# 哥德尔机

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A **Gödel machine** is a hypothetical self-improving computer program that solves problems in an optimal way.模板:Clarify It uses a recursive self-improvement protocol in which it rewrites its own code when it can prove the new code provides a better strategy.^{[1]}^{[2]} The machine was invented by Jürgen Schmidhuber (first proposed in 2003^{[3]}), but is named after Kurt Gödel who inspired the mathematical theories.^{[4]}

A Gödel machine is a hypothetical self-improving computer program that solves problems in an optimal way. It uses a recursive self-improvement protocol in which it rewrites its own code when it can prove the new code provides a better strategy. The machine was invented by Jürgen Schmidhuber (first proposed in 2003), but is named after Kurt Gödel who inspired the mathematical theories.

哥德尔机是一种假设的自我改进的计算机程序，它以最佳的方式解决问题。它使用一个递归的自我改进协议，当它能够证明新代码提供了一个更好的策略时，它重写自己的代码。这种机器是由尔根·施密德胡伯 Jürgen Schmidhuber 发明的(2003年首次提出) ，但是它是以库尔特 · 哥德尔 Kurt Gödel的名字命名的，他启发了数学理论。

The Gödel machine is often discussed when dealing with issues of meta-learning, also known as "learning to learn." Applications include automating human design decisions and transfer of knowledge between multiple related tasks, and may lead to design of more robust and general learning architectures.^{[5]} Though theoretically possible, no full implementation has been created.^{[6]}

The Gödel machine is often discussed when dealing with issues of meta-learning, also known as "learning to learn." Applications include automating human design decisions and transfer of knowledge between multiple related tasks, and may lead to design of more robust and general learning architectures. Though theoretically possible, no full implementation has been created.

在处理元学习(也称为“学会学习”)问题时，经常讨论哥德尔机器应用程序包括自动化人类设计决策和多个相关任务之间的知识转移，并可能导致设计更健壮和通用的学习架构。虽然在理论上是可能的，但是还没有完全实现。

The Gödel machine is often compared with Marcus Hutter's AIXItl, another formal specification for an artificial general intelligence. Schmidhuber points out that the Gödel machine could start out by implementing AIXItl as its initial sub-program, and self-modify after it finds proof that another algorithm for its search code will be better.^{[7]}

The Gödel machine is often compared with Marcus Hutter's AIXItl, another formal specification for an artificial general intelligence. Schmidhuber points out that the Gödel machine could start out by implementing AIXItl as its initial sub-program, and self-modify after it finds proof that another algorithm for its search code will be better.

人们经常将哥德尔机器与 Marcus Hutter 的 AIXItl 相提并论，AIXItl 是人工智能的另一个形式规范。斯米德胡贝尔指出，哥德尔机器可以从实现 AIXItl 作为其最初的子程序开始，并在找到证据证明其搜索代码的另一个算法会更好之后进行自我修改。

## Limitations

## Limitations

# = 限制 =

Traditional problems solved by a computer only require one input and provide some output. Computers of this sort had their initial algorithm hardwired.^{[8]} This doesn't take into account the dynamic natural environment, and thus was a goal for the Gödel machine to overcome.

Traditional problems solved by a computer only require one input and provide some output. Computers of this sort had their initial algorithm hardwired. This doesn't take into account the dynamic natural environment, and thus was a goal for the Gödel machine to overcome.

计算机解决的传统问题只需要一个输入，并提供一些输出。这类计算机最初的算法是硬接线的。这没有考虑到动态的自然环境，因此是哥德尔机器要克服的一个目标。

The Gödel machine has limitations of its own, however. According to Gödel's First Incompleteness Theorem, any formal system that encompasses arithmetic is either flawed or allows for statements that cannot be proved in the system. Hence even a Gödel machine with unlimited computational resources must ignore those self-improvements whose effectiveness it cannot prove.^{[3]}

The Gödel machine has limitations of its own, however. According to Gödel's First Incompleteness Theorem, any formal system that encompasses arithmetic is either flawed or allows for statements that cannot be proved in the system. Hence even a Gödel machine with unlimited computational resources must ignore those self-improvements whose effectiveness it cannot prove.

然而，哥德尔机器有其自身的局限性。根据哥德尔的第一不完备性定理，任何包含算术的形式系统要么有缺陷，要么允许在系统中无法证明的陈述。因此，即使是具有无限计算资源的哥德尔机器也必须忽略那些它无法证明其有效性的自我改进。

## Variables of interest

## Variables of interest

# = 感兴趣的变量 =

There are three variables that are particularly useful in the run time of the Gödel machine.^{[3]}

There are three variables that are particularly useful in the run time of the Gödel machine.

