Untyped Constrained Lambda Calculus
举例lambda类测度论
举例lambda类测度论Lambda类测度论是一种用于度量函数或集合的不同性质的数学理论。
它是一种广义的度量方法，可以用来描述不同函数或集合之间的相似性或差异性。
下面是一些关于Lambda类测度论的例子：1. 例子1：假设有两个函数f(x)和g(x)，它们在某个区间[a, b]上定义。
我们可以使用Lambda类测度论来度量这两个函数之间的相似性。
如果它们在这个区间上的Lambda类测度接近于0，那么说明这两个函数在该区间上非常相似。
反之，如果Lambda类测度远离0，则说明它们在该区间上差异很大。
2. 例子2：Lambda类测度论还可以应用于集合的相似性度量。
假设有两个集合A和B，我们可以使用Lambda类测度来度量它们之间的相似性。
如果它们的Lambda类测度接近于0，那么说明这两个集合非常相似。
反之，如果Lambda类测度远离0，则说明它们之间差异很大。
3. 例子3：Lambda类测度论还可以用于度量函数的连续性。
假设有一个函数f(x)，我们可以使用Lambda类测度来度量它在某个区间上的连续性。
如果它在该区间上的Lambda类测度接近于0，那么说明这个函数在该区间上非常连续。
反之，如果Lambda类测度远离0，则说明它在该区间上的连续性较差。
4. 例子4：Lambda类测度论还可以用于度量函数的平滑性。
假设有上的平滑性。
如果它在该区间上的Lambda类测度接近于0，那么说明这个函数在该区间上非常平滑。
反之，如果Lambda类测度远离0，则说明它在该区间上的平滑性较差。
5. 例子5：Lambda类测度论还可以用于度量函数的振荡性。
假设有一个函数f(x)，我们可以使用Lambda类测度来度量它在某个区间上的振荡性。
如果它在该区间上的Lambda类测度接近于0，那么说明这个函数在该区间上非常稳定。
反之，如果Lambda类测度远离0，则说明它在该区间上的振荡性较大。
6. 例子6：Lambda类测度论还可以用于度量函数的周期性。
康托尔、哥德尔、图灵——永恒的金色对角线
康托尔、哥德尔、图灵——永恒的金色对角线我看到了它，却不敢相信它。
——康托尔哥德尔的不完备性定理震撼了20世纪数学界的天空，其数学意义颠覆了希尔伯特的形式化数学的宏伟计划，其哲学意义直到21世纪的今天仍然不断被延伸到各个自然学科，深刻影响着人们的思维。
图灵为了解决希尔伯特著名的第十问题而提出有效计算模型，进而作出了可计算理论和现代计算机的奠基性工作，著名的停机问题给出了机械计算模型的能力极限，其深刻的意义和漂亮的证明使它成为可计算理论中的标志性定理之一。
丘齐，跟图灵同时代的天才，则从另一个抽象角度提出了lambda算子的思想，与图灵机抽象的倾向于硬件性不同，丘齐的lambda算子理论是从数学的角度进行抽象，不关心运算的机械过程而只关心运算的抽象性质，只用最简洁的几条公理便建立起了与图灵机完全等价的计算模型，其体现出来的数学抽象美开出了函数式编程语言这朵奇葩，Lisp、Scheme、Haskell…这些以抽象性和简洁美为特点的语言至今仍然活跃在计算机科学界，虽然由于其本质上源于lambda算子理论的抽象方式不符合人的思维习惯从而注定无法成为主流的编程语言[2]，然而这仍然无法妨碍它们成为编程理论乃至计算机学科的最佳教本。
而诞生于函数式编程语言的神奇的Y combinator至今仍然让人们陷入深沉的震撼和反思当中…然而，这一切的一切，看似不很相关却又有点相关，认真思考其关系却又有点一头雾水的背后，其实隐隐藏着一条线，这条线把它们从本质上串到了一起，而顺着时光的河流逆流而上，我们将会看到，这条线的尽头，不是别人，正是只手拨开被不严密性问题困扰的19世纪数学界阴沉天空的天才数学家康托尔，康托尔创造性地将一一对应和对角线方法运用到无穷集合理论的建立当中，这个被希尔伯特称为“谁也无法将我们从康托尔为我们创造的乐园中驱逐出去”、被罗素称为“19世纪最伟大的智者之一”的人，他在集合论方面的工作终于驱散了不严密性问题带来的阴霾，仿佛一道金色的阳光刺破乌云，19世纪的数学终于看到了真正严格化的曙光，数学终于得以站在了前所未有的坚固的基础之上；集合论至今仍是数学里最基础和最重要的理论之一。
计算理论6章 Lambda演算模型
1 Lambda演算理论
• Lambda变换规则
• 变换（应用规则）： (x.E) E0 E[E0/x]
• 例： (x.xy)z zy (x.yz)x yz (x.(y.yx)z)t (y.yt)z zt (x.(y.yx)z)t (x.zx)t zt
2 纯Lambda演算实例
• 整数上的运算: 令 plus m n 表示 m+n
time m n 表示 m×n 则
• plus = x.y. a.b. (x a)((y a)b) • time = x.y.a. x (y a)
2 纯Lambda演算实例
• plus m n = a.b. (m a)((n a)b) = ab. ((ab. amb)a)(((ab.anb)a)b) = ab. (b. amb)(anb) = ab. amanb = ab. am+nb
• 逻辑运算
• 布尔值
T = a.b.a F = a.b.b
• 布尔运算 and = x.y. x y F or = x. y. x T y not = x. x F T
2 纯Lambda演算实例
• and T T = T T F = (ab. a) T F =T
• or T F = T T F = (ab. a) T F =T
• E ::= x | E1 E2 | x:T. E | (E) T ::= Int | T T
• 类型系统 (表示类型上下文)
• x: T ├ x: T
•
├ x: T1，├ y:T2
， y:T2 ├ x：T1
•
├ x: T1, ├ E: T2 ├ x:T. E : T1 T2
兰姆达演算法
兰姆达演算法
兰姆达演算法（Lambda calculus）是一种形式系统，以数理逻
辑为基础，用于描述计算过程和函数的抽象概念。
它由阿隆佐·丘鲁斯基（Alonzo Church）在1930年代中期发展起来，被认为是计算理论的基础之一。
兰姆达演算法的核心思想是将函数抽象为匿名函数，没有名称、参数名或返回类型的函数。
它使用变量、抽象和应用这三个基本元素构建函数，并通过递归应用和代换规则来实现计算。
在兰姆达演算法中，函数的定义通过抽象实现。
例如，用λ符号表示抽象，x表示变量，(λx.x)表示一个接收一个参数并返
回该参数的函数。
函数应用通过用一个表达式替换另一个表达式中的变量来实现。
例如，(λx.x)(y)将变量x替换为y，得到
结果y。
兰姆达演算法最初是作为一个纯粹的数理逻辑系统而引入的，它能够表示和计算递归函数、自然数和简单的数学运算。
尽管它不直接与计算机或编程语言相关，但其抽象计算能力以及与实际计算模型之间的等价关系使其成为计算理论的一个重要工具。
兰姆达演算法对于理解函数式编程语言、计算复杂性理论以及计算机科学的基本原理都有很大的意义。
十二个不可积分函数
十二个不可积分函数1. Dirichlet函数：定义在实数集上的函数，对于有理数为1，对于无理数为0。
2. Thomae函数：定义在实数集上的函数，对于有理数1/n（n为正整数），为1/n，对于无理数为0。
3. Riemann函数：定义在实数集上的函数，对于有理数1/n（n为正整数），为1/n，对于无理数为14. Thomae反例函数：定义在实数集上的函数，对于有理数，如果该有理数可以表示为a/b（a、b互质且b>0），则f(x)=1/b；对于无理数，f(x)=0。
5.欧拉函数：定义在正整数集上的函数，表示小于等于n且与n互质的正整数的个数。
6.莫比乌斯函数：定义在正整数集上的函数，根据n的素因子分解形式确定。
如果n有平方因子，则f(n)=0；如果n是不同素数的乘积且素数个数为奇数，则f(n)=-1；如果n是不同素数的乘积且素数个数为偶数，则f(n)=17. Sierpinski函数：定义在实数集上的函数，对于有理数，如果该有理数可以表示为a/b（a、b互质且b>0），则f(x)=1/b^2；对于无理数，f(x)=0。
8. Weierstrass函数：定义在实数集上的函数，为以2^(-n)cos(3^n x)的无穷和。
9. Cantor函数：定义在实数集上的函数，是一个实数x在Cantor集合中的特征函数。
10.不连续开关函数：定义在实数集上的函数，当x为有理数时为1，当x为无理数时为0。
11.阶梯函数：定义在实数集上的函数，在n为整数的区间[n,n+1)上取常数值n。
12. Riemann定积分不可积函数：定义在实数集上的函数，只在一列分割区间中有限个点的函数。
lamda准则
lamda准则
Lambda 准则是一种常用于分类问题中的准则，特别是在决策树分类器中。
Lambda 准则的主要思想是：在任何节点分裂时，应尽可能使分裂后的信息增益最小。
Lambda 准则定义如下：设D 是样本集，G 是Gini 指数，则节点的λ 值为λ = 1 - Gini(D)，其中Gini(D) 是样本集D 的Gini 指数。
当样本集D 中所有样本都属于同一类别时，Gini(D) 最小，为0；当样本集D 中两个类别的样本数量相等时，Gini(D) 最大，为0.5。
因此，λ 的值域为[0,1]。
Lambda 准则选择分裂信息最大的属性作为最优划分属性，即选取能使分裂后的Gini 值最小（即λ 值最大）的属性作为划分属性。
这样做的目的是为了提高分类的准确性。
需要注意的是，Lambda 准则仅适用于分类问题，并且是在决策树算法中使用的。
在其他机器学习算法中，可能会有其他的准则和优化方法。
mutivariable calculus 与calculus的区别
mutivariable calculus 与calculus的区别Multivariable calculus, also known as multivariate calculus, is a branch of calculus that deals with functions of more than one variable. It extends the concepts of calculus, which are primarily used for studying functions of a single variable, to functions that depend on multiple variables. In this article, we will discuss the key differences between multivariable calculus and calculus.1. Number of Variables:The most obvious difference between multivariable calculus and calculus is the number of variables involved. Calculus typically deals with functions of a single variable, such as finding derivatives and integrals of functions like f(x) or g(t). On the other hand, multivariable calculus deals with functions that depend on multiple variables, such as h(x, y) or k(x, y, z).2. Vectors and Vector Functions:Multivariable calculus often incorporates vectors and vector functions into its analysis. It introduces the concept of vector-valued functions, which map a set of variables to a vector. This is in contrast to calculus, where vectors are rarely used. Understanding vector calculus is crucial in fields like physics, engineering, and computer science, where many problems involve multi-dimensional quantities.3. Partial Derivatives:A key concept in multivariable calculus is partial derivatives. When dealing with functions of multiple variables, we can take the derivative with respect to one variable while keeping other variables constant. These partial derivatives provide informationabout how the function changes when a specific variable changes, while others remain fixed. In calculus, only total derivatives are considered, which capture the change in a function with respect toa single variable.4. Multiple Integration:Integration in multivariable calculus