19 - 19 Decidable problems An example of decision problems Does a given polynomial p over variable x have an integral root The equivalent language

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Unformatted text preview: 19. Decidable problems An example of decision problems: Does a given polynomial p over variable x have an integral root? The equivalent language membership problem: D = { “ p ” | p is a polynomial over x with an integral root } Is a given encoding of a polynomial, “ p ” , ∈ D ? The decision problem is solvable if and only if language D is recursive. The language D is recursively renumerable because we can con- struct a TM that, given an encoding of a polynomial p over variable x , the TM evaluates p by setting x successively to 0,-1, 1, -2, 2, and so on. If at any point the polynomial evaluates to 0, the TM accepts “p”. The language D is also recursive because it is known that the roots of a single-variable polynomial are bounded within ± k c max c 1 , where k is the number of terms in the polynomial, c max is the coefficient with largest absolute value, and c 1 is the coefficient of the highest order term. Hence, if no integral root has been found within these bounds, the TM rejects “p”. 1 The following language is not recursive: D 1 = { “ p ” | p is a multivariable polynomial with an integral root } Proving that this problem is algorithmically unsolvable requires a formal definition of algorithm. In 1936, Alan Turing intro- duced TM computational model and Alonzo Church introduced the equivalent λ-calculus, which are formal definitions of algo- rithm. Decision problem ≡ Language Algorithm ≡ Turing machine that halts on all inputs An algorithm for a problem ≡ A Turing machine that decides the corresponding language Problem is solvable (decidable) ≡ Language is recursive 2 Decision problem: Given a DFA M and a string w , does M accept w ? Equivalent language membership problem: L = { “ M ”“ w ” : M is a DFA that accepts input string w } We will prove that L is recursive. Proof: We construct a Turing machine M D (in fact, a UTM) that de- cides L . The input to M D is “ M ”“ w ” where “ M ” is an encoding of a DFA. One reasonable way to encode M is simply to adapt the encoding for TM to encode DFA, i.e., encode Q, Σ , δ, s, F . Be- fore starting the simulation, M D first checks if the input string consitutes an encoding of a DFA and an encoding of an input string to a DFA....
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This note was uploaded on 09/22/2011 for the course COMP 272 taught by Professor Prof.tai during the Spring '10 term at HKUST.

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19 - 19 Decidable problems An example of decision problems Does a given polynomial p over variable x have an integral root The equivalent language

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