Linearizing Intuitionistic Implication

Patrick Lincoln^{∗} Andre Scedrov^{†} Natarajan Shankar^{‡}

Abstract

An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cut-free proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspace-hardness of imall. It exploits several proof-theoretic properties of intuitionistic implication that analyze the use of resources in iil proofs.

Linear logic is a refinement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard’s words [12], “linear logic is a logic behind logic.”

A convenient way to present linear logic is by modifying the traditional Gentzen-style sequent calculus axiomatization of classical logic (see, e.g., [15, 22]). The modification may be briefly described in three steps.

The first step is to remove two structural rules, contraction and weakening, which manipulate the use of hypotheses and conclusions in classical proofs. For expository purposes let us concentrate on the treatment of hypotheses. The contraction rule states that if a property follows from two

∗Lincoln@CS.Stanford.EDU Department of Computer Science, Stanford University, Stanford, CA 94305, and the Computer Science Laboratory, SRI International, Menlo Park, CA 94025. Supported by AT&T Bell Laboratories Doctoral Scholarship, and SRI internal funding.

†Andre@CIS.Upenn.EDU Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395. Partially supported by NSF Grants CCR-87-05596 and CCR-91-02753, by ONR Grants NOOO14-88-K-0635 and N00014-92-J-1916, and by the 1987 Young Faculty Award from the Natural Sciences Association of the University of Pennsylvania. Work begun while on sabbatical leave at the Computer Science Department and Center for the Study of Language and Information, Stanford University.

‡Shankar@CSL.SRI.COM Computer Science Laboratory, SRI International, Menlo Park, CA 94025. Supported by SRI internal funding.

assumptions of a formula, then that property can be derived just from a single assumption of that formula. In effect, this means that any assumption, once stated, may then be reused as often as desired. The weakening rule makes it possible to use dummy assumptions, i.e., it allows us to carry out a deduction without using all of the hypotheses. Because contraction and weakening together make it possible to use an assumption as often or as little as desired, these rules are responsible for what one may see in hindsight as a loss of control over resources in classical (and intuitionistic) logic. This realization is the starting point of linear logic. Removing the rules of contraction and weakening produces a linear system in which each assumption must be used exactly once. In the resulting linear logic, formulas indicate finite resources that cannot necessarily be discarded or duplicated without effort.

The second step in deriving linear logic involves the propositional connectives. Briefly, the removal of structural rules just mentioned leads naturally to two forms of conjunction, one called multiplicative and the other additive, and correspondingly to two forms of disjunction. Viewing the hypotheses as resources, the proof of a multiplicative conjunction as a conclusion forbids any sharing between the resources used to establish each conjunct, whereas the additive conjunction requires the sharing of all of the resources. We note that unlike this distinction between the two forms of conjunction and disjunction, the quantifier rules are the same as in classical logic.

The third step in the presentation of linear logic involves adding a kind of modality: a storage or reuse operator, ! . Intuitively, the hypothesis !A provides unlimited use of the resource A. A computational metaphor that describes the meaning of !A quite well is that “the datum A is stored in the memory and may be referenced an unlimited number of times”. (There is also a dual modal operator, ? , which is definable from ! using negation. Intuitively, while !A provides unlimited creation of A, the formula ?B allows the unlimited consumption of B.) However, since the basic framework remains linear, such an unbounded use is allowed “locally”, only at formulas specifically marked with ! . The resulting logic is remarkably natural and well-structured, both from proof-theoretic and computational standpoints. The logic is constructive, has the cut elimination property [12] and interesting semantics [12, 13, 8, 32, 7], and admits a Curry-Howard style interpretation of proofs as programs [12, 23, 16, 1, 2, 17, 18, 24].

An important impetus toward formulation of linear logic was Girard’s discovery that in coherent domains, function type A ⇒ B could be decomposed as

A ⇒ B = !A−∘B ,

where ! is the reuse operator just mentioned above, and −∘ is linear implication, which provides the type of functions that “use” their argument exactly once. Subsequently, after formulating the syntax of linear logic, Girard recast the decomposition mentioned above as a conservative, syntactic translation from intuitionistic logic into linear logic, compositional on subformulas and subproofs. Girard’s translation covers the full spectrum of the propositions-as-types paradigm: the level of formulas, the level of proofs, and the level of proof reduction (cut elimination steps). Furthermore, this translation extends naturally to first order and second order logic [12].

