Untyped Functional Programs Research Articles

$ \lambda $-DEFINABILITY OF BUILT-IN McCARTHY FUNCTIONS AS FUNCTIONS WITH INDETERMINATE VALUES OF ARGUMENTS

The built-in functions of programming languages are functions with indeterminate values of arguments. The built-in McCarthy functions $ car $, $ cdr $, $ cons $, $ null $, $ atom $, $ if $, $ eq $, $ not $, $ and $, $ or $, are used in all functional programming languages. In this paper we show the $ \lambda $-definability of the built-in McCarthy functions as functions with indeterminate values of arguments. This result is necessary when translating typed functional programming languages into untyped functional programming languages.

ON INTERPRETATION OF TYPED AND UNTYPED FUNCTIONAL PROGRAMS

In this paper the interpretation algorithms for typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $ \leq 1 $, where constants of order 1 are strongly computable, monotonic functions with indeterminate values of arguments. The basic semantics of the typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The interpretation algorithms of typed functional programs are based on substitutions, $ \beta $-reduction and canonical $ \delta $-reduction. The basic semantics of the untyped functional program is the untyped $ \lambda $-term, which is defined by means of the fixed point combinator. The interpretation algorithms of untyped functional programs are based on substitutions and $ \beta $-reduction. Interpretation algorithms are examined for completeness and comparability. It is investigated how the “behavior” of the interpretation algorithm changes after translation of typed functional program into untyped functional program.

ON TRANSLATION OF TYPED FUNCTIONAL PROGRAMS INTO UNTYPED FUNCTIONAL PROGRAMS

In this paper typed and untyped functional programs are considered. Typed functional programs use variables of any order and constants of order $\le$1, where constants of order 1 are strong computable, $\lambda$-definable functions with indeterminate values of arguments. The basic semantics of a typed functional program is a function with indeterminate values of arguments, which is the main component of its least solution. The basic semantics of an untyped functional program is an untyped $\lambda$-term, which is defined by means of a fixed point combinator. An algorithm that translates typed functional program $P$ into untyped functional program $P'$ is suggested. It is proved that the basic semantics of the program $P'$ $\lambda$-defines the basic semantics of the program $P$.

Modules over relative monads for syntax and semantics

We give an algebraic characterization of the syntax and semantics of a class of untyped functional programming languages.To this end, we introduce a notion of 2-signature: such a signature specifies not only the terms of a language, but also reduction rules on those terms. To any 2-signature (S, A) we associate a category of ‘models’. We then prove that this category has an initial object, which integrates the terms freely generated by S, and which is equipped with reductions according to the rules given in A. We call this initial object the programming language generated by (S, A). Models of a 2-signature are built from relative monads and modules over such monads. Through the use of monads, the models – and in particular, the initial model – come equipped with a substitution operation that is compatible with reduction in a suitable sense.The initiality theorem is formalized in the proof assistant Coq, yielding a machinery which, when fed with a 2-signature, provides the associated programming language with reduction relation and certified substitution.

On basic semantics of untyped functional programs

The paper is devoted to untyped functional programs, which are defined as systems of equations with separating variables in the untyped ?-calculus [1, 2]. Basic semantics of such programs is usually defined by means of the fixed-point combinator Y. A theorem on semantics invariance with respect to the fixed-point combinator is proved.

Semantics of Untyped Functional Programs

The paper is devoted to procedural semantics that use interpretation algorithms on which interpreters of untyped functional programming systems are based (or can be based). The aim of this study is to describe such algorithms and compare procedural semantics based on them both with each other and with the basic semantics defined by means of the fixed-point combinator Y.