2. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Analysis of Busy Beaver Machines via Induction Proofs James Harland jah@cs.rmit.edu.au www.cs.rmit.edu.au/~jah School of CS & IT RMIT University

3. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Introduction New twist on an old problem More questions than answers! Innocuous class of machines generate huge numbers Involves termination analysis and constraint programming Frustrating to the point of obsession …

4. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Busy Beaver Turing Machines Two-way infinite tape Only tape symbols are 0 and 1 Deterministic Blank on input Question: What is the largest number of 1’s that can be printed by a terminating n-state machine?

5. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Known Values (n states, 2 symbols) nbb(n)ff(n) 111 246 3621 413107 5≥ 4098≥ 47,176,870 6≥ 1.29×10 865 ≥ 3×10 1730 7!!!!!!!!????

6. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Known Values (n states, m symbols) StatesSymbolsbb(n)ff(n) 23938 24≥ 2,050≥ 3,932,964 33≥ 95,524,079≥ 4,345,166,620,336,565 25≥ 1.7×10 11 ≥ 7.1×10 21 26????

7. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Known Beaver Machines

8. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Known Beaver Machines This can be represented in around 60 bits … 10 865 takes about 2,800 bits …

9. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Busy Beaver function Non-computable Grows faster than any computable function Various mathematical bounds known Seems hopeless for n ≥ 7 Values for n = 5 seem settled 3, 4, 5, 6 symbol versions are popular

10. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Contribution of this work Use execution history for non-termination conjectures Evaluate conjectures on a “hypothetical” engine Automate the search as much as possible

11. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Example 11{C}1 → 11{C}111 → 11{C}11111 … Conjecture is 11{C} 1 (11) N → 11{C} 111(11) N Start engine in 11{C} 1 (11) N Terminate with success if we reach 11{C} 111 (11) N (or 11{C} 11 (11) N 1 or …)

12. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Killer Kangaroos 1 6 {D}0 → 1 18 {D}0 → 1 42 {D}0 (!!!) → 1 90 {D}0 1 30 {D}0 does not occur … 1 N {D}0 → 1 2N+6 {D}0 or alternatively 1 N {D}0 → (11) N 111111{D}0 Then execute on engine as before

13. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Engine Design L {S}I N R → ???? Run L {S}I R and look for “repeatable” parts L {S}I R → L O {S} R wild wombat L {S}I R → L’ O {S} R slithery snake L {S}I R → L’ O {S} R’ maniacal monkey slithery snake → resilient reptile when |I| < |O|

14. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Engine State Around 4,000 lines of Ciao Prolog Available on my web page Includes all three heuristics Some killer kangaroos still escape … Analysis does not terminate for all machines (yet!) At least one further heuristic needed

15. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Addictive Adders 1111{C}11 1011{C}111 110{C}1111 111111{C}11 101111{C}111 11011{C}1111 1110{C}11111 11111111{C}11

16. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Addictive Adders Conjecture is 1 N 1111{C}11 → 1 N 111111{C}11 “Secondary” induction of the form 1 N 0(11) K {C} 1 M → 1 N+1 0(11) K-1 {C} 1 M+1 The forthcoming observant otter heuristic will evaluate this as 1 N+K 0 {C} 1 M+K

17. CATS’07 1/2/2007James Harland Analysis of Busy Beaver machines via Induction Proofs Conclusions & Further Work Plenty of interesting questions … Decidability (?) of n = 4 case Complete analysis of n = 5 case Engine improvements 2x3, 2x4, 2x5, 3x3 cases as well Placid platypus and other aspects “mine” cases for 3,4,5 for attempt on n = 6 (aka quest for the demon duck of doom)