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Hypercomputation and the
Human Mind
Selmer Bringsjord
Professor of Logic, Philosophy, Cognitive Science, and Computer Science
Rensselaer AI & Reasoning (RAIR) Laboratory
Department of Cognitive Science
Department of Computer Science
Rensselaer Polytechnic Institute (RPI)
Troy NY 12180 USA
AFRL-R 8.11.05
[email protected]
http://www.rpi.edu/~brings
Next-Generation “Weak” Logic-based AI
• Isolate and dissect human ingenuity.
• Mathematize a weak correlate to this
ingenuity courtesy of advanced logical systems.
• Implement this correlate in working
programs.
• Augment the correlate with machine-specific
power.
The Rensselaer AI & Reasoning Lab
(The RAIR Lab)
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are nee ded to s ee this picture.
Wargaming
 Cracking Project; “Superteaching”
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TIFF (Uncompressed) decompressor
are needed to see this picture.
A while
back,
RPI
Strategic
Investment
hypothesis generation; AI in support of IA
Slate (Intelligence
Analysis)
Item generation
(theorem proving-based
generation)
synthetic characters/psychological time
Cognitive Science and artificial
intelligence (AI) assume that
cognition is computation.
Newell’s “20 Questions” Paper
• The study of the mind
is painfully
fragmentary, but at
any rate...
• Man is a computer.
Pinker’s How the Mind Works
• “The mind is a system
of organs of
computation designed
by natural selection to
solve problems faced
by our evolutionary
ancestors in their
foraging way of life.”
Anderson’s Recent “Newell Test”
Paper in BBS
• The study of the mind
is less fragmentary
now, and certainly...
• Humans are computers.
Are Newell & Simon & Co. correct?
Are human persons computers?
No. They’re wrong. And they can be wrong...
even if we are fundamentally information processors.
If they’re right, this is what, at bottom, we are!
If R5 is non-empty,
take a bead away.
,
,
,
R1 R2 R3 R4 R5 R6 R7 R8 R9
“Humble” Circle of Computation
Abaci
Register machines
(modern high-speed
digital computers)
Recursive functions
Turing machines
“Humble” Circle of Computation
Abaci
Recursive functions
Register machines
(modern high-speed
digital computers)
-calculus
Turing machines
“Humble” Circle of Computation
Abaci
Recursive functions
programming languages
(Lisp, C++, ...)
-calculus
Register machines
(modern high-speed
digital computers)
Turing machines
“Humble” Circle of Computation
Abaci
Recursive functions
programming languages
(Lisp, C++, ...)
Register machines
(modern high-speed
digital computers)
-calculus
logic programming
(in first-order logic)
Turing machines
But we’re not mere carbon-based
creatures somewhere in this circle
(of mere animal life)!
The Mathematical Landscape
Space of all information processes
Hypercomputation,
now exploding
with activity
revolutionizing
computer science,
mathematics,
physics, etc.
Analog Chaotic Neural
Networks, Zeus Machines,
Weyl Machines, P/Poly
Machines, …
Turing machines, abaci, programs, …
}
The “Turing Limit”
AI & Cog Sci
What are we?
Phenomena not
capturable in any thirdperson scheme.
e.g., phenomenal
consciousness
(e.g., that which it
feels like to carve
ski turns at 40mph)
Superminds!
Analog Chaotic Neural
Networks, Zeus Machines,
Weyl Machines, P/Poly
Machines, …
“Turing
Limit”
Turing machines, abaci, programs, …
So what are these
hypercomputers you’re talking
about?
We know what computers are.
What’s a hypercomputer?
Bertrand Russell (1936):
“Ambrose says it is logically impossible [for a man] to run through
the whole expansion of . I should have said it was medically
impossible. ... Might not a man's skill increase so fast that he
performed each operation in half the time required for its predecessor?
In that case, the whole infinite series would take only twice as long
as the first operation.”
A Hypercomputer: Zeus Machine
7th Grade Math Problem,
Anticipating Elementary Calculus
1
lim n  0
n  4

John Anderson:
“But Godel shot down Hilbert’s Program
with a proof! You’re just claiming.”
