16.2. Bit Strings with Substring '000'

In question 2 you have to find out the recurrence relation for the number of bit strings of length n that contain the substring '000'. It might have more than 1 occurrence of '000' as well. In fact, a string of length n consisting of all 0's is a valid string. So let us try to find this by formulating a recurrence equation. So let S(n) denote the number of bit strings of length n containing the substring '000'.

In this portion, we set the stage by focusing on the task given in the question, which is to find the recurrence relation for bit strings of a specified length that include the substring '000'. To simplify our approach, we will denote this count with the function S(n). By defining S(n) clearly, we could then work towards building formulas based on the structure of the bit strings we are interested in.

Imagine you are planning to create a sequence of lights in a row where a '0' indicates that a light is off and a '1' indicates that a light is on. We want to include a specific sequence of lights—three consecutive lights turning off (000)—at least once in our row of lights. Like organizing our sequence of lights, creating the equation S(n) will help us know how many different arrangements we can have while ensuring '000' appears.

Instead of counting the number of strings containing '000', we will rather count the number of strings of length n not containing '000'. This will be relatively easier to formulate a recurrence equation for the number of bad strings. Here, the term bad strings denotes strings that do not have an occurrence of '000'. So I denote the sequence of all n length bit strings not containing '000' by T(n) and T(n) is the n-th term of such sequence.

Here we shift our focus away from directly counting the valid strings that contain '000'. Instead, we start counting the 'bad' strings that do not contain this substring. This is a strategic choice; it often proves easier to determine the characteristics of strings that are disallowed (bad strings) rather than figuring out all variations of allowed strings. By using T(n) to represent bad strings, we can derive insights connecting it back to S(n) easily.

Think of a game where you have to construct tower blocks. The rule is you need to balance the blocks perfectly without forming a section that has three blocks straight on one side (which represents '000'). To better assess your options, instead of counting how many valid towers you can make, you first list out how many configurations would fail to meet this rule (bad strings), making it easier to derive the acceptable ones.

Now it is easy to see that as per the definition of S and T, S(n) = 2^n - T(n). This is because 2^n is all possible n bit strings. 2^n denotes all possible n bit strings which have an occurrence of '000' and which do not have an occurrence of '000'. And you subtract from such strings all such strings which do not have an occurrence of '000' we will get the number of strings of length n which has an occurrence of '000'.

Now that we have defined the total possible arrangements of bit strings (2^n), we can easily express S(n) in relation to T(n). It becomes clear that S(n) represents the total count minus the count of bad strings. Therefore, the formula S(n) = 2^n - T(n) follows logically from the definitions we’ve established, linking our interest in valid strings directly to our understanding of invalid arrangements.

Let’s relate this back to our tower blocks. Imagine you have a total of 16 distinct ways (2^n) to stack blocks of various heights. However, 4 specific formations (T(n)) are illegal because they form a three-block straight structure. Therefore, to find valid formations (S(n)), you simply take the total (16) and subtract those illegal structures (4), leaving you with 12 permissible stacking configurations.

It turns out that there are 3 disjoint categories of bit strings of length n not containing '000'. Let us look into those 3 categories. Category 1 consists of those strings of length n that start with 1. If it starts with 1 then in order that the overall string does not have any occurrence of '000', the remaining portion, namely the remaining portion of length n - 1 bit, should not have any occurrence of '000'.

In this section, we start to classify the bad strings into specific categories to make counting and deriving their occurrences clearer. The first category specifies that if a string starts with '1', then the following bits (n - 1 in length) must also avoid the '000' sequence. This crucial classification allows us to develop a structured way to approach counting these bad strings in chunks.

Imagine a situation where you are organizing a seating arrangement at an event. If the first seat is occupied by an influential figure (the '1'), then the remaining seats must be allocated without placing that influential figure's disliked group (the sequence '000') immediately after them. This approach encourages a systematic way to determine valid arrangements, just like with bad strings.

Category 2 of bad strings are those that start with '01'. In this case, the remaining n - 2 bit positions should not have any occurrence of '000'. And category 3 of bad strings where the first two positions are 0, and in that case, the third position cannot be 0 because if the third position has a 0 then it is not a bad string. The definition of bad string is that it should not have any occurrence of '000'.

Continuing our enumeration of bad strings, Category 2 denotes strings that start with '01', meaning we have further restrictions on the following bits to remain valid. In Category 3, we observe the constraint where two zeros cannot be followed by another zero. Each category introduces specific conditions that must be met, reinforcing the definitions we’re working with. The careful consideration of these cases forms the backbone of our recurrence relation.

Returning to our seating arrangement analogy, say the second seat must only allow a certain type of person (the '01'). Therefore, the seats behind them must fit certain criteria to ensure harmony. Alternatively, a two-person block should not precede another member of that block. This structured seating leads to organized classifications, allowing for clear pathways to follow in exploring possibilities.

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