math discovery
Gerry Wolff
gerry at sees.bangor.ac.uk
Fri Jun 18 02:44:23 PDT 1999
Chuck Carlson wrote:
>
> Hello all,
>
> Does anyone have any comment or know of resources regarding automated
> math discovery
> involving the principles of compression and/or entropy?
>
> I know about Douglas Lenat's AM program but that started with a
> knowledge of sets
> and used lots of heuristics.
>
> I'd like to use genetic algorithms with a fitness function involving
> compression or
> entropy to evolve structures that would probably start out representing
> simple maths
> like addition and subtraction.
>
> Thanks,
>
> Chuck Carlson
I've done a search in the Science Citation Index for the years 1981-1999
using the search pattern math* & (induc* OR discover* OR learn*) but the
results are a bit disappointing. One difficulty is that 'mathematical
induction' has a meaning which is different from the induction of
mathematical functions.
I know that H Simon has worked in this area (one reference in the list
below) and Pat Langley edited a book on it some time ago but I can't
find the details.
A selection of the most relevant of the search results is below.
Gerry
-----------
(1) TI: Parallel processing and OOP as an analogy for the discovery of
certain mathematical proofs
AU: Hennefeld_J
NA: CUNY BROOKLYN COLL,DEPT MATH,BROOKLYN,NY,11210
JN: COMPUTERS & MATHEMATICS WITH APPLICATIONS, 1997, Vol.33, No.6,
pp.91-97
IS: 0898-1221
DT: Article
AB: This paper presents a heuristic methodology that can be used
to
discover (and/or better understand) proofs of some
mathematical
theorems, when the statement of the theorem involves a set for
which every element should be ''processed.'' This heuristic,
which has a number of interesting connections with recent
trends in computer program design, is called the Method of
Uniform Parallel Object-Modules, after the concepts of
modularization, parallel processing, and object oriented
programming.
WA: heuristic, object-oriented, parallel processing, proof
(2) TI: A computational approach to George Boole's discovery of
Mathematical Logic
AU: deLedesma_L, Perez_A, Borrajo_D, Laita_LM
NA: UNIV POLITECN MADRID,FAC INFORMAT,BOADILLA MONTE
28660,MADRID,SPAIN
UNIV CARLOS III MADRID,ESCUELA POLITECN SUPER,LEGANES
28911,MADRID,SPAIN
JN: ARTIFICIAL INTELLIGENCE, 1997, Vol.91, No.2, pp.281-307
IS: 0004-3702
DT: Article
AB: This paper reports a computational model of Boole's discovery
of Logic as a part of Mathematics. George Boole (1815-1864)
found that the symbols of Logic behaved as algebraic symbols,
and he then rebuilt the whole contemporary theory of Logic by
the use of methods such as the solution of algebraic
equations.
Study of the different historical factors that influenced this
achievement has served as background for our two main
contributions: a computational representation of Boole's Logic
before it was mathematized; and a production system, BOOLE2,
that rediscovers Logic as a science that behaves exactly as a
branch of Mathematics, and that thus validates to some extent
the historical explanation. The system's discovery methods are
found to be general enough to handle three other cases: two
versions of a Geometry due to a contemporary of Boole, and a
small subset of the Differential Calculus. (C) 1997 Published
by Elsevier Science B.V.
KP: HEURISTICS
(3) TI: Analyzing mathematical models with inductive learning networks
AU: Steiger_DM, Sharda_R
NA: UNIV N CAROLINA,DEPT INFORMAT SYST & OPERAT
MANAGEMENT,GREENSBORO,NC,27412
OKLAHOMA STATE UNIV,DEPT MANAGEMENT,STILLWATER,OK,74078
JN: EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, 1996, Vol.93, No.2,
pp.387-401
IS: 0377-2217
DT: Article
AB: After building and validating a mathematical model, the
decision maker frequently solves (often many times) a slightly
different version of the model. That is, by changing various
input parameters and re-running different model instances, he
tries to develop insight(s) into the workings and tradeoffs of
the complex system represented by the model. However, very
little research has been devoted to helping the decision maker
in this important model analysis endeavor. This paper
investigates the application of two inductive learning
technologies, backpropagation neural networks and the group
method of data handling, to the analysis of multiple instances
of a mathematical model. Specifically, these two techniques
are
compared in the analysis tasks of identifying key factors and
determining key relations between uncertain and/or unknown
model parameters and the associated objective function values.
KP: SENSITIVITY ANALYSIS, NEURAL NETWORK, OPTIMIZATION, OUTPUT
WA: neural networks, group method of data handling (GMDH),
inductive learning networks, model analysis, decision support
systems
(4) TI: Automated mathematical induction
AU: Bouhoula_A, Kounalis_E, Rusinowitch_M
NA: INRIA LORRAINE,F-54600 VILLERS LES NANCY,FRANCE
CRIN,F-54600 VILLERS LES NANCY,FRANCE
LAB INFORMAT I3S,F-06560 VALBONNE,FRANCE
JN: JOURNAL OF LOGIC AND COMPUTATION, 1995, Vol.5, No.5,
pp.631-668
IS: 0955-792X
DT: Article
AB: Proofs by induction are important in many computer science and
artificial intelligence applications, in particular, in
program
verification and specification systems. We present a new
method
to prove (and disprove) automatically inductive properties.
