Proposal
summary:
The structures of self-reference in reflexive theories will be used in
the design of intelligent systems, as models of indexicals, performatives
and self-awareness. Stronger formal models of reflection will be investigated,
constructed as reflexive progressions of reflexive theories.
Proposed timetable and results
1996: improving the preliminary results of
using formal
self-referential
structures in modelling the semantics of indexicals and performatives;
analysis of the uses of Gödel's theorems in the theory of artificial
intelligence
The main result was that algorithmic insight
into one's own workings cannot be deepened by anticipating and
incalculating its effects. That is, although there are Gödel
operators of any degree of reflexivity, they are only
"extensional", internally performing the steps of
reflexion they appear to skip; there are no
"intensional"
operators which would only
represent (as opposed to
perform) the action of a Gödel
operator and then improve upon it.
1997: analysis of the Gödel sentence as a logic
meta-program and definition
of a reflexive Gödel operator and the corresponding progressions of
(reflexive) theories
An interpretation of self-reference as self-recognition
was also constructed.
Not yet completed; work continues on assesing the usefulness of Kleene's
recursion theorems
to these ends.
As a side-effort, it was proved
that Gödel's mapping also holds
in non-standard models. On the analytical side, self-referential
aspects of experience were used to analyse and reject a dualist
structure of experience.
1998:
definition and implementation of a reflexive
meta-circular interpreter
and tests of its usefulness in tasks requiring reflection
(learning)
Team members were also engaged in (implementation) work on agents
and in the publication of the book Mind
versus Computer, IOS Press.
Not completed, in view of the preliminary negative results in 1996. Work
on the
implications
of Kleene's theorem for computational reflection
continued, partly relocated to the project
Reflection
in machine learning.
Work on modelling
reflection
in consciousness also continues.
Project team
Background
The background of the project is the growing role of
self-referential structures in the development of (meta-level)
computer architectures (Yonezawa & Smith 92), especially in
logic programming (meta-circular interpretation) (Abramson &
Rogers 89) and artificial intelligence (reflective systems,
meta-knowledge, learning) (Maes & Nardi 88). In computer
science in general, these structures appear in connection with
the specifically computational problem of regulating the flow of
control information (Smith 82, Yonezawa & Smith 92), and in
artificial intelligence in the design of introspective systems,
which can reason about themselves and their behavior (Maes 88).
On the other hand, in artificial intelligence and cognitive
science, discussions reappear of the relevance of Gödel's
and others' limitative theorems for the possibility of
artificial
intelligence, especially the possibility of developing formal
models of consciousness (Sloman 92, Penrose 94). The structures
of formal self-reference, familiar from Gödel's theorems
(Smullyan 92) are also the more specific, unified logical
background of this research, as opposed to the different forms of
self-reference which appear in the problems of control regulation
in various computational architectures (Silver 86). These formal
structures (Gödel, Smullyan) can be used as models of
various self-referential aspects of intelligence, within language
(Lucy 93) eg. talk of oneself by using indexicals such as "I",
"now", performatives such as "I promise",
"I
apologize", or outside of language (self-awareness, consciousness
of oneself) (Baars 88).
The effects of self-reference in formal theories - the construction of sentences which "talk" about themselves and the theory to which they belong - are not limited to those theories themselves. If the effect of such a sentence, eg. Gödel's sentence, is incompleteness of the theory, this opens up the possibility of its extension, in different directions. The principles of extending reflexive theories, formulated so far (Gödel, Turing, Feferman) have been limited to incremental, linear advance along the progression of (transfinite) ordinals (Giunchiglia & Smaill 89). Such advance is, however, non-reflexive: the usual extension operator does not take into account its own role in the process of extension: it only repeatedly reproduces the basis for its application. A reflexive extension operator would not do away with the incompleteness of a reflexive theory, but it could extend it in longer leaps along the progression of ordinals. Such reflexive progressions of reflexive theories could be a better model of the kind of reflection which is peculiar to consciousness and which is usually considered to surpass the reflexivity of reflexive formal theories.
Baars, B.J., A cognitive theory of consciousness,
Cambridge Univ. press
1988
Baars, B., Banks, W.,
Consciousness
and cognition, an international journal,
Academic press
Bartlett, S.,
Suber, P. (ed.),
Self-reference - reflections on reflexivity,
Martin Nijhoff 1987; includes A Bibliography of
Works on Reflexivity
Bartlett, S. J. (ed.), Reflexivity:
A Source-Book in Self-Reference, North-Holland 1992
Bojadziev,
D., papers
on self-reference.
Bratko, I.,
Prolog programming for AI, Addison-Wesley 1990
Case, J.,
Infinitary
self-reference in learning theory, J. Exprt. Theor.
AI 6, 1994
Feferman, S.,
Transfinite recursive progressions of axiomatic theories,
J. Symbolic logic 27, 1962
George, F., Models of thinking,
Allen & Unwin 1970
Giunchiglia,
F., Smaill,
A., Reflection in constructive and
non-constructive automated reasoning, in H.
Abramson, M.H.
Rogers (ed), Meta-Programming in
logic programming, MIT Press 1989
Kiczales,
G., Rivieres,
J., Bobrow, D.,
The art of meta-object protocol,
MIT press 1992
Lucy, J.A. (ed.),
Reflexive language,
Cambridge Univ. press
1993
Maes,
P., Nardi, D. (ed.),
Meta-level Architectures and
Reflection, North-Holland 1988
Penrose,
R., The Emperor's New Mind, Oxford Univ. Press 1989
----, Shadows of the Mind, Oxford Univ. Press 1994
Pizzi, R.,
Self-observation in artificial reality,
Note di software
48/49
Sanchis, L., Reflexive
structures, Springer 1988, and Recursive
functionals, Springer 1992
Silver, B.,
Meta-level inference, North-Holland 1986
Sloman, A..
The emperor's real mind: review
of Roger Penrose's The Emperor's
New Mind, Artificial Intelligence 56 (1992) 355-396
Smullyan, R.M.,
Gödel's
incompleteness theorems, Oxford Univ. press
1992
Smith, B.,
Reflection and semantics in a procedural language,
MIT TR-272, 1982
----, Varieties of Self-reference, in
J.Y. Halpern (ed.),
Theoretical Aspects of Reasoning about Knowledge,
Proceedings
of the 1986 Conference, Morgan Kaufmann 1986
Smorynsky, C.,
Self-reference in modal logic,
Springer 1985
Stefik, M., Smoliar, S.,
Four reviews of "The Society of
Mind",
Artificial intelligence 48/3, 1991
Turing, A.,
Systems of logic based on ordinals,
Proc. London math. soc.
45, 1939
Webb,
J. (1980), Mechanism,
Mentalism and Metamathematics - An Essay on Finitism, D.
Reidel Publ. Co.
Yonezawa,
A., Smith, B. (ed)., Reflection
and Meta-level Architectures,
Proc. Int. Worskhop on New Models for Software
Architecture, Tokyo 1992
References
Abramson, H,
Rogers, M.H. (ed), Meta-Programming in
logic programming, MIT Press 1989