involves integrating over regions in space rather than just along a single axis. Multiple integrals involve integrating a function over a specified region of two or more variables. This concept is not studied in calculus, where only single-variable integrals are considered. Multiple integrals have applications in areas such as finding areas, volumes, and evaluating probabilities in statistics.5. Gradient, Divergence, and Curl:Multivariable calculus introduces vector operations such as gradient, divergence, and curl, which are not present in calculus. The gradient of a function is a vector that points in the direction of steepest increase of the function. Divergence is a measure of how much a vector field spreads out or converges to a point, while curl describes the rotational behavior of a vector field. These concepts are crucial in fields like fluid dynamics and electromagnetism.6. Taylor Series Expansion:In calculus, the Taylor series expansion is often used to approximate functions near a point using a polynomial. Multivariable calculus extends this concept to functions of multiple variables, allowing approximation in higher dimensions. Taylor series expansions in multivariable calculus are used to understand the behavior of functions at a point and derive higher-orderderivatives.7. Optimization in Multiple Variables:Optimization problems involving multiple variables are another key difference between multivariable calculus and calculus. Multivariable calculus allows us to find the maximum or minimum values of functions of multiple variables using techniques such as partial derivatives and the Hessian matrix. This is not explored in calculus, where optimization is typically limited to functions of a single variable.In conclusion, multivariable calculus explores functions of multiple variables, delving into concepts like partial derivatives, multiple integration, vector functions, and optimization in higher dimensions. These concepts extend the principles of calculus and are essential in fields such as physics, engineering, economics, and computer science where problems involve multiple variables and dimensions.。
lambda的物理公式
lambda的物理公式
Lambda是一个物理学中的概念，它是指波长，也可以用来表示衰减常数。
lambda的物理公式可以用来计算波长或衰减常数，这些参数对于我们理解光、电磁波等现象非常重要。
在物理学中，波长是指在介质中传播的波的长度。
波长是光谱学和光学中一个非常重要的概念，它可以用来描述光的颜色和频率。
波长的计算公式是λ = c / f，其中λ表示波长，c表示光速，f表示频率。
在这个公式中，波长与频率成反比。
频率越高，波长越短，频率越低，波长越长。
衰减常数是指波沿垂直方向传播时的衰减率。
衰减常数与波长也有一定的关系。
在电磁波传播中，衰减常数可以表示为α = 4πλ / λ0^2，其中α表示衰减常数，λ表示波长，λ0表示自由空间中的波长。
衰减常数与波长成正比。
波长越长，衰减常数越小，波长越短，衰减常数越大。
Lambda的物理公式在许多物理学和电子学领域都有应用。
例如，在光学中，波长是描述颜色和频率的重要参数。
在无线通信中，波长可以用来计算信号传输的距离和衰减程度。
在电子学中，波长可以用来计算电子束的聚焦和散射。
总之，lambda的物理公式是一个非常重要的公式，在物理学和电子学领域有广泛的应用。
通过这个公式，我们可以计算波长和衰减常数，更好地理解和描述光、电磁波等现象。
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【英文资料】有趣的悖论
Some interesting paradoxesA paradox is a statement or event that seems to contradict itself, though it is still true, or at least makes sense.It is a true statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth . Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction. It is also used to describe situations that are ironic.Here’s a quote:"All animals are equal, but some are more equal than others." ——George Orwell Animal farm With this quote, you've probably got an idea about what a paradox is. You've probably heardthe word paradox being used a lot, but understanding its meaning may seem a little difficult. Though it appears quite complicated on the surface, below the surface is a very simple expression that is trying to make a clear point. Taking the aforementioned quote as an example, we all know that fiction is unreal. However, as the statement goes, there's always some reality behind fiction. In essence, fiction is inspired from the truth, so it can't be completely unreal. This is the paradox here, that actual truth differs from the perceived truth, and both are explained in the same statement. Here, we will explain exactly what is a paradox with the help of some more examples of paradox, all of which will help you understand this surprisingly simple literary device.Here are some well-known paradoxes:* Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences.* What the Tortoise Said to Achilles "Whatever Logic is good enoughto tell me is worth writing down...", also known as Carroll's paradox, not to be confused with the physical paradox of the same name.* Crocodile Dilemma: If a crocodile steals a child and promises itsreturn if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?* Catch-22 (logic): In need of something which can only be had by not being in need of it.* Drinker paradox: In any pub there is a customer such that, if he or she drinks, everybody in the pub drinks.* Paradox of entailment: Inconsistent premises always make an argument valid.* Horse paradox: All horses are the same color.* Lottery paradox: It is philosophically justifiable to believe that every individual lottery ticket won't win, but not justifiable to believe that no lottery ticket will win.* Raven paradox (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.* Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The Bottle Imp paradox uses similar logic.* Barber paradox: An adult male barber shaves all men who do not shave themselves, and no one else. Can he shave himself? (Russell's popularization of his set theoretic paradox.)* Berry paradox: The phrase "the first number not nameable in under eleven words" appears to name it in nine words.* Paradox of the Court: A law student agrees to pay his teacher after winning his first case. The teacher then sues the student (who has not yet won a case) for payment.* Curry's paradox: "If this sentence is true, the world will end in a week."* Epimenides paradox: A Cretan says: "All Cretans are liars".* Exception paradox: "If there is an exception to every rule, then every rule must have at least one exception; the exception to this one being that it has no exception." "There's always an exception to the rule, except to the exception of the rule — which is, in of itself, an accepted exception of the rule."* Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterological word? (Another close relative of Russell's paradox.)* Kleene–Rosser paradox: By formulating an equivalent to the liarparadox, untyped lambda calculus is shown to be inconsistent.* Liar paradox: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this question no?" And"I'm lying."o Card paradox: "The next statement is true. The previous statementis false." A variant of the liar paradox that does not use self-reference.o Quine's paradox: "'yields a falsehood when appended to its own quotation' yields a falsehood when appended to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.o Yablo's paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are false. Uses neither self-reference nor circular reference.* Opposite Day: "It is opposite day today."* Petronius's Paradox: "Practice moderation in all things. Including moderation."* Richard's paradox: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.* Socratic paradox: "I know that I know nothing at all."* Bonini's paradox: models or simulations that explain the workingsof complex systems are seemingly impossible to construct: As a modelof a complex system becomes more complete, it becomes less understandable; for it to be more understandable it must be less complete and therefore less accurate. When the model becomes accurate, it is just as difficult to understand as the real-world processes it represents.* Code-talker paradox: how can a language both enable communication and block communication?* Ship of Theseus (a.k.a. George Washington's axe or Grandfather'sold axe): It seems like you can replace any component of a ship, andit will still be the same ship. So you can replace them all, one at a time, and it will still be the same ship. But then you can take allthe original pieces, and assemble them into a ship. That, too, is the same ship with which you started.* Sorites paradox (also known as the paradox of the heap): One grainof sand is not a heap. If you don't have a heap, then adding only onegrain of sand won't give you a heap. Then no number of grains of sand will make a heap. Similarly, one hair can't make the differen。
lambda-calculus
Lambda-演算在形式语义学的中应用1傅庆芳1.形式语义学与组合原则自Chomsky创立生成语法（generative grammar）以来，人们开始普遍承认语法是可以按组合原则逐步生成的。
生成语法的一个基本出发点是：自然语言中存在无穷多的句子，同时大脑的功能是有限的，所以我们的语言能力必须包含某种可以描述这些无穷多句子类型的有穷机制。
也就是说，通过这些有穷的机制，我们可以不断地对已有的元素进行组合，并生成无穷多的句子或语法类型。
因此，可以用形式化的方法分析句子的语法及其生成过程。
但是，在很长一段时间里，Chomsky反对把组合原则和形式化方法应用到语义学的研究中。
这是因为语义问题不像语法结构那样有比较清楚的结构，尤其是，很难有与语法相对应的组合生成的语义解释。
然而，组合原则对于形式语义学是至关重要的，如果不能在自然语言的语义研究中很好地贯彻组合原则，那么也就没有形式语义学这一研究领域。
这成为形式语义学发展的一个瓶颈。
Chomsky对形式语义学基本持怀疑态度。
在很长的时间里，MIT一直没有研究形式语义学的人，直到1989年Chomsky才聘请了Irene Heim2去MIT工作。
这时的形式语义学已经发展得比较成熟。
形式语义学的想法是，对应于语法中的组合原则，语义解释也应该是可以通过应用组合原则逐步生成的，即：一个语言的说话者可以知道无穷多句子的意义，也能够理解或表达他从未听说过意思。
因此，对于语义学来说，也一定存在某种有穷的机制可以让人们理解任何一种自然语言的无穷多的句子。
因此，形式语义学的一个主要原则是语法与语义之间的关系是组合的。
任何一个自然语言的语法和语义各自是按组合原则生成的，同时这两个组合原则又是相互对应的。
这类似于一阶逻辑这种形式语言，其句法和语义都要遵循组合原则，且相互之间存在一种组合的对应关系。
也就是说，语法与语义之间并非都是随意联系在一起的，而是可以遵循某些规则逐步组合而成的。
怎么写一个解释器
怎样写一个解释器这是一篇解释器的入门教程。
虽然我试图从最基本的原理讲起，尽量让这篇文章不依赖于其它知识，但是这篇教程并不是针对编程的入门知识，所以我假设你已经学会了最基本的Scheme 和函数式编程。
我不是很推崇函数式编程，但它里面确实包含了很重要的一些方法。
如果你完全不了解这些，可以读一下SICP 的第一，二章（或者接下去读The Little Schemer)。
当然你也可以继续读这篇文章，有不懂的地方再去查资料。
我在这里也会讲递归和模式匹配的原理。
如果你已经了解这些东西，这里的内容也许可以加深你的理解。
解释器是一种简单却又深奥的东西，以至于好多人都不会写，或者自认为会写却又不真正的会写。
在这个领域里有一些历史遗留下来的误解，以至于很少有人真正的知道如何写出正确的解释器。
很多“语言专家”或者“逻辑学家”的解释器代码里面有各种各样的错误，却又以谬传谬，搞得无比复杂。
这误解的渊源之深，真是一言难尽。
你必须从最简单的语言开始，逐步增加语言的复杂度，才能构造出正确的解释器。
这篇文章就是告诉你如何写出一个最简单的语言(lambda calculus) 的解释器，并且带有基本的的算术功能，可以作为一个高级计算器来使用。
一般的程序语言课程往往从语法分析(parsing)开始，折腾lex 和yacc 等麻烦却不中用的工具，解决一些根本不需要存在的问题。
Parsing 的作用其实只是把字符串解码成程序的语法树（AST）结构。
麻烦好久得到了AST 之后，真正的困难才开始！而很多人在写完parser 之后就已经倒下了。
鉴于这个原因，这里我用所谓的“S-expression”来表示程序的语法树（AST）结构。
S-expression （加上Lisp 对它发自“本能”的处理能力）让我们可以直接跳过parse 的步骤，进入关键的主题：语义(semantics)。
这里用的Scheme 实现是Racket。
为了让程序简洁，我使用了Racket 的模式匹配（pattern matching）。
lambda演算 参考书
lambda演算参考书以下是一些关于lambda演算的参考书目：1. "Lambda-Calculus and Combinators: An Introduction" by JohnH. Conway and Simon B. Kochen - 这本书是一个很好的入门教材，由两位数学家编写，并提供了对lambda演算和组合子的详细介绍。
2. "The Lambda Calculus: Its Syntax and Semantics" by Henk Barendregt - 这本书是一个受欢迎的教材，提供了广泛的关于lambda演算和它的语法和语义的覆盖。
3. "Lambda Calculi: A Guide for Computer Scientists" by Chris Hankin - 这本书提供了一个针对计算机科学家的全面介绍，涵盖了lambda演算的各个方面以及其在编程语言和计算模型中的应用。
4. "The Lambda Calculus: Its Syntax, Semantics and Implementation" by Henk Barendregt and Willem-Paul de Roever- 这本书提供了关于lambda演算的形式语法和语义的详细介绍，并深入讨论了其在编程语言实现中的应用。
5. "Lambda-Calculus and Computation: An Introduction" by Hindley J.R. and Seldin J.P. - 这本书是关于lambda演算和计算的综合介绍，涵盖了其在函数式编程和计算理论中的应用。
请注意，以上的参考书目仅供参考，可以根据自己的需求和兴趣选择适合自己的书籍。
mutivariable calculus 与calculus的区别
mutivariable calculus 与calculus的区别摘要：1.多变量微积分与单变量微积分的概念区分2.多变量微积分的基本概念和主要应用领域3.多变量微积分与单变量微积分在求解问题上的异同4.我国在多变量微积分领域的研究和发展正文：mutivariable calculus，即多变量微积分，是与calculus（单变量微积分）相对应的数学分支。
在许多领域，如工程、自然科学、经济学等，微积分发挥着重要作用。
了解多变量微积分与单变量微积分的区别，有助于我们更好地应用这些知识解决实际问题。
首先，从概念上区分多变量微积分与单变量微积分。
多变量微积分涉及多个变量的函数及其导数、偏导数、梯度等概念，而单变量微积分主要研究单一变量的函数及其导数、微分等。
简单来说，多变量微积分是研究多个变量之间关系的数学工具，而单变量微积分则是研究单一变量关系的数学工具。
其次，多变量微积分的基本概念和主要应用领域。
多变量微积分主要包括多元函数的求导、偏导、梯度、多元积分等。
这些概念在实际应用中具有重要意义，如在优化问题、微分方程、数值计算等领域。
相比之下，单变量微积分的主要应用领域包括物理、工程、经济学等，其基本概念如导数、微分、积分等。
在求解问题上，多变量微积分与单变量微积分有一定的异同。
两者都遵循微积分的基本原理，如牛顿-莱布尼茨公式、微分中值定理等。
然而，多变量微积分涉及到更多的变量和更高阶的导数，因此在求解问题时，多变量微积分通常更加复杂。
例如，在求解多变量函数的极值问题时，需要利用多元微分的知识，而单变量微积分则只需涉及单一变量的导数。
最后，谈谈我国在多变量微积分领域的研究和发展。
近年来，我国在多变量微积分领域取得了显著的成果，包括理论研究、教育改革和实践应用。
许多科研机构和高校在多变量微积分的研究中做出了突出贡献，如中国科学院、北京大学、清华大学等。
此外，我国还积极参与国际多变量微积分学术交流，推动了该领域的发展。
lambda方法的基本原理
lambda方法的基本原理
Lambda方法是C++11标准中新增的语法糖，也称为lambda表达式或匿名函数。
Lambda方法具有距离近、简洁、高效和功能强大的特点。
Lambda表达式的语法如下：参数列表、参数列表是可选的，类似于普通函数的参数列表，如果没有参数列表，括号可以省略不写。
Lambda函数不能有默认参数，所有参数必须有参数名，且不支持可变参数。
Lambda函数可以用后置的方法书写返回类型，类似于普通函数的返回类型，如果不写返回类型，编译器会根据函数体中的代码推断出来。
Lambda表达式的原理是，它会被编译生成为当前类的一个私有方法。
Lambda表达式内直接引用局部变量本质是一种隐式传参，编译时会自动将引用的局部变量放到参数列表中（Lambda方法多了个参数），而引用的实例变量并不需要放到参数列表，因为方法内可以直接引用。