有三个变量在哥德尔机器的运行时特别有用。

- At some time [math]\displaystyle{ t }[/math], the variable [math]\displaystyle{ \text{time} }[/math] will have the binary equivalent of [math]\displaystyle{ t }[/math]. This is incremented steadily throughout the run time of the machine.

- At some time t, the variable \text{time} will have the binary equivalent of t. This is incremented steadily throughout the run time of the machine.

- 在某个时刻
*t*，变量**time**将具有与*t*等价的二进制值。在整个机器的运行时间中，这个值逐步递增。

- Any input meant for the Gödel machine from the natural environment is stored in variable [math]\displaystyle{ x }[/math]. It is likely the case that [math]\displaystyle{ x }[/math] will hold different values for different values of variable [math]\displaystyle{ \text{time} }[/math].

- Any input meant for the Gödel machine from the natural environment is stored in variable x. It is likely the case that x will hold different values for different values of variable \text{time}.

- 自然环境中用于哥德尔机器的任何输入都存储在变量
*x*中。可能的情况是，对于变量**time**的不同值，*x*将持有不同的值。

- The outputs of the Gödel machine are stored in variable [math]\displaystyle{ y }[/math], where [math]\displaystyle{ y(t) }[/math] would be the output bit-string at some time [math]\displaystyle{ t }[/math].

- The outputs of the Gödel machine are stored in variable y, where y(t) would be the output bit-string at some time t.

- 哥德尔机的输出存储在变量 y 中，其中 y (t)是某个时刻的输出位串 t。

At any given time [math]\displaystyle{ t }[/math], where [math]\displaystyle{ (1 \leq t \leq T) }[/math], the goal is to maximize future success or utility. A typical *utility function* follows the pattern [math]\displaystyle{ u(s, \mathrm{Env}) : S \times E \rightarrow \mathbb{R} }[/math]:

At any given time t, where (1 \leq t \leq T), the goal is to maximize future success or utility. A typical utility function follows the pattern u(s, \mathrm{Env}) : S \times E \rightarrow \mathbb{R}:

在任何给定的时间 t，其中(1 leq t leq t) ，目标是最大化未来的成功或效用。典型的效用函数遵循 u (s，mathrm { Env }) : s 乘以 e，right tarrow mathbb { r } :

- [math]\displaystyle{ u(s, \mathrm{Env}) = E_\mu \Bigg[ \sum_{\tau=\text{time}}^T r(\tau) \mid s, \mathrm{Env} \Bigg] }[/math]

- u(s, \mathrm{Env}) = E_\mu \Bigg[ \sum_{\tau=\text{time}}^T r(\tau) \mid s, \mathrm{Env} \Bigg]

- u (s，mathrm { Env }) = e _ mu Bigg [ sum { tau = text { time } ^ t r (tau) mid s，mathrm { Env } Bigg ]

where [math]\displaystyle{ r(t) }[/math] is a real-valued reward input (encoded within [math]\displaystyle{ s(t) }[/math]) at time [math]\displaystyle{ t }[/math], [math]\displaystyle{ E_\mu [ \cdot \mid \cdot ] }[/math] denotes the
conditional expectation operator with respect to some possibly unknown distribution [math]\displaystyle{ \mu }[/math] from a
set [math]\displaystyle{ M }[/math] of possible distributions ([math]\displaystyle{ M }[/math] reflects whatever is known about the possibly probabilistic reactions of the environment), and the above-mentioned [math]\displaystyle{ \text{time} = \operatorname{time}(s) }[/math] is a function of state [math]\displaystyle{ s }[/math] which uniquely identifies the current cycle.^{[3]} Note that we take into account the possibility of extending the expected lifespan through appropriate actions.^{[3]}

where r(t) is a real-valued reward input (encoded within s(t)) at time t, E_\mu [ \cdot \mid \cdot ] denotes the conditional expectation operator with respect to some possibly unknown distribution \mu from a set M of possible distributions (M reflects whatever is known about the possibly probabilistic reactions of the environment), and the above-mentioned \text{time} = \operatorname{time}(s) is a function of state s which uniquely identifies the current cycle. Note that we take into account the possibility of extending the expected lifespan through appropriate actions.

