INTEGRATING LOGICAL AND SUB-SYMBOLIC CONTEXTS OF
REASONING
∗
Tarek Besold and Stefan Mandl
Chair of Computer Science 8: Artificial Intelligence, University of Erlangen-Nuremberg
Haberstraße 2, D-91058 Erlangen, Germany
Keywords:
Information integration, Knowledge-based systems, Nonmonotonic reasoning, Knowledge representation.
Abstract:
We propose an extension of the heterogeneous multi-context reasoning framework by G. Brewka and T. Eiter,
which, in addition to logical contexts of reasoning, also incorporates sub-symbolic contexts of reasoning. The
main findings of thepaper area simpleextension of the concept of bridge rules to the sub-symbolic case and the
concept of bridge rule models that allows for a straightforward enumeration of all equilibria of multi-context
systems. We illustrate our approach with two examples from the fields of text and image classification.
1 INTRODUCTION
One of the important problems in knowledge repre-
sentation and knowledge engineering is the impossi-
bility of writing globally true statements about realis-
tic problem domains. A circumstance that is also doc-
umented by the use of contexts and micro-theories in
CYC ((Lenat, 1995)). Multi-context systems (MCS)
are a formalization of simultaneous reasoning in mul-
tiple contexts . Different contexts are inter-linked
by bridge rules which allow for a partial mapping
between formulas/concepts/information in different
contexts. Recently there have been a number of inves-
tigations of MCS reasoning (for instance, see (Roelof-
sen and Serafini, 2005) or (Brewka et al., 2007)),
with (Brewka and Eiter, 2007) being one of the lat-
est contributions. There, the authors describe reason-
ing in multiple contexts that may use different log-
ics locally. Logical reasoning on the one hand is a
special case of symbolic reasoning where, according
to (Kurfess, 2002), entities of the application domain
are represented by symbols. In sub-symbolic reason-
ing on the other hand domain entities are represented
by (micro-)features.
∗
This is a short version of the paper. The full ver-
sion, containing a short introduction to the notion of MCS
and another more sophisticated application example from
the domain of image classification, as well as proofs for
the propositions stated, a categorization of our work and
a comparison to similar approaches in the field can be
found under http://www8.informatik.uni-erlangen.
de/inf8/Publications/bridging_mcs_original.pdf
.
There is no strict boundary between symbolic
and sub-symbolic: what in one example are micro-
features can be declared as entities and be symbol-
ically reasoned about in another example (and vice
versa). In this paper, we integrate contexts of logi-
cal reasoning and contexts of sub-symbolic reasoning
into a single MCS. Possible applications of such rea-
soners are numerous. I.e. shortcomings of statistical
methods could be remedied with declarative knowl-
edge and vice versa.
2 INTEGRATING LOGICAL AND
SUB-SYMBOLIC CONTEXTS
OF REASONING
We now generalize concepts from (Brewka and Eiter,
2007) to be applicable for both logical and sub-
symbolic reasoners.
Observation 1. The concept of ‘logic’ as defined
in (Brewka and Eiter, 2007) is – besides its name –
also a valid characterization of sub-symbolic reason-
ing: The knowledge base consists of the available ev-
idence (the input of the sub-symbolic reasoner), the
set of possible belief sets is the set of possible results
of the sub-symbolic reasoner, and the function ACC
defines the actual reasoning. (It is ACC that is often
generated from training examples.)
In fact the only conceptual generalization that we
make is to make no assumptions about the form of
494
Besold T. and Mandl S. (2010).
INTEGRATING LOGICAL AND SUB-SYMBOLIC CONTEXTS OF REASONING.
In Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Artificial Intelligence, pages 494-497
DOI: 10.5220/0002704704940497
Copyright
c
 SciTePress

the inputsof the reasoner which in the original work
were assumed to be sets. Instead, we require test and
update functions. This idea leads to our definition of
a reasoner.
Definition 1. A reasoner is a 5-tuple R =
(Inp
R
, Res
R
, ACC
R
, Cond
R
, Upd
R
) where Inp
R
is the set of possible inputs to the reasoner, Res
R
is the set of possible results of the reasoner,
ACC
R
: Inp
R
7→ 2
Res
R
defines the actual reasoning
(assigning each input a set of results in a decidable
manner), Cond
R
is a set of decidable conditions on
inputs and results, cond
R
: Inp
R
×Res
R
7→ {0, 1},
and Upd
R
is a set of update functions for inputs,
upd
R
: Inp
R
7→ Inp
R
.