造成直接引用的局部变量需要final修饰的原因应该和这种隐式传参有关。
以上内容仅供参考，建议查阅C++书籍或咨询编程专家，以获取更准确的信息。
lambda演算 参考书
lambda演算参考书摘要：mbda 演算的定义与背景mbda 演算的基本概念和符号mbda 演算的应用领域4.参考书籍推荐正文：lambda 演算是一种计算模型，由Alonzo Church 于1936 年提出，旨在解决数学中的函数定义和计算问题。
这种演算基于函数定义和计算，以简洁的形式描述了函数的抽象概念。
在计算机科学和数学领域，lambda 演算有着广泛的应用。
mbda 演算的定义与背景Lambda 演算是一种形式化的计算模型，通过一些简单的语法规则和符号表示，来描述和实现计算过程。
在lambda 演算中，有两个基本的符号：λ和μ。
其中，λ表示匿名函数，μ表示函数应用。
通过这两个符号，可以表示和计算各种复杂的数学函数。
mbda 演算的基本概念和符号在lambda 演算中，函数被看作是一个表达式，它接受一些输入参数，并返回一个输出结果。
函数的定义和计算过程都遵循一定的规则。
例如，一个简单的函数定义可以表示为：λx.f(x)，表示一个接受一个参数x 的函数，其计算结果为f(x)。
而函数的计算过程则通过μ符号表示，如：μx.f(x) 表示将函数应用到参数x 上，即计算f(x) 的结果。
mbda 演算的应用领域Lambda 演算在计算机科学和数学领域有着广泛的应用。
在编程语言的设计和实现中，lambda 演算提供了一种简洁的方法来表示匿名函数，如Python 中的lambda 函数。
此外，lambda 演算还为函数式编程提供了理论基础，如Haskell 等函数式编程语言。
在数学领域，lambda 演算被用于研究函数的性质和计算复杂性等问题。
λ演算学习——精选推荐
λ演算学习最近为了学习Lambda表达式，特地学习了λ演算，函数式编程教程等许多⽂章，⽹络上的资料还是蛮多的，收获颇深，特此做了⼀个⼩总结λ演算（英语：lambda calculus，λ-calculus）lambda演算是由Alonzo Church设计的⼀个正式的数学系统，⽤于探讨函数，函数应⽤和递归(recursion)1.定义形式化地，我们从⼀个标识符（identifier）的可数⽆穷集合开始，⽐如{a, b, c, ..., x, y, z, x1, x2, ...}，则所有的lambda表达式可以通过下述以BNF范式表达的上下⽂⽆关⽂法描述：1. <表达式> ::= <标识符>2. <表达式> ::= (λ<标识符>.<表达式>)3. <表达式> ::= (<表达式> <表达式>)前两条规则⽤来⽣成函数，⽽第三条描述了函数是如何作⽤在参数上的在λ演算中，函数是通过λ表达式匿名定义的。
例如：f (x) = x + 2 在λ演算中表⽰为λ x. x + 2注意，如果⼀个λ表达式定义了⼀个函数，尽管这个函数的参数没有被赋值，这个表达式也是有值的，这个值就是这个表达式本⾝,在λ演算中,值的概念并不仅仅说⼀个数字是⼀个值，⼀个表达式也可以是⼀个值.λ演算只有三类表达式,⽽每条表达式就代表⼀个函数,⽽函数只有⼀个参数,也就是单参,并且返回⼀个值,函数是通过λ表达式匿名地定义的,⽐如函数f(x)=x+1,对应的λ表达式为λx.(x + 1)通常对于括号,如果不产⽣歧义的时候是可以被省略的,因为函数的"作⽤"是左结合的⽐如 (λx.x)(λx.1-x),依次代 (λx.1-x)进函数(λx.x),就是λx.1-x并且只要λ表达式操作符操作符被绑定到它后⾯的整个表达式例如:1. 表达式λx.(x+1),可以简写为λx.x+12. 表达式 (λx.x x)(λy.y),可以简写成λ(x.x x) λy.y3. 表达式λx. (λy. ( x+y)),可以简写为λx. λy. ( x+y)类似λx.(x y)这样的lambda表达式,是未定义⼀个函数的,因为变量y的出现是⾃由的,即它并没有被绑定到表达式中的任何⼀个λ上,所以⼀个lambda表达式的⾃由变量的集合是通过下述规则定义的：1. 在表达式V中，V是变量，则这个表达式⾥⾃由变量的集合只有V,⽐如表达式x,则⾃由变量的集合就是x2. 在表达式λV.E中（V是变量，E是另⼀个表达式），⾃由变量的集合是E中⾃由变量的集合减去变量V。
lambda演算主要语法规则、演算规则
lambda演算主要语法规则、演算规则Lambda演算是一种计算模型，也是一种形式系统，它以函数的定义和应用为基本操作，用于研究计算和计算机科学的基本概念。
本文将介绍Lambda演算的主要语法规则和演算规则。
一、主要语法规则1. 变量：Lambda演算中的变量是用小写字母表示的，例如x、y、z等。
2. 抽象：抽象是Lambda演算中定义函数的方式。
它的语法形式为λx.E，其中λ表示抽象的开始，x表示函数的参数，E表示函数体。
例如，λx.x表示一个将参数x映射到自身的函数。
3. 应用：应用是Lambda演算中调用函数的方式。
它的语法形式为E1 E2，其中E1是函数，E2是函数的参数。
例如，(λx.x) y表示将参数y应用到函数λx.x上。
二、演算规则1. α-变换规则：α-变换是指改变抽象中的参数名，以避免命名冲突。
例如，λx.E可以通过α-变换规则变为λy.E[y/x]，其中E[y/x]表示将E中的所有x替换为y。
2. β-规约规则：β-规约是指应用函数时，将函数体中的参数替换为实际参数的过程。
例如，(λx.E) v可以通过β-规约规则变为E[v/x]，其中E[v/x]表示将E中的所有x替换为v。
3. η-规约规则：η-规约是指将抽象中的函数体改写为更简洁的形式。
例如，λx.(E x)可以通过η-规约规则变为E，其中E表示函数体中的表达式。
4. 自由变量：自由变量是指在函数体中未被绑定的变量。
在进行演算时，需要注意自由变量的约束。
Lambda演算的语法规则和演算规则非常简洁和精确，能够描述和推导各种计算和计算机科学问题。
以下是一个简单的例子来说明Lambda演算的应用。
假设有一个函数f，它将一个数字加1。
在Lambda演算中，可以使用抽象和应用的方式定义这个函数。
首先，定义一个函数addOne，表示将参数x加1。
它的Lambda表示形式为λx.(x + 1)。
然后，定义函数f，它将参数y应用到addOne函数上，即f = addOne y。
lambda公式
lambda公式Lambda公式是指使用lambda函数（也称为匿名函数）进行表达的一种数学公式。
lambda函数是一种简洁的函数表达方式，可以在不定义函数名称的情况下直接使用函数。
lambda公式在数学、计算机科学等领域具有广泛的应用，能够简化代码，提高效率。
在计算机科学中，lambda函数通常用于函数式编程。
函数式编程是一种以函数为基础的编程范式，强调使用函数来进行计算和操作数据。
使用lambda函数可以方便地定义和传递函数，使代码更加简洁和易于理解。
Lambda公式的格式如下：lambda 参数: 表达式其中，参数是函数的输入，表达式是函数的输出。
lambda公式可以接受多个参数，用逗号分隔。
表达式可以是任意合法的Python表达式，可以是简单的计算、判断语句或者函数调用等。
lambda公式的特点之一是匿名性，即没有函数名称。
这意味着lambda函数只在定义的地方可见，并且没有与之关联的名称。
因此，lambda函数通常用于一次性的简单操作或作为其他函数的参数传递。
在实际应用中，lambda公式可以用于各种场景，例如列表排序、筛选、映射等。
下面将介绍一些常见的应用示例。
1. 列表排序：假设有一个包含一组整数的列表numbers，我们可以使用lambda 公式对其进行排序。
例如，可以按照从小到大的顺序对列表进行排序：```numbers = [5, 2, 8, 1, 6]sorted_numbers = sorted(numbers, key=lambda x: x)print(sorted_numbers) # 输出：[1, 2, 5, 6, 8]```在这个示例中，lambda函数接受一个参数x，表示列表中的每个元素，然后返回x本身。
sorted函数根据lambda函数的返回值对列表进行排序。
2. 筛选列表：lambda公式还可以用于筛选列表中符合条件的元素。
例如，可以筛选出所有大于5的元素：```numbers = [5, 2, 8, 1, 6]filtered_numbers = list(filter(lambda x: x > 5, numbers))print(filtered_numbers) # 输出：[8, 6]```在这个示例中，filter函数根据lambda函数的返回值对列表进行筛选，只保留大于5的元素。
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Untyped Constrained Lambda Calculus John N.Crossley,Luis Mandel and Martin WirsingLudwig-Maximilians-Universität MünchenInstitut für InformatikBericht9318September1993Untyped Constrained Lambda CalculusJohn N.CrossleyMonash University,Melbourne,Australiajnc@.auLuis MandelFORWISS,München,Germanymandel@forwiss.tu-muenchen.deMartin WirsingLudwig-Maximilians-Universität,München,Germanywirsing@informatik.uni-muenchen.deAbstractA calculus which extends the untyped-calculus by constraints is presented.The constraints can be used for two purposes:in a passive way for restricting the range of variables and in an active way for computing solutions of goals.Rules for the constrained-calculus are presented and a complete proof of the weak Church Rosser property is given.Keywords:-Calculus,Constraints.1IntroductionLogic programming languages,such as Prolog,use the Herbrand universe as compu-tation domain.Only uninterpreted functors are considered,implying that unification solves equations in an empty theory.It does not make use of the properties of particular application domains.Therefore in order to solve problems over e.g.the real numbers one has to include an axiom system for some part of the theory of the real numbers. This makes for great inefficiency because one then has to use the system to prove theorems from these axioms.Variables are replaced by terms in order to make them syntactically equal and that is all. Ground terms which are semantically equal cannot be unified.For example,the term 1and the term fact(where fact is the factorial functor),although mathematically equal,cannot be unified.The same is true for the terms2and1+1,and so on. Prolog is the most well-known example of a logical programming language.Given a query,Prolog returns a unification(or the answer“yes”if the query has no free variables)if the query is satisfiable in the Herbrand universe,or no answer(the query does not terminate),or it simply returns“no”,or does not terminate.The output of Prolog is quite poor in the sense that one cannot obtain answers such as“your query is right if”.In general,Prolog implementations have an arithmetic“interface”whose purpose is to reduce terms such as1+1.New versions of Prolog,such as Prolog