The possibility of a dramatic improvement over Girard’s translation is indicated by the recent results in [25]. One of these results establishes the undecidability of (the provability in) propositional linear logic. The other result in [25] bearing on the present discussion is the pspace-completeness of propositional linear logic without the modalities ! ,?, i.e., of the multiplicative additive fragment, mall. Statman [33] has shown that propositional intuitionistic logic, indeed, even its purely implicational fragment is pspace-complete. Hence a natural question arises whether (beyond an immediate Turing reduction) there exists another “logical” embedding of intuitionistic logic into linear logic that does not rely on the modalities.

Let us be realistic. One cannot hope to have such an embedding which would be too “logical”, because on the one hand, first order multiplicative additive linear logic without function symbols is decidable, basically because of a linear bound on the depth of cut free proofs. On the other hand, first order intuitionistic logic is undecidable (even without function symbols). This is an immediate corollary of the negative interpretation of classical logic in intuitionistic logic and the undecidability of classical first order logic (even without function symbols), both of which are standard and may be found, e.g., in [22]. Therefore it is impossible to have a desired embedding for first order quantifiers.

Another, more subtle obstruction to obtaining a very “logical” embedding is the discrepancy in complexity on the level of cut elimination (proof normalization). Already for the purely implicational fragment of propositional intuitionistic logic, cut elimination is hyperexponential (the equivalent fact about normalization in the simple typed lambda calculus is usually one of the first exercises in a graduate course in the subject). In contrast, cut elimination for mall is known to be much lower, at most exponential. In fact, this is true not just in the propositional case, but also for first order and for second order mall. The required bounds are given by the Small Normalization Theorem in [12], see also [27]. Hence a translation that preserves normalization of proofs with cut would have to be hyperexponential. (However, it may be possible to use an optimized presentation of intuitionistic logic such as [20] to give a triple exponential translation that preserves normalization of optimized proofs with cut.)

These results do leave open the possibility of an efficient syntactic translation of propositional intuitionistic logic into mall so that such a translation does preserve cut free proofs of a certain optimized form. In this paper we construct such a translation. Our translation is an instance of what Girard terms an “asymmetrical interpretation,” that is, positive occurrences of formulas are translated differently from negative occurrences [15]. It can therefore only be viewed as a translation on cut free proofs, unlike Girard’s symmetric translation of intuitionistic logic into linear logic.

In precise technical terms, the target of our translation is an “intuitionistic” version of mall, presented by two-sided sequents with at most one consequent formula. Similar “intuitionistic” versions of various fragment of linear logic are considered in relationship to computer science, e.g., in [14, 23, 6, 11, 16, 1, 24].

Apart from the foundational interest, we believe that the result of this paper, which is theoretical in nature, contributes to the understanding of the role of linear logic as an expressive and natural framework for describing control structure of logic programs. This logic programming perspective is based on [29]; related work is in [19, 4, 5, 3]. Furthermore, our result addresses the issue of replacing copying and reuse by sharing as discussed below.

A first indication that copying and reuse of hypotheses in intuitionistic logic might be replaceable by sharing is the contraction free formulation of intuitionistic logic, given by the system G3 in Section 80 of [22] (see also the formulation of a purely implicational fragment considered in Section E6, Chapter 5 of [9]). A similar formulation is given in Section 2 below. We concentrate on the purely implicational fragment, which suffices because of the reduction discussed in [33]. However, a central role in our approach is played by a further reformulation of intuitionistic logic suggested by the methods used in [34] and [31]. The corresponding calculus, presented in Section 3 below, is the actual source calculus for our translation of cut free proofs into the “intuitionistic” version of multiplicative additive linear logic.

Our translation is exponential in the implication depth, but polynomial on formulas of bounded implication depth. (In fact, it suffices to consider only implications of depth at most 2, see, e.g., [30]. This depth reduction dates back to [35].) A preliminary version of this work was reported in [26].

