derivation-mko-finite-automaton-01 · ocme-new142-reader-adoption-20261021
The specified DFA rejects 1011 and accepts epsilon.
A full deterministic total Q-by-alphabet table and finite-word traces, not language equivalence from examples.
calcG1 — Q={E,O}, alphabet={0,1}, start=E, finals={E}; delta(E,0)=E,delta(E,1)=O,delta(O,0)=O,delta(O,1)=E; word=1011.\delta^* — iterated transition on a wordi — number of consumed symbolsSome rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.
calc · prerequisite — Assume finite sets, finite strings and indices, stepwise function updates and induction on natural numbers; epsilon denotes the length-zero word.mko_definition · cited — Completely check that each Q×Sigma input has exactly one valid output, then trace each complete finite word in order. The empty word is accepted exactly when the initial state is accepting. One finite trace does not establish language equivalence or regularity of an unspecified target language. · mko-finite-automatonBefore
Q=\{E,O\},\quad\Sigma=\{0,1\}After
\begin{array}{c|cc}\delta&0&1\\\hline E&E&O\\O&O&E\end{array}Conditions: none
Rule: calc · Uses: G1
Leaf basis: calc
Each of the four state-symbol pairs has exactly one output in Q, completely checking totality and determinism.
Before
w=1011,\quad q_0=E
After
(q,i)=(E,0)
Conditions: none
Rule: calc · Uses: G1, S01
Leaf basis: calc
Initialize at the unique start; position zero means no symbol has been read.
Before
(E,0),\quad w_1=1
After
(O,1)
Conditions: none
Rule: calc · Uses: G1, S01, S02
Leaf basis: calc
Reading the first 1 uses delta(E,1)=O and advances position by one.
Before
(O,1),\quad w_2=0
After
(O,2)
Conditions: none
Rule: calc · Uses: G1, S01, S03
Leaf basis: calc
The second symbol 0 keeps O unchanged while still consuming that symbol.
Before
(O,2),\quad w_3=1
After
(E,3)
Conditions: none
Rule: calc · Uses: G1, S01, S04
Leaf basis: calc
The third symbol 1 returns to E, with one final 1 still unread.
Before
(E,3),\quad w_4=1
After
(O,4),\quad4=|w|
Conditions: none
Rule: calc · Uses: G1, S01, S05
Leaf basis: calc
The final 1 sends E to O, consuming all four symbols in order.
Before
\delta^*(E,1011)=O,\quad F=\{E\}After
O\notin F\Rightarrow1011\text{ rejected}Conditions: none
Rule: mko_definition · Uses: G1, S06
Leaf basis: mko_definition
The unique complete run ends outside the final set, rejecting this word.
Before
w=\varepsilon,\quad|w|=0
After
\delta^*(E,\varepsilon)=E\in F\Rightarrow\varepsilon\text{ accepted}Conditions: none
Rule: calc · Uses: G1, S01, S07
Leaf basis: calc
The empty word makes no transition, so the start is final and epsilon is accepted.
No substeps in this guide.
S08 · premise_audit · G1, S01, S07 — No epsilon edge is added; an empty word differs from an epsilon transition.S07 · premise_audit · G1, S06 — This certifies the specified word, not equivalence with an arbitrary target language.S06 · premise_audit · G1, S01, S05 — One cannot stop at E in S05 and discard the last symbol.S05 · premise_audit · G1, S01, S04 — Reaching a final state midway does not yet accept the whole word.S04 · premise_audit · G1, S01, S03 — An unchanged state is not an epsilon move; the position advances from 1 to 2.S03 · premise_audit · G1, S01, S02 — There is only one successor; remaining at E is not a choice.S02 · premise_audit · G1, S01 — An accepting start does not allow ignoring the remaining nonempty input.S01 · premise_audit · G1 — There is no epsilon transition or multivalued cell.All four state-symbol cells need one legal output.
Check acceptance only after position equals word length.
reject — The final 1 remains; the full terminal state is O.reject — Every DFA input transition consumes one symbol and advances position.reject — The empty word takes no edge and start E is final.The specified DFA rejects 1011 and accepts epsilon.
0 declared checks for this guide; only the listed expressions are covered.
b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9fc631c60e2976f05f20ff82cd04318bc5c870bf60c51efcfb4bf34202ec0524b11b4e2cc34b16b6214d69dfde3f776b2089abe1dec7d7ab3534bf2cf53e8d3aaeSCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEWSame-source full JSON (formulas, ASTs, controls and revision log)