Masolo, C., 1996. A pointless theory of space based on strong connection and congruence. In: L.C. Ailello, J. Doyle and S. C. Shapiro eds. Principles of knowledge representation and reasoning (KR96): Proc. 5th international conference. Boston: Morgan Kaufmann, MA, 220–229. Boumans, M., 2005. Measurement outside the laboratory. Philosophy of science, 72, 850–863. Casati, R., Varzi, A., 1999. Parts and places: The structures of spatial representation. Cambridge: MIT Press. Diestel, R., 2000. Graph theory. 2nd ed. Graduate texts in mathematics 173. Heidelberg: Springer. Ebbinghaus, H.D., Flum, J., 2005. Finite model theory. Springer monographs in mathematics. Berlin: Springer.

November 13, 2009 tions˙revisedendversion 18

20:22

Journal

of

Geographical

Information

Science

Junc-

REFERENCES

Gangemi, A., Guarino, N., Masolo, C., Oltramari, A, 2001. Understanding top-level ontological distinctions. In: H. Stuckenschmidt ed. Proc. IJCAI 2001 workshop on ontologies and information sharing. Gibson, J.J., 1986. The ecological approach to visual perception. Hillsdale, NJ: LEA. Guarino, N., Welty, C., 2000. Identity, unity, and individuality: Towards a formal toolkit for ontological analysis. In: W. Horn ed. ECAI 2000 proceedings (Frontiers in artificial intelligence and applications). Amsterdam: IOS Press, 219–223. Harnad, S., 1990. The symbol grounding problem. Physica, D 42, 335–346. Hayes, P.J., 1988. The second naive physics manifesto. In: J.R. Hobbs and R.C. Moore, eds. Formal theories of the commonsense world, Norwood, NJ: Ablex, 1–36. Klien, E., 2008. Semantic annotation of geographic information. Thesis (PhD). University of M¨ unster, Germany. Kuhn, W., 2005. Geospatial semantics: Why, of what and how? In: S. Spaccapietra and E. Zim´ anyi, eds. Journal on data semantics III. LNCS 3534. Berlin: Springer, 1–24. Kuhn, W., 2007. An image-schematic account of spatial categories. In: S. Winter, ed. Spatial information theory. Proc. 8th international conference, COSIT 2007, Melbourne, Australia, 19–23 September 2007. LNCS 4736. Berlin: Springer, 152– 168. Lakoff, G., 1990. Women, fire and dangerous things. Chicago: University of Chicago Press. Langacker, R.W., 2005. Dynamicity, fictivity, and scanning. The imaginative basis of logic and linguistic meaning. In: D. Pecher and A. Zwaan eds. Grounding cognition. The role of perception and action in memory, language and thinking. Cambridge: Cambridge University Press, 164–197. Marr, D., Nishihara, H.K., 1978. Representation and recognition of the spatial organization of three dimensional structure. Proc. R. Soc. Lond. B, 200 (1140), 269–294. Mathes, A., 2004. Folksonomies - Cooperative classification and communication through shared metadata. Computer mediated communication, LIS590CMC, Illinois. Mau´e, P., Duchesne, P., Schade, S., 2008. Semantic annotations in OGC standards. OGC Discussion Paper 08-167. May, M. et al., 2008. Pedestrian flow prediction in extensive road networks using biased observational data. In: W.G. Aref et al. eds. 16th ACM SIGSPATIAL international conference on advances in geographic information systems (ACM GIS 2008), Irvine, CA, US, 5–7 November 2008. New York: ACM, 471–480. Mitchell, T.M., Keller, R.M. and Kedar-Cabelli, S.T., 1986. Explanation-based generalisation: A unifying view. Machine Learning, 1 (1), 47–80. Quine, W.V.O., 1960. Word and object. Cambridge: MIT Press. Rodegerdts, L.A. et al., 2004. Signalized intersections: Informational guide. Federal highway administration (FHWA) Report No. FHWA-HRT-04-091. Available from: http://www.tfhrc.gov/safety/pubs/04091/ [Accessed 01 April 2009]. Scheider, S. and Schulz, D., 2007. Specifying essential features of street networks. In: S. Winter, ed. Spatial information theory. Proc. 8th international conference, COSIT 2007, Melbourne, Australia, 19–23 September 2007. LNCS 4736. Berlin: Springer, 169–186. Scheider, S. and Kuhn, W., 2008. Road networks and their incomplete representation by network data models. In: T.J. Cova et al., eds. Proc. 5th international conference on Geographic Information Science, Park City, UT, US, September 2008. LNCS 5266. Berlin: Springer, 290–307.

November 13, 2009 tions˙revisedendversion

20:22

Journal

of

Geographical

Information

REFERENCES

Science

Junc19

Scheider, S., Janowicz, K. and Kuhn, W., 2009. Grounding geographic categories in the meaningful environment. In: K.S. Hornsby et al., eds. Spatial information theory, 9th international conference, COSIT 2009, Aber Wrach, France, September 21-25, 2009 proceedings. LNCS 5756. Berlin: Springer, 69–87. Talmy, L., 1996. Fictive motion in language and ”ception”. In: P. Bloom et al. eds. Language and space. Cambridge, MA: MIT Press, 211–276. Turner, P., 2005. Affordance as context. Interacting with computers, 17 (6), 787– 800. Winter, S., 2002. Modeling costs of turns in route planning. GeoInformatica, 6 (4), 345–360. Wrobel, S., 1996. Inductive logic programming. In: G. Brewka, ed. Principles of knowledge representation. Stanford, CA, USA: Center for the study of language and information, 153–189.