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McCarthy’s LISP and Basis
for Theory of Computation
Hong-Yi Dai
University of Edinburgh, UK
It is well-known that LISP is an algebraic list-
processing language designed for artificial in-
telligence, dedicated to symbolic computation,
powered by function orientation, enhanced by
program manipulation, and underpinned by
garbage collection. Ever since it was born, LISP
has been celebrated by practitioners as a genu-
ine high-level, functional- and meta-program-
ming language for potentially hard problems
involving symbol manipulation. Largely overlo-
oked is that LISP can also serve as a formalism
for theory of computation. Indeed, it embraces
and integrates ideas from all three major cha-
racterizations of computation, namely lambda
calculus, general recursive function, and Tu-
ring machine. Unfortunately, both LISP itself
and the formal basis distilled from it were not
well received for theory of computation. 50
years later, given prominent and promising
developments of the programming-language
approach to computability and complexity, it
is time to give McCarthy’s pioneering work a
reappraisal. We proceed as follows.
First, we review the development of
LISP.We start with McCarthy’s motivation for
designing a programming language for artifi-
cial intelligence, and the inspirations behind
the initial set of features: algebraic notation
and list processing. Of other language features
outlined in the 1st AI memo [7], we pay par-
ticular attention to two, namely conditional
expression and recursive function definition
(making use of the former), both later be-
coming the key components of McCarthy’s
“basis for a mathematical theory of compu-
tation” [12]. We note two revisions to the
language documented in two subsequent AI
memos – Memo 3 [8] introducing garbage
collection and refining list operations, Memo 4
[9] adopting Church’s -notation (to facilitate
higher-order functions) and naming the lan-
guage LISP (short for LISt Processor)–which
simplified it to such an extent that McCarthy
agilely realized its capability to describe com-
putable functions in a much neater way “than
the Turing machines or general recursive defi-
nitions used in [computable]1 function theo-
ry” [15, p.219]. We highlight McCarthy’s cast
of LISP “both as a programming language and
as a formalism for doing [computable] fun-
ction theory” [15, p.219] in the 8th AI memo
[10] (draft of the landmark paper [11]) where
he showed two similarities between LISP and
Turing machine: one shallow by simulating Tu-
ring machine in LISP, the other deep by giving
a universal function in LISP for LISP.
Second, we revisit McCarthy’s computa-
tion-theoretical basis [12] distilled from LISP
by parameterizing the computation domain.
We briefly mention some elementary results in
traditional theory of computation McCarthy
reproduced in the formalism obtained by in-
1 Following Soare’s recommendation [17], we
substitute the word computable for recursive in
the term recursive function theory.