The example below shows that our Definition 1 com-
prises logics in the sense of (Brewka and Eiter, 2007)
and sub-symbolic reasoners like Neural Nets, which
originally were not covered. Hence Definition 1 is a
generalization.
Example 1. A logic L defined over a signature Σ
(as of Definition 1) is a reasoner with Inp
R
=
KB
L
, Res
R
= BS
L
, ACC
R
= ACC
L
, Upd
R
= {fn
x
:
kb → kb∪{x}|∀x ∈
S
k∈KB
L
k, ∀kb ∈ KB
L
}, Cond
R
=
{fn
x
: (·, b) → 1 iff x ∈ b, 0 else|x ∈
S
bs∈BS
L
bs}
Example 2. A standard feed-forward neural network
N with n real valued inputs and m real valued out-
puts is a reasoner with Inp
R
= R
n
, Res
R
= R
m
and
ACC
R
(inp) = {N (inp)} where inp ∈ Inp
R
. In this
case, ACC
R
maps to singleton sets. Cond
R
is a set of
indicator functions on feature vectors, Upd
R
is a set
of update functions, each performing an update for a
certain value of a component of a feature vector.
The following definitions adapt the basic concepts of
multi-context reasoning given in (Brewka and Eiter,
2007) for the use with reasoners as of Definition 1.
In order to adapt the concept of bridge rules, we have
to take into account the fact that the assumption of
the reasoner inputs being sets is not made for general
reasoners. Instead we have to use the defined test and
update functions.
Definition 2. Let R = {R
1
, . . . , R
n
} be a set of reason-
ers. An R
k
-bridge rule over R, 1 ≤ k ≤ n, containing
m conditions, is of the form
u ← (r
1
: c
1
), . . . , (r
j
: c
j
),
not(r
j+1
: c
j+1
), . . . , not(r
m
: c
m
)
(1)
where j ≤ m, 1≤ r
k
≤ n and c
k
is a condition of inputs
and results of some R
r
k
and u is an element of Upd
k
.
Definition 3. A generalized multi-context system
M = (C
1
, . . . ,C
n
) consists of a collection of contexts
C
i
= (R
i
, inp
i
, br
i
), where R
i
= (Inp
i
, Res
i
, ACC
i
,
Cond
i
, Upd
i
) is a reasoner, inp
i
an input (an ele-
ment of Inp
i
), and br
i
is a set of R
i
-bridge rules over
{R
1
, . . . , R
n
} as of equation (1).
Concerning the belief states, we require input-output
pairs instead of belief sets in every context.
Definition 4. Let M = (C
1
, . . . ,C
n
) be a generalized
MCS. A generalized belief state is a sequence S =
(S
1
, . . . , S
n
) such that each S
i
is of the form (inp
i
, res
i
)
with inp
i
∈ Inp
i
and res
i
∈ Res
i
.
We say a bridge rule r of form (1) is applicable
in a generalized belief state S = (S
1
, . . . , S
n
) iff for
1 ≤ i ≤ j : c
i
(inp
i
, res
i
) = 1 in S
i
and for j + 1 ≤ k ≤
m : c
k
(inp
k
, res
k
) = 0 in S
k
. Now we prepare for the
concept of equilibrium in the generalized setting.
Definition 5. The set of (context local) update func-
tions with respect to a corresponding element S
i
of a
belief state S is given by US
i
(MCS, S) = {head(r)|r ∈
br
i
applicable in S}, where br
i
denotes the set of
bridge rules of S
i
’s corresponding context C
i
.
In general, a set of update functions may yield dif-
ferent results when the functions are applied multiple
times or in different orders. We do not allow such sets
of update functions.
Definition 6. An applicable set of update functions
US
i
(MCS, S) is stationary for an input inp
i
iff the
following two conditions hold: ∀u ∈ US
i
(MCS, S) :
u(inp
i
) = u
m
(inp
i
) for m ≥ 1 (i. e. idempotency), and
∀u, u
′
∈ US
i
(MCS, S) : u(u
′
(inp
i
)) = u
′
(u(inp
i
)) (i. e.
commutativity).
Definition 7. The update of a belief state element
S
i
of a belief state S, with respect to a set of up-
date functions US
i
with k elements, is given by
u
1
(u
2
(. . . u
k
(inp
i
). . .)) if US
i
is stationary for inp
i
,
and undefined otherwise.