其中 r (t)是时间 t 的实值奖赏输入(编码在 s (t)内) ，e _ mu [ cdot mid cdot ]表示条件期望算子对于一组可能的分布集合 m 中某些可能未知的分布 μ (m 反映了关于环境可能的概率反应的任何已知) ，上述文本{ time } = 操作者{ time }(s)是唯一标识当前周期的状态函数。请注意，我们考虑到了通过适当的行动延长预期寿命的可能性。

## Instructions used by proof techniques

## Instructions used by proof techniques

# = 证明技术使用的指令 =

The nature of the six proof-modifying instructions below makes it impossible
to insert an incorrect theorem into proof, thus trivializing proof verification.^{[3]}

The nature of the six proof-modifying instructions below makes it impossible to insert an incorrect theorem into proof, thus trivializing proof verification.

下面的六个证明修改指令的性质使得不可能在证明中插入一个错误的定理，从而使证明验证变得无足轻重。

### get-axiom(*n*)

### get-axiom(n)

# = = get-axiom (n) =

Appends the *n*-th axiom as a theorem to the current theorem sequence. Below is the initial axiom scheme:

Appends the n-th axiom as a theorem to the current theorem sequence. Below is the initial axiom scheme:

将第 n 公理作为一个定理附加到当前定理序列。下面是最初的公理方案:

**Hardware Axioms**formally specify how components of the machine could change from one cycle to the next.**Reward Axioms**define the computational cost of hardware instruction and the physical cost of output actions. Related Axioms also define the lifetime of the Gödel machine as scalar quantities representing all rewards/costs.**Environment Axioms**restrict the way new inputs*x*are produced from the environment, based on previous sequences of inputs*y*.**Uncertainty Axioms/String Manipulation Axioms**are standard axioms for arithmetic, calculus, probability theory, and string manipulation that allow for the construction of proofs related to future variable values within the Gödel machine.**Initial State Axioms**contain information about how to reconstruct parts or all of the initial state.**Utility Axioms**describe the overall goal in the form of utility function*u*.

- Hardware Axioms formally specify how components of the machine could change from one cycle to the next.
- Reward Axioms define the computational cost of hardware instruction and the physical cost of output actions. Related Axioms also define the lifetime of the Gödel machine as scalar quantities representing all rewards/costs.
- Environment Axioms restrict the way new inputs x are produced from the environment, based on previous sequences of inputs y.
- Uncertainty Axioms/String Manipulation Axioms are standard axioms for arithmetic, calculus, probability theory, and string manipulation that allow for the construction of proofs related to future variable values within the Gödel machine.
- Initial State Axioms contain information about how to reconstruct parts or all of the initial state.
- Utility Axioms describe the overall goal in the form of utility function u.

- 硬件公理正式指定机器的组件如何从一个周期改变到下一个周期。
- 奖励公理定义硬件指令的计算成本和输出动作的实际成本。相关公理还将哥德尔机的寿命定义为表示所有报酬/成本的标量。
- 环境公理限制了基于先前输入序列 y 从环境中产生新输入 x 的方式。
- 不确定性公理/字符串操作公理是算术、微积分、概率论和字符串操作的标准公理，允许在哥德尔机器中构造与未来变量值相关的证明。
- 初始状态公理包含如何重构部分或全部初始状态的信息。
- 效用公理以效用函数 u 的形式描述总体目标。

### apply-rule(*k*, *m*, *n*)

### apply-rule(k, m, n)

# = = apply-rule (k，m，n) =

Takes in the index *k* of an inference rule (such as Modus tollens, Modus ponens), and attempts to apply it to the two previously proved theorems *m* and *n*. The resulting theorem is then added to the proof.

Takes in the index k of an inference rule (such as Modus tollens, Modus ponens), and attempts to apply it to the two previously proved theorems m and n. The resulting theorem is then added to the proof.

引入推理规则的索引 k (如否定式、否定式) ，并试图将其应用于前面已证明的两个定理 m 和 n。然后将得到的定理加到证明中。

### delete-theorem(*m*)

### delete-theorem(m)

# = delete-theorem (m) =

Deletes the theorem stored at index *m* in the current proof. This helps to mitigate storage constraints caused by redundant and unnecessary theorems. Deleted theorems can no longer be referenced by the above * apply-rule* function.

Deletes the theorem stored at index m in the current proof. This helps to mitigate storage constraints caused by redundant and unnecessary theorems. Deleted theorems can no longer be referenced by the above apply-rule function.