Given a set of axioms, a well-suited induction scheme is
constructed automatically. We call such an induction scheme a
test set. Then, for proving a property, we just instantiate it
with terms from the test set and apply pure algebraic
simplification to the result. This method needs no completion
and explicit induction. However it retains their positive
features, namely, the completeness of the former and the
robustness of the latter. It has been implemented in the
theorem-prover SPIKE.
KP: TEST SETS, TERMINATION, PROOF
WA: theorem proving, mathematical induction, term rewriting
systems
(5) TI: Automated mathematical induction - Preface
AU: Zhang_H
NA: UNIV IOWA,IOWA CITY,IA,52242
JN: JOURNAL OF AUTOMATED REASONING, 1996, Vol.16, No.1-2, p.1
IS: 0168-7433
DT: Editorial
(6) TI: Building intelligent alarm systems by combining mathematical
models and inductive machine learning techniques
AU: Muller_B, Hasman_A, Blom_JA
NA: EINDHOVEN UNIV TECHNOL,DEPT MED ELECT ENGN,POB 513,5600 MB
EINDHOVEN,NETHERLANDS
MAASTRICHT UNIV,DEPT MED INFORMAT,6200 MD
MAASTRICHT,NETHERLANDS
JN: INTERNATIONAL JOURNAL OF BIO-MEDICAL COMPUTING, 1996, Vol.41,
No.2, pp.107-124
IS: 0020-7101
DT: Article
AB: In this article a technique is described to develop knowledge-
based alarm systems for ventilator therapy, using mathematical
modeling and machine learning. With a mathematical model
airway
pressure, expiratory gas flow and CO2 concentration at the
endotracheal tube are simulated for patients, undergoing
volume-controlled ventilation with constant ventilator
settings, during normal functioning of the breathing circuit
and during breathing circuit mishaps (leaks and obstructions).
Simulations were performed for 94 physiologically different
'patients', by varying airway resistance and lung/thorax
compliance values in the model. Each simulated breath was
described by a set of derived signal features and a label that
constituted during which event (normal function or mishap) the
breath was recorded. With an inductive machine learning
algorithm rules, linking signal feature values to breathing
circuit events, were created from data of 54 of the simulated
patients. The resulting set of rules was able to classify 99%
of events in the data of the remaining 40 patients correctly.
Of signals, measured at a ventilated lung simulator, 100% of
events were classified correctly.
KP: ALGORITHM
WA: anesthesia, inductive machine learning, intelligent alarm
systems, mathematical modeling, mechanical ventilation,
respiratory mechanics
(7) TI: TOWARDS A MATHEMATICAL-THEORY OF MACHINE DISCOVERY FROM FACTS
AU: MUKOUCHI_Y, ARIKAWA_S
NA: KYUSHU UNIV 33,FUNDAMENTAL INFORMAT SCI RES INST,FUKUOKA
812,JAPAN
JN: THEORETICAL COMPUTER SCIENCE, 1995, Vol.137, No.1, pp.53-84
IS: 0304-3975
DT: Article
AB: This paper intends to give a theoretical foundation of machine
discovery from facts. We point out that the essence of a
computational logic of scientific discovery or a logic of
machine discovery is the refutability of the entire spaces of
hypotheses. We discuss this issue in the framework of
inductive
inference of length-bounded elementary formal systems (EFSs),
which are a kind of logic programs over strings of characters
and correspond to context-sensitive grammars in Chomsky
hierarchy.
First we present some characterization theorems on inductive
inference machines that can refute hypothesis spaces. Then we
show differences between our inductive inference and some
other
related inferences such as in the criteria of reliable
identification, finite identification and identification in
the
limit. Finally we show that for any n, the class, i.e.
hypothesis space, of length-bounded EFSs with at most n axioms
is inferable in our sense, that is, the class is refutable by
a
consistently working inductive inference machine. This means
that sufficiently large hypothesis spaces are identifiable and
refutable.
KP: INDUCTIVE INFERENCE, IDENTIFICATION
(8) TI: ON THE DISCOVERY OF MATHEMATICAL CONCEPTS
AU: EPSTEIN_SL
NA: CUNY HUNTER COLL,DEPT COMP SCI,695 PK AVE,NEW YORK,NY,10021
JN: INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, 1988, Vol.3,
No.2, pp.167-178
IS: 0884-8173
DT: Article
(9) TI: A COMPUTER-INSPIRED MATHEMATICAL DISCOVERY
AU: FREITAG_HT
NA: HOLLINS COLL,MATH,HOLLINS COLL,VA,24020
JN: ABACUS-NEW YORK, 1985, Vol.2, No.4, p.36 et seq.
IS: 0724-6722
DT: Article
(10) TI: COMPUTER MODELING OF SCIENTIFIC AND MATHEMATICAL DISCOVERY
PROCESSES
AU: SIMON_HA
NA: CARNEGIE MELLON UNIV,DEPT PSYCHOL,PITTSBURGH,PA,15213
JN: BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 1984, Vol.11,
No.2, pp.247-262
IS: 0273-0979
DT: Article
(11) TI: MATHEMATICAL INDUCTION - A COMPUTER APPROACH
AU: AUSTIN_AK
NA: UNIV SHEFFIELD,DEPT PURE MATH,SHEFFIELD S10 2TN,S
YORKSHIRE,ENGLAND
JN: MATHEMATICAL GAZETTE, 1984, Vol.68, No.443, pp.46-48
IS: 0025-5572
DT: Note
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