Theory of Computation: Question Set – 30

What is the difference between P and NP?

P is the class of decision problems that can be solved in polynomial time by a deterministic Turing machine, while NP is the class of decision problems that can be solved in polynomial time by a non-deterministic Turing machine. It is currently unknown whether P equals NP, but the question is one of the most important open problems in computer science.

What is the difference between a Turing machine and a register machine?

A Turing machine is a theoretical model of a computer that consists of a tape, a head that can read and write symbols on the tape, and a finite set of states that control the machine’s behavior. A register machine is a theoretical model of a computer that has a finite number of registers, each of which can store a finite number of integers, and a finite set of instructions that can manipulate the registers. While Turing machines can simulate register machines, register machines cannot simulate Turing machines in general.

What is the difference between a recursive function and a recursive enumerable function?

A recursive function is a function that can be computed using a Turing machine that always halts, while a recursive enumerable function is a function that can be computed using a Turing machine that may not always halt. Recursive functions are a subset of recursive enumerable functions, as any function that can be computed using a Turing machine that always halts can also be computed using a Turing machine that may not always halt.

What is the difference between a context-sensitive grammar and a context-free grammar?

A context-sensitive grammar is a formal system that generates context-sensitive languages, which are a proper superset of context-free languages. Context-sensitive grammars use rules that can modify symbols in the input string based on the context in which they appear, while context-free grammars use rules that replace nonterminal symbols with sequences of terminal and nonterminal symbols. Context-sensitive grammars are more powerful than context-free grammars, as they can generate languages that cannot be generated by context-free grammars.

What is the difference between a recursive language and a recursively enumerable language?

A recursive language is a language that can be recognized by a Turing machine that always halts with a correct output, while a recursively enumerable language is a language that can be recognized by a Turing machine that may not always halt with a correct output. Recursive languages are a subset of recursively enumerable languages, as any language that can be recognized by a Turing machine that always halts with a correct output can also be recognized by a Turing machine that may not always halt with a correct output.

What is the difference between a universal Turing machine and a standard Turing machine?

A universal Turing machine is a theoretical model of a computer that can simulate any other Turing machine on any input, while a standard Turing machine is a theoretical model of a computer that can only recognize the languages that can be recognized by a finite automaton. Universal Turing machines are more powerful than standard Turing machines, as they can simulate any algorithm that can be performed on any other Turing machine.

What is the difference between a pushdown automaton and a finite automaton?

A pushdown automaton is a theoretical model of a computer that has a finite control unit, a tape, and a stack, while a finite automaton is a theoretical model of a computer that only has a finite control unit and a tape. Pushdown automata can recognize context-free languages, which are a proper superset of regular languages, while finite automata can only recognize regular languages.

What is the difference between a one-way function and a trapdoor function?

A one-way function is a function that is easy to compute in one direction, but computationally infeasible to compute in the opposite direction, without knowledge of some secret information. A trapdoor function is a one-way function that also has a secret “trapdoor” that can be used to efficiently compute the function in the opposite direction. Trapdoor functions are used in many cryptography applications, such as public-key encryption.

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