III(see[Col87]and[Col90])or CHIP(see [Din86])have solved some of these problems.These implementations have typed variables over well-known computational domains,such as real numbers,rational numbers,booleans,lists,etc.In some steps of the resolution some clauses are solved using a symbolic simplex-like solver or boolean unification,instead of traditional resolution.The constraint logic programming scheme of Jaffar and Lassez[JL87]gives a general approach for incorporating constraints in a logic programming language.This has been implemented in the language CLP(IR)[JMSY92]which contains a simplex algorithm for solving constraints over the real numbers.In-Prolog(see[Mil91]and[MNS87]) functional and logic programming are unified using a typed-calculus.Logic programs are sets of Harrop-formulae which may contain-terms.In fact,each formula is a -term of propositional type.Constraint functional programming was also introduced by Darlington et al.in[DGP92]. FALCON[GP91]is a functional programming language with constraints.In this lan-guage constraints are used in a passive way,that is,the constraints are used only to restrict the applicability of a function.There are no inferences from the constraints.1The aim of the present work is to develop an enhancement of the traditional untyped -calculus in a general setting.For that purpose constraints are added to the calculus. The result of the reduction of a term can be a term with constraints.In this way “incomplete information”can be handled.The constraints are used in two ways.On the one hand constraints are used in a passive way,to restrict the universe of validity of a term.On the other,inferences that are made out of constraints cause the replacing of subterms in the target term;this is the so-called active use of constraints.Similar to Constraint Logic Programming[JL87],our approach provides a scheme for constraint functional programming.The constraint language is considered as a black-box and we get different-calculi for different solvers.For this purpose the chosen constraint domain has to satisfy a number of requirements such as the existence of canonical names and the presence of the equality relation.The usual-reduction is extended by associativity rules for constrained-terms.The key rules are the three-reductions.They allow us to add new inferences to the constraints,to propagate constraints and to pass values obtained by inferences from the constraints into the functional part of the program.As our main result we get that the calculus satisfies the weak Church Rosser property if the underlying constraint solver produces unique normal forms in reducing the constraints.In section2a brief explanation of the language is presented.In section3the term formation rules and some examples are given.In section4the reduction rules are given.In section5the semantics of the language is given.In section6a complete example is presented.In section7the proof of the weak Church Rosser property is presented.The conclusions and future research directions are given in section8.2The Constraint LanguageThe term constraint—roughly speaking—can be defined as a relationship existing among some entities.Wefix a domain,the constraint domain,for example the real numbers with addition,multiplication,equality and perhaps other functions and pred-icates.We use a language(not necessarilyfirst-order logic)called the constraint language with appropriate functions and predicate letters for the constraint domain.A constraint language is formally defined as follows:Definition2.1(Constraint Language)A constraint language is a4-tuple, ,,where:=is a countable set of individual constants,2=is a countable set of variables,=is a countable set of function letters,and=is a countable,non-empty set of predicate letters.For a function letter,the natural number indicates the arity.For a predicate letter ,the natural number indicates the arity.Definition2.2(Constrained Term,Atomic Constraint)The set of constrained terms over a constraint language,,,is inductively defined as fol-lows:;;and.where is the set of all the constraints definable over£.If is a predicate letter and are constrained terms,then is an atomic constraint.Definition2.3(Constraint Formula,Constraint)A well formed formula of the con-straint language will be called a constraint formula.A constraint is afinite conjunction of constraint formulae written as a set.We write for the set of all possible constraints over a constrained language.We shall use the notation,,,etc.,for constraints.Constraints could be seen as“quantifiers”over the free variables of each constraint formula.The best notation should be where are the free variables in.In order to simplify the notation we write only.Definition2.4(Occurrence of a Constraint Formula in a Constraint)A constraint formula occurs in a constraint if there exists an index such and.We make a number of requirements on the theory of the constraint domain.1.Each element of the domain should have a canonical nameuniquely determines.Note that in the case of,for example, the real numbers any program will invoke only afinite or countable set of real number names,so there are no problems even for uncountable domains caused by cardinality considerations.2.As usual we shall assume that the constraint theory has axioms which are truein the constraint domain.We also assume a provability relation:in the constraint theory the formula is provable from the hypotheses.This provability relation is assumed to be monotonic:if.3.If we want to ensure we have the weak Church Rosser property,then it isnecessary to impose the condition thatis only possible ifsuch that is unique.This namethen for every variable and for every term MM M)and the case of no solution().Later we will study the case of multiple solutions(i.e.).Example2.6An example of a constraint domain is the real numbers.As predicate letters we have the usual equality and.As function letters we have+,-, *,/,and abbreviates.Examples of constraints are:1.,42.,3..In