From 667253f48d20f76aff54f7ce9464b13c169ad96e Mon Sep 17 00:00:00 2001 From: j-james Date: Mon, 14 Nov 2022 23:55:42 -0800 Subject: Add cursed Fibonacci implementations --- fib/tm/fib-desc.txt | 219 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 219 insertions(+) create mode 100644 fib/tm/fib-desc.txt (limited to 'fib/tm/fib-desc.txt') diff --git a/fib/tm/fib-desc.txt b/fib/tm/fib-desc.txt new file mode 100644 index 0000000..b98b324 --- /dev/null +++ b/fib/tm/fib-desc.txt @@ -0,0 +1,219 @@ +The following is a low-level description of a Turing machine that will write +the Fibonacci sequence (represented in binary, separated by $) without halting. + +A Turing machine is a 7-tuple T = (Q,Σ,Γ,δ,qI,qA,qR) where: +- Q is the set of states; non-empty and finite +- Σ is the input alphabet; non-empty and finite +- Γ is the tape alphabet; non-empty and finite +- δ is the transition function: δ(Q, Γ) -> (Q, Γ, {L, R}) +- qI ∈ Q is the initial state +- qA ∈ Q is the accept state +- qR ∈ Q is the reject state + +- Q: { + ca, cb, cc, cd, ce, cf, cg, ch, ci, + sa, sb, sc, sd, + aa, ab, ac, + aaa, aab, aac, aad, aae, aaf, + aba, abb, abc, abd, abe, abf, + ba, bb, bc, + baa, bab, bac, bad, bae, baf, + bba, bbb, bbc, bbd, bbe, bbf, + za, zb, zc, zd +} +- Σ: not relevant as we entirely disregard the input to begin with. +- Γ: {_, $, X, 0, 1, 0*, 1*} (_ means the blank symbol) +- δ: described below. a note on syntax: + - no entry in the output parameter means do not write a character to the tape. + - similarly, no entry in the position parameter means do not move the tape head. + - numerous "possible" transition functions are not stated. those are thought by the author to be inaccessible in normal operation of this machine (and if they are, it is probably a bug). +- qI: the initial state is ca. +- qA: the machine does not accept. +- qR: the machine does not reject. + +Overview: +1. We clear the tape and write our initial state: the first three Fibonacci numbers +2. We write a number of Xs to the end of the tape for convenience +3. We add the last two numbers of the sequence digit-by-digit to create another number + - Effort/pain is taken to ensure carrying works: including when the next number is a digit larger +4. Upon having created the new Fibonacci number, we write a $ and repeat from step 3, ad nauseam + +states c*: clearing the initial tape +- clear the tape: write $0$1$1$, then write _ until reading a _ + +δ(ca, Γ) -> (cb, $, R) +δ(cb, Γ) -> (cc, 0, R) +δ(cc, Γ) -> (cd, $, R) +δ(cd, Γ) -> (ce, 1*, R) +δ(ce, Γ) -> (cf, $, R) +δ(cf, Γ) -> (cg, 1, R) +δ(cg, Γ) -> (ch, $, R) +δ(ch, Γ \ _) -> (ci, _, R) +δ(ch, _) -> (sa, , ) + + +states s*: making space for the next number +- from the end: append XX...X where len(XX...X) = len(previous_number) + +δ(sa, {_,X}) -> (sa, , L) +δ(sa, $) -> (sb, , L) + +δ(sb, {0*,1*}) -> (sb, , L) +δ(sb, 0) -> (sc, 0*, R) +δ(sb, 1) -> (sc, 1*, R) +δ(sb, $) -> (sd, , R) + +δ(sc, Γ \ _) -> (sc, , R) +δ(sc, _) -> (sa, X, L) + +δ(sd, {$,X}) -> (sd, , R) +δ(sd, 0*) -> (sd, 0, R) +δ(sd, 1*) -> (sd, 1, R) +δ(sd, _) -> (aa, , L) + + +states a*: add the last digit of both numbers without carrying +- read left until $, then read left until: + - 0: replace with 0*, goto state (aaa) + - 1: replace with 1*, goto state (aba) + - $: do not replace, read right until _, write $, goto state (s) + +δ(aa, {X,0,1}) -> (aa, , L) +δ(aa, $) -> (ab, , L) + +δ(ab, {0*,1*}) -> (ab, , L) +δ(ab, 0) -> (aaa, 0*, L) +δ(ab, 1) -> (aba, 1*, L) + +δ(ab, $) -> (ac, , R) +δ(ac, Γ \ _) -> (ac, , R) +δ(ac, _) -> (sa, $, R) + +states