Please note that stationarity is only required for the
set of update functions that is actually applied to be-
lief state elements at a time. We now can give the
definition of the generalized concept of equilibrium.
Definition 8. A generalized belief state
S = ((inp
1
, res
1
), . . . , (inp
n
, res
n
)) of M is an equilib-
rium iff, for 1 ≤ i ≤ n, the following condition holds:
update(inp
i
, US
i
) = inp
i
and res
i
∈ ACC(inp
i
)
where update(inp
i
, US
i
) denotes the update of S
i
with respect to US
i
, which in turn has to be stationary
for the corresponding inp
i
.
Please note that inputs for which the update is non-
stationary are not part of any equilibrium.
Proposition 1. Definitions 1 to 8 are a generalization
of Definitions 1 to 5 in (Brewka and Eiter, 2007).
INTEGRATING LOGICAL AND SUB-SYMBOLIC CONTEXTS OF REASONING
495

3 COMPUTING EQUILIBRIA
FOR FINITE MCS
For a belief state being an equilibrium only means that
all the bridge rules are respected. As local reasoners
may be non-monotonic and, furthermore, the bridge
rules are non-monotonic also, there may be several
equilibria for a given MCS. In general it is not clear
which one constitutes the desired one.
When no external knowledge about preferences
(e. g. a preference function which induces an order on
equilibria) is available, in the field of computational
logic, there exists the principle of minimality. Sadly
minimality in general has no straightforward transla-
tion to sub-symbolicreasoning contexts (e. g. for vec-
tor valued sub-symbolic reasoning contexts we would
need some kind of metric).
As for a deterministic sub-symbolic reasoner,
given a set of inputs, there is exactly one correspond-
ing set of results, one may nonetheless try to carry
over minimality from the symbolic-only background.
Using the notion of C*-minimality as introduced by
in (Brewka and Eiter, 2007), minimality may be de-
manded for the symbolic contexts of a generalized
MCS, which may in this case be composed of sym-
bolic and deterministic sub-symbolic reasoners. As
the deterministic sub-symbolic reasoners only yield
exactly one set of results for a given set of inputs (and
no phenomena as self-sustaining equilibria are possi-
ble), the C*-minimality generalizes to a global prop-
erty of the equilibrium.
The remainder of this section describes a proce-
dure to compute all equilibria of a finite MCS, based
on complete enumeration. Thus criteria as e. g. min-
imality may be applied to the set of equilibria after-
wards. Part of future research will be to construct
more specialized algorithms, already exploiting the
properties of ordering relations during the computa-
tion.
Definition 9. An MCS M = (C
1
, . . . ,C
n
) is said to be
finite, iff for 1 ≤ i ≤ n, following condition holds:
|ACC(inp
i
)| < ∞ and |br
i
| < ∞.
For the implementation, we consider finite MCS
only.
Definition 10. Let Br be a set of n bridge rules of
an MCS. A bridge rule model is an assignment Br 7→
{0, 1}
n
that represents for each bridge rule in Br
whether it is active or not.
Proposition 2. For each equilibrium there is exactly
one bridge rule model.
For a given bridge rule model and an MCS we first
apply all the bridge rules activated in the bridge rule
model yielding inp
′
1
...inp
′
n
. Then we compute the set
of results for each context i given inp
′
i
by applying
ACC(inp
′
i
), yielding a set of results res
j
i
for each i,
being of finite cardinality as MCS was said to be fi-
nite. Thus, testing whether (inp
i
, res
j
i
) is an equilib-
rium for all j, we obtain the set of equilibria for the
given bridge rule model. Iterating the procedure over
the (finite) set of all bridge rule models and joining
the resulting sets of equilibria finally yields the set of
all equilibria.
Definition 11. Given an MCS with a (global) set
of bridge rules br =
S
i
br
i
. A set of bridge
rules br
j
⊆ br is called update-monotonic iff
for all belief states S, S’ the following condi-
tion holds: S
′
= update(MCS, S) ⇒ VC(MCS, S) ⊆
VC(MCS, S
′
) where VC(MCS, S) =
S
i
{cond
i
∈
R
i
|cond
i
(inp
i
, res
i
) = 1} and update(MCS, S) is the
(global) update over all S
i
∈ S.
As bridge rules in the update-monotonic subset
of bridge rules of the MCS are guaranteed to remain
active after any update, the update-monotonic bridge
rules that are initially active in the MCS when search-
ing for equilibria have to be active in any equilibrium.
