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... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...

... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...

Propositions as types

... 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 behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...

... 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 behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...

Notes

... does it. This is a function that takes a pair of functions as its argument and returns their composition. The proof tree that establishes the typing of this function is essentially an intuitionistic proof of the transitivity of implication. Here is another example. Consider the formula ∀P, Q, R . (P ...

... does it. This is a function that takes a pair of functions as its argument and returns their composition. The proof tree that establishes the typing of this function is essentially an intuitionistic proof of the transitivity of implication. Here is another example. Consider the formula ∀P, Q, R . (P ...

Continuous Model Theory - Math @ McMaster University

... We fix a language L and a complete theory T in this language. For a tuple of sorts S from L, we define the set SS (T ) to be all complete types defined on FS . The logic topology on SS (T ) is the restriction of the weak-* topology on the dual space of FS . Equivalently, the collection of sets {p ∈ ...

... We fix a language L and a complete theory T in this language. For a tuple of sorts S from L, we define the set SS (T ) to be all complete types defined on FS . The logic topology on SS (T ) is the restriction of the weak-* topology on the dual space of FS . Equivalently, the collection of sets {p ∈ ...

A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL

... Before defining our class of models, we first review the notions of a frame and a boolean prelattice, in terms of which these models will be specified. We take a (Henkin) frame to be a type-indexed family of sets S = hSA i such that SB A is a (possibly proper) subset of the set of functions from SA ...

... Before defining our class of models, we first review the notions of a frame and a boolean prelattice, in terms of which these models will be specified. We take a (Henkin) frame to be a type-indexed family of sets S = hSA i such that SB A is a (possibly proper) subset of the set of functions from SA ...

Tsinghua Software Day Program

... operational). After graduating from the Ecole Normale Sup□□ rieure of Paris, he worked with G□□ rard Berry on a mathematical account of the notion of sequential program, where programs as functions and programs as algorithms get closer. His most widely-known contributions are outcomes of this early ...

... operational). After graduating from the Ecole Normale Sup□□ rieure of Paris, he worked with G□□ rard Berry on a mathematical account of the notion of sequential program, where programs as functions and programs as algorithms get closer. His most widely-known contributions are outcomes of this early ...

The superjump in Martin-Löf type theory

... Martin-Löf, in 1975 [11] and in his 1984 monograph [12] on an intuitionistic theory of types, gave a framework for a theory of constructive types or sets. The role of universes in this type theory is to allow for the formation of sets of sets which are themselves closed under certain set forming op ...

... Martin-Löf, in 1975 [11] and in his 1984 monograph [12] on an intuitionistic theory of types, gave a framework for a theory of constructive types or sets. The role of universes in this type theory is to allow for the formation of sets of sets which are themselves closed under certain set forming op ...

Propositions as [Types] - Research Showcase @ CMU

... types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK interpretation to the standard category-theoretic interpretation of proposi ...

... types of Maietti [Mai98], in a suitable setting. Palmgren [Pal01] formulated a BHK interpretation of intuitionistic logic and used image factorizations, which are used in the semantics of our bracket types, to relate the BHK interpretation to the standard category-theoretic interpretation of proposi ...

Notes - Cornell Computer Science

... easily able to see some of the deepest ideas in action. For example, you already know that in some sense, OCaml is a universal programming language. It is surprising how small the universal core of OCaml is. 3. We are studying the functional programming paradigm, algorithm design, and precise proble ...

... easily able to see some of the deepest ideas in action. For example, you already know that in some sense, OCaml is a universal programming language. It is surprising how small the universal core of OCaml is. 3. We are studying the functional programming paradigm, algorithm design, and precise proble ...

Proof Theory in Type Theory

... the reference [1, 4], one should expect that it is not needed). This should come from an analysis of the given proof. For instance, it seems that we are really working in the fragment of (S0 ) with only positive sequents, and that we never really need to do an induction on a tree with uncountable br ...

... the reference [1, 4], one should expect that it is not needed). This should come from an analysis of the given proof. For instance, it seems that we are really working in the fragment of (S0 ) with only positive sequents, and that we never really need to do an induction on a tree with uncountable br ...

Dependent Types In Lambda Cube

... mathematician’s intuition. The intuitionism is sometimes mistaken for constructivism, but the intuitionism is just one kind of constructivism. On the other hand, it could be useful for our purposes not to distuinguish between these two philosophies. The main reason is that the true of a statement in ...

... mathematician’s intuition. The intuitionism is sometimes mistaken for constructivism, but the intuitionism is just one kind of constructivism. On the other hand, it could be useful for our purposes not to distuinguish between these two philosophies. The main reason is that the true of a statement in ...

An introduction to functional programming using Haskell

... Functions are values too! Hence you can pass functions as arguments to other functions (called higher order functions) ...

... Functions are values too! Hence you can pass functions as arguments to other functions (called higher order functions) ...

Jun 2004 - University of Malta

... Define a function fromTo, which takes two integers and returns the list of integers starting from the first number and finishing with the second. For example, fromTo 2 5 would return [2,3,4,5]. Define a function dots which takes an integer parameter n and returns a string consisting of n dots. For e ...

... Define a function fromTo, which takes two integers and returns the list of integers starting from the first number and finishing with the second. For example, fromTo 2 5 would return [2,3,4,5]. Define a function dots which takes an integer parameter n and returns a string consisting of n dots. For e ...