A Proof Theory for Generic Judgments: An extended abstract
... their intensional nature and guarantee of newness or freshness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus [15], nonces in security protocols [1], reference locations in imperative programming [2, 16], and constructors hidden within abstract data-type ...
... their intensional nature and guarantee of newness or freshness in proof search, eigenvariables have been used to encode name restrictions in the π-calculus [15], nonces in security protocols [1], reference locations in imperative programming [2, 16], and constructors hidden within abstract data-type ...
slides
... Medical researchers announced today that MRI studies reveal that the structure of the human brain is significantly different in adolescence ...
... Medical researchers announced today that MRI studies reveal that the structure of the human brain is significantly different in adolescence ...
Adjointness in Foundations
... The poset reflection expresses the idea of “there exists a deduction A → B” in C/X, and PX serves as a system of “proof-theoretic propositions” about elements of X. In case C is actually a topos, there is a natural map PX → PC (X) to the usual subobject lattice, defined by taking the image of any A ...
... The poset reflection expresses the idea of “there exists a deduction A → B” in C/X, and PX serves as a system of “proof-theoretic propositions” about elements of X. In case C is actually a topos, there is a natural map PX → PC (X) to the usual subobject lattice, defined by taking the image of any A ...
Morley`s number of countable models
... Proof. Consider any two finite sequences of elements ha0 , . . . , an−1 i and hb0 , . . . , bn−1 i of A and B respectively. Either they have the same Lα -type for each α < ω1 or there is a least α such that they have different Lα -types. Since there is only a countable number of pairs of finite sequ ...
... Proof. Consider any two finite sequences of elements ha0 , . . . , an−1 i and hb0 , . . . , bn−1 i of A and B respectively. Either they have the same Lα -type for each α < ω1 or there is a least α such that they have different Lα -types. Since there is only a countable number of pairs of finite sequ ...
curried functions - Universitatea "Politehnica"
... (syntactic terms) to yield values (abstract entities that we regard as answers). Every value has an associated type. 5 :: Integer 'a' :: Char inc :: Integer -> Integer [1,2,3] :: [Integer] ('b',4) :: (Char,Integer) ...
... (syntactic terms) to yield values (abstract entities that we regard as answers). Every value has an associated type. 5 :: Integer 'a' :: Char inc :: Integer -> Integer [1,2,3] :: [Integer] ('b',4) :: (Char,Integer) ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... way of putting it is that R(α) is the result P α (∅) of iterating the PowerSet operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upo ...
... way of putting it is that R(α) is the result P α (∅) of iterating the PowerSet operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upo ...
Chapter 7: Functional Programming Languages
... defined function symbol; in the functional language, it can be any expression (for instance, a lambda abstract or an application). Thus the evaluation of f is not simply a look-up in the function table. But we can just replace the look-up by a step of evaluating the expression f . This evaluation re ...
... defined function symbol; in the functional language, it can be any expression (for instance, a lambda abstract or an application). Thus the evaluation of f is not simply a look-up in the function table. But we can just replace the look-up by a step of evaluating the expression f . This evaluation re ...
Lecture 8: Back-and-forth - to go back my main page.
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
MoggiMonads.pdf
... programming language and categories with a monad satisfying the mono requirement. For other programming languages we will give only their translation in a suitable extension of the metalanguage. In this way, issues like call-by-value versus call-by-name affect the translation, but not the metalangua ...
... programming language and categories with a monad satisfying the mono requirement. For other programming languages we will give only their translation in a suitable extension of the metalanguage. In this way, issues like call-by-value versus call-by-name affect the translation, but not the metalangua ...
Functional Programming
... Every expression must have a valid type, which is calculated prior to evaluating the expression by a process called type inference; ...
... Every expression must have a valid type, which is calculated prior to evaluating the expression by a process called type inference; ...
Links> Web programming without tiers
... it permits its users to boldly go where no progamming language has gone before. ...
... it permits its users to boldly go where no progamming language has gone before. ...
A Brief Introduction to the Intuitionistic Propositional Calculus
... Problem 1 Prove that α ⇒ (β ⇒ γ) `I (α ∧ β) ⇒ γ. Problem 2 Show that α ⇒ β 6`I ¬α ∨ β by demonstrating that there exists a Kripke model K = (W, ≤, |=) and a world w ∈ W such that w |= α ⇒ β, but w 6|= ¬α ∨ β. Problem 3 Show that world w1 in the simple Kripke model in Section 4 does not satisfy Peirc ...
... Problem 1 Prove that α ⇒ (β ⇒ γ) `I (α ∧ β) ⇒ γ. Problem 2 Show that α ⇒ β 6`I ¬α ∨ β by demonstrating that there exists a Kripke model K = (W, ≤, |=) and a world w ∈ W such that w |= α ⇒ β, but w 6|= ¬α ∨ β. Problem 3 Show that world w1 in the simple Kripke model in Section 4 does not satisfy Peirc ...
Type Class
... – symbol always represents the same value – Equational reasoning (equals can be substituted by equals) • easy mathematical manipulation, parallel execution, etc. ...
... – symbol always represents the same value – Equational reasoning (equals can be substituted by equals) • easy mathematical manipulation, parallel execution, etc. ...
Chapter 11 - Functional Programming, Part II: ML, Delayed
... We say what is needed for a type a to be in a class. In this case, we need == defined over a. In other words, a ->a->Bool class Eq a where (==) :: a-> a-> Bool Members of a type class are called its instances. Functions from Int->Int are NOT of type Eq since there is no algorithm to decide if two fu ...
... We say what is needed for a type a to be in a class. In this case, we need == defined over a. In other words, a ->a->Bool class Eq a where (==) :: a-> a-> Bool Members of a type class are called its instances. Functions from Int->Int are NOT of type Eq since there is no algorithm to decide if two fu ...
Chapter 11 - Functional Programming, Part II: ML, Delayed
... We say what is needed for a type a to be in a class. In this case, we need == defined over a. In other words, a ->a->Bool class Eq a where (==) :: a-> a-> Bool Members of a type class are called its instances. Functions from Int->Int are NOT of type Eq since there is no algorithm to decide if two fu ...
... We say what is needed for a type a to be in a class. In this case, we need == defined over a. In other words, a ->a->Bool class Eq a where (==) :: a-> a-> Bool Members of a type class are called its instances. Functions from Int->Int are NOT of type Eq since there is no algorithm to decide if two fu ...
Haskell Summary Functions • A function takes 1 or more parameter
... • List comprehensions : lists of the form [x|x satisfies a given property] Function application Function application is denoted using spaces to separate the arguments, and a multiplication is denoted using ∗: Example: f a b + c * d Also, functions have precedence over all other operators. Hence, we ...
... • List comprehensions : lists of the form [x|x satisfies a given property] Function application Function application is denoted using spaces to separate the arguments, and a multiplication is denoted using ∗: Example: f a b + c * d Also, functions have precedence over all other operators. Hence, we ...
Lecture 9
... allEqual 1 2 3 which returns False, because Int is an Eq type and therefore allEqual :: Int -> Int -> Int -> Bool • we can check pairs of Integers and Characters: allEqual (1,’a’) (1,’a’) (1’a’) which returns True, ...
... allEqual 1 2 3 which returns False, because Int is an Eq type and therefore allEqual :: Int -> Int -> Int -> Bool • we can check pairs of Integers and Characters: allEqual (1,’a’) (1,’a’) (1’a’) which returns True, ...