Hence, when iterating over all bridge rule models,
only those bridge rule models that comply with the
initially active update-monotonic bridge rules have to
be considered.
As a downside computing the update-monotonic
subset of the bridge rules depends on the idiosyn-
crasies of the reasoners involved, condition test and
update functions and therefore cannot be performed in
general. Another inconvenienceis the fact that if there
are no update-monotonic bridge rules all elements of
the entire set of bridge rule models have to be tested
for representing an equilibrium.
Definition 12. A reasoner R =
(Inp
R
, Res
R
, ACC
R
, Cond
R
, Upd
R
) is determin-
istic iff ACC
R
(x) is a singleton set for every x ∈ Inp
R
.
Proposition 3. For an MCS with deterministic rea-
soners only, there exists at the most one equilibrium
for each bridge rule model.
Applying the proposition to the algorithm
sketched above, one may reduce the number of pairs
(inp
i
, res
j
i
) to be tested for being an equilibrium, by
testing each pair (inp
i
, res
j
i
) directly after it was gen-
erated, and switching to the next bridge rule model
after having found an equilibrium for this very model,
as there may be one at the most.
ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence
496

4 EXAMPLE
We present an example application for multi-context
reasoning which involves logical and sub-symbolic
contexts of reasoning. As logical reasoners, we
assume a propositional logical reasoner R
pl
. As
sub-symbolic reasoners, we assume a maximum-
likelihood reasoner using the Naive Bayes assumption
R
nb
.
4.1 Text Classification
This example is taken from the domain of (statistical)
text classification. Text may be categorized into two
different classes: ‘music event’ and ‘political event’.
We assume a binomial text model (see (Manning
et al., 2008) for more details) and – for the sake of
simplicity – a vocabulary of only two terms: Queen,
Elizabeth.
Term t P(mus. ev.|t) P(pol. ev.|t)
Queen 4.1× 10
−3
2.0× 10
−3
Elizabeth 3.0× 10
−3
1.9× 10
−2
Query q Naive Bayes probability of q
Queen Elizabeth 1.2× 10
−5
3.8× 10
−5
Figure 1: Prior probabilities of terms and Naive Bayes prob-
ability of a query for the classes ‘music event’ and ‘political
event’.
Figure 1 shows the prior probabilities of the
classes given the terms, and the Naive Bayes prob-
abilities given the combined query Queen Elizabeth.
The priors have been obtained by querying a web
search engine, but for the example the actual source
of the priors is not of much interest. Those prob-
abilities define the function ACC
nb
of R
nb
. When
stating the query ‘Queen’ to the reasoner (inp
nb
=
{Queen}), the result (via maximum likelihood) is
res
nb
= {‘music event’}.
We would like to improve the reasoner R
nb
by providing specific knowledge about the British
Royals. Hence, we use a proposition logi-
cal reasoner R
pl
together with a knowledge base
inp
pl
= Elizabeth ← Queen of relevant information.
In order to link the two reasoners, we define bridge
rules br
pl
= {add Queen ← nb : has input Queen} and
br
nb
= {add to query Elizabeth ← pl : holds Elizabeth},
where the condition and update functions have the
obvious meaning. Taking the query ‘Queen’ into ac-
count, the MCS for reasoning in R
nb
and R
pl
is given
by M
t
= {C
nb
,C
pl
} with C
nb
= (R
nb
, {Queen}, br
nb
),
C
pl
= (R
pl
, inp
pl
, br
pl
).
Then, the belief state
(({Queen, Elizabeth}, {‘political event’}),
(inp
pl
∪ {Queen}, inp
pl
∪ {Queen, Elizabeth}))
is the only equilibrium of M
t
. Hence, with multi con-
text reasoning, the result for the query ‘Queen’ has
been changed from ‘music event’ to ‘political event’.
5 CONCLUSIONS
The paper presents a generalization of heterogeneous
multi-context systems that allows for the use of sub-
symbolic contexts of reasoning alongside logical con-
texts of reasoning. An exhaustive algorithm for enu-
merating all equilibria of an MCS is given.
Still, the lack of a conceptual notion of minimality
or stability for sub-symbolic beliefs poses a challenge
for future research, which we are confident to handle
in the near future.
On the pragmatic side, the illustrative examples
demonstrate that a more powerful language to de-
scribe updates and conditions on reasoner inputs and
results, respectively, has to be developed in order to
allow for concise definitions of bridge rules.
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