Department of Computer Science,
Queen Mary and Westfield College,
University of London.
The Dynamo Project, (An investigation of the model-based
implementation of pragmatic reasoning) is a three-year
EPSRC-funded research project investigating the model-building
approach to implementing nonmonotonic reasoning, which has been
running for just over six months.
Pragmatic reasoning (commonsense reasoning, nonmonotonic
reasoning) is concerned with context-dependent inference; with
what follows from the premises in a given context. Typically
the context is incompletely specified. In such cases we make
assumptions about what is normal or typical or conventional
given the context and its limitations. For example, an agent
planning to catch a bus might reason that the bus will come on
time because it normally does and there is no reason to think
otherwise; that is, there is nothing about the context which
leads the agent to think otherwise. Defeasible inference of
this kind is essential in daily life and its formalisation and
implementation is a central concern of AI.
We are expert pragmatic reasoners. We are able to produce
reasonable conclusions quickly and with little or no effort. By
contrast we tend to find semantic reasoning (context-free
deductive reasoning of the kind traditionally studied)
difficult. However, attempts at providing proof-theoretical
formalisations of pragmatic reasoning, in nonmonotonic logics
such as Default Logic
[Reiter 1980], suggest paradoxically that
pragmatic reasoning is harder, both conceptually and
technically, than semantic reasoning. For example, the
application of a default rule is, in comparison with typical
deduction rules, hard to grasp, and the task of determining
whether a given sentence is in an extension of even a
semi-normal default theory is, because of the consistency
check, NP-hard in the propositional case and undecidable in the
[Kautz and Selman 1989]. One response is to
attempt to find tractable subsets of such logics.
The model-building approach
However, it is possible to use model theory to formalise
pragmatic reasoning, and we propose to resolve the paradox of
Section 2 by implementing the model theory of pragmatic logics.
A general theory has been outlined
[Bell 1995]. This recommends
replacing mathematical models by tractable computational models
in a principled way (model schemas, model circumscription), and
then building (dynamic model theory) and evaluating the
relevant computational models (evaluative confirmational proof
In [Bell forthcoming]
the programme recommended by the theory
is carried out in the case of causal theories of the kind
defined by Shoham [Shoham 1988]
and extended by [Bell 1991]. To
begin with, causal theories are reformulated on the basis of
Kleene three-valued logic
[Kleene 1952]. This simplifies the
model theory, as it has the effect of removing possible-worlds
structures from the models. As the interpretation of terms will
be fixed by each causal theory, the uniqueness theorem
guarantess that we need build only a single relevant model of
it, and furthermore the theorem shows how this model can be
constructed. In order to make such models tractable, we require
that they have finite domains. Futhermore, it is not necessary
to construct complete models. It is sufficient that for any
time point t we can construct the time-bounded model M/t.; that
is, the initial part of the model M "up to t". The construction
starts with an initial time point t0 and model M/t0. Each M/t
is then extended to M/t+1 by "temporal forward chaining" on the
causal rules using the facts added at t as triggers. This is
model building, rather than theorem proving, as the antecedents
of causal rules are evaluated in M/t, and if these are true,
their consequents are used to extend M/t. In particular a
conjunct 'not p' occuring in the antecedent of a rule is simply
evaluated in M/t; whereas establishing it proof-theoretically
requires a (theoretically intractable) meta-level proof that p
is not provable. The rules can thus be regarded as detailed
instructions for building models. In static model theory, rules
classify models. In dynamic model theory rules build models. In
the class of models considered, the cost of building M/t is, in
practice, polynominal, when the cost of applying a cuasal rule
(making a causal inference in) M/t is linear.
Ongoing and Future Work
We are extending the work in Section 4 to more complex theories
of causal reasoning, and aim to extend this work still further
and apply the theory to teleological reasoning; in particular,
to the persistence of mental states and rational agency [Bell
1995a]. The major problem is to restrict the number of
computational models it is necessary to build and evaluate.
This is can be done by extending the above idea of a model
schema (one partial model representing a class of complete
models) further; and many interesting ideas are being explored.
The aim is to maintain mathematical integrity while achieving
something like parity with human reasoners (natural
J. Bell (1991)
Extended causal theories, Artificial
Intelligence 48, 211-224.
J. Bell (1995)
Pragmatic reasoning; a model-based theory. In:
Applied Logic: How, What and Why? A selection of papers from
the Applied Logic Conference, Amsterdam 1992.
M. Masuch and L.
Polos (eds.), Kluwer Academic Publishers, pp. 1-27.
PostScript version available.
J. Bell (1995a) Changing Attitudes. In:
M.J. Wooldridge and N.R. Jennings (eds.). Springer Lecture
Notes in AI, No 890. Springer: Berlin, pp. 40-55.
J. Bell (forthcoming)
A model-based approach to predictive
causal reasoning. In:
Partiality, Modality and Nonmonotonicity. P.
Doherty (ed.). To appear in the Studies in Logic, Language and
Information series, CSLI Press: Stanford.
H. Kautz and B. Selman (1989)
Hard problems for simple default
logics, KR'89, 189-197.
S. Kleene (1952)
Introduction to Metamathematics,
R. Reiter (1980)
A logic for default reasoning, Artificial
Intelligence 13, 81-132.
Y. Shoham (1988),
Reasoning About Change, MIT Press: Cambridge MA.
J. Bell and W. Hodges, Building Models of Prediction Theories,
Working Paper, 1997.