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Context-Dependent Thinning 17

knows(Sam, loves(John, Mary)). (8.1)

Using the Holographic Reduced Representations of Plate, it can be represented as:

L1 = love + loveagt*john + loveobj*mary, (8.2)

L2 = know + knowagt*sam + knowobj*L1, (8.3)

where * stands for binding operation, and + denotes addition. In our representation:

L1 = ²(love ∨ ∖loveagt ∨ john) ∨ ∖loveobj ∨ mary)), (8.4)

L2 = ⁴(know ∨ ³(knowagt ∨ sam) ∨ ³(knowobj ∨ L1))∙ (8.5)

8.3.2. Predicate-arguments structure

Let us consider representation of relational instances loves(John, Mary) and loves(Tom, Wendy) by the
predicate-arguments (or symbol-argument-argument) structure (Halford, Wilson, & Phillips, in press):
loves*John*Mary + loves*Tom*Wendy. (8.6)

Using our representation, we obtain:

²(1(loves_0 ∨ John_1 ∨ Mary_2) ∨ \loves_0 ∨ Tom_1 ∨ Wendy_2)). (8.7)

Let us note that this example may be represented using the role-filler scheme of HRRs as

L1 = loves + lover*Tom + loved*Wendy, (8.8)

L2 = loves + lover*John + loved*Mary, (8.9)

L = L1 + L2. (8.10)

Under such a representation, the information about who loves whom is lost in L (Plate, 1995; Halford,
Wilson, & Phillips, in press). In our representation, this information is preserved even using the role-
filler scheme:

L1 = ²(loves ∨ ¹(lover ∨ Tom) ∨ ¹ (loved ∨ Wendy)), (8.11)

L2 = ²(loves ∨ ¹(lover ∨ John) ∨ ¹ (loved ∨ Mary)), (8.12)

L = (L1 ∨ L2). (8.13)

Another example of relational instance from Halford, Wilson, & Phillips (in press):

cause(shout-at(John ,Tom),hit(Tom, John)). (8.14)

Using our representation scheme, it may be represented as

²(cause_0 ∨ \shout-at_0 ∨ John_1 ∨ Tom_2)_1 ∨ \hit_0 ∨ Tom_1 ∨ John_2)_2). (8.15)

8.3.3. Tree-like structure

An example of bracketed binary tree adapted from Pollack (1990):

((d (a n)) (v (p (d n)))). (8.16)

If we do not take the order into account, but use only the information about the grouping of constituents,
our representation may look as simple as:

⁴(³(d ∨ ²(a ∨ n)) ∨ ³(v ∨ ²(p ∨ ¹(d ∨ n)))). (8.17)

8.3.4 Labeled directed acyclic graph

Sperduti & Starita (1997), Frasconi, Gori, & Sperduti (1997) provide examples of labeled directed
acyclic graphs. Let us consider

F( a, f(y), f(y, F(a, b)) ). (8.18)

Using our representation, it may look as

³(F_0 ∨ a_1 ∨ ²O ∨ y_1)_2 ∨ ²O ∨ y_1 ∨ ¹(F_0 ∨ a_1 ∨ b_2)_2)_3 ). (8.19)

9. Related work and discussion

The procedures of Context-Dependent Thinning allow construction of binary sparse representations of
complex data structures, including nested compositional structures or part-whole hierarchies. The basic
principles of such representations and their use for data handling were proposed in the context of the

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