We would like to thank Jean-Yves Girard, Vincent Danos, Yves Lafont, Grigori Mints, and John Mitchell for very stimulating discussions. We are also grateful to Grigori Mints for help in investigating the literature and to Vincent Danos for pointing out an oversight in the proof of Lemma 5.5 in an earlier version of the paper.

1 Overview

We only consider propositional systems of intuitionistic and linear logic. We use the following notations that are common to both the intuitionistic and linear formalisms:

l_{i}, p_{i}, q_{i}, r_{i}  Propositional literals
A, B, C  Arbitrary formulas
Σ, Γ, Δ  Arbitrary finite multisets of formulas
Γ ⊢ Σ  Sequent with antecedent Γ

and consequent Σ

We often use the word context instead of antecedent. The linear negation operation of mall is omitted entirely. Note that a sequent is represented in terms of two multisets, not sets, of formulas. For the intuitionistic sequent calculi, the consequent multiset is either a singleton or is empty. When we speak of a formula in a sequent, we are really referring to an occurrence of the formula. A reduction is the process of applying a rule to a sequent matching the conclusion of the rule in order to generate the corresponding premises. The principal formula of the rule is then said to be reduced by the reduction. The occurrence of an instance of a rule in a proof is said to be an inference. The proper subformulas of a principal formula of a rule that appear in the premises of the rule are called the side formulas. A proof is represented as a tree rooted at its conclusion sequent at the bottom and with the leaves at the top. Given this orientation, the notion of a rule occuring above or below another rule should be clear.

The main result of this paper is an efficient embedding of the implicational fragment of propositional intuitionistic logic (iil) in the intuitionistic fragment of multiplicative-additive linear logic (imall). We provide a transformation of an iil sequent σ to an imall sequent ρ so that imall proves ρ exactly when iil proves σ. The sequents σ and ρ are then said to be equiprovable. The system iil is given by a fairly standard sequent formulation of the intuitionistic implicational logic shown in Figure 5 in Section 2. These rules are similar to those of Kleene’s G3 [22]. The target system, imall, is shown in Figure 10 in Section 4. Note that the rules for negation, par, and the constant 0 are absent. Because the presentation is in terms of two sided sequents, cut-elimination for imall holds despite these omissions. Cut-elimination is of course a crucial tool in many of our proofs.

The main distinction between iil and imall is in their treatment of the structural rules. iil has an explicit rule of contraction and the rule of weakening is implicitly built into the I rule. Furthermore, the principal formula of an L ⊃ rule is copied into the premise sequents of each iil rule. imall, on the other hand, has neither contraction nor weakening, and expressly forbids the copying of the principal formula of any rule into a premise. What imall does allow is the sharing of the non-principal formulas between the two premises of an additive inference rule. The cut rule and the contraction rule of iil can be shown to be eliminable. In order to further bridge the gap between these two systems, it is important to establish control over the use of structural rules in iil proofs so that any copying of the principal formulas into the premises is made inessential. Consider the iil proof of the sequent Σ ⊢ r in Figure 1, where Σ denotes l ⊃ r, (p ⊃ q) ⊃ l, (q ⊃ r) ⊃ q.

One clear difficulty in translating that proof into imall is that the multiset Σ appears in every sequent in the proof. In imall, a formula can appear as the principal formula of at most one inference along any branch of the proof. In the above proof, the copying of the principal formula of an inference into the premises seems essential. The formulas (p ⊃ q) ⊃ l and l ⊃ r appear twice as principal formulas, and in both cases, these duplicate occurrences are along the same branch of the proof. We can deal with the duplicate use of l ⊃ r by rearranging the above proof as in Figure 2.