thefirst example,if the values of and are known,we can interpret the constraint as an assignment to the variable.The second example is the well-known triangular inequality where,,and are the lengths of each side of a triangle.In the last example we combine equalities and inequalities.3Term Formation RulesIn ordinary-calculus[Bar84],terms can be identified with programs,and program execution corresponds to reduction of the term.We now extend the usual set of formation rules by three additional ones.We add all the terms of the underlying constraint language as well as the constant as terms of the language,where denotes a contradiction.A further rule is one which constrains a term with a formula of the constraint language.Definition3.1(Constrained Lambda Term)The set of-terms is defined induc-tively from a set of variables,a set of constraints and a set of function letters as follows:1.;(Const)2.;(Var)3.M M M M;(Functors)4.M M;(Abstr)5.M N M N;(Applic)6.M M.(Constr) Using BNF notation this is:5-termvariable-term-termvariable-term-term-termconstraint-termwhere constraint is as defined in2.3and variable.We extend the constraint language(and theory)by adding all-terms as terms in the constraint language,which recursively extends the definition3.3above.We use M,N,for-terms.We adopt the usual conventions omitting brackets where there is no ambiguity and multiple abstraction as in[Bar84].Example3.2Here we show examples of-terms using the real numbers as constraint domain.If and are interpreted as the sides of a right-angled triangle that are not the hypotenuse then the following term computes the area of the given triangle..For an arbitrary triangle,we divide it into two right-angled ones.Given the sides, and,we draw the perpendicular from to its opposite vertex,and calculate its length .We compute the distance between the foot of the perpendicular and the vertex opposite to,and then:cbhp6Remark3.3Note that every-term can be written in the form M whereis a(possibly empty)constraint and M begins with a variable or a left bracket.Thus,in the extended constraint language we have terms of the form M(with)and atomic formulae of the form M(with),which intuitively are equivalent to M respectively M.These equivalences will be introduced in the form of reduction rules in section4.We need the definition of the scope of a constraint and the definition of bound,con-strained and free occurrences of a variable.Furthermore,we introduce a variable convention as in[Bar84],by which unintended variable capturing is avoided.Definition3.4(Subterm)M is a subterm of N(written M N)if M N or if M Sub N,where Sub N is the collection of all proper subterms of N inductively defined as follows:1.Sub;2.Sub;3.Sub Sub Sub;4.Sub M Sub M;5.Sub M M Sub M Sub M M M;6.Sub M Sub M.Remark3.5Note that the subterms of(in the usual sense)which could have-terms are not in Sub M.Definition3.6(Occurrence of a Variable)The occurrence of a variable in a-term is inductively defined as follows:1.does not occur in;2.occurs in iff;3.occurs in iff there exists such that occurs in;4.occurs in M iff or occurs in M;75.occurs in M M iff occurs in M or occurs in M;6.occurs in M iff occurs in or occurs in M.Definition3.7(Scope)Given a-term M,the scope of the constraint is the term M.Given a-term M the scope of the abstractor is M. Abstraction is said to bind the free variables in M.Symmetrically constraining is said to constrain variables in M.Every occurrence of a variable in M is bound in M. If the variable occurs free in,every non-bound occurrence of in M is said to be constrained in M.An occurrence of that is not bound is said to be free.As usual one has to make conventions so as to avoid the unintended binding of free variables(see e.g.[CS93]for details).Definition3.8(Free Variable,Set of Free Variables)An occurrence of a variable is free in a-term if it does not occur in the scope of a.Otherwise the occurrence is bound.FV M is the set of all free variables in M.It is defined inductively as follows:1.FV;2.FV;3.FV FV FV;4.FV M FV M;5.FV M N FV M FV N;6.FV M FV M FV.where FV corresponds to the underlying notion of free variable in the constraint language,as required in2.6.A term N is said to be closed if FV N.Definition3.9(Constraint-Free Variable,Constrained Variable)A constraint-free occurrence of variable in a term M is defined inductively as follows:1.is constraint-free in;2.is constraint-free in;83.is constraint-free in if for all,is constraint-free in;4.is constraint-free in M if is constraint-free in M;5.is constraint-free in M N if is constraint-free in M and is constraint-freein N;6.is constraint-free in M if is constraint-free in M and does not occurfree in.An occurrence of a variable is said to be constrained in a term M if occurs in M and is not constraint-free in M.If a variable occurs in a constraint and occurs free in a term M then this variable will be interpreted as existentially quantified.Definition3.10(Variable Convention)If the-terms M M occur in a certain context,then in these terms all free variables are chosen to be different from constrained variables and from bound variables.Remark3.11A variable can be bound and constrained simultaneously,as for example in the term.The following term violates the variable convention:is bound here,occurs free here,because occurs free and bound simultaneously.In the term M N,occurrences of in may constrain occurrences of in N,but do not constrain occurrences of in M.In the definition below substitution is defined as usual for-terms with the addition of an extra clause for terms involving constraints.We write M N for the result of substituting N for at all free occurrences of in M,whether in constraint formulae or in constrained-terms,avoiding unintended binding of variables.Definition3.12(Substitution of Variables by Terms)Substitution of variables by terms is inductively defined as follows:1.N is;92.(a)N is N,and(b)N is if is not identical with;3.M M N is M N M N;4.