A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus

Abstract

We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction

Call-by-Value Non-determinism in a Linear Logic Type Discipline

“…We would like to design type systems characterizing convergence properties in these systems. First steps have been done in [2,3].…”

Non determinism through type isomorphism

“…In such a formalism, there is no possibility to tie terms with different types: if r and s have both type A, then α.r + β .s have type (α + β ).A, however if the types of r and s differ, the previous term cannot be typed. That weakness is solved in [2], where a more powerful system is introduced, with a type system also allowing for linear combination of types, just like for terms. In both these systems, while powerful, it is hard to establish a connection with a well-known logic.…”

“…As mentioned in Section 4, the calculus λ + has implicit scalars on it, which can convert this nondeterministic setting into a probabilistic one. The original motivation behind λ lin [3] and its vectorial type system [2] was to encode quantum computing on it. A projection depending on scalars could lead to a measurement operator in a future design-after other questions like deciding orthogonality [25] have been addressed in that setting.…”

The probability of non-confluent systems

“…One way to interpret such a linear combination is that it represents a term which is the term r with probability α, or the term s with probability β . However, endowing such a calculus with a non-restrictive type system is a challenge [3,4].…”