Supporting language learners with the

International Journal of Artificial Intelligence & Applications (IJAIA) Vol. 6, No. 3, May 2015
DOI : 10.5121/ijaia.2015.6301 1
SUPPORTING LANGUAGE LEARNERS WITH THE
MEANINGS OF CLOSED CLASS ITEMS
Hayat Alrefaie 1
and Prof. Allan Ramsay 2
1
The University of Manchester, Manchester, United Kingdom
2
The University of Manchester, Manchester, United Kingdom
ABSTRACT
The process of language learning involves the mastery of countless tasks: making the constituent sounds of
the language being learned, learning the grammatical patterns, and acquiring the requisite vocabulary for
reception and production.
While a plethora of computational tools exist to facilitate the first and second of these tasks, a number of
challenges arise with respect to enabling the third. This paper describes a tool that has been designed to
support language learners with the challenge of understanding the use of ‘closed-class’ lexical items.
The process of learning the Arabic for ‘office’ is ‫مكتب‬ (mktb) is relatively simple and should be possible by
means of simple repetition of the word. However, it is much more difficult to learn and correctly use the
Arabic equivalent of the word ‘on’. The current paper describes a mechanism for the delivery of diagnostic
information regarding specific lexical examples, with the aim of clearly demonstrating why a particular
translation of a given closed-class item may be appropriate in certain situations but not others, thereby
helping learners to understand and use the term correctly.
KEYWORDS
CALL tool, Diagnostic, Meaning postulates, logical form & Arabic Preposition.
1. INTRODUCTION
This study seeks to assist learners of Arabic to correctly understand and utilise closed-class items.
The particular focus of this study is to support learners in the correct use of spatial prepositions.
The process of learning vocabulary is often extremely labour intensive, as fluency in a language
relies upon the acquisition of an extremely comprehensive individual lexicon (for example, an
adult speaker will typically have active knowledge of 10-40K words). Although the acquisition of
these words may require substantial investment of time and effort, it often involves the
memorisation of large numbers of pairs and can therefore be as straightforward as learning that
the equivalent of ‘office’ is ‫مكتب‬ (mktb), the equivalent of ‘man’ is ‫رجل‬ (rgl), or theequivalent of
‘boy’ is ‫ولد‬ (wld).
Closed-class items constitute a different type of challenge. By definition, closed-class items are
relatively uncommon, with English and Arabic each having a few dozen prepositions. However,
their meanings are often somewhat fluid and so choosing a direct equivalent for one preposition
can be difficult, if not impossible.
Typically, in order to learn the meaning of a word, common practice is to check the meaning in a
dictionary. However, this is often ineffective for words like ‘in’, which is defined as a word ‘used

2
to indicate location or position within something’. Checking the word‘within’ in the same
dictionary explains that it means ‘inside something’, and then the definition of ‘inside’ is
provided as ‘an interior or internal part or place: the part within’. These circular definitions are
often unhelpful for language learners and fail to offer clear advice on functional usage, such as
whether ‘in’ or ‘on’ is correct in a particular case (why do people get in a car but on a bus? Why
is my office in the computer science building but on the second floor? . . . ).
Understanding the function of prepositions work requires the knowledge that they express a
relationship between some parts of some facets of the entities being related. The latter half of this
idea is informed by Pustejovsky’s [30] notion of ‘qualia structure’ or Dowty’s [15] more general
notion of ‘logical space’:
There will be as many axes of logical space as there are kinds of measurement. [ . . . ] Each axis
might have a different mathematical structure according to the discriminations that can be
appropriately made in each case [p. 126].
Consider the examples in (1):
(1) a. I read it in the park.
b. I read it in the Guardian.
c. I read it in January.
Each of these examples describes a relationship between a reading event and another thing
through the use of the same preposition: (1a) relates the reading event to a place; (1b) expresses a
relationship between the reading event and a source of information; and (1c) links the reading
event with an interval. It can be difficult to understand that an entity can be linked to such
different objects through the same relationship. Therefore, an assumption can be made with
respect to the varied ways in which these objects can be perceived, so the preposition begins by
determining compatible views of the figure and ground. This allows (1a) to be concerned with the
physical act of looking at an object and using received light to create a picture of that object on
your retina; (1b) to describe the abstract transferral of information from an external source into
your mind; and (1c) to focus on the temporal view, with something that begins at one point in
time and ends at a later point in time.
