Lecture 10 Notes

... Understanding the atomic propositions and their “evidence semantics” is more subtle. One way it will be clarified is by looking at first-order and higher-order logic and type theory. Those logics refine and explicate our notion or a proposition. Type theory also shows promise in providing a semantic ...

... Understanding the atomic propositions and their “evidence semantics” is more subtle. One way it will be clarified is by looking at first-order and higher-order logic and type theory. Those logics refine and explicate our notion or a proposition. Type theory also shows promise in providing a semantic ...

PDF

... that we can think of ¬φ as corresponding to a function τ → void. 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 ...

... that we can think of ¬φ as corresponding to a function τ → void. 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 ...

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 ...

PDF

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

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

12 Towards a Theory of Document Structure

... in an unrestricted natural language), or not feasible (for instance if we present a proof of a mathematical proposition with the word true, then it would be necessary to find the proof in order to build its abstract syntax tree). If the document is built using a WYSIWYG structure editor, one is temp ...

... in an unrestricted natural language), or not feasible (for instance if we present a proof of a mathematical proposition with the word true, then it would be necessary to find the proof in order to build its abstract syntax tree). If the document is built using a WYSIWYG structure editor, one is temp ...

Document

... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...

... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...

Dependent Types In Lambda Cube

... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...

... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...

A Calculus for Type Predicates and Type Coercion

... wherever the syntax calls for a term of a supertype. To make it clear that we mean this notion of type, we will talk about the static type of a term. – When we evaluate a term using some interpretation, we get an element of the domain. Every element of the domain has exactly one type. Our semantics ...

... wherever the syntax calls for a term of a supertype. To make it clear that we mean this notion of type, we will talk about the static type of a term. – When we evaluate a term using some interpretation, we get an element of the domain. Every element of the domain has exactly one type. Our semantics ...

pdf

... A binary relation is a set of pairs. Let rel range over binary relations. If hx, yi ∈ rel then we sometimes write it x rel y. Let dom(rel ) = {x | hx, yi ∈ rel} and ran(rel ) = {y | hx, yi ∈ rel}. We write rel ∗ for the reflexive and transitive closure of the relation rel (see the first line of Figu ...

... A binary relation is a set of pairs. Let rel range over binary relations. If hx, yi ∈ rel then we sometimes write it x rel y. Let dom(rel ) = {x | hx, yi ∈ rel} and ran(rel ) = {y | hx, yi ∈ rel}. We write rel ∗ for the reflexive and transitive closure of the relation rel (see the first line of Figu ...

classden

... interest is in stating and proving metatheorems on completeness, decidability etc. Such aims are best served with logics with a limited expressivity, as increase in expressivity generally leads to loss of metalogical properties. On the other hand, the goal of providing a general logical framework f ...

... interest is in stating and proving metatheorems on completeness, decidability etc. Such aims are best served with logics with a limited expressivity, as increase in expressivity generally leads to loss of metalogical properties. On the other hand, the goal of providing a general logical framework f ...

PDF (216 KB)

... According to Markov’s CRM approach, all objects are algorithms, where algorithms are understood as ﬁnite strings in a ﬁnite alphabet. All logical connectives are understood in a constructive way. That is, a statement is true if and only if there exists an algorithm that produces a witness of this st ...

... According to Markov’s CRM approach, all objects are algorithms, where algorithms are understood as ﬁnite strings in a ﬁnite alphabet. All logical connectives are understood in a constructive way. That is, a statement is true if and only if there exists an algorithm that produces a witness of this st ...

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 ...

Martin-Löf`s Type Theory

... It is also possible to view a set as a problem description in a way similar to Kolmogorov’s explanation of the intuitionistic propositional calculus [25]. In particular, a set can be seen as a specification of a programming problem; the elements of the set are then the programs that satisfy the spec ...

... It is also possible to view a set as a problem description in a way similar to Kolmogorov’s explanation of the intuitionistic propositional calculus [25]. In particular, a set can be seen as a specification of a programming problem; the elements of the set are then the programs that satisfy the spec ...

Supplemental Reading 1

... P . In axiomatic set theory the existence of this set is guaranteed by the separation axiom. The idea is that the predicate P separates a subset of A as in the example of say the prime numbers, fx :N j prime(x)g. To understand this type, we need to know something about predicates. In axiomatic set t ...

