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... statement about the existence of certain types of generics (a bit less than weakly 2generics) that can be proven by a finite injury priority argument and that is conservative over both IΣ1 and IΣ2 but not over BΣ2 , as it implies IΣ2 over BΣ2 . This seems to be the first example of such a theorem. T ...
ppt - TAMU Computer Science Faculty Pages
ppt - TAMU Computer Science Faculty Pages

... •Sentences are called statements, expressions, terms, commands, and so on •Words are called tokens •Grammar rules describe both tokens and statements • Often, grammars alone cannot capture exactly the set of valid programs. Grammars combined with additional rules are a common approach. ...
Notes on Simply Typed Lambda Calculus
Notes on Simply Typed Lambda Calculus

... languages that express both computational and logical information. Computational information in that they can be see as functional programming languages, or more realistically, a solid core on which to build a functional language. Logical information in two ways. First, typed λ-calculi can be used d ...
Example
Example

... It follows that the last line in the definition of ExtAutomaton contains a list of tuples (state, tokenUnit), with state being a final state where a lexical unit is recognised and sent out in tokenUnit. Every tokenUnit is a tuple where the first component is a code (an integer value used by the pars ...
Document
Document

... • The Haskell type IO a is the type of I/O actions of type a, e.g., the elements of a are somehow wrapped into an I/O container; the monad IO. • An expression of type IO a is a program which will do some I/O and then return a value of type a. • One way of looking at the I/O a types is that they prov ...
The Natural Order-Generic Collapse for ω
The Natural Order-Generic Collapse for ω

In: x - UCF Complex Adaptive Systems Laboratory
In: x - UCF Complex Adaptive Systems Laboratory

... • python: to start a Python shell and ctrl-D to exit ...
Lecture Notes on Stability Theory
Lecture Notes on Stability Theory

Temporal Logic with “Until”, Functional Reactive Programming with
Temporal Logic with “Until”, Functional Reactive Programming with

... Because of the Curry–Howard correspondence between our FRP language and our temporal logic, we can take the categorical semantics from the previous subsection as a semantics for FRP. Let (T, 6, B) be a fan category. Since B is a CCCC, it models a system of ordinary types that comprises the type cons ...
A general introduction to Functional Programming using Haskell
A general introduction to Functional Programming using Haskell

... where keyword; it is a function that is applied to sequences of values • if the sequence to which sum_seq is applied is empty, the result is 0 • otherwise, the result of sum_seq is given by adding the first element of the sequence to the sum of the other elements ...
A Conditional Logical Framework *
A Conditional Logical Framework *

... holds iff “all free variables occurring in M belong to a subterm which can be typed in the derivation with o”. This is precisely what is needed to encode it correctly, provided o is the type of propositions. Indeed, if all the free variables of a proof term satisfy such condition, it is clear, by in ...
Chapter 11 - Functional Programming, Part II: ML, Delayed
Chapter 11 - Functional Programming, Part II: ML, Delayed

... ML and Scheme obey the standard "applicativeorder" evaluation rule (as do C and Java): all parameters (arguments) are evaluated prior to the execution of a call. It is surprising how often we need a different rule: (define (my-if a b c) (if a b c)) This definition can't work because both b and c are ...
CMSC330 Summer 2010—Midterm #2
CMSC330 Summer 2010—Midterm #2

... exists, then put will just update the value. You can use helper functions if necessary. let rec put helper (x,y) a m = match m with Nil –> Map((x,y),a) | Map((k,v),t) –> if k = x then put helper (x,y) a t else put helper (x,y) (Map((k,v),a)) t;; ...
PPT
PPT

... • Functions have their own local state • Objects can send and receive messages • Objects can refer to themselves • Object Oriented Programming is a programming language paradigm that facilitates defining, handling and coordinating objects. ...
Inductive Types in Constructive Languages
Inductive Types in Constructive Languages

... such objects, and inductive types are types whose objects are generated by production rules. The purpose of this dissertation is twofold. First, I am searching for languages in which the mathematician can express his inspirations well structured, correct, and yet as freely as possible. Secondly, I w ...
- Free Documents
- Free Documents

... By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that to ...
overview on declarative programming
overview on declarative programming

... must be equal. For example, map length [ "Haskell", "Curry" ] is a valid application of map because the a in the type signature of map can be instantiated with String which is defined as [ Char ] and matches the argument type [ a ] of length. The type b is instantiated with Int and, therefore, the r ...
Slide 1
Slide 1

... (* infix *) ...
Document
Document

The Development of Categorical Logic
The Development of Categorical Logic

... concept of classifying topos for a first-order theory, that is, a topos obtained by freely adjoining a model of the theory to the topos of constant sets. The roots of this idea lie in the work of the Grothendieck school and in Lawvere’s functorial semantics, but it was Joyal and Reyes (see Reyes 197 ...
On Sets of Premises - Matematički Institut SANU
On Sets of Premises - Matematički Institut SANU

popl13
popl13

... • In C and Pascal, identifiers of type int are mutable, but identifiers of function type are constants so that they cannot be changed to other values (functions). In other words, in C and Pascal, whether variables are constants or not depends on the types (whether they are functions or not). ...
The Dedekind Reals in Abstract Stone Duality
The Dedekind Reals in Abstract Stone Duality

... is related to open subspaces in the way that compactness is to closed ones. As far as this paper is concerned, overtness is simply part of the axiomatic structure, but we shall show in following work [J] that it explains the situations in which equations f x = 0 for f : R → R can or cannot be solved ...
Introduction to Functional Programming
Introduction to Functional Programming

... The principle of mathematical induction says exactly that every natural number is generated by starting with 0 and repeatedly adding one, i.e. applying the successor operation S (n) = n + 1. If we regard the natural numbers as a set or a type, then we may say that it is generated by the constructors ...
FUNCTIONAL PEARL Data types `a la carte
FUNCTIONAL PEARL Data types `a la carte

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Intuitionistic type theory

Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher, in 1972. Martin-Löf has modified his proposal a few times; his 1971 impredicative formulation was inconsistent as demonstrated by Girard's paradox. Later formulations were predicative. He proposed both intensional and extensional variants of the theory. For more detail see the section on Martin-Löf type theories below.Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs. This identification is usually called the Curry–Howard isomorphism, which was originally formulated for intuitionistic logic and simply typed lambda calculus. Type theory extends this identification to predicate logic by introducing dependent types, that is types which contain values.Type theory internalizes the interpretation of intuitionistic logic proposed by Brouwer, Heyting and Kolmogorov, the so-called BHK interpretation. The types in type theory play a similar role to sets in set theory but functions definable in type theory are always computable.
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