Download presentation

Presentation is loading. Please wait.

Published byShirley Wolffe Modified over 3 years ago

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)

Similar presentations

OK

1 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt Wants.

1 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt 10 pt 15 pt 20 pt 25 pt 5 pt Wants.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Free ppt on uses of water Ppt on areas related to circles download Plain backgrounds for ppt on social media Ppt on bluetooth broadcasting hardware Ppt on obesity prevention Ppt on malignant bone tumors Ppt on non destructive testing Doc convert to ppt online viewer Ppt on satellite orbital speed Ppt on time management presentation