(a)M N is M;(b)M N is M N;if is not identical with where is a new variable;5.M M N is M N M N;6.M N is N M N.where N corresponds to the underlying notion of substitution in the con-straint language,as required in2.6,extended by-terms.We have identified terms that differ only in the names of bound variables.An alternative is to add to the-calculus the following axiom scheme:M M(-conversion) provided that does not occur in M.We prefer the version of the theory in which the identifications are made on the syntactic level.These identifications are made in our minds and not on paper.For example: Example3.13M MM MDefinition3.14(Occurrence of a Constraint Formula in a Term)A constraint formula occurs in a term is defined as follows:1.does not occur in;2.does not occur in;3.occurs in M M iff there exists such that occurs M;104.occurs in M iff occurs in M;5.occurs in M M iff occurs in M or occurs in M;6.occurs in M iff occurs in the constraint or occurs in M.4Reduction RulesThe set of rules is divided in two.On the one hand we have,as in the traditional -calculus,-reduction,as well as a new one called:-reduction.This-reduction is divided into three rules:Inference from a constraint,propagation and replacement.On the other hand we have associativity rules for constrained-terms and bottom rules for inconsistencies.We present also two structural rules:the P-rule and the F-rule.Definition4.1(Reduction Rules for Constrained Lambda Terms)The following are the rules of the calculus:()M N M N;()The-rule is defined as the union of the following three rules:()(Inf.)IfM;()(Prop.)If does not occur in then:M N M N;()(Repl.)If is not a canonical name then:(P)(Pred.)If M M N then:M M NM M N;(F)(Func.)If M M then:M MM M.In the previous definition it was globally assumed that is consistent.Sometimes we call F an inconsistent constraint,where F denotes“false.”The idea behind the associative rule is the following.If the is the constraint over M and M is the scope of the constraint,intuitively one can say that both and constrain M.We call the rule result of unifying,and.It is motivated as follows.If),the constraint theory is constraining the value of to a specific,unambiguous,value holds and that in evaluating M we can take the value of to beM Mprovided is not a canonical nameIn general,given a reduction rule,it induces a binary relation.We writeto denote“-reduces to”in one step,and for the reflexive-transitive closure of.Remark4.3It is assumed that both predicates letters and function letters of the con-straint language are strict.That is,if is a predicate letter of the constraint languagethen if there exists such that.The samefor any function letter.5SemanticsWe give an informal explanation of the semantics of the terms of the constrainedlambda calculus.The term M may be read as“if the constraint is satisfied, then evaluate M”.While it is true that the“if then”construction is available in -calculus and programming languages(such as Lisp)based on it,the constrained version allows us to use a special purpose constraint solver to solve the condition.In the traditional-calculus we would have to write extra code which would,in essence, compute the satisfiability of from axioms and this could be very uneconomical. As a specific example,if we take the constraint domain to be the real numbers then, as in CLP(IR)(see[JMSY92]),linear programming techniques can be used to solve constraints.In the course of reducing a-term some terms will reduce to ground terms.In this case the constraint theory will prove,for such a term,that.Then the reduction rule applies,allowing to be replaced by6ExamplesIn the following example we show how to calculate the area of a triangle.Wefix the set of real numbers as the constraint domain.As function letters we have+,-,*,/,(for square).As predicate letters we have the usual equality and the usual relation symbols <,>,etc.We assume the constraint solver is able to solve both linear and quadratic equations.The triangle is given by the lengths of its three sides,which have to satisfy the triangle constraint.We divide the given triangle into two right-angled triangles.The area of the given triangle is the sum of the areas of both right-angled triangles.We will use the functions defined in the example3.2.We give only the most important reduction steps.cbhpUsing the F-rule we get:14We assume that the constraint solver can solve the system on and:By-reduction our term reduces in two steps to:Performing the propagation and the substitution of the values of and we get:and thenwhich is thefinal result.We now calculate the area of a triangle where the side is unknown but restricted to satisfy the constraint.We can take advantage of the possibility of computing functions with incomplete information as follows:By-reduction we get:15Applying the-rule twice we get:Using the F-rule we have:By use of the P-rule we have:We assume that the constraint solver can solve the system on and returning values which are terms of the contraint language.From the constraint solver we then get:Writing=of the calculus is locally confluent in the calculus.Graphically this is:MM M ....................MIn subsection 7.2we demonstrate that given two different rules of the calculus,theyare locally confluent in the calculus.That is,if then:MM M ....................MThese two facts are enough to ensure that the calculus satisfies the weak Church Rosserproperty.Theorem 7.1(Knuth-Bendix)If for any term M,M M and M M implies there is an M with M M and M M ,then the -calculus satisfies the weak Church Rosser property (i.e.,it is locally confluent).Proof:See [KB70],[Hue80].7.1Local ConfluenceIn this subsection we demonstrate that each rule is locally confluent in the theory.Formally this means that for every term M,if MM and M M ,then there exists M such that MM and M M .Lemma 7.2If MM and MM ,then there exists M such thatMM and M M .Proof:The property is graphically demonstrated by the commutation of the followingdiagram:18MM MMTherefore the property is satisfied by taking