Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic
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Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic∗
Patrick Baillot 0
Anupam 0
0 Univ Lyon , CNRS, ENS de Lyon, UCB Lyon 1, LIP, France Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIP , France
We prove a general form of 'free-cut elimination' for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing theorem for a fragment of this arithmetic via the 'witness function method', showing that the provably convergent functions are precisely the polynomial-time functions. The programs extracted are implemented in the framework of 'safe' recursive functions, due to Bellantoni and Cook, where the ! modality of linear logic corresponds to normal inputs of a safe recursive program.
and phrases proof theory; linear logic; bounded arithmetic; polynomial time computation; implicit computational complexity
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1998 ACM Subject Classification F.4.1 Mathematical Logic
Introduction
Free-cut elimination1 is a normalisation procedure on formal proofs in systems including
nonlogical rules, e.g. the axioms and induction rules in arithmetic, introduced in [26]. It
yields proofs in a form where, essentially, each cut step has at least one of its cut formulas
principal for a nonlogical step. It is an important tool for proving witnessing theorems in
first-order theories, and in particular it has been extensively used in bounded arithmetic
for proving complexity bounds on representable functions, by way of the witness function
method [9].
Linear logic [14] is a decomposition of both intuitionistic and classical logic, based on
a careful analysis of duplication and erasure of formulas. It has been useful in
proofs-asprograms correspondences, proof search [1] and logic programming [24]. By controlling
structural rules with designated modalities, the exponentials, linear logic has allowed for a
fine study of complexity bounds in the Curry-Howard interpretation, inducing variants with
polynomial-time complexity [17] [16] [18].
In this work we explore how the finer granularity of linear logic can be used to control
complexity in first-order theories, restricting the provably convergent functions rather than
the typable terms as in the propositional setting. We believe this to be of general interest,
in particular to understand the effect of substructural restrictions on nonlogical rules, e.g.
induction, in mathematical theories. Some related works exist, e.g. the naïve set theories
of Girard and Terui [15] [27], but overall it seems that the first-order proof theory of linear
logic is still rather undeveloped; in particular, to our knowledge, there seems to be no general
form of free-cut elimination available in the literature (although special cases occur in [22]
and [3]). Thus our first contribution, in Sect. 3, is to provide general sufficient conditions on
nonlogical rules for a first-order linear logic system to admit free-cut elimination.
We illustrate the usefulness of this result by proving a witnessing theorem for an arithmetic
in linear logic, showing that the provably convergent functions are precisely the
polynomialtime computable functions (Sects. 6 and 7), henceforth denoted FP. Our starting point is
an axiomatisation A21 from [7], based on a modal logic, already known to characterise FP.
This approach, and that of [20] before, differs from the bounded arithmetic approach since it
does not employ bounds on quantifiers, but rather restricts nonlogical rules by substructural
features of the modality [7] or by ramification of formulas [21]. The proof technique employed
in both cases is a realisability argument, for which [20] operates directly in intuitionistic logic,
whereas [7] obtains a result for a classical logic via a double-negation translation, relying on
a higher-type generalisation of safe recursion [6].
We show that Buss’ witness function method can be employed to extract functions directly
for classical systems similar to A21 based in linear logic, by taking advantage of free-cut
elimination. The De Morgan normal form available in classical (linear) logic means that the
functions we extract remain at ground type, based on the usual safe recursive programs of
[6]. A similar proof method was used by Cantini in [11], who uses combinatory terms as the
model of computation as opposed to the equational specifications in this work.2
1
Our result holds for an apparently weaker theory than A2, with induction restricted
to positive existential formulas in a way similar to Leivant’s RT 0 system in [21] (see also
[23]), but the precise relationship between the two logical settings is unclear. We conclude
in Sect. 8 with a survey of related work and some avenues for further applications of the
free-cut elimination result.
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