chinmay bhagat
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Introduction to Theory of Computation
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Abstract
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The Theory of Computation course aims to explore the mathematical foundations of computation, addressing key questions about algorithms, computability, and the limitations of computing systems. It covers three main areas: Complexity Theory, Computability Theory, and Automata Theory, providing students with a formal understanding of computational systems and their capabilities. Through theoretical analyses and practical exercises, students engage with mathematical models that reflect real-world computing scenarios.
References (22)
- w : 110 is not a substring of w}
- w : w contains at least five 1s}
- w : w contains the substring 1011}
- w : w contains at least two 1s and at most two 0s}
- w : w contains an odd number of 1s or exactly two 0s}
- w : w begins with 1 and ends with 0}
- w : every odd position in w is 1}
- w : w has length at least 3 and its third symbol is 0} 10. {ǫ, 0}
- 2 For each of the following languages, construct an NFA, with the specified number of states, that accepts the language. In all cases, the alphabet is {0, 1}.
- The language {w : w ends with 10} with three states.
- The language {w : w contains the substring 1011} with five states.
- The language {w : w contains an odd number of 1s or exactly two 0s} with six states.
- 3 For each of the following languages, construct an NFA that accepts the language. In all cases, the alphabet is {0, 1}. 1. {w : w contains the substring 11001}
- w : w has length at least 2 and does not end with 10} 3. {w : w begins with 1 or ends with 0}
- 4 Convert the following NFA to an equivalent DFA. 1. 2 ≤ a < x, 2. 2 ≤ b < x, and 3. x = ab.
- Clearly, these three conditions can be verified in polynomial time. Moreover, the length of this solution, i.e., the total number of bits in the binary rep- resentations of a and b, is polynomial in the number of bits in the binary representation of x. Languages having the property that the correctness of a proposed "solu- tion" can be verified in polynomial time, form the class NP: Definition 6.3.6 A language A belongs to the class NP, if there exist a polynomial p and a language B ∈ P, such that for every string w, w ∈ A ⇐⇒ ∃s : |s| ≤ p(|w|) and w, s ∈ B. In words, a language A is in the class NP, if for every string w, w ∈ A if and only if the following two conditions are satisfied: 1. There is a "solution" s, whose length |s| is polynomial in the length of w (i.e., |s| ≤ p(|w|), where p is a polynomial).
- In polynomial time, we can verify whether or not s is a correct "solu- tion" for w (i.e., w, s ∈ B and B ∈ P).
- Hence, the language B can be regarded to be the "verification language": B = { w, s : s is a correct "solution" for w}. We have given already informal proofs of the fact that the languages kColor, HC , SOS , and NPrim are all contained in the class NP. Below, we formally prove that NPrim ∈ NP. To prove this claim, we have to specify the polynomial p and the language B ∈ P. First, we observe that NPrim = { x : there exist a and b in N such that 2 ≤ a < x, 2 ≤ b < x and x = ab }. p(n) := n + 2, and the language B as B := { x, a, b : x ≥ 2, 2 ≤ a < x, 2 ≤ b < x and x = ab}.
- A ≤ P A, and 2. if A ≤ P B and B ≤ P C, then A ≤ P C. The following properties hold: 1. Every context-free language is in P. (We did not prove this).
- Every language in P is also in NP.
- It is not known if there exist languages that are in NP, but not in P. 4. Every language in NP is decidable.
- We have introduced reductions to define the notion of a language B to be "at least as hard" as a language A. A language B is called NP-complete, if 1. B belongs to the class NP, and 2. B is at least as hard as every language in the class NP. We have seen that NP-complete exist. The figure below summarizes the relationships among the various classes of languages.