Unlike Pustejovsky, this study makes no claim about the specific number of potential views in
natural language. Instead, the notion is implemented through the use of a simple and efficient
semi-partitioned type lattice. We will write T1 ∼ T2 to state that T1 and T2 are compatible with
regards to the type lattice.
Once compatible views of the figure and ground have been found, the preposition then provides
information on the relationship between given views of the two entities being related. For
example, the word ‘in’ states that the (selected view of the) ground has an interior which encloses
some region in the relevant space, and that every part of the (compatible view of the) figure is
contained in this region. In contrast, the word ‘on’ says that the (selected view of the) ground has
an orientable surface, and that the (compatible view of the) figure has a set of contiguous points
which lie an arbitrarily small distance from the surface.
Reducing ‘F in G’ to (S ∼ S )&(view(F, S) ⊆ interior(view(G, S )) avoids the circularity of saying
that ‘in’ means ‘inside’ because we can give precise definitions of the interior of some entity in a
given space, e.g.

3
• If X has an embedding E into Rn, then interior(X) is the set of members p of X such that E(p) is
not a member of X but there are members p1, p2of X such that E(p) lies on the line from E(p1) to
E(p2). Insofar as our everyday notion of physical space can be captured by mapping points in
space to R3, this enables us to talk about the interiors of physical objects.
• IfX is a dense partially ordered set of points, an interval has a set of lower bounds (points which
are not preceded by any other point in the set) and upper bounds (defined likewise). The interior
of an interval is then the interior of the set consisting of its lower and upper bounds. Most of the
commonsense approaches to time discussed by authors such as [32]; [33]; [4] and linguistic
approaches discussed by authors such as [35]; [23] involve at least dense partial orderings, and
hence can be covered by this definition.
• The interior of a simple unstructured set is the set itself. To say that someone is in my class is
simply to say they are a member of the set of people who I am teaching.
In other words, if we have two items placed in some abstract space with an appropriate structure
(distance if we want to use ‘at’, region hood if we want to use ‘in’, vertical direction if we want to
use ‘on’, . . .) then we can locate them with respect to each other in that space, using clear
definitions that exploit the structure of the space. Of course, the two items have to exist in the
same space–it is not possible to say that a physical object is contained in a temporal space (*
‘London is in January’), or that a temporal object is contained in an abstract space (* ‘January is
in my best paper on syntax’). Hence the first task when interpreting a preposition is to find views
of the entities in question which are in the same space.
Not every item in a given space, however, has the properties required by a given preposition. It is
perfectly possible to have a physical object which does not have a non-empty interior (* ‘my
office is in the second floor’), or an interval that does not have upper and lower bounds (* ‘I have
been knowing how to play tennis since I was 12’ is problematic because once you know
something, you don’t stop knowing it, so the interval denoted by the verb ‘know’ has no (natural)
upper bound). For a preposition to locate one entity with respect to another, the entities in
question have to have the right properties as well as being of the right general kinds. We will
return to this in Section 2.
1.1. English and Arabic prepositions
The main aim of the current paper is to provide English learners of Arabic with diagnostic
information about their use of essentially spatial prepositions by exploiting the notions described
above. The problem is that while a given English preposition may have a ‘typical’ translation into
Arabic, there will be numerous exceptions to this standard translation, and the causes of these
exceptions are hard to discern.
Consider (2), where ‫(في‬fy) and‫(على‬ElY) are normally taken mean ‘in’ and ‘on’ respectively.
(2) a. My office is on the second floor.
b.*.‫الثاني‬ ‫الطابق‬ ‫على‬ ‫(مكتبي‬mktby Ely AlTAbq AlvAny.)
c. ‫الثاني‬ ‫الطابق‬ ‫في‬ ‫مكتبي‬(mktby fy AlTAbq AlvAny.)