Ok, fair enough.
I have proofs, or at least formal
arguments. Seventeen of them, in fact.
Today, one intuitive, one precise
(synopsis only, of course). For more, see
‘For Further Reading’ @ the end.
The Mountaineering “Argument”
Imagine a race of creatures:
ipas (ih-pah-s). Ipas are
information-processing
creatures. They live on the High
Peaks of Hypercomputation, and
are capable of phenomenal feats
of information processing (such
as cracking the halting problem).
We know this because some humans
are able to climb the mountains
and study the ipas. In fact, these
humans have proved all kinds
of things about the capability of
ipas, and in general know what
the essence of ipa cognition is.
Ipas
Humans
And yet it is said that no humans
are capable of ipa mentation, but
are forever limited to living in the
Valley of Turing Machines.
Argument from Infinitary Reasoning
(from Superminds)
Proof. Assume that all human reasoning is at or beneath
the Turing Limit. Then all chains of human reasoning
(e.g., proofs) are identical to some chain of reasoning
expressed in first-order logic. But there are many chains
of human reasoning in infinitary logics, and we know
that such chains in infinitary logic cannot possibly be
expressed in first-order logic. Contradiction! So, by
indirect proof, the starting assumption (which has led to
absurdity) is false. I.e., it’s not true that all human reasoning
is at or beneath the Turing Limit. QED
If we are superminds, then, by
the way, the received view
concerning our origins
explodes...
A Proof that People Didn’t Evolve
(1) If people are the product of evolution, they were
produced by an algorithmic process (= by a
process at or below the Turing Limit).
(2) Theorem Ã: No algorithmic process can
produce a device capable of hypercomputation.
(3) People are capable of hypercomputation.
(4)  People were not produced by an algorithmic
process. (from 2 and 3)
(5)  People aren’t the product of evolution.
(from 4 and 1)
If we are superminds, how then
should we proceed, in concrete
terms?
RAIR Lab’s Push Against the Turing Limit
http://www.cs.rpi.edu/~kelleo/busybeaver/
For Further Reading...
For Further Reading
New Proof (one of the seventeen) that Minds Hypercompute: Bringsjord, S., Shilliday, A.,
Taylor, J., Bram van Heuveln, B., Baumes, J. & Ross, K. (forthcoming) “A New Gödelian
Argument for Hypercomputing Minds Based on the Busy Beaver Problem,” Applied
Mathematics and Computation. Preprint available at top level of Bringsjord’s web site.
Recent Proof that Minds Hypercompute: Bringsjord, S. & Arkoudas, K. (2004) “The Modal
Argument for Hypercomputing Minds” Theoretical Computer Science 317: 167-190.
Offprint available from top level of Bringsjord’s web site.
Argument #1 Against Pinkerian View that We are Evolved Computers: Bringsjord, S.
(2001) “Are We Evolved Computers? A Critical Review of Steven Pinker’s How the Mind
Works” Philosophical Psychology 14.2:227-243.
Paper Exhibiting Human Infinitary Reasoning: Bringsjord, S. & van Heuveln, B. (2003)
“The Mental Eye Defense of an Infinitized Version of Yablo’s Paradox,” Analysis 63.1: 6170.
Book Explaining and Defending View That We’re “Superminds”: Bringsjord, S. & Zenzen,
M. (2003) Superminds (Dordrecht, The Netherlands: Kluwer Academic Publishers).
Anderson’s “Newell Test” paper: Anderson, J. & Lebiere, C. (2003) “The Newell Test for a
Theory of Cognition,” Behavioral and Brain Sciences 26: 587-601.
Zeus Machines Introduced and Contextualized, and the Humble Circle Fully Explained:
Boolos, G. & Jeffrey, R. (1989) Computability and Logic (Cambridge, UK: Cambridge
University Press).
Phenomenal Consciousness and Cognitive Science: Block, N. (1995) “On a Confusion
About a Function of Consciousness,” Behavioral and Brain Sciences 18: 227-247.
END