aa* +- read left until $, then read left until: + - 0*: replace with 0, read right until _, then read left until X and replace with 0 + - 1*: replace with 1, read right until _, then read left until X and replace with 1 + +δ(aaa, {0,1}) -> (aaa, , L) +δ(aaa, $) -> (aab, , L) + +δ(aab, 0*) -> (aac, 0, R) +δ(aac, Γ \ _) -> (aac, , R) +δ(aac, _) -> (aad, , L) +δ(aad, {0,1}) -> (aad, , L) +δ(aad, X) -> (aa, 0, L) + +δ(aab, 1*) -> (aae, 1, R) +δ(aae, Γ \ _) -> (aae, , R) +δ(aae, _) -> (aaf, , L) +δ(aaf, {0,1}) -> (aaf, , L) +δ(aaf, X) -> (aa, 1, L) + +δ(aab, {0,1}) -> (aab, , L) + +states ab* +- read left until $, then read left until: + - 0*: replace with 0, read right until _, then read left until X and replace with 1 + - $: do not replace, read right until _, then read left until X and replace with 1 + - 1*: replace with 1, read right until _, then read left until X and replace with 0, goto state (b) + +δ(aba, {0,1}) -> (aba, , L) +δ(aba, $) -> (abb, , L) + +δ(abb, 0*) -> (abc, 0, R) +δ(abc, Γ \ _) -> (abc, , R) +δ(abc, _) -> (abd, , L) +δ(abd, {0,1}) -> (abd, , L) +δ(abd, X) -> (aa, 1, L) + +δ(abb, 1*) -> (abe, 1, R) +δ(abe, Γ \ _) -> (abe, , R) +δ(abe, _) -> (abf, , L) +δ(abf, {0,1}) -> (abf, , L) +δ(abf, X) -> (ba, 0, L) + +δ(abb, $) -> (abc, , R) + +δ(abb, {0,1}) -> (abb, , L) + + +states b*: add the last digit of both numbers while carrying a one +- read left from end of tape, after reading $, until reading: + - 0: replace with 0*, goto state (baa) + - 1: replace with 1*, goto state (bba) + - $: do not replace, read right until _, goto state (za) + +δ(ba, {X,0,1}) -> (ba, , L) +δ(ba, $) -> (bb, , L) + +δ(bb, {0*,1*}) -> (bb, , L) +δ(bb, 0) -> (baa, 0*, L) +δ(bb, 1) -> (bba, 1*, L) + +δ(bb, $) -> (bc, , R) +δ(bc, Γ \ _) -> (bc, , R) +δ(bc, _) -> (za, , L) + +states ba* +- read left until $, then read left until: + - 0*: replace with 0, read right until _, then read left until X and replace with 1, then goto state (aa) + - 1*: replace with 1, read right until _, then read left until X and replace with 0, then goto state (ba) + +δ(baa, {0,1}) -> (baa, , L) +δ(baa, $) -> (bab, , L) + +δ(bab, 0*) -> (bac, 0, R) +δ(bac, Γ \ _) -> (bac, , R) +δ(bac, _) -> (bad, , L) +δ(bad, {0,1}) -> (bad, , L) +δ(bad, X) -> (aa, 1, L) + +δ(bab, 1*) -> (bae, 1, R) +δ(bae, Γ \ _) -> (bae, , R) +δ(bae, _) -> (baf, , L) +δ(baf, {0,1}) -> (baf, , L) +δ(baf, X) -> (ba, 0, L) + +δ(bab, {0,1}) -> (bab, , L) + +states bb* +- read left until $, then read left until: + - 0*: replace with 0, read right until _, then read left until X and replace with 0, goto state (ba) + - 1*: replace with 1, read right until _, then read left until X and replace with 1, goto state (ba) + +δ(bba, {0,1}) -> (bba, , L) +δ(bba, $) -> (bbb, , L) + +δ(bbb, 0*) -> (bbc, 0, R) +δ(bbc, Γ \ _) -> (bbc, , R) +δ(bbc, _) -> (bbd, , L) +δ(bbd, {0,1}) -> (bbd, , L) +δ(bbd, X) -> (ba, 0, L) + +δ(bbb, 1*) -> (bbe, 1, R) +δ(bbe, Γ \ _) -> (bbe, , R) +δ(bbe, _) -> (bbf, , L) +δ(bbf, {0,1}) -> (bbf, , L) +δ(bbf, X) -> (ba, 1, L) + +δ(bbb, $) -> (bbc, , R) + +δ(bbb, {0,1}) -> (bbb, , L) + +states c*: scooting over the computed number to make space for a carried digit +- read left until $, then: + - read right, noting the current number and writing the previous number + - then go to state (sa) + +δ(za, {0,1}) -> (za, , L) +δ(za, $) -> (zb, , R) + +δ(zb, 0) -> (zc, 1, R) +δ(zb, 1) -> (zb, 1, R) +δ(zb, _) -> (zd, 1, R) + +δ(zc, 0) -> (zc, 0, R) +δ(zc, 1) -> (zb, 0, R) +δ(zc, _) -> (zd, 0, R) + +δ(zd, _) -> (sa, $, R) -- cgit v1.2.3-70-g09d2