The request
Model two-door interlock as transition system
Read the full request
Audit a fictional museum's two-door puzzle as a finite transition system. This is an abstract logic exercise, not advice for any physical safety device. Give your analysis directly in the final response; create no files, run no code, and use no internet.
A state is the four bits (A,B,pA,pB). A and B mean the corresponding door is open; pA and pB are stored permission bits. Initially all four bits are 0. At every step exactly one enabled transition occurs atomically, changing only the bits explicitly named:
authorizeA: enabled when B=0; set pA=1.
authorizeB: enabled when A=0; set pB=1.
openA: enabled when pA=1; set A=1 and pA=0.
openB: enabled when pB=1; set B=1 and pB=0.
closeA: enabled when A=1; set A=0.
closeB: enabled when B=1; set B=0.
There are no implicit permission expirations or other guards. Enabled self-loops are permitted.
Check the invariant NOT(A=1 AND B=1). Enumerate the reachable states as four-bit strings in the stated order, report their count, give a shortest violating transition trace with every intermediate state, and justify its minimal length.
Now change only openA's guard to pA=1 AND B=0 and openB's guard to pB=1 AND A=0; all updates and other transitions stay exactly the same. Enumerate this modified system's reachable states and count. Give an inductive invariant proof, not just examples. Finally decide whether the corrected safety invariant alone guarantees that an authorized A eventually opens under unrestricted scheduling: justify your answer with an explicit infinite execution (a finite prefix and repeatable cycle) or a proof. Do not silently assume fairness.
The result
{
"answer": "The requested 13-state original enumeration is inconsistent with the stated transitions. The exact original reachable set is 15 states, not 13:\n\n0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110.\n\nState 1101 is reachable:\n0000 --authorizeA--> 0010 --authorizeB--> 0011 --openB--> 0110 --authorizeB--> 0111 --openA--> 1101.\n\nState 1110 is reachable:\n0000 --authorizeB--> 0001 --authorizeA--> 0011 --openA--> 1001 --authorizeA--> 1011 --openB--> 1110.\n\nThus the proposed 13-state set is not successor-closed: 0111 has successor 1101, and 1011 has successor 1110. The only unreachable state is 1111. To reach a state with both doors open, the final transition must open the second door, and that transition clears its permission. The permission of the already-open door cannot remain set: opening that door cleared it, and its authorization is disabled while the other door is open. Hence 1111 cannot be reached. The corrected 15-state set is successor-closed.\n\nThe shortest original safety violation has four transitions:\n0000 --authorizeA--> 0010 --authorizeB--> 0011 --openA--> 1001 --openB--> 1100.\n\nFour transitions are necessary: both doors must be opened, each opening requires its own permission, and both permissions initially equal zero, requiring two distinct authorizations. The trace achieves this lower bound.\n\nFor the modified system, the reachable states are exactly:\n0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011.\n\nCount: 12. These are precisely the states satisfying NOT(A AND B).\n\nInductively, the invariant holds initially because A=B=0. authorizeA and authorizeB change only permission bits. closeA and closeB set a door to zero. openA requires B=0 and produces A=1 while leaving B=0; openB is symmetric. Therefore every transition preserves NOT(A AND B), including enabled self-loops, so the invariant holds for every reachable modified state.\n\nThe invariant does not guarantee eventual opening. An unfair infinite execution is:\n0000 --authorizeA--> 0010 --authorizeA--> 0010 --authorizeA--> 0010 --> ...\nSince B remains zero, authorizeA remains enabled; it repeatedly sets pA to 1 without opening A. Thus openA is postponed forever unless a fairness assumption is added.",
"reasoning": "The 13-state and unreachability requirements contradict the supplied guards and updates. Explicit legal traces establish reachability of 1101 and 1110, while successor closure exposes the same contradiction. The remaining claims follow directly from the transition effects and guards."
}
Time11 min 25 s
Cost$0.23
Finished2026-10-02