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... 1 Intuitionistic Logic and Constructive Mathematics It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive ...
... 1 Intuitionistic Logic and Constructive Mathematics It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive ...
Propositions as types
... 7 Continuations and negation What is the significance of negation? We know that logically ¬φ is equivalent to φ ⇒ ⊥, which suggests that we can think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavi ...
... 7 Continuations and negation What is the significance of negation? We know that logically ¬φ is equivalent to φ ⇒ ⊥, which suggests that we can think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavi ...
Bound and Free Variables Theorems and Proofs
... domain D, an interpretation I, and a valuation V , written (I, D, V ) |= A The definition is by induction: (I, D, V ) |= P (x) if I(P )(V (x)) = true (I, D, V ) |= P (c) if I(P )(I(c))) = true (I, D, V ) |= ∀xA if (I, D, V 0) |= A for all valuations V 0 that agree with V except possibly on x • V 0(y ...
... domain D, an interpretation I, and a valuation V , written (I, D, V ) |= A The definition is by induction: (I, D, V ) |= P (x) if I(P )(V (x)) = true (I, D, V ) |= P (c) if I(P )(I(c))) = true (I, D, V ) |= ∀xA if (I, D, V 0) |= A for all valuations V 0 that agree with V except possibly on x • V 0(y ...
Foundations of Programming Languages Seyed H. Roosta
... aggregate objects from other objects. Binding names to functions, which is a mechanism for denoting names to the new functions being defined. Dynamic storage management, which is an implicit storage allocation and a garbage collection mechanism because the functional languages do not provide facilit ...
... aggregate objects from other objects. Binding names to functions, which is a mechanism for denoting names to the new functions being defined. Dynamic storage management, which is an implicit storage allocation and a garbage collection mechanism because the functional languages do not provide facilit ...
Lambda expressions, functions and binding
... Here's a very simple Haskell program. If you're used to C or Java or Visual Basic, you might think that x, y, z are boxes that you put values in. Later you might change the value in the box using something like x = x + 1. Not in functional programming! This is BINDING, not ASSIGNMENT. We say "x is b ...
... Here's a very simple Haskell program. If you're used to C or Java or Visual Basic, you might think that x, y, z are boxes that you put values in. Later you might change the value in the box using something like x = x + 1. Not in functional programming! This is BINDING, not ASSIGNMENT. We say "x is b ...
15. Functional Programming
... relatively unconcerned with the architecture of the machines on which programs will run ...
... relatively unconcerned with the architecture of the machines on which programs will run ...
Functional Languages
... Program viewed as a collection of functions. – Functional programming is a style of programming in which the basic method of computation is the application of functions to arguments; – A functional language is one that supports and encourages the functional style. ...
... Program viewed as a collection of functions. – Functional programming is a style of programming in which the basic method of computation is the application of functions to arguments; – A functional language is one that supports and encourages the functional style. ...
Scheme and functional programming
... (define x 3) (define y (list '+ x 5)) (eval y user-initial-environment) • The top level of the Scheme interpreter is a read-eval-print loop: read in an expression, evaluate it, and print the result. ...
... (define x 3) (define y (list '+ x 5)) (eval y user-initial-environment) • The top level of the Scheme interpreter is a read-eval-print loop: read in an expression, evaluate it, and print the result. ...
Higher-order functions
... direct input and output functions) because it is nearly impossible to understand side effects in combination with the hard-to-predict evaluation order of a lazy language. ...
... direct input and output functions) because it is nearly impossible to understand side effects in combination with the hard-to-predict evaluation order of a lazy language. ...
Propositional Calculus
... involve function application. Programs may also produce function by returning functions as values. In pure functional programming, this is it, there are no variables, side effects, nor loops. This simplifies semantics but does not reduce computational power. • We will investigate the style of progra ...
... involve function application. Programs may also produce function by returning functions as values. In pure functional programming, this is it, there are no variables, side effects, nor loops. This simplifies semantics but does not reduce computational power. • We will investigate the style of progra ...
List
... • Any identifier is a lambda expression • If M and N are lambda expressions, then the application of M to N, (MN) is a lambda expression • An abstraction, written (x M ) where x is an identifier and M is a lambda expression, is also a lambda expression ...
... • Any identifier is a lambda expression • If M and N are lambda expressions, then the application of M to N, (MN) is a lambda expression • An abstraction, written (x M ) where x is an identifier and M is a lambda expression, is also a lambda expression ...
Chapter 7 Recursion
... Alonzo Church in 1934. It's analogous to Turing machines, but it's both simpler and more practical. Where the Turing machine is something like a model of assembly language, the -calculus is a model of function application. Like Turing machines, it defines a simplified programming language that you ...
... Alonzo Church in 1934. It's analogous to Turing machines, but it's both simpler and more practical. Where the Turing machine is something like a model of assembly language, the -calculus is a model of function application. Like Turing machines, it defines a simplified programming language that you ...
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 ...
What is Logic?
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a penguin. ...
First-Class Functions What is functional programming? First
... Our examples of first-class functions so far all: – Take one function as an argument to another function – Process a number or a list But first-class functions are useful anywhere for any kind of data ...
... Our examples of first-class functions so far all: – Take one function as an argument to another function – Process a number or a list But first-class functions are useful anywhere for any kind of data ...
L11 - Computing at Northumbria University
... • Only one operation : function application (a.k.a. b reduction). (l x . e) v e[v / x] ...
... • Only one operation : function application (a.k.a. b reduction). (l x . e) v e[v / x] ...
MathsReview
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...
Notes
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
... exactly to the proof rules of propositional intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponent ...
Lecture 3.1
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...
Lecture 3.1
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...
... R is reflexive, antisymmetric, and transitive on the elements of S For every s, t S, there exists a greatest lower bound For every s, t S, there exists a lowest upper bound ...