The natural way for an English person to think of a floor is as a flat surface, with no interior, so
that if you want to locate a room with respect to a floor you have to say that the room is ‘on’ the
floor. Arabic speakers, on the other hand, conceptualize a‫طابق‬ (TAbq) as a container, like a very

4
large room, and hence want to say the ‫مكتب‬ (mktb) is inside the ‫طابق‬ (TAbq)rather than on its
surface. Why speakers of one language have different ideas of what various everyday objects are
like is beyond the scope of the work reported here. It just seems to be the case. What matters is
that it does seem to be the case, and it is this underlying conceptual framework that drives the
choice of particular prepositions for linking different kinds of figures and grounds.
2. MEANINGS, MODELS AND INFERENCE
To make any computational sense of this idea, we have to give precise characterisations of how
the various prepositions work, and we have to describe the entities that they can link. We use a
framework in which natural language texts are translated, through compositional rules, into some
suitable logic; we then write meaning postulates to capture the consequences of using particular
combinations of words; and finally we carry out abductive inference to investigate the use of
different prepositions for linking the same figure and ground in two languages.
2.1 Logical forms
The framework used here follows the tradition started by Montague[26]; [16] of annotating
grammatical rules with semantic interpretations which can be used to ‘construct an interpretation
by combining the interpretations of the parts’. This process lets you construct formal paraphrases
(‘logical forms’, LFs) in some suitable logic of any sentences that you can parse. There have been
many developments in this area since Montague’s original presentation of this idea: in particular,
the choice of a suitable logic has been extended to cover the notion that natural language is
situated, i.e. that utterances have to be interpreted with respect to the context in which they are
produced [21]; [6]; [19]; [18] and to take account of the fact that natural languages make
extensive use of constructions such as defaults [34]; [5] and propositional attitudes which require
highly expressive logics [37]; [1]; [2]. The general principle remains the same: if you have a
structural analysis of a text which is obtained by applying the rules of a grammar, you can
annotate the rules of that grammar with semantic composition rules which will let you produce an
LF. And if you have an LF in some suitable logic, and you have an inference engine for that
logic, then you can reason about what has been said.
There are, clearly, two limiting factors to be considered here:
The structural analysis has to be obtained by using the rules of a grammarModern data
driven parsers often have no explicit grammar [25]; [28]; [27], or have a very (very) large
context-free grammar induced from a treebank [11]; [9]. Clearly, it is not possible to annotate the
rules of the grammar being used if there is in fact no grammar; and it is infeasible to do so in
cases where the induced grammar contains tens of thousands of rules. Bos?Bos:08a describes a
system which produces semantic interpretations using a fairly wide-coverage grammar, but
grammar-based parsers can become very slow when faced with long (40+ words) sentences. In
the present context, however, we are working with comparatively short sentences, since the task
is to compare interpretations of English sentences set as translation exercises with interpretations
of putative translations into Arabic. As such we can control the length of the sentences under
consideration, thus making it possible to use an HPSG-like grammar and a fairly orthodox chart
parser. The grammar we are using has an underlying set of principles, together with a small set of
language specific constraints. We can thus use essentially the same grammar for parsing the
English and Arabic texts. This has the enormous advantage that exactly the same machinery can
be used for constructing LFs for both languages, which makes it comparatively easy to compare
them. Fig 1 shows the LFs for (2a) and (2b).
These LFs are fairly orthodox. The main points to note are

5
• The LFs contain the surface utterance type–claim, query, . . . There are two reasons for
this: (i) the utterance type is part of the meaning of the utterance, and hence it naturally
belongs in the LF; and(ii) in some cases, notably the English determiner ‘any’, there are
quantifiers which are naturally seen as out scoping the utterance type: these simply
cannot be accommodated without including the utterance type in the LF.
• Referring terms in the original text are represented as reference terms of the form ref(lX
(. . . X . . .)). Such terms have to be anchored with respect to the context in order to
determine the significance of the utterance[6].
