An application of PER models to program extraction

1993 ◽  
Vol 3 (3) ◽  
pp. 309-331 ◽  
Author(s):  
Stefano Berardi
Keyword(s):  
Equivalence Relation ◽  
Type Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Constructive Proof ◽  
Program Extraction ◽  
Relation Model

Type theory allows us to extract from a constructive proof that a specification is satisfiable a program that satisfies the specification. Algorithms for optimization of such programs are currently the object of research.In this paper we consider one such algorithm, which was described in Beeson (1985) and which we will call ‘Harrop’. This algorithm greatly simplifies programs extracted from proofs in the Pure Construction Calculus. We use a Partial Equivalence Relation model for higher order lambda calculus, to check that t and Harrop(t) return the same outputs from the same inputs, i.e. that they are extensionally equal.As a corollary, we show that it is correct (and, of course, useful) to replace a program t with Harrop(t). Such a correctness result has already been proved by Möhring (Möhring 1989a, 1989b) using realizability semantics, but we obtain it as a corollary of a new result, the extensional equality between t and Harrop(t). Also the semantic method we use is interesting in its own right.

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

2021 ◽  
pp. 1-20
Author(s):  
Ernesto Copello ◽  
Nora Szasz ◽  
Álvaro Tasistro
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Proof Assistant ◽  
First Order ◽  
Constructive Type Theory ◽  
Fundamental Part ◽  
Constructive Type ◽  
Free Variables ◽  
The Church ◽  
Theoretical Results

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

Lambda Calculus, Type Theory, and Natural Language II

2007 ◽  
Vol 18 (2) ◽  
pp. 203-203
Author(s):  
C. Fox ◽  
M. Fernandez ◽  
S. Lappin
Keyword(s):  
Natural Language ◽  
Type Theory ◽  
Lambda Calculus

Tychonoff's theorem in the framework of formal topologies

1997 ◽  
Vol 62 (4) ◽  
pp. 1315-1332 ◽  
Author(s):  
Sara Negri ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
Constructive Proof ◽  
Topological Product ◽  
Essential Difference ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Inductive Property ◽  
Formal Topology

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

The calculus of dependent lambda eliminations

2017 ◽  
Vol 27 ◽  
Author(s):  
AARON STUMP
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Denotational Semantics ◽  
Prototype Implementation ◽  
Constructive Type Theory ◽  
Typed Lambda Calculus ◽  
Constructive Type

AbstractModern constructive type theory is based on pure dependently typed lambda calculus, augmented with user-defined datatypes. This paper presents an alternative called the Calculus of Dependent Lambda Eliminations, based on pure lambda encodings with no auxiliary datatype system. New typing constructs are defined that enable induction, as well as large eliminations with lambda encodings. These constructs are constructor-constrained recursive types, and a lifting operation to lift simply typed terms to the type level. Using a lattice-theoretic denotational semantics for types, the language is proved logically consistent. The power of CDLE is demonstrated through several examples, which have been checked with a prototype implementation called Cedille.

Efficient self-interpretation in lambda calculus

1992 ◽  
Vol 2 (3) ◽  
pp. 345-364 ◽  
Author(s):  
Torben Æ. Mogensen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Lambda Calculus ◽  
The Self ◽  
Constructive Proof ◽  
Compact Representation ◽  
Direct Execution

AbstractWe start by giving a compact representation schema for λ-terms, and show how this leads to an exceedingly small and elegant self-interpreter. We then define the notion of aself-reducer, and show how this too can be written as a small λ-term. Both the self-interpreter and the self-reducer are proved correct. We finally give a constructive proof for the second fixed point theorem for the representation schema. All the constructions have been implemented on a computer, and experiments verify their correctness. Timings show that the self-interpreter and self-reducer are quite efficient, being about 35 and 50 times slower than direct execution using a call-by-need reductions strategy

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