The λ – Calculus

... So Square( x) x x the corresponding lambda expression would be ( x x x) • An instance of a lambda expression to a particular value ‘a’ takes the form: (( x x x) a) a a a 2 • So, for example: (( x x x)3) 3 3 9 ...

... So Square( x) x x the corresponding lambda expression would be ( x x x) • An instance of a lambda expression to a particular value ‘a’ takes the form: (( x x x) a) a a a 2 • So, for example: (( x x x)3) 3 3 9 ...

Notes

... no computation remaining. Note that no closure is created, and that no computation is performed in body e. • Variable x does not evaluate to anything and immediately results in an error. For this to happen, x must have been free in the expression that is being evaluated. Since it was free, there is ...

... no computation remaining. Note that no closure is created, and that no computation is performed in body e. • Variable x does not evaluate to anything and immediately results in an error. For this to happen, x must have been free in the expression that is being evaluated. Since it was free, there is ...

ppt

... alpha-renaming, beta reduction, eta conversion, applicative and normal evaluation orders, ChurchRosser theorem, combinators, booleans ...

... alpha-renaming, beta reduction, eta conversion, applicative and normal evaluation orders, ChurchRosser theorem, combinators, booleans ...

LambdaCalculus

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following -calculus syntax: ...

... represents the same f function, except it is anonymous. To represent the function evaluation f(2) = 4, we use the following -calculus syntax: ...

Lambda Calculus and Lisp

... lambda, period, parentheses and numbers. • M -> x | (M M) | x.M • If F and A are both expressions then so is (F A) and indicates the application of the function F with A as its parameter. • If F is a expression then so is x.F This is a function definition and is also called an ABSTRACTION beca ...

... lambda, period, parentheses and numbers. • M -> x | (M M) | x.M • If F and A are both expressions then so is (F A) and indicates the application of the function F with A as its parameter. • If F is a expression then so is x.F This is a function definition and is also called an ABSTRACTION beca ...

Functional Programming

... • Currying: transform a function with multiple arguments into a chain of single-argument functions. • Partial function application: apply only a subset of the arguments of a curried function (from left to right) to yield another function taking the remaining arguments. • Reduction order: eager (redu ...

... • Currying: transform a function with multiple arguments into a chain of single-argument functions. • Partial function application: apply only a subset of the arguments of a curried function (from left to right) to yield another function taking the remaining arguments. • Reduction order: eager (redu ...

Lecture 15: The Lambda Calculus

... integer (α → bool) “abc” → False identity (α → α) 456 → 456 identity (α → α) “fred” → “fred” ...

... integer (α → bool) “abc” → False identity (α → α) 456 → 456 identity (α → α) “fred” → “fred” ...

Document

... a) when Y is the abstraction λx.W then [M/x]Y => λx.W b) when Y is the abstraction λy.W and there are no free occurrences of x in W then [M/x]Y is λy.W c) when Y is the abstraction λy.W and there are no free occurrence of y in M then [M/x]Y is λy.[M/x]W d) when Y is the abstraction λy.W and there is ...

... a) when Y is the abstraction λx.W then [M/x]Y => λx.W b) when Y is the abstraction λy.W and there are no free occurrences of x in W then [M/x]Y is λy.W c) when Y is the abstraction λy.W and there are no free occurrence of y in M then [M/x]Y is λy.[M/x]W d) when Y is the abstraction λy.W and there is ...

Lambda calculus

... be reduced. However, N and P are assumed to be in normal form, so they don’t allow for any further reductions. The only possible interpretation is that N = P = Q. ...

... be reduced. However, N and P are assumed to be in normal form, so they don’t allow for any further reductions. The only possible interpretation is that N = P = Q. ...

PPT

... Formal system with three parts • Notation for function expressions • Proof system for equations • Calculation rules called reduction ...

... Formal system with three parts • Notation for function expressions • Proof system for equations • Calculation rules called reduction ...

Functional Programming

... performed by a left-to-right reduction sequence. If the expression is reducible, the reduction sequence will terminate. If the expression is irreducible the reduction will not terminate and also, any other reduction sequence will not terminate as well. The left-to-right reduction corresponds to the ...

... performed by a left-to-right reduction sequence. If the expression is reducible, the reduction sequence will terminate. If the expression is irreducible the reduction will not terminate and also, any other reduction sequence will not terminate as well. The left-to-right reduction corresponds to the ...

C311 First Class Objects

... Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call Constructable at runt ...

... Expressible as an anonymous literal value Storable in variables Storable in data structures Having an intrinsic identity (independent of any given name) Comparable for equality with other entities Passable as a parameter to a function Returnable as the result of a function call Constructable at runt ...

Chapter 2 - Lambda Calculus - Rensselaer Polytechnic Institute

... f(x) = x2 , g(x) = x+1 f g is the composition of f and g: f g (x) = f(g(x)) f g (x) = f(g(x)) = f(x+1) = (x+1)2 = x2 + 2x + 1 g f (x) = g(f(x)) = g(x2) = x2 + 1 Function composition is therefore not commutative. Function composition can be regarded as a (higher-order) function with the followi ...