• The terms in the Arabic LF make use of predicates with names like‫كتب‬office, where we
use the English gloss to distinguish between distinct terms with the same root (the stem
‫(مكتب‬mktb) is derived from the root ‫(كتب‬ktb), so since we use roots rather than stems in
our lexicon it is the root that appears in the LF).
utt(claim,
on(ref(lEown(ref(lF(speaker(F))), E)&office(E)),
ref(lGfloor(G)&second(G, lH(floor(H))))))
utt(claim,
ElY(ref(lDowner(ref(lE(speaker(E))),D)&‫كتب‬office(D)),
ref(lF‫طابق‬floor(F) ‫&ﺛاني‬second(F))))
Figure 1. Logical forms for (2)
You have to have an inference engine for the chosen logic There is no point in constructing
any kind of meaning representation unless you are going to do something with it. In general, what
you want to do when confronted with an utterance is to reason about it: to see what follows from
it, to think about why the speaker produced, to try to decide what you should do in response to it,
. . .
This means that you have to have some kind of engine that is capable of carrying out inference
over formulae in the formalism that you have chosen as your representation. This is true no matter
whether the meaning representation is just the original string of words and the inference engine is
string-edit distance between sequences of words; or the meaning representation is the dependency
tree for the text and the inference engine is some form of tree-edit distance [39]; [3], as is
commonly employed for textual entailment (Dagan et al., 2005); or whether it is a translation into
first-order logic and the inference engine is a theorem prover for first order logic [7]. Simple
string/tree-based representations have two advantages: (i) they are robust, since to construct the
representation you simply invoke a morphological analyser (for string-based representation) or
some kind of dependency parser (for tree-based ones). It doesn’t even matter that the analyser
may not be particularly accurate, since you can get some information out of a parse tree even if
contains some mistakes. (ii) the inference engines are fairly easy to implement–string edit
distance algorithms which exploit lexical relations when decided the cost of substituting one term
for another are straightforward and the more complex ones (such as [39]) are widely
downloadable. They do not, however, support detailed chains of reasoning, and they do not
generally pay much attention to whether the reasoning that is carried out is sound. Indeed, typical
measures of success for shallow inference report on precision and recall values, where precision
means ‘How often are the entailments discovered by the system valid?’ In other words, such
systems recognise from the outset that the inference that they are carrying out may not be sound.
Approaches based on translation into logic have the obverse advantages: (i) they are sound, since
the target logics have well-defined proof and model theories, so the steps that can be carried out
are guaranteed to lead to true conclusions if the premises are true. (ii) They are efficient, and
hence can be used to carry out long chains of inference. They also have the obverse problems. In

6
particular, it is difficult to construct them, since the standard approach involves parsing the input
text using a grammar which has been annotated with semantic interpretations, and as noted above
most systems for parsing free text with long sentences use either a grammar which has been
inferred from a tree bank, and which is very hard to annotate,or no grammar at all. For the current
task, the need to carry out detailed chains of inference outweighs the difficulties that occur when
you try to parse long natural texts. For our particular task we have a great deal of control over
what the learner will write, because their task is to choose the right prepositions to translate a set
of target texts. If the target texts are fairly simple (say up to ten or twelve words) then the texts
that users generate will also be fairly simple–you would hardly translate ‘I am on the train’ by a
sentence with 40 words in it. We therefore use translations into property theory[37]; [10],which
allows properties and propositions to be treated as arguments and hence allows for finer-grained
distinctions than are permitted in first-order logic, as our target representations; and we use a
version of the theorem prover described in [31] as our inference engine.
2.2 Meaning postulates
In order to reason about the consequences of some utterance, you have to have some knowledge
to reason with. If you have a perfect translation from English or Arabic into some formalism and
you have a theorem prover for that formalism, you will still not be able to do anything unless you
also have an appropriate body of background knowledge.
The knowledge that we require involves describing the circumstances under which some
preposition can use for locating one entity with respect to another. We write these down as
meaning postulates in property theory–typical rules for describing the use of the preposition ‘in’
and ‘on’ are given in Fig 2.
