Miranda * A Functional Language
... Basic Themes of Miranda • Miranda is purely functional - there are no side effects or imperative features of any kind • A program, called a “script”, contains a collection of equations defining various functions and data structures • Changing the order of equations in the script does not change the ...
... Basic Themes of Miranda • Miranda is purely functional - there are no side effects or imperative features of any kind • A program, called a “script”, contains a collection of equations defining various functions and data structures • Changing the order of equations in the script does not change the ...
type - ktuce
... (1) Try out the above slides using Hugs. (2) Fix the syntax errors in the program below, and test your solution using Hugs. ...
... (1) Try out the above slides using Hugs. (2) Fix the syntax errors in the program below, and test your solution using Hugs. ...
Theorems for free! - Computing Science
... a function; this has already been explored to a certain extent by Sheeran [She89]. So there is reason to believe that further research will further extend the applicability of this method. Many functional languages, including Standard ML [Mil84, Mil87], Miranda1 [Tur85], and Haskell [HW88], are bas ...
... a function; this has already been explored to a certain extent by Sheeran [She89]. So there is reason to believe that further research will further extend the applicability of this method. Many functional languages, including Standard ML [Mil84, Mil87], Miranda1 [Tur85], and Haskell [HW88], are bas ...
19th Century Logic and 21st Century Computing
... A and B one can deduce B ∧ A, and then applies modus ponens. One might reasonably expect that such a proof could be simplified. A major contribution of Gentzen’s 1934 paper was the subformula property: he showed that any proof of a statement Γ ` A can be simplified so that the only propositions it m ...
... A and B one can deduce B ∧ A, and then applies modus ponens. One might reasonably expect that such a proof could be simplified. A major contribution of Gentzen’s 1934 paper was the subformula property: he showed that any proof of a statement Γ ` A can be simplified so that the only propositions it m ...
A Calculus for Type Predicates and Type Coercion
... the corresponding type predicate literal, if we forbid spurious applications of the γ-rule for non-normalized terms. Lemma 1. Let H be a saturated branch obtained by a derivation in which the < A ∈ H. Let φ ∈ H contain a subterm γ rule is only applied for terms tA with t − < A. tA , i.e. either f A ...
... the corresponding type predicate literal, if we forbid spurious applications of the γ-rule for non-normalized terms. Lemma 1. Let H be a saturated branch obtained by a derivation in which the < A ∈ H. Let φ ∈ H contain a subterm γ rule is only applied for terms tA with t − < A. tA , i.e. either f A ...
First-Order Logic with Dependent Types
... ΣB Σ0 . . . Σd is an LF signature, and let Σ n abbreviate ΣB Σ0 . . . Σn for n ≤ d. Σ is called a DFOL signature if 1. only sort, function or predicate symbols are declared in Σ0 . . . Σd , 2. all sort symbol declarations in Σn have only arguments from Σ n−1 , 3. the target of a function symbol decl ...
... ΣB Σ0 . . . Σd is an LF signature, and let Σ n abbreviate ΣB Σ0 . . . Σn for n ≤ d. Σ is called a DFOL signature if 1. only sort, function or predicate symbols are declared in Σ0 . . . Σd , 2. all sort symbol declarations in Σn have only arguments from Σ n−1 , 3. the target of a function symbol decl ...
Martin-Löf`s Type Theory
... One of the basic ideas behind Martin-Löf’s type theory is the Curry-Howard interpretation of propositions as types, that is, in our terminology, propositions as sets. This view of propositions is closely related to Heyting’s explanation of intuitionistic logic [21] and will be explained in detail b ...
... One of the basic ideas behind Martin-Löf’s type theory is the Curry-Howard interpretation of propositions as types, that is, in our terminology, propositions as sets. This view of propositions is closely related to Heyting’s explanation of intuitionistic logic [21] and will be explained in detail b ...
connections to higher type Recursion Theory, Proof-Theory
... using the given bijective pairing of numbers. However, since we are interested in higher type computations, as given by the HPEF, we need also other kinds of higher type objects, such as exponentiations, in the category. Unfortunately, there is no general way to enumerate the set of morphisms of two ...
... using the given bijective pairing of numbers. However, since we are interested in higher type computations, as given by the HPEF, we need also other kinds of higher type objects, such as exponentiations, in the category. Unfortunately, there is no general way to enumerate the set of morphisms of two ...
Higher-Order Modal Logic—A Sketch
... Just as in the classical setting there are standard higher-order modal models and non-standard ones. Because of space limitations I’ll only sketch the standard version, and say a few words later on about the non-standard one. A higher-order modal model is a structure M = hG, R, D, Ii, and we spend m ...
... Just as in the classical setting there are standard higher-order modal models and non-standard ones. Because of space limitations I’ll only sketch the standard version, and say a few words later on about the non-standard one. A higher-order modal model is a structure M = hG, R, D, Ii, and we spend m ...
The next Haskell will be strict
... nothing now, but it catches up with you in the end Enforced purity is like paying up front: painful on Day 1, but usually worth it But we made one big mistake... ...
... nothing now, but it catches up with you in the end Enforced purity is like paying up front: painful on Day 1, but usually worth it But we made one big mistake... ...
Getting Started With . . . Haskell for Knowledge Representation
... Examples: [Int] is the type of lists of integers; [Char] is the type of lists of characters, or strings. • By pair- or tuple-formation: if a and b are types, then (a,b) is the type of pairs with an object of type a as their first component, and an object of type b as their second component. If a, b ...
... Examples: [Int] is the type of lists of integers; [Char] is the type of lists of characters, or strings. • By pair- or tuple-formation: if a and b are types, then (a,b) is the type of pairs with an object of type a as their first component, and an object of type b as their second component. If a, b ...
A static analysis for Bulk Synchronous Parallel ML to avoid
... it is easy to see that not every instantiations from a variable type are in one of these kinds of types. Take for example, the simple type (α par)→ int or the simple type (α par) and the instantiation α =int par: it leads to a nesting of parallel vectors. To remedy this problem, we will use constrai ...
... it is easy to see that not every instantiations from a variable type are in one of these kinds of types. Take for example, the simple type (α par)→ int or the simple type (α par) and the instantiation α =int par: it leads to a nesting of parallel vectors. To remedy this problem, we will use constrai ...
Introduction, Scheme basics (expressions, values)
... • A procedure type is a composite type, as it is composed of the types of its inputs (domain) and output (range) • In fact, the procedure type can be instantiated with any type for domain and range, resulting in a different type for the procedure (=data) • Such types are called polymorphic – Another ...
... • A procedure type is a composite type, as it is composed of the types of its inputs (domain) and output (range) • In fact, the procedure type can be instantiated with any type for domain and range, resulting in a different type for the procedure (=data) • Such types are called polymorphic – Another ...
Lecture Notes
... div is enclosed in back quotes, not forward; x `f` y is just syntactic sugar for f x y. So this is just saying “div (sum ns) (length ns) ...
... div is enclosed in back quotes, not forward; x `f` y is just syntactic sugar for f x y. So this is just saying “div (sum ns) (length ns) ...
Lecture Slides
... The Standard Prelude: List Madness! Haskell comes with a large number of standard library functions. In addition to the familiar numeric functions such as + and *, the library also provides many useful functions on lists. Select the first element of a list: ...
... The Standard Prelude: List Madness! Haskell comes with a large number of standard library functions. In addition to the familiar numeric functions such as + and *, the library also provides many useful functions on lists. Select the first element of a list: ...