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The goal is to see if there is a copy of this string starting at a position that is a multiple of. The machine starts in state and tries to find the correct string as input.

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If it does, then it reaches the accepting state. If while doing this it gets a wrong input, then it switches to states that have stopped looking for the input. After seeing three inputs the machine reaches and then moves back to the start state.

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Lemma 2 Suppose and. Then all but primes satisfy. Proof: Consider the quantity for not equal to. Call a prime bad if it divides this quantity. This quantity can be divisible by at most primes. So there are at most bad primes in total. We need some definitions about strings. Let be the length of the string.

Also let be the number of occurrences of in. A string has the period provided. A string is periodic provided it has a period that is less than half its length. Lemma 3 For any string either or is not periodic. Proof: Suppose that is periodic with period where is a single character. Let the length of equal. So by definition,. So it follows that. This shows that and cannot both be periodic, since. Lemma 4 Suppose that is not a periodic string.

Then the number of copies of in a string is upper bounded by where. Proof: The claim follows once we prove that no two copies of in can overlap more than where is the length of. This will immediately imply the lemma. If has two copies in that overlap then clearly.

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  8. This says that has the period. Since is not periodic it follows that. This implies that the overlap of the two copies of are at most length. Thus we have shown that they cannot overlap too much. Say an automaton finds the occurrence of in provided it enters a special state after scanning the last bit of this occurrence.

    Lemma 5 Let be a string of length and let be a non-periodic string. Then, there is an automaton with at most states that can find the occurrence of in where. Here allows factors that are fixed powers of. This lemma is the main insight of Robson and will be proved later. I am still confused a bit about his stronger theorem, to be honest. Proof: Since and are distinct we can assume that starts with the prefix and starts with the prefix for some string.

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    If the length of is less than order the theorem is trivial. Just construct an automaton that accepts and rejects. So we can assume that for some strings and where the latter is order in length. By lemma we can assume that is not periodic. So by lemma we get that. Then by lemma we are done. Proof: Let have length and let be a non-periodic string in of length. Also let. By the overlap lemma it follows that is bounded by.

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    Let occur at locations. Suppose that we are to construct a machine that finds the copy of. By the hashing lemma there is a prime so that. Note we can also assume that. If it is congruent to another value the same argument can be used. This follows by having the machine initially skip a fixed amount of the input and then do the same as in the congruent to case. The automaton has states and for. The machine starts in state and tries to get to the accepting state.

    The transitions include:. This means that the machine keeps checking the input to see if it is scanning a copy of. If it gets all the way to the accepting state , then it stops.

    Indian scientist offers proof for P=NP riddle

    The second group means that if a wrong input happens, then moves to. Finally, the state resets and starts the search again by going to the start state with an epsilon move. Clearly this has the required number of states and it operates correctly. The open problem is: Can the SWP be solved with a better bound? The lower bound is still order.

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    So the gap is exponential. Another exponential gap in complexity theory? Jeffrey Shallit is a famous researcher into many things, including number theory and being a skeptic. He has a colorful website with an extensive quotation page —one of my favorites by Howard Aiken is right at the top:.

    See also his talk. They say in their introduction:. Imagine a computing device with very limited powers. What is the simplest computational problem you could ask it to solve? It is not the addition of two numbers, nor sorting, nor string matching—it is telling two inputs apart: distinguishing them in some way. Let and be two distinct long strings over the usual binary alphabet. What is the size of the smallest deterministic automaton that can accept one of the strings and reject the other?

    That is, how hard is it for a simple type of machine to tell apart from? Suppose that and are distinct binary words of length. Define to be the number of states of the smallest automaton that accepts and rejects or vice-versa.