B C: {in(B, C)}
D E F: {view(B, F) &type(F,D) &dim(F, E)}
G H: {view(C,H)
&type(H,G)&(D ∼G) &dim(H, E)}
I: {interior(H, I)}
F I & located(B, C)
B C: {on(B, C)}
D E : {view(B, E) &type(E,D)}
F C: {view(C,G) &type(G, F) &D ∼F}
H: {bottom(E,H)}
I: {surface(G, I)}
touching(H, I)&located(B, C)
Figure 2: Meaning postulates for ‘in’ and ‘on’
The first rule in Fig 2 says that if the preposition ‘in’ has been used to link to entities B and C then
B will be successfully located with respect to C if there are views F and H of B and C where F
and H have compatible types D and G and F has an interior I with the same dimensionality E as
H. Under those circumstances F is a subset of I, and we have located B with respect to C. In other
words ‘in’ works for locating a figure entity with respect to a ground if the ground has a suitable
interior. Similarly, the second rule in Fig 2 says that the preposition ‘on’ may be used to locate a
figure with respect to a ground if the ground has a view which has a surface, and the figure has to
have a compatible view which has a bottom.

7
Given that Arabic has prepositions, ‫(في‬fy) and ‫على‬ (ElY), which seem to carry the same meanings
as ‘in’ and ‘on’, we can write parallel meaning postulates for them. The Arabic spatial
prepositions seem to mean almost exactly the same as the English ones, and hence we use exactly
the same rules for them. We also need meaning postulates to tell us what views, with what
properties, particular words induce. The two languages do differ at this point: to continue with our
running example, ‘building’ and‫(مبنى‬mbnY) both denote entities with interiors; but although
‘floor’ and ‫طابق‬ (TAbq) are roughly equivalent, a floor is a 2D surface, with no interior, whereas a
‫طابق‬is a 3D space with an interior and no surface. To capture these observations, we need some
general rules about what kinds of entities have interiors, and surfaces, and so on
B C D: {embedding(C, B,RD)}
$Einterior(B, E)&$F(embedding(F, E,RD))
B C: {embedding(C,B,R3)}
D : {type(B,D)}&($Etop(B,E) &type(E,D))&$Fbottom(B, F) &type(F,D)
B C D: {embedding(C, B,R2) &orientable(C)}
$E(surface(B, E))
. . .
Figure 3: Rules about RN
The first rule in Fig. 3 says that if there is an embedding C of B into RN
then B has an interior
which also has an embedding into this set. This is not the only kind of entity that can have an
interior, but it is a very common one. The second rule says that if there is an embedding of B into
RN
then B has members which are top and bottom points. Again, there are other kinds of entity that
can have top and bottom points, but this is a common one. The third says that if there is an
orientable embedding of B in to R2
then B has a surface. The rules in Fig. 3 are themselves
derivable from basic properties of RN and the definitions of interior, top and bottom for these
spaces. We are not, in this paper, concerned with deriving such rules from first principles (see
[29] for work along these lines). What we need is to be able to say what kinds of things have
interiors and so on. We then need to say things about particular entities. It is at this point that the
nuances of different languages emerge.
B :{office(B)}
type(B,physical)&$C(embedding(C, B,R3
))
B :{building(B)}
type(B, physical)&$C(embedding(C, B,R3
))
B :{floor(B)}
type(B, physical)
&$Cembedding(C, B,R2
)&orientable(C)
B :{‫كتب‬office(B)}
))
B :{‫مبنى‬building(B)}
))
B :{ ‫طابق‬floor(B)}
type(B, physical) &$C(embedding(C, B,R3
))
Figure 4: Buildings, offices & floors
Fig. 4 says that the English and Arabic words for building and office denote entities with the
same spatial properties, whereas the words ‘floor’ and‫(طابق‬TAbq), despite being rough
translations of one another, differ when you look at them in detail–a ‘floor’ is an oriented 2D
surface, a ‫طابق‬ (TAbq) is a 3D container.

8
Given these descriptions, it is easy to see why English and Arabic speakers use a preposition
(‘in’,‫ف‬ (fy))which talks about interiors for locating an office with respect to a building, but
English speakers use one (‘on’) which talks about surfaces for locating an office with respect to a
floor while Arabic speakers continue to use ‫(في‬fy) for these kinds of objects. ‘in’ just won’t work
for locating the office with respect to the floor, because a floor doesn’t have an interior.