M=M.Lemma7.3If M M and M M,then there exists M such thatM M and M M.Proof:The rule satisfies the Church Rosser property(see[Bar84, pages364–365]).The rules and have no overlap and they are terminating,so and satisfies the Church Rosser property.We will show that the-rule satisfies the weak diamond property.The proof is made by case analysis.Recall F denotes a contradiction in the constraint language.Case1:vs.Case1.1:M F NM FMCase1.2:M F NMMCase1.3:M F N NM F NM N19Case2:vs.Case2.1:M F NM F NMCase3:vs.Case3.1:MMMCase4:vs.Case4.1:MMMCase5:vs.Case5.1:M FMM FIn each case of overlapping the-reduction satisfies the weak diamond property.Also -reductions decrease the length of a term,hence they are terminating.By Newman’s lemma(see[New42])the-reduction satisfies the Church Rosser property.Therefore in every case if M M and M M then there exists M such that MM and M M.Lemma7.4If M M and M M,then there exists M such thatM M and M M.20Proof:There are two cases of overlapping.Thefirst one is when one has a term of the form M and both.The second one is when the term M has a subterm N with.These are graphically demonstrated as follows:MMMM(MMTherefore the property is satisfied.Lemma7.5If M M and M M,then there exists M such that MM and M M.Proof:There are two overlapping cases:1.M M;2.M M M.Case1:M M MM M MM MM M MM M MM MM M M21Case2:M M M MM M M MM M M M M M M MM M M M In both cases the reduction is confluent.That is,the property is satisfied. Lemma7.6If M M and M M,then there exists M such that M M and M M.Proof:There is only one case of overlapping,which is multiple.Lemma7.8If M M and M M,then there exists M such thatM M and M M.Proof:There are only two cases of overlapping and they are multiple.1.M andandM M.MCase2:.23In both cases the result holds.Lemma7.9If M M and M M,then there exists M such thatM M and M M.Proof:There are two cases of overlapping where the residuals of a rule affect the redex of the other.1.M;2.M.Case1:MMMCase2:MMMIn both cases the result holds.Corollary7.10The-rule is locally confluent in the theory.24Proof:As the-rule was defined as the union of the,and rules,there are six cases to consider:1.M M and M M2.M M and M M3.M M and M M4.M M and M M5.M M and M M6.M M and M MThe threefirst cases are the local confluence of the rules,and and they were demonstrated in7.4,7.5and7.6.The other three cases were demonstrated in7.7,7.8 and7.9respectively.Therefore the property is satisfied.Lemma7.11If M M and M M,then there exists M such that MM and M M.Proof:There are two cases of overlapping,where we omit the unaffected arguments of,:1.M M N;2.M M M N.They are graphically demonstrated as follows:Case1:M M NM M NM M N M M NM M N25Case2:M M M NM M M NM M M N M M M NM M M NSo,the result holds.Lemma7.12If M M and M M,then there exists M such thatM M and M M.Proof:There are two cases of overlapping,where we omit the unaffected arguments of,:1.M;2.M M.They are graphically demonstrated as follows:Case1:MMF M FMCase2:M MM MF M M FM MCase1:F M F MF Mand monotonicityCase2:F(M)F MF Mand monotonicityTherefore the property is satisfied.Lemma7.14If M M and M M,then there exists M such thatM M and M M.Proof:We can have a term M,where the-redex occurs in or in.In such a case,the residuals of one rule remain unaffected by the application of the other, that is,there is no“real”overlapping,and the lemma holds trivially.Discarding such a case,there are other three cases to consider:Case1:M -redexM N-redexM M NM N M M M NM N MCase2:M -redexM-redexN28M M NM N M M M NM N MCase3:M-redexM N-redexM M NM N MM M NM N MTherefore the property holds.Lemma7.15If M M and M M,then there exists M such that M M and M M.Proof:We perform a case analysis.Case1:M -redexM-redexwithMM MM M(by monotonicity of we haveM MCase2:M-redexM-redexwithMM MM M(by monotonicity of we haveM MTherefore the property is satisfied.Lemma7.16If M M and M M,then there exists M such thatM M and M M.Proof:The residuals of one rule do not affect the redex of the other.Therefore there is no real overlapping and the property is satisfied.Lemma7.17If M M and M M,then there exists M such thatM M and M M.Proof:There is only one overlapping case which isMM MMPMTherefore the property is satisfied.Corollary7.18The-rule and the-rule locally commute in the theory.30Proof:As the-rule was defined as the union of the,and rules,the property is satisfied by the lemmata7.15,7.16and7.17.Lemma7.19If M M and M M,then there exists M such thatM M and M M.Proof:There are two cases of overlapping(where we omit the irrelevant arguments).1.N M;2.N M.They are graphically demonstrated as follows:Case1:(N M)N MPN M N MPN MCase2:N MThis case is symmetrically to the previous one.Therefore the property is satisfied.Lemma7.20If M M and M M,then there exists M such thatM M and M M.Proof:There are two cases of overlapping(where we omit the irrelevant arguments).1.M;2.M.They are graphically demonstrated as follows:31Case1:MMMFMFMCase2:MMMFMFMTherefore the property is satisfied.Lemma7.21If M M and M M,then there exists M such thatM M and M M.Proof:The commutation of the and rule was already demonstrated in[Bar84, pages364–365].We have to demonstrate only the overlapping cases of and. There are two cases of overlapping:1.M;2.M F N.They are graphically demonstrated as follows:Case1:M32MM M:1.;2.M.They are graphically demonstrated as follows:33Case1:Case2:MMMTherefore the property is satisfied.Lemma7.23If M M and M M,then there exists M such thatM M and M M.Proof:The residuals of one rule do not affect the redex of the other.Therefore there is no real overlapping and the property is satisfied.Lemma7.24If M M and M M,then there exists M such thatM M and M M.Proof:We show one overlapping case,the proofs of the others are similar.Therefore the property is satisfied.Corollary7.25The-rule and the-rule locally commute in the theory.Proof:As the-rule was defined as the union of the,and rules,by lemmata 7.22,7.23and7.24the property is satisfied.Lemma7.26If M M and M M,then there exists M such thatM M and M M.34。