删除当前证明中存储在索引 m 处的定理。这有助于减轻由冗余和不必要的定理引起的存储约束。上述应用规则函数不能再引用删除定理。

### set-switchprog(*m*, *n*)

### set-switchprog(m, n)

# = set-switchprog (m，n) =

Replaces *switchprog* *S ^{ p}_{m:n}*, provided it is a non-empty substring of

*S*.

^{ p}Replaces switchprog S pm:n, provided it is a non-empty substring of S p.

替换 switchprog s pm: n，前提是它是 s p 的非空子字符串。

### check()

### check()

# = 谢谢观赏 =

Verifies whether the goal of the proof search has been reached. A target theorem states that given the current axiomatized utility function *u* (Item 1f), the utility of a switch from *p* to the current switchprog would be higher than the utility of continuing the execution of *p* (which would keep searching for alternative switchprogs).^{[3]} This is demonstrated in the following description of the decoded check() function for the Gödel Machine:

Verifies whether the goal of the proof search has been reached. A target theorem states that given the current axiomatized utility function u (Item 1f), the utility of a switch from p to the current switchprog would be higher than the utility of continuing the execution of p (which would keep searching for alternative switchprogs). This is demonstrated in the following description of the decoded check() function for the Gödel Machine:

验证证明搜索的目的是否已经达到。一个目标定理指出，给定当前公理化效用函数 u (条目1f) ，从 p 切换到当前切换的效用将高于继续执行 p 的效用(继续搜索替代切换项)。下面对 Gödel Machine 的 decoded check ()函数的描述说明了这一点:

- [math]\displaystyle{ D_{KA} = \frac{ d_\text{pore}} 3 u = \frac{d_\text{pore}} 3 \sqrt{\frac{8\kappa NT}{\pi M_A}} }[/math]

- D_{KA} = \frac{ d_\text{pore}} 3 u = \frac{d_\text{pore}} 3 \sqrt{\frac{8\kappa NT}{\pi M_A}}

3 u = frac { d _ text { pore }3 sqrt { kappa NT }{ pi m _ a }

### state2theorem(*m*, *n*)

### state2theorem(m, n)

# = = 状态2theorem (m，n) =

Takes in two arguments, *m* and *n*, and attempts to convert the contents of *S _{m:n}* into a theorem.

Takes in two arguments, m and n, and attempts to convert the contents of Sm:n into a theorem.

接受两个参数，m 和 n，并试图将 Sm: n 的内容转换成一个定理。

## Example applications

## Example applications

# = 应用实例 =

### Time-limited NP-hard optimization

### Time-limited NP-hard optimization

# = = 时间有限的 np 难优化 = =

The initial input to the Gödel machine is the representation of a connected graph with a large number of nodes linked by edges of various lengths. Within given time *T* it should find a cyclic path connecting all nodes. The only real-valued reward will occur at time *T*. It equals 1 divided by the length of the best path found so far (0 if none was found). There are no other inputs. The by-product of maximizing expected reward is to find the shortest path findable within the limited time, given the initial bias.^{[3]}

The initial input to the Gödel machine is the representation of a connected graph with a large number of nodes linked by edges of various lengths. Within given time T it should find a cyclic path connecting all nodes. The only real-valued reward will occur at time T. It equals 1 divided by the length of the best path found so far (0 if none was found). There are no other inputs. The by-product of maximizing expected reward is to find the shortest path findable within the limited time, given the initial bias.

哥德尔机器的初始输入是一个连通图的表示，其中包含大量由不同长度的边连接起来的节点。在给定的时间 t 内，它应该找到一个连接所有节点的循环路径。唯一真正有价值的奖励将发生在 t 时刻。它等于1除以迄今为止找到的最佳路径的长度(如果没有找到，则为0)。没有其他的输入。最大化期望报酬的副产品是在给定初始偏差的情况下，在有限时间内找到最短路径。

### Fast theorem proving

### Fast theorem proving

# = = 快速定理证明 = =

Prove or disprove as quickly as possible that all even integer > 2 are the sum of two primes (Goldbach’s conjecture). The reward is 1/*t*, where *t* is the time required to produce and verify the first such proof.^{[9]}

Prove or disprove as quickly as possible that all even integer > 2 are the sum of two primes (Goldbach’s conjecture). The reward is 1/t, where t is the time required to produce and verify the first such proof.