3 MODELS VS ABDUCTION
Turning this observation into a CALL tool is not straightforward. There are two plausible ways to
proceed: we can either use the theorem prover as a model builder, and compare the models that
are generated from an English utterance and the English meaning postulate and an Arabic
utterance and the Arabic meaning postulates; or we can adapt the theorem prover to work
abductively, trying to fill in the gaps in the use of the prepositions.
3.1 Model building
Given a set of axioms, a tableau-style theorem prover can be used as a model builder. In order to
use a tableau engine to prove that a goal G follows from a set of axioms A you have to negate the
gaol and show that every branch of the resulting tableau is closed. The key point here is that a
branch of a tableau consists of a set of literals: if the branch is closed, then it contains some
directly contradictory pair of literals, and hence cannot have a model. If every branch is closed,
then there is no model for the initial set of propositions. Since the initial set of propositions was G
∪ ሼ࢔࢕࢚(࡭)ሽ, this means that it is not possible for G to be true and A to be false, and hence that A
is true whenever G is. Just before the closing literal is added, however, the set of literals that
make up a branch are consistent, and hence have a model. Indeed, the set of positive literals in
this set is a Herbrand model.
Suppose, then, that we apply the machinery of a tableau theorem prover to a consistent set of
clauses. If at some point we find that we have an open branch, and that there is nothing further
that can be added to that branch. Then again the set of positive literals is a model.
Suppose we take two sentences, one in English and Arabic, and try to build models for them
along with an axiomatisation of the background knowledge relating to the terms that they contain.
If the two sentences mean the same, then we might expect the models to be equivalent. If they are
not, then it is a reasonable supposition that the differences between them reflect different
connections between the terms that they contain and the relevant background knowledge. It may
then be possible to provide feedback about these differences, and hence to help the learner to see
how terms that appears to be mutually equivalent differs when used in different contexts.
To make this concrete, imagine that our learner has been asked to translate ‘My office is on the
second floor.’, and they have done so by producing the sentence. ‫الثاني‬ ‫الطابق‬ ‫على‬ ‫(مكتبي‬mktby Ely
AlTAbq AlvAny.)This is not an unreasonable translation, since ‫(على‬Ely) is often taken as the
translation of ‘on’. Unfortunately, as we have just seen, it is wrong, because the‫طابق‬ (TAbq) is
conceptualised as a container rather than a surface. The models for these two sentences are shown
in Fig. 5. The two models in Fig. 5 contain a substantial amount of information that is not present
in the original LFs for the two sentences. That is as it should be–a model contains information
that arises when you combine an interpretation with a body of background knowledge. If you
look carefully, it is possible to see that the English model contains the fact that the office, #3228,
has been located with respect to the floor, #3229, and that this has happened because the floor has
an oriented embedding #3274 into R2
and hence has a surface, #3265, whereas the‫طابق‬ floor,
#3655, in the Arabic model has an embedding into R3
rather than R2
, and hence has no surface for
the ‫كتب‬office, # 3654, to rest on. The problem is that you have to look very carefully in order to

9
spot the relevant differences between the two models. Given that we are concerned with the use
of prepositions for locating one entity with respect to another, it makes sense to look for the
entries in the models that correspond directly to the prepositions, and to then look to see whether
the entities referred to in these entries are located with respect to one another.
Figure 5: Models for (2a) and (2b)
3.2 Abductive reasoning
There are, however, some problems associated with using comparison between models as a
source of feedback. Firstly, you have to be able to link items in the two models, despite the fact
that they are described using different terms (drawn, as they are, from source texts in different
languages). This requires a set of bilingual equivalents, so that an item described as a floor in one
model can be matched to an item described as a‫طﺎﺑق‬‫طﺎﺑق‬‫طﺎﺑق‬‫طﺎﺑق‬ in the other. This can be tricky, since the
conditions under which a term can be given a particular translation are complex (if they were not,
then they would not cause problems for learners). Secondly, a single consistent set of formulae
can have multiple models. There is no easy guarantee that the first model that is produced for one
sentence is the best match for some model of the second. If the English sentence has models ME
1,
ME
2,,ME
3, … and the Arabic sentence has models MA
1, MA
2,,MA
3, … then it may be that the best
match is between ME
5 andMA
7. But trying to compare every model of the English with every
model of the Arabic to find out which ones have the smallest number of mismatches is going to
be very time-consuming.