... f(x) = x2 , g(x) = x+1 f g is the composition of f and g: f g (x) = f(g(x)) f g (x) = f(g(x)) = f(x+1) = (x+1)2 = x2 + 2x + 1 g f (x) = g(f(x)) = g(x2) = x2 + 1 Function composition is therefore not commutative. Function composition can be regarded as a (higher-order) function with the followi ...

Compiling Functional Programming Languages (FPLs) λ

... Amazingly, Y can be defined as a lambda abstraction, i.e., WITHOUT RECURSION. It’s called the fixpoint combinator: ...

... Amazingly, Y can be defined as a lambda abstraction, i.e., WITHOUT RECURSION. It’s called the fixpoint combinator: ...

List

... – The list is the fundamental data structure – Developed by John McCarthy in the 60’s • Used for symbolic data processing • Example apps: symbolic calculations in integral and differential calculus, circuit design, logic, game playing, AI • As we will see the syntax for the language is extremely sim ...

... – The list is the fundamental data structure – Developed by John McCarthy in the 60’s • Used for symbolic data processing • Example apps: symbolic calculations in integral and differential calculus, circuit design, logic, game playing, AI • As we will see the syntax for the language is extremely sim ...

Breck, Hartline

... evaluate the function. As an aside, this class is quite a bit about how to get from an intensional representation, an algorithm, to the extension, meaning, or eﬀect of a function. Real programming languages such as Lisp, Scheme, Haskell and ML are very much based on lambda calculus, although there a ...

... evaluate the function. As an aside, this class is quite a bit about how to get from an intensional representation, an algorithm, to the extension, meaning, or eﬀect of a function. Real programming languages such as Lisp, Scheme, Haskell and ML are very much based on lambda calculus, although there a ...

Handout 10 from Models of Computation

... It may be that a λ-term offers many opportunities for reduction at the same time. In order for the whole calculus to make sense, it is necessary that the result of a computation is independent from the order of reduction. We would like to express this property for all terms, not just for those which ...

... It may be that a λ-term offers many opportunities for reduction at the same time. In order for the whole calculus to make sense, it is necessary that the result of a computation is independent from the order of reduction. We would like to express this property for all terms, not just for those which ...

PowerPoint - School of Computing Science

... Different reduction strategies are based on where in a term it is possible to apply beta reduction. We will concentrate on call by value reduction, which means adding the following rules: e e' e1 e'1 ve ve' e1e2 e'1 e2 Values v are -abstractions, and any other values which we might incorpor ...

... Different reduction strategies are based on where in a term it is possible to apply beta reduction. We will concentrate on call by value reduction, which means adding the following rules: e e' e1 e'1 ve ve' e1e2 e'1 e2 Values v are -abstractions, and any other values which we might incorpor ...

Functional Programming

... want to pass it only once (defun consval (x) (cons val x)) An obvious solution is just to pass the function’s body (mapcar ‘(cons val x) L) It is ambiguous : we do not know which names are parameters and which ones are global “lambda expressions” come in hand (mapcar ‘(lambda (x) (cons val x)) ...

... want to pass it only once (defun consval (x) (cons val x)) An obvious solution is just to pass the function’s body (mapcar ‘(cons val x) L) It is ambiguous : we do not know which names are parameters and which ones are global “lambda expressions” come in hand (mapcar ‘(lambda (x) (cons val x)) ...

Advanced Formal Methods

... Exercise 4: Define ”Church lists”. A list [x,y,z] can be thought of as a function taking two arguments c (for cons) and n (for nil), and returns c x (c y (c z n)), similar to the Church numerals. Write a function nil representing the empty list, and a function cons that takes arguments h and (a func ...

... Exercise 4: Define ”Church lists”. A list [x,y,z] can be thought of as a function taking two arguments c (for cons) and n (for nil), and returns c x (c y (c z n)), similar to the Church numerals. Write a function nil representing the empty list, and a function cons that takes arguments h and (a func ...

Lecture 10

... Map, filter and reduce Python provides this as • Map – transform everything in the iterable • Filter – filter those that return the value true • Reduce – a two argument function ls1 = map(lambda x: x +2, a) ls2 = filter(lambda x: x%2 == 0, a) ls3 = reduce(lambda x, y : x+y, a ) ...

... Map, filter and reduce Python provides this as • Map – transform everything in the iterable • Filter – filter those that return the value true • Reduce – a two argument function ls1 = map(lambda x: x +2, a) ls2 = filter(lambda x: x%2 == 0, a) ls3 = reduce(lambda x, y : x+y, a ) ...

Functional Programming COMP2003

... 1. When the function definition is specific to a particular situation, and so can be defined and used in just one place. Usually the motive for forming the computation into a function is that it is being passed in as argument to some other functions which will use that computation in various ways, p ...

... 1. When the function definition is specific to a particular situation, and so can be defined and used in just one place. Usually the motive for forming the computation into a function is that it is being passed in as argument to some other functions which will use that computation in various ways, p ...