Owen Kellett

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Toward Conquering the?-Cracking (Busy Beaver)
Problempresenters OK, BvH, SB28 January 2004

• Owen Kellett
• Kyle Ross
• Bram van Heuveln
• Selmer Bringsjord
• Kostas Arkoudas
• Marc Destefano
• Boleshaw Szymanski
• Carlos Varela
• Shailesh Kelkar
• Department of Cognitive Science
• Department of Computer Science (SB, Chair)
• Rensselaer AI Reasoning (RAIR) Lab (SB,
  Director)
• RPI Troy NY 12180 USA

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The Rensselaer AI Reasoning Lab(The RAIR Lab)
Wargaming
? Cracking Project Intelligent Tutoring
Systems (mathematical logic)
Over 1million internal seeding
Slate (Intelligence Analysis)
Item generation (theorem proving-based generation)
synthetic characters/psychological time
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The Busy Beaver ProblemGeneral Version

• In general, the busy beaver problem is to find
  the most productive Turing machine with a given
  state and symbol set.
• The productivity of a Turing machine can be
  defined in many ways
• The number of symbols written
• The number of steps taken
• The number of cells moved away from the starting
  cell
• The number of non-blank symbols on tape
• Etc.
• Note Non-halting machines will set almost any of
  these numbers to infinity, so non-halting
  machines will be excluded from consideration.

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Standard Settings

• Some standard settings for the Busy Beaver
  Problem are
• The alphabet consists of a blank and a non-blank
• The Turing Machine starts on an empty tape.
• Productivity is the number of non-blank symbols
  left on the tape

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Formalizing the Problem

• n number of states
• M(n) set of TMs with n states and binary
  alphabet
• Prod(M) number of non-blank symbols left on tape
  by machine M, when started on empty tape.
• Busy Beaver Problem Find BB(n) max Prod(M)
  M ? M(n)
• Any machine M ? M(n) for which Prod(M) BB(n) is
  called a Busy Beaver.
• Rado defined the problem and proved that BB(n) is
  uncomputable (1962).
• However, we can find individual values BB(n) for
  small n.

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Variations of the Problem

• We can still define a variety of Busy Beaver
  problems
• Do we use quadruple or quintuple machines?
• How does the machine come to a halt?
• Are there any restrictions on the output
  configuration?
• Standard configuration head positioned at
  leftmost 1 (non-blank) of consecutive string of
  1s on otherwise empty tape
• Anything goes

quadruple formulation
quintuple formulation
explicit halt
implicit halt
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Taxonomy of BB Problems
BB
Quintuples
Quadruples
Anything goes
Standard config.
Anything goes
?
?
Explicit Halt State
Explicit Halt State
Implicit Halt State
Explicit Halt State
Implicit Halt State
?
B(n) (Boolos Jeffrey, Turings World)
R(n)
P(n) (Pereira et al.)
O(n) (Oberschelp et al.)
?(n) (Rado)
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Previous Known Results
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Problems in determining BB(n)

• Turing unsolvable
• Large search space
• Implicit halt M(n)(4n1)2n
• Explicit halt M(n)(4n4)2n
• 4 possible actions for each of n next states
• For implicit machines 1 no-action transition to
  halt-state
• For explicit machines 4 possible actions to
  halt-state
• 2n possible transitions

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Inefficiency Isomorphisms
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Inefficiency Unused Transitions
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Solution Tree Normalization
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Solution Tree Normalization
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Solution Tree Normalization
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Improvement from Normalization
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Inefficiency Empty Tape Machine

• machine reaches an empty tape after 1 or more
  shifts
• any machine that does not write 1 as its first
  action is such a machine

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(Partial) Solution Force First Write
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(Partial) Solution Force First Write
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Inefficiency Mirror Machines
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Solution Force First Move
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Solution Force First Move
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Solution Force First Move
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Solution Force First Move
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Solution Force First Move
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Improvement from Optimizations
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Inefficiency Nonproductive Transitions
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General Algorithm thus far

• Begin with root machine (single state with no
  transitions) on stack
• Initialize best machine as root machine
• While stack is not empty
• Pop machine m off of stack
• If m is a non-halter, discard m and continue at
  the beginning of the loop
• Run m until it halts
• If m halts in the halt state (for the purposes of
  this algorithm we will consider an implicit
  machine to have a halt state where no action is
  performed when transitioning to it)
• Compare ms productivity to best machine and
  update if necessary
• Else
• Generate ms children in the optimized fashion
  described
• Push ms children onto the stack

Problem What if m is a non-halter?
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Solution Non-halt detection

• Step Limit
• Run a machine for a fixed number of steps.
  Problem if it hasnt halted, it may still halt
  at some later point.
• Also, Halting problem
• There is no generalized algorithm that will take
  as input machine m and return whether or not m
  halts
• Fortunately, algorithms can be designed to test
  for specific non-halting behaviors
• By adding such detection routines to the
  tree-normalization algorithm from the previous
  slide, we obtain an algorithm A based is such
  that for any machine m
• A eventually declares that m halts (m is a
  halter)
• A eventually declares that m does not halt (m is
  a non-halter)
• A eventually declares that it doesnt know
  whether m halts or not (m is a holdout)

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Backtracking
q0x1 ? halt 10
q3x1 ? q0xL 113
q2x0 ? q3xR 021
101
013
Local tapes match, continue
Does not match! Non-halter!
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Subset Loops

• A Turing Machine M is classified as a subset loop
  if
• There is a set of states S such that every
  possible transition from each state in S is
  defined
• Every transition defined from a state in S is a
  transition to another state in S
• During execution, at some point the machine
  enters one of the states in S

