Lambda-calculus, types and models
Lambda-calculus, types and models
Complete Sets of Reductions for Some Equational Theories
Journal of the ACM (JACM)
The differential Lambda-calculus
Theoretical Computer Science
Mathematical Structures in Computer Science
Theoretical Computer Science
A Computational Definition of the Notion of Vectorial Space
Electronic Notes in Theoretical Computer Science (ENTCS)
Linear-algebraic λ-calculus: higher-order, encodings, and confluence.
RTA '08 Proceedings of the 19th international conference on Rewriting Techniques and Applications
Algebraic Totality, towards Completeness
TLCA '09 Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications
Mathematical Structures in Computer Science
Intuitionistic differential nets and lambda-calculus
Theoretical Computer Science
Probabilistic coherence spaces as a model of higher-order probabilistic computation
Information and Computation
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We define an extension of λ-calculus with linear combinations,endowing the set of terms with a structure of R-module, where R is a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then extend β-reduction on those algebraic λ-terms as follows: at+u reduces to at′+u as soon as term t reduces to t′ and a is a non-zero scalar. We prove that reduction is confluent. Under the assumption that the set R of scalars is positive (i.e. a sum of scalars is zero iff all of them are zero), we show that this algebraic λ-calculus is a conservative extension of ordinary λ-calculus. On the other hand, we show that if R admits negative elements, then every term reduces to every other term. Preliminary Definitions and Notations. Recall that a rig (or "semi-ring with zero and unit") is the same as a ring, without the condition that every element admits an opposite for addition. Let R be a rig. We write R• for R \ {0}. We denote by letters a, b, c the elements of R, and say that R is positive if, for all a, b λ R, a + b = 0 implies a = 0 and b = 0. An example of positive rig is N, the set of natural numbers, with usual operations. Also, we write application of λ-terms à la Krivine: (s) t denotes the application of term s to term t.