32
R. ROSEN
7. KLEENE, S. c.: In: C. E. SHANNON, and J. MCCARTHY, (Eds.), Automata
Studies, p. 1. Princeton University Press 1956.
8. MCCULLOCH, W. S., and W. PITTS: Bull. Math. Biophys. 5, 115 (1943).
9. MYHILL, J.: AF. Technical Documentary Report, Contract AF 30 (602)-3754,
48 (1966).
10. RASHEVSKY, N.: Mathematical Biophysics, 3rd. edition. New York: Dover
Publications 1960.
11. ROSEN, R.: Bull. Math. Biophys. 24, 375 (1962).
12. - J. Theor. BioI. 15, 282 (1967).
13. - International Reviews of Cytology 23, 25 (1968).
14. SCARPELLINI, A.: Z. ftir Math. Logik 9, 265 (1963).
15. SIMON, Z.: J. Theor. BioI. 8, 258 (1965).
16. STAHL, W. R.: J. Theor. BioI. 8, 371 (1965).
17. - and H. E. GOHEEN: J. Theor. BioI. 5,266 (1963).
18. SUGITA, M.: J. Theor. BioI. 1, 415 (1961).
19. - J. Theor. BioI. 4, 179 (1963).
20. - J. Theor. BioI. 13, 330 (1966).
21. THOM, R.: Stabilite Structurelle et Morphogenese. Benjamin, to appear.
22. ZADEH, L., and C. A. DESOER: Linear System Theory - The State Space
Approach. New York: McGraw-Hill 1963.
WALTER:
Discussion
Could you describe again how this correspondence principle applies to transitions from limit cycles (which still may be time-dependent) rather than
stable points. How is the proper "new basin" chosen?
ROSEN:
The equivalence classes of transients which correspond to the discrete input
symbols are defined pointwise on the phase space and the definitions work in
all cases, including the one you cite.
BREMERMANN:
The abstract notion of "computability" is concerned with questions that have
arisen in mathematical logic. TURING machines are ideal abstract machines unhampered by limitation of memory and time. Even in this setting there are
considerable problems and uncomputable functions, etc.-There are problems
that are trivial in this context, but uncomputable in a more restricted sense:
for example, the game of chess. The problem to decide optimal moves is a
finite one and hence trivial in abstract computability theory. However, the
number of cases to be considered is so large that physical computation is
impossible, while the solution through insight (that famous players seem to
have) has not yet been achieved by computers.-The question to determine the
maximal ease of computation of a given task may be undecidable.
ROSEN:
I feel you are quite right in believing that computability plays at best a small
role in the design of biological systems. As I said in my talk, it seems to me that
criteria related to ease of computation are likely to be far more important.
However, computability is a typical attribute of discrete systems, one which has
no single continuous analog and which has been raised in the literature in
connection with biological design. Therefore I thought it appropriate to use
computability as an illustration of how the correspondence principle works in
transferring ideas from the discrete to the continuous case.
R. ROSEN
7. KLEENE, S. c.: In: C. E. SHANNON, and J. MCCARTHY, (Eds.), Automata
Studies, p. 1. Princeton University Press 1956.
8. MCCULLOCH, W. S., and W. PITTS: Bull. Math. Biophys. 5, 115 (1943).
9. MYHILL, J.: AF. Technical Documentary Report, Contract AF 30 (602)-3754,
48 (1966).
10. RASHEVSKY, N.: Mathematical Biophysics, 3rd. edition. New York: Dover
Publications 1960.
11. ROSEN, R.: Bull. Math. Biophys. 24, 375 (1962).
12. - J. Theor. BioI. 15, 282 (1967).
13. - International Reviews of Cytology 23, 25 (1968).
14. SCARPELLINI, A.: Z. ftir Math. Logik 9, 265 (1963).
15. SIMON, Z.: J. Theor. BioI. 8, 258 (1965).
16. STAHL, W. R.: J. Theor. BioI. 8, 371 (1965).
17. - and H. E. GOHEEN: J. Theor. BioI. 5,266 (1963).
18. SUGITA, M.: J. Theor. BioI. 1, 415 (1961).
19. - J. Theor. BioI. 4, 179 (1963).
20. - J. Theor. BioI. 13, 330 (1966).
21. THOM, R.: Stabilite Structurelle et Morphogenese. Benjamin, to appear.
22. ZADEH, L., and C. A. DESOER: Linear System Theory - The State Space
Approach. New York: McGraw-Hill 1963.
WALTER:
Discussion
Could you describe again how this correspondence principle applies to transitions from limit cycles (which still may be time-dependent) rather than
stable points. How is the proper "new basin" chosen?
ROSEN:
The equivalence classes of transients which correspond to the discrete input
symbols are defined pointwise on the phase space and the definitions work in
all cases, including the one you cite.
BREMERMANN:
The abstract notion of "computability" is concerned with questions that have
arisen in mathematical logic. TURING machines are ideal abstract machines unhampered by limitation of memory and time. Even in this setting there are
considerable problems and uncomputable functions, etc.-There are problems
that are trivial in this context, but uncomputable in a more restricted sense:
for example, the game of chess. The problem to decide optimal moves is a
finite one and hence trivial in abstract computability theory. However, the
number of cases to be considered is so large that physical computation is
impossible, while the solution through insight (that famous players seem to
have) has not yet been achieved by computers.-The question to determine the
maximal ease of computation of a given task may be undecidable.
ROSEN:
I feel you are quite right in believing that computability plays at best a small
role in the design of biological systems. As I said in my talk, it seems to me that
criteria related to ease of computation are likely to be far more important.
However, computability is a typical attribute of discrete systems, one which has
no single continuous analog and which has been raised in the literature in
connection with biological design. Therefore I thought it appropriate to use
computability as an illustration of how the correspondence principle works in
transferring ideas from the discrete to the continuous case.