We therefore also plan to investigate the use of ‘abductive’ inference. Suppose that our learner
has written a sentence using the preposition ‫على‬as a translation of the English preposition
‘on’.We have a collection of rules that describe the conditions under which some preposition will
successfully locate the figure with respect to the ground, as in Fig. 2. Given a model, we can look
for places where someone has used a preposition without locating the relevant entities by looking
at each rule that has located(B,C) has its consequent where the first element of the antecedent
(which will always specify that some preposition has been used) is satisfied but the consequent is
not. We can then use the theorem prover abductively, by allowing proofs with missing steps,
where we can prespecify the number of steps that can be omitted. In order to find the minimal set
of missing steps, we call the theorem prover once allowing one missing step, then allowing two,
then . . . until a proof is found. This does increase the time taken for finding a suitable proof, but
if we simply allow proofs with unbounded numbers of missing steps then we are likely to be led
to incorrect diagnoses. Thus if we try to carry out an abductive proof that two items mentioned in
the model of (2b) are located, we will find that the rule using ‫على‬ (Ely) would have worked if we

10
had been able to prove surface(#3655, B) for some B. So the primary reason why (2b) failed to
locate the ‫(مكتب‬mktb) is that the ‫طابق‬ (TAbq) does not have a surface. We could, therefore, provide
a message to that effect to the learner as the first round of feedback–‘You tried to use ‫على‬ (Ely) as
the translation of ‘on’, but it doesn’t work in this case because although ‫طابق‬ (TAbq) is the correct
translation of ‘floor’, a ‫طابق‬ (TAbq) does not have a surface’. We can follow this up with more
detailed feedback. If the learner is not satisfied with this as a comment, he or she can ask why a
‫طابق‬ (TAbq) does not have a surface. An abductive proof of surface(#3655, B) produces the
response that this would have worked if we had been able to show embedding
(B,#3655,R2)&orientable(B). It is harder to see how this could be turned into a comprehensible
diagnostic message for the learner, but the general principle is simple. The reason why some
attempted translation fails to produce a target consequence can be found by carrying an abductive
proof of the target and recording the missing steps. In many cases, these missing steps can be
converted to meaningful diagnostic messages; and if the learner wants to know more about what
went wrong, then abductive proofs of the missing steps themselves can be derived, though as this
process gets deeper the rules that are being used become more and more abstract, to the point
where they will not in fact produce output will help the learner to understand what they have done
wrong.
4 FUTURE WORKS
The transformation of the work described here into a functional CALL tool requires the
completion of two principal tasks:
1. Meaning postulates need to be written for a considerably wider set of open-class words. Rules
have already been prepared for the major special prepositions (Fig. 2), as well as for general uses
like the interior, surface or ceiling of an object (Fig. 3). In order to make the tool suitable for use
within the classroom context, however, it will be necessary to prepare a large set of entities able
to enter into these relationships, as illustrated by the examples in Fig. 4.
2. The output of the abductive reasoner, which constitutes a set of missing steps, must be
converted into useful, comprehensible diagnostic messages. At this time, it is expected that
canned phrases with slots in will be suitable for this function. However, the missing steps contain
Skolem constants, which will be unknown to language learners, meaning that care must be taken
when using these steps to derive suitable referring expressions [13]; [38].
The first task, writing a larger set of open-class words, will be time consuming but relatively
simple. The second is likely to be more challenging, although we believe that the outcome should
offer richer and more useful information about the use of particular prepositions than is currently
available through the use of statistical approaches, such as those proposed by [14]; [17]; [36].
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[39] Zhang, K. and D. Shasha (1989). Simple fast algorithms for the editing distance between trees and
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Authors
1. Hayat M A. Alrefaie
Postgraduate Student
School of Computer Science
University of Manchester
2. Prof Allan Ramsay
Professor ofFormal Linguistics
School of Computer Science
University of Manchester

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