The Impact of the Lambda Calculus in Logic and Computer Science

@article{Barendregt1997TheIO,
  title={The Impact of the Lambda Calculus in Logic and Computer Science},
  author={HENK P. Barendregt},
  journal={Bulletin of Symbolic Logic},
  year={1997},
  volume={3},
  pages={181 - 215}
}
  • H. Barendregt
  • Published 1 June 1997
  • Computer Science
  • Bulletin of Symbolic Logic
Abstract One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand. 
Comparing Mathematical Provers
TLDR
This work compares fifteen systems for the formalizations of mathematics with the computer based on the size of their library, the strength of their logic and their level of automation.
Foundations of Mathematics from the Perspective of Computer Verification
TLDR
It is argued that most philosophical views over-emphasize a particular aspect of the mathematical endeavor.
Computability Via The Lambda Calculus with Patterns
TLDR
It is shown that, with the introduction of the new congruence, all the basic properties of the original lambda calculus with patterns still hold, including the Church-Rosser theorem.
Concrete Abstractions: An Introduction to Computer Science Using Scheme
TLDR
This text covers the basics of programming and data structures, and gives first-time computer science students the opportunity to not only write programs, but to prove theorems and analyze algorithms as well.
The sound of lambda
TLDR
The CodeKlavier’s Ckalcuλator: an arithmetic calculator for the piano following lambda calculus principles is discussed, which adds a conceptual, creative and performative dimension to otherwise simple arithmetic operations.
The Calculus of Natural Calculation
The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus
Linearization by Program Transformation
We identify a restricted class of terms of the lambda calculus, here called weak linear, that includes the linear lambda-terms keeping their good properties of strong normalization, non-duplicating
Programming in the λ-Calculus: From Church to Scott and Back
  • J. M. Jansen
  • Computer Science
    The Beauty of Functional Code
  • 2013
TLDR
This paper shows how to convert programs written in functional programming languages like Clean and Haskell to closed λ-expressions by using the Scott-encoding for Algebraic Data Types instead of the more common Church encoding to obtain an encoding that is better comprehensible and also more efficient.
Models of the Lambda Calculus : An Introduction
The λ -calculus is a symbolic formalism for describing and calculating with functions. To give meaning to expressions in the λ -calculus they must be interpreted in terms of standard mathematical
...
...

References

SHOWING 1-10 OF 255 REFERENCES
Highlights of the History of the Lambda-Calculus
  • J. Rosser
  • Mathematics
    Annals of the History of Computing
  • 1984
This paper gives an account of both the lambda-calculus and its close relative, the combinatory calculus, and explains why they are of such importance for computer software. The account includes the
The lambda calculus - its syntax and semantics
  • H. Barendregt
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1985
The Mechanical Evaluation of Expressions
TLDR
It is shown how some forms of expression in current programming languages can be modelled in Church's X-notation, and a way of "interpreting" such expressions is described, which suggests a method of analyzing the things computer users write.
Background : computational structures
J.W. Klop: Term rewriting systems H.P. Barendregt: Lambda calculi with types D.M. Gabbay: Elements of algorithmic proof Lawrence C. Paulson: Designing a theorem prover Colin Stirling: Modal and
A Theory of Type Polymorphism in Programming
  • R. Milner
  • Computer Science
    J. Comput. Syst. Sci.
  • 1978
Automatic Synthesis of Typed Lambda-Programs on Term Algebras
Origins of Recursive Function Theory
  • S. Kleene
  • Mathematics
    Annals of the History of Computing
  • 1981
TLDR
The notion of "?-definability" was the first of what are now accepted as equivalent exact mathematical descriptions of the class of the functions for which algorithms exist and traces the investigation in 1931-1933 by which the notion was quite unexpectedly so accepted.
A Self-Interpreter of Lambda Calculus Having a Normal Form
TLDR
The notion of a canonical algebraic term rewriting system can be interpreted in the lambda calculus by the Bohm — Piperno technique in such a way that strong normalization is preserved and allows us to improve some recent results of Mogensen concerning efficient godelizations.
Theoretical Pearls: Self-interpretation in lambda calculus
  • H. Barendregt
  • Computer Science
    Journal of Functional Programming
  • 1991
TLDR
Self-interpretation will be shown here to be possible in lambda calculus to represent functional programs including their input.
...
...