尽快证明或证伪所有大于2的偶数都是两个素数之和(哥德巴赫猜想)。奖励是1/t，其中 t 是出示和验证第一份证据所需的时间。

### Maximizing expected reward with bounded resources

### Maximizing expected reward with bounded resources

# = = 用有界资源最大化期望报酬 = =

A cognitive robot that needs at least 1 liter of gasoline per hour interacts with a partially unknown environment, trying to find hidden, limited gasoline depots to occasionally refuel its tank. It is rewarded in proportion to its lifetime, and dies after at most 100 years or as soon as its tank is empty or it falls off a cliff, and so on. The probabilistic environmental reactions are initially unknown but assumed to be sampled from the axiomatized Speed Prior, according to which hard-to-compute environmental reactions are unlikely. This permits a computable strategy for making near-optimal predictions. One by-product of maximizing expected reward is to maximize expected lifetime.^{[3]}

A cognitive robot that needs at least 1 liter of gasoline per hour interacts with a partially unknown environment, trying to find hidden, limited gasoline depots to occasionally refuel its tank. It is rewarded in proportion to its lifetime, and dies after at most 100 years or as soon as its tank is empty or it falls off a cliff, and so on. The probabilistic environmental reactions are initially unknown but assumed to be sampled from the axiomatized Speed Prior, according to which hard-to-compute environmental reactions are unlikely. This permits a computable strategy for making near-optimal predictions. One by-product of maximizing expected reward is to maximize expected lifetime.

一个每小时至少需要1升汽油的认知机器人与部分未知的环境互动，试图找到隐藏的、有限的汽油库，偶尔给油箱加油。它被按照寿命的比例给予奖励，最多100年后死亡，或者在它的水箱空了或者掉下悬崖时死亡，等等。概率环境反应最初是未知的，但假定是从公理化速度优先采样，根据难以计算的环境反应是不可能的。这使得一个可计算的策略可以做出接近最优的预测。最大化期望报酬的一个副产品是最大化期望寿命。

## See also

Gödel's incompleteness theorems

Gödel's incompleteness theorems

= = 也见

- 哥德尔的不完备性定理

## References

- ↑ Mahmud, M. M. Hassan (2008).
*Universal Transfer Learning*. pp. 16–18. ISBN 9780549909880. - ↑ Anderson, Michael L.; Tim Oates (Spring 2007). "A review of recent research in metareasoning and metalearning".
*AI Magazine*.**28**(1): 7. - ↑
^{3.0}^{3.1}^{3.2}^{3.3}^{3.4}^{3.5}^{3.6}^{3.7}^{3.8}Schmidhuber, Jürgen (December 2006).*Gödel Machines: Self-Referential ¨ Universal Problem Solvers Making Provably Optimal Self-Improvements*. ftp://ftp.idsia.ch/pub/juergen/gm6.pdf. Retrieved 10 November 2014. - ↑ "Gödel machine".
- ↑ Schaul, Tom; Schmidhuber, Juergen (2010). "Metalearning".
*Scholarpedia*.**5**(6): 4650. Bibcode:2010SchpJ...5.4650S. doi:10.4249/scholarpedia.4650. Retrieved 10 November 2014. - ↑ Steunebrink, Bas R.; Schmidhuber, Jürgen (2011).
*A Family of Gödel Machine Implementations*.**6830**. pp. 275–280. doi:10.1007/978-3-642-22887-2_29. ISBN 978-3-642-22886-5. - ↑ Schmidhuber, Jürgen (5 March 2009). "Ultimate Cognition à la Gödel" (PDF).
*Cognitive Computation*.**1**(2): 177–193. CiteSeerX 10.1.1.218.3323. doi:10.1007/s12559-009-9014-y. Retrieved 13 November 2014. - ↑ Schmidhuber, Jürgen (5 March 2009). "Ultimate Cognition à la Gödel" (PDF).
*Cognitive Computation*.**1**(2): 177–193. CiteSeerX 10.1.1.218.3323. doi:10.1007/s12559-009-9014-y. Retrieved 13 November 2014. - ↑ Schmidhuber, Jürgen (5 March 2009). "Ultimate Cognition à la Gödel".
*Cognitive Computation*.**1**(2): 177–193. CiteSeerX 10.1.1.218.3323. doi:10.1007/s12559-009-9014-y.

## External links

- Goedel machines home page
- Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements

- Goedel machines home page
- Goedel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements

# = 外部链接 =

- Goedel Machines 主页
- Goedel Machines: Self-Referential Universal Problem Solvers Making provedable Optimal Self-Improvements

Category:Artificial intelligence

类别: 人工智能

This page was moved from wikipedia:en:Gödel machine. Its edit history can be viewed at 哥德尔机/edithistory