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Simple Loops

• A machine is classified as a simple loop if
  (given words of arbitrary length X, Y, V, and C
  and state s)
• The following tape configuration is reached
  0YsXC0
• -and one of the following-
• 1 The same tape configuration is reached at a
  later point
• -or-
• 2 The following tape configuration is reached at
  a later point 0YsXVC0
• Between these points, the read head never moves
  past the right edge of the initial X
• The corresponding mirror of the above
  specification also identifies a simple loop

•        0      State 0       1      State
  1       10     State 2       100    State
  3       1000   State 4       10000  State
  5       10000  State 5       10000  State
  5       10000  State 5       10000  State
  5      010000  State 3      010000  State
  4      010000  State 2      010000  State
  3      010000  State 1     0010000  State
  0     1010000  State 1     1010000  State
  2     1010000  State 3     1010000  State
  1     1010000  State 0    01010000  State
  4    01010000  State 5    01010000  State
  3    01010000  State 4    01010000  State
  2    01010000  State 3    01010000  State
  1   001010000  State 0   101010000  State
  1   101010000  State 2   101010000  State
  3   101010000  State 1   101010000  State 0

0 Y X C 0
0 Y X V C 0
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Christmas Trees

• In the general sense, a christmas tree non-halter
  sweeps back and forth across the tape in a
  repeatable manner

         1101010            State
1         1101010            State
2         1101010            State
0         1111010            State
1         1111010            State
2         1111010            State
0         1111110            State
1         1111110            State
2         1111110            State
0         1111111            State
1         11111110           State
2         11111110           State
3         11111110           State
0         11111010           State
3         11111010           State
3         11111010           State
0         11101010           State
3         11101010           State
3         11101010           State
0         10101010           State
3         10101010           State
3        010101010           State
0        110101010           State
1        110101010           State
2        110101010           State
0        111101010           State
1        111101010           State
2        111101010           State
0        111111010           State
1        111111010           State
2        111111010           State
0        111111110           State
1        111111110           State
2        111111110           State
0        111111111           State
1        1111111110          State 2
            0               State
0            1               State
1            10              State
2            10              State
3           010              State
0           110              State
1           110              State
2           110              State
0           111              State
1           1110             State
2           1110             State
3           1110             State
0           1010             State
3           1010             State
3          01010             State
0          11010             State
1          11010             State
2          11010             State
0          11110             State
1          11110             State
2          11110             State
0          11111             State
1          111110            State
2          111110            State
3          111110            State
0          111010            State
3          111010            State
3          111010            State
0          101010            State
3          101010            State
3         0101010            State 0
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Christmas Trees
 11111110    State 2 11111110    State
3 11111110    State 0 11111010    State
3 11111010    State 3 11111010    State
0 11101010    State 3 11101010    State
3 11101010    State 0 10101010    State
3 10101010    State 3010101010    State
0110101010    State 1110101010    State
2110101010    State 0111101010    State
1111101010    State 2111101010    State
0111111010    State 1111111010    State
2111111010    State 0111111110    State
1111111110    State 2111111110    State
0111111111    State 11111111110   State 2
0UXXVs0
0UYYV0
0UZZV0

0UXXXV0
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Leaning Christmas Trees
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Multi-sweep Christmas Trees

• The machine transforms the tape much like a
  normal Christmas tree, however, it takes more
  than one sweep across the tape to complete a cycle

Christmas Trees
Double-sweep Christmas Trees
0UXXXV0
0UXXXV0
0UYYYV0
0UYYYV0
0UZZZV0
0UZZZV0

0UXXXXV0
0UMMMV0
0UNNNV0

0UXXXXV0
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Counters
0         State 01         State
110        State 2100       State
3101       State 0101       State
1101       State 1101       State
2101       State 3100       State
21000      State 31001      State
01001      State 11001      State
11001      State 11001      State
21001      State 31011      State
01011      State 11011      State
11011      State 21011      State
31001      State 21001      State
31000      State 210000     State 3
10001     State 010001     State
110001     State 110001     State
110001     State 110001     State
210001     State 310101     State
010101     State 110101     State
110101     State 210101     State
310001     State 210001     State
310011     State 010011     State
110011     State 110011     State
110011     State 210011     State
310111     State 010111     State
110111     State 110111     State
210111     State 310011     State 2
Note this machine generates binary numbers that
read from right to left rather than the
conventional left to right
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Results non-halters
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Updated Records
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Future Work/Goals

• Certification of our records (Athena)
• Continued development of non-halt detection
  routines
• Christmas tree variations
• Counter variations (base 3, base 4, etc..)
• With only 221 holdouts left, explicit
  confirmation of BB(5) is imminent
• novel techniques for detecting non-halting
• Additional tree normalization optimizations
• Identical state machines?
• Visual Reasoning (relates to new representations
  in support of novel non-halter detection)
• Treat issues in Logico-Mathematical Foundations
• Distributed computing (SALSA)
• Working implementation using the developmental
  language SALSA to allow the potential for massive
  distributed computation

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Visual Reasoning

• The behavior of certain machines can sometimes be
  generalized by observing them in action.
• Indeed, notice that we have named certain classes
  of machines, such as Christmas Trees, exactly
  by observing a graphical display of their
  bevhavior.
• Maybe a system component can be created that can
  be used to reason about the behavior of a Turing
  machine based on a visual representation of (the
  behavior of) that Turing machine rather than a
  more traditional linear symbolic representation.

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The Mathematical Landscape
Space of all information processes
Analog Chaotic Neural Networks, Zeus Machines,
Weyl Machines, P/Poly Machines,
Hypercomputation
engineering on the busy beaver?
The Turing Limit
Turing Machines, Algorithms, Programs,