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It seems impossible for me to find a pattern for integers $n$ such that $$n \equiv 1 \quad(\bmod 7)$$ in order to construct this DFA.

Is there smart way to do it without a machine which has memory or completes operations? A hint would be greatly appreciated.

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    $\begingroup$ How are your integers $n$ encoded? Try to be as explicit as possible, since the answer would heavily depend on the details of the encoding. $\endgroup$ – Yuval Filmus Oct 25 '18 at 20:52
  • $\begingroup$ A DFA has memory. It's called the "state". You need 7 states, just below 3 bits. $\endgroup$ – gnasher729 Oct 27 '18 at 10:12
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I assume you are talking about integers encoded in some base $b \geq 2$, and we're reading the integer from left to right.

An inductive hint: say we have read the number $n$ so far, and our inductive assumption is that we are in state $n \bmod 7$ (so we have 7 states). We read the digit $d$. Our next state should be $bn + d \bmod 7$. How do we correctly determine what state to go to, since we don't know $n$?

Well the trick is that while you don't know $n$, you do know $n \bmod 7$. And:

$$bn + d \equiv b(n \bmod 7) + d\mod 7$$

So for example, in base $10$, if we are in the state $3$ (meaning $n \equiv 3 \mod 7$) and we just read the digit $8$, our next state will be $10\cdot 3 + 8 \equiv 3 \mod 7$. In other words, state $3$ points to itself on reading digit $8$ in base $10$.

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Place 7 nodes. Number them from 0 to 6. In Any time, in whatever node we are, represents the current number module 7. So starting node is 0. And note 0 is the accepting node.

For each node, there are 7 outgoing edges (from 0 to 6.) Say we are in node i; outgoing edge j from i, connects the node i to node (i*10+j) mod 7.

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