Introduction

In a previous post, I explored the idea that Montague Grammar could be applied to neuro-symbolic AI systems. Montague Grammar offers a framework for converting natural language (especially English) into formal lambda expressions. If we can evaluate those expressions in Prolog, then we can directly execute semantic representations produced by the grammar.

This post presents a lambda calculus reducer implemented in Prolog — a step toward integrating natural language semantics and logic programming.

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What Is Lambda Calculus?

Lambda calculus is one of the most fundamental theoretical models in computer science. It served as a major inspiration for John McCarthy when he designed the Lisp programming language. The original formulation was introduced by Alonzo Church, who, along with Turing, is a pioneer of computability theory.

Lambda Syntax in Prolog

Church's traditional notation (e.g., λx.x) is not well-suited to Prolog, so I use a predicate-logic-friendly syntax.

Lambda Calculus Prolog Representation
λx.x                 l(x, x)
(λx.x) a             [l(x, x), a]

Here, l(X, Body) represents a lambda abstraction, and [Function, Argument] represents application.

Beta Reduction

In lambda calculus, function application is achieved entirely through substitution — known as beta reduction.

For example:

?- reduce([l(x, x), a], Y).
Y = a.

The variable x in the body is replaced with a, and thus the entire expression reduces to a.

Alpha Conversion

To avoid variable name collisions in nested scopes, we need alpha conversion, which renames bound variables.

For example:

l(x, l(x, x)) → l(x, l(x0, x0))

Here, the inner x shadows the outer one. Alpha conversion clarifies the binding structure by renaming inner variables as necessary.

Examples of Reduction

The following examples show the behavior of the reduce/2 predicate implemented in Prolog:

% Identity function
?- reduce([l(x, x), a], Y).
Y = a.

% Constant function (returns y regardless of input)
?- reduce([l(x, y), a], Y).
Y = y.

% Nested lambda (returning x)
?- reduce([l(x, l(y, x)), a], Y).
Y = l(y1, a).

% Double application
?- reduce([[l(x, l(y, x)), a], b], Y).
Y = a.

% Constants remain unchanged
?- reduce(a, a).

% Lambda expressions are returned as-is if not applied
?- reduce(l(x, x), Y).
Y = l(x, x).

% Application where the argument is itself an application
?- reduce([l(x, [x, z]), [l(y, y), a]], Y).
Y = [[l(y, y), a], z].

Conclusion

With this implementation, lambda expressions produced via Montague Grammar can now be executed directly in Prolog. This brings us closer to an integrated system that unifies natural language semantics with symbolic reasoning — a key objective in neuro-symbolic AI.

This work also highlights Prolog's expressive power for modeling substitution-based computation, and its suitability for representing logical and functional semantics.

full code https://github.com/sasagawa888/nprolog/blob/master/library/lambda.pl