... P . In axiomatic set theory the existence of this set is guaranteed by the separation axiom. The idea is that the predicate P separates a subset of A as in the example of say the prime numbers, fx :N j prime(x)g. To understand this type, we need to know something about predicates. In axiomatic set t ...

Post Systems in Programming Languages Pr ecis 1 Introduction

... of a premise. If a variable appears twice in a premise, then any instance of the production must insert the same string in place of all occurrences of the variable. A single production in a grammar cannot achieve the same result (although the same eect can be achieved in a grammar through a complex ...

... of a premise. If a variable appears twice in a premise, then any instance of the production must insert the same string in place of all occurrences of the variable. A single production in a grammar cannot achieve the same result (although the same eect can be achieved in a grammar through a complex ...

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 ...

A Concurrent Logical Framework: The Propositional Fragment Kevin Watkins , Iliano Cervesato

... We write {A} for the monad type, which in lax logic would be written A [33]. But which types should be available inside the monad? They must be expressive enough to represent the state after a computation step in the concurrent object language. This is most naturally represented by the multiplicati ...

... We write {A} for the monad type, which in lax logic would be written A [33]. But which types should be available inside the monad? They must be expressive enough to represent the state after a computation step in the concurrent object language. This is most naturally represented by the multiplicati ...

Subset Types and Partial Functions

... This paper develops a unified approach to partial functions and subset types, which does not suffer from this anomalous behavior. We begin with a higherorder logic that allows functions to be undefined on some arguments. We extend this logic’s type system to include subset types, but we retain deci ...

... This paper develops a unified approach to partial functions and subset types, which does not suffer from this anomalous behavior. We begin with a higherorder logic that allows functions to be undefined on some arguments. We extend this logic’s type system to include subset types, but we retain deci ...

Interactive Theorem Proving in Coq and the Curry

... In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). O ...

... In Coq, type-checking is done with respect to an environment, determined by the declaration and definitions that were executed earlier. A declaration is used to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). O ...

A Verified Compiler for Probability Density Functions

... type rather than represented as unit + unit. – The type double is called real and it represents a real number with absolute precision as opposed to an IEEE 754 floating point number. In the following subsections, we give the precise syntax, typing rules, and semantics of both our source language and ...

... type rather than represented as unit + unit. – The type double is called real and it represents a real number with absolute precision as opposed to an IEEE 754 floating point number. In the following subsections, we give the precise syntax, typing rules, and semantics of both our source language and ...

... type rather than represented as unit + unit. – The type double is called real and it represents a real number with absolute precision as opposed to an IEEE 754 floating point number. In the following subsections, we give the precise syntax, typing rules, and semantics of both our source language and ...

Structural Types for the Factorisation Calculus

... calculus similar to combinatory logic, introduces the factorisation combinator F , whose reduction behaviour is different depending on whether its first argument is an operator or an application. This enables F to identify and decompose the internal structure of a term, and test for intensional equa ...

... calculus similar to combinatory logic, introduces the factorisation combinator F , whose reduction behaviour is different depending on whether its first argument is an operator or an application. This enables F to identify and decompose the internal structure of a term, and test for intensional equa ...

Rewriting in the partial algebra of typed terms modulo AC

... happens in each dimension of the vector is dissociated from the other components, petri nets possess subtle parallelism facilities. On the other hand the expressiveness over control flow is quiet weak (e.g. it is impossible to encode a stack). The fundamental result over petri nets is the problem of ...

... happens in each dimension of the vector is dissociated from the other components, petri nets possess subtle parallelism facilities. On the other hand the expressiveness over control flow is quiet weak (e.g. it is impossible to encode a stack). The fundamental result over petri nets is the problem of ...

Syntax and Semantics of Dependent Types

... We will henceforth freely suppress type annotations if this increases readability. For instance, we may write x: :M or even x:M instead of x: :M . We sometimes omit a prevailing context ? and thus write ` J instead of ? ` J . We write ` J if we want to emphasise that a judgement holds in th ...

... We will henceforth freely suppress type annotations if this increases readability. For instance, we may write x: :M or even x:M instead of x: :M . We sometimes omit a prevailing context ? and thus write ` J instead of ? ` J . We write ` J if we want to emphasise that a judgement holds in th ...