vaelii.impl.orientation — com.vaelii/vaelii 0.8.0

Cardinal-direction reasoning — a relation algebra over the generic constraint network in vaelii.impl.qcn, and the companion to the RCC-8 topology in vaelii.impl.space (vaelii.impl.interval is the third, over time). RCC-8 says whether two regions touch or nest; this says where one place lies relative to another. (Orientation is the standard qualitative-spatial name for this family — topology, orientation, distance — and it keeps the namespace clear of a rule's chaining :direction, which is an unrelated thing.)

The nine base relations are the projection-based cardinal direction calculus. A direction decomposes into two independent one-dimensional point relations, one per axis — east-west on x, north-south on y — each :lt / :eq / :gt:

:n [:eq :gt] :ne [:gt :gt] :e [:gt :eq] :se [:gt :lt] :s [:eq :lt] :sw [:lt :lt] :w [:lt :eq] :nw [:lt :gt] :eq [:eq :eq] (the same place)

That decomposition is the whole trick, and it is why there is no 9×9 table here: composition is computed, not transcribed. Composing two directions is composing their x projections and their y projections separately through the three-relation point algebra, then reading the result pairs back as directions. Since the nine directions are exactly the nine [x y] combinations, that read-back is total and the composition of any two base directions is a non-empty set — so the table nobody wrote down cannot disagree with itself, and north-then-east is northeast because :eq;:gt is :gt on x and :gt;:eq is :gt on y.

Directions are stored as ordinary sentexes — the nine named binary predicates (northOf, southeastOf, sameLocationAs, …), plus four derived predicates (northwardOf, eastwardOf, …) that each name a disjunction of them: a northerly component, whatever the east-west one. Places are ordinary individuals; nothing about them is special.

The calculus reads every asserted direction visible from a context into a qualitative constraint network — {[a b] → #{possible base directions}}, an unrecorded pair meaning "unknown", i.e. all nine — and qcn/path-consistent tightens it to a fixpoint. the prover then answers a goal (P a b) by entailment: it holds iff every direction still possible from a to b satisfies P, possible ⊆ denotation(P). So (northeastOf A B) and (northeastOf B D) entail (northeastOf A D), and with it the weaker (northwardOf A D) and (eastwardOf A D).

An emptied constraint anywhere means the asserted directions are unsatisfiable, and then no direction goal is answered — an inconsistent theory should not be mined for conclusions.

Soundness. Path consistency is sound but not in general complete, so an entailment reported here is real while a non-entailment means "not provable", never "provably false" — the same open-world reading argIsa and exceptWhen take.

The vocabulary ships in kb/upper/CxSpace.txt beside the RCC-8 one — both are about space, so CxCore keeps only the grammar they are declared in. The prover is opt-in on top of it: register it with vaelii.core/add-prover, and until then a KB stores and retrieves directions as ordinary facts without paying for the network.

all-directions clj

The universe of the algebra: the constraint on a pair nothing is known about.

base-direction-predicate clj

Base direction keyword → the binary predicate a fact about it is stored under.

cardinal clj

The cardinal directions as a vaelii.impl.qcn-kb calculus — the algebra, the vocabulary, and the two caches. Everything below delegates to the shared glue, which is the same code RCC-8 and the interval algebra run.

compose clj

The composition of two direction SETS: every direction possible from x to z when the direction x→y is in s1 and y→z is in s2. Computed axis-wise — see vaelii.impl.projection.

converse-set clj

The converse of a direction set — how an asserted (P a b) constraint is read backwards as the constraint on (b a). North becomes south and northeast southwest; coincidence is its own converse.

definite-direction clj

(definite-direction kb context a b)

The single base direction from a to b when path consistency pins it down; :inconsistent when the network contradicts itself, :unknown when two or more directions remain possible.

direction->xy clj

Each base direction as its [x-relation y-relation] projection — where the target lies on the east-west axis, and where on the north-south one. The nine base directions are these keys: the eight compass points and coincidence, jointly exhaustive and pairwise disjoint, so exactly one holds of any two places.

This table is the whole of what distinguishes the cardinal directions from the relative frame (vaelii.impl.relative); everything algebraic about them is vaelii.impl.projection's, computed from here.

direction-algebra clj

The cardinal directions as a vaelii.impl.qcn relation algebra — universe, identity, composition and converse, all derived from the projection above.

direction-denotation clj

Every direction predicate — the nine base ones and the four derived ones — mapped to the set of base directions it denotes. A goal (P a b) is entailed exactly when the directions still possible from a to b are a subset of P's denotation, so a derived predicate is entailed by more networks than a base one, and the base predicates are the singletons.

The four derived predicates each constrain one axis and leave the other open: northwardOf is anything with a northerly component, whatever it does east-west. None of them contains :eq, so none holds of a place and itself.

orientation-prover clj

(orientation-prover)

The cardinal-direction entailment prover, to register with vaelii.core/add-prover.

possible-directions clj

(possible-directions kb context a b)

The base directions still possible from place a to place b given everything believed in context — #{} when the network is inconsistent.