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# 数学代考| Decidability 离散数学代写

## 数学代写| Decidability 离散代考

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## 离散数学代写

Mathematics is decidable if the truth or falisty of any mathematical proposition may be determined by an algorithm.

Church and Turing independently showed this to be impossible in 1936. Church developed the lambda calculus in the 1930 s as a tool to study computability, ${ }^{3}$ and he showed that anything that is computable is computable by the lambda calculus. Turing showed that decidability was related to the halting problem for Turing machines, and that therefore if first-order logic were decidable then the halting problem for Turing machines could be solved. However, he had already proved that there was no general algorithm to determine whether a given Turing machine halts or not. Therefore, first-order logic is undecidable.

The question as to whether a given Turing machine halts or not can be formulated as a first-order statement. If a general decision procedure exists for first-order logic, then the statement of whether a given Turing machine halts or not is within the scope of the decision algorithm. However, Turing had already proved that the halting problem for Turing machines is not computable: i.e. it is not possible algorithmically to decide whether or not any given Turing machine will halt or not. Therefore, since there is no general algorithm that can decide whether any given Turing machine halts, there is no general decision procedure for first-order logic. The only way to determine whether a statement is true or false is to try to solve it. However, if one tries but does not succeed this does not prove that an answer does not exist.

There are first-order theories that are decidable. However, first-order logic that includes Peano’s axioms of arithmetic (or any formal system that includes addition and multiplication) cannot be decided by an algorithm. That is, there is no algorithm to determine whether an arbitrary mathematical proposition is true or false. Propositional logic is decidable as there is a procedure (e.g. using a truth table) to determine whether an arbitrary formula is valid ${ }^{4}$ in the calculus.

Gödel (Fig. 13.2) proved that first-order predicate calculus is complete. i.e. all truths in the predicate calculus can be proved in the language of the calculus.
Definition 13.2 (Completeness)
A formal system is complete if all the truths in the system can be derived from the axioms and rules of inference.

Gödel’s first incompleteness theorem showed that first-order arithmetic is incomplete; i.e. there are truths in first-order arithmetic that cannot be proved in the language of the axiomatization of first-order arithmetic. Gödel’s second incompleteness theorem showed that that any formal system extending basic arithmetic cannot prove its own consistency within the formal system.
Definition $13.3$
(Consistency) A formal system is consistent if there is no formula A such that A and $\neg \mathrm{A}$ are provable in the system (i.e. there are no contradictions in the system).

## 图论代考

Mathematics is decidable if the truth or falisty of any mathematical proposition may be determined by an algorithm.

Church and Turing independently showed this to be impossible in 1936. Church developed the lambda calculus in the 1930 s as a tool to study computability, ${ }^{3}$ and he showed that anything that is computable is computable by the lambda calculus. Turing showed that decidability was related to the halting problem for Turing machines, and that therefore if first-order logic were decidable then the halting problem for Turing machines could be solved. However, he had already proved that there was no general algorithm to determine whether a given Turing machine halts or not. Therefore, first-order logic is undecidable.

The question as to whether a given Turing machine halts or not can be formulated as a first-order statement. If a general decision procedure exists for first-order logic, then the statement of whether a given Turing machine halts or not is within the scope of the decision algorithm. However, Turing had already proved that the halting problem for Turing machines is not computable: i.e. it is not possible algorithmically to decide whether or not any given Turing machine will halt or not. Therefore, since there is no general algorithm that can decide whether any given Turing machine halts, there is no general decision procedure for first-order logic. The only way to determine whether a statement is true or false is to try to solve it. However, if one tries but does not succeed this does not prove that an answer does not exist.

There are first-order theories that are decidable. However, first-order logic that includes Peano’s axioms of arithmetic (or any formal system that includes addition and multiplication) cannot be decided by an algorithm. That is, there is no algorithm to determine whether an arbitrary mathematical proposition is true or false. Propositional logic is decidable as there is a procedure (e.g. using a truth table) to determine whether an arbitrary formula is valid ${ }^{4}$ in the calculus.

Gödel (Fig. 13.2) proved that first-order predicate calculus is complete. i.e. all truths in the predicate calculus can be proved in the language of the calculus.
Definition 13.2 (Completeness)
A formal system is complete if all the truths in the system can be derived from the axioms and rules of inference.

Gödel’s first incompleteness theorem showed that first-order arithmetic is incomplete; i.e. there are truths in first-order arithmetic that cannot be proved in the language of the axiomatization of first-order arithmetic. Gödel’s second incompleteness theorem showed that that any formal system extending basic arithmetic cannot prove its own consistency within the formal system.
Definition $13.3$
(Consistency) A formal system is consistent if there is no formula A such that A and $\neg \mathrm{A}$ are provable in the system (i.e. there are no contradictions in the system).

## 密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码