The next step is to deal with the copying of the formula (p ⊃ q) ⊃ l. For this purpose, we modify the L ⊃ rule of iil to the following two rules:

\begin{matrix}\underline{Γ ⊢ p Γ, B ⊢ C} & L ⊃ 1 \\ Γ, (p ⊃ B) ⊢ C \end{matrix}

\begin{matrix}\underline{Γ, (B ⊃ C) ⊢ (A ⊃ B) Γ, C ⊢ D} & L ⊃ 2 \\ Γ, ((A ⊃ B) ⊃ C) ⊢ D \end{matrix}

We also discard the cut and contraction rules and call the resulting system iil*. The advantage of iil* is that there is no copying of principal

\underline{\begin{matrix}\overline{Σ, p, q, p ⊢ q} & I \\ \overline{Σ, p, q ⊢ p ⊃ q} & R ⊃ \end{matrix} \overline{Σ, p, q, l ⊢ l}^{I}}_{L ⊃}

Σ, p, q ⊢ l Σ, p, q, r ⊢ r I L ⊃

Σ, p, q ⊢ r

\overline{Σ, p ⊢ q ⊃ r}^{R ⊃}
\begin{matrix}\underline{⋮ ; Σ, p ⊢ q ⊃ r  \overline{Σ, p, q ⊢ q}^{I}} & L ⊃ \\ Σ, p ⊢ q \\ \overline{Σ ⊢ p ⊃ q} & R ⊃ \end{matrix} \overline{Σ, l ⊢ l}^{I} L ⊃

Σ ⊢ l \overline{Σ, r ⊢ r}^{I} L ⊃

Σ ⊢ r

Figure 1: Proof of Σ ⊢ r in iil where Σ is l ⊃ r, (p ⊃ q) ⊃ l, (q ⊃ r) ⊃ q

\begin{matrix}\overline{Σ, p, q, p ⊢ q} & I \\ \overline{Σ, p, q ⊢ p ⊃ q} & R ⊃ \end{matrix} \underline{\overline{Σ, p, q, l ⊢ l}^{I} \overline{Σ, p, q, l, r ⊢ r}^{I}}_{L ⊃} ; Σ, p, q, l ⊢ r L ⊃

Σ, p, q ⊢ r

\overline{Σ, p ⊢ q ⊃ r}^{R ⊃} \overline{Σ, p, q ⊢ q}^{I} L ⊃

Σ, p ⊢ q

\overline{Σ ⊢ p ⊃ q}^{R ⊃}

\underline{\overline{Σ, l ⊢ l}^{I} \overline{Σ, l, r ⊢ r}^{I}} Σ, l ⊢ r
Σ ⊢ r

Figure 2: Modified Proof

L ⊃

\underline{\underline{\overline{A, B, p, q ⊢ q}^{I} \underline{\overline{B, p, q, l ⊢ l}^{I} \overline{B, p, q, l, r ⊢ r}^{I}}_{L ⊃ 1} ; A, B, p, q, l ⊢ r}_{L ⊃ 1} ; A, D, B, p, q ⊢ r ; \overline{A, D, B, p ⊢ q ⊃ r}^{R ⊃} ; \overline{A, D, p, q ⊢ q}^{I}}_{L ⊃ 2}
A, D, (q ⊃ r) ⊃ q, p ⊢ q
\overline{A, D, (q ⊃ r) ⊃ q ⊢ p ⊃ q}^{R ⊃}

\underline{\overline{C, l ⊢ l}^{I} \overline{C, l, r ⊢ r}^{I}}_{L⊃1} A, C, l ⊢ r

L⊃2
Σ ⊢ r

Figure 3: “Linearized” Proof in iil* where A is l ⊃ r, B is r ⊃ q, C is (q ⊃ r) ⊃ q, D is q ⊃ l.

formulas.^{1} An antecedent principal formula of the form (A ⊃ B) ⊃ C is replaced by the simpler formula B ⊃ C in one of the premises of the L ⊃ 2 rule. Let A, B, C, and D label the formulas l ⊃ r, r ⊃ q, (q ⊃ r) ⊃ q, and q ⊃ l, respectively. With these new rules, the above proof can be transformed to an iil* proof as in Figure 3.

The absence of contraction and the absence of copying of principal formulas in iil* along with the restriction on weakening make it possible to embed iil* in imall. This translation is asymmetric, i.e., positive occurrences of formulas in sequents are treated differently from negative occurrences. The basic idea is that a left implication rule of iil* is translated by a block in imall consisting of a −∘L rule (which accounts for the principal formula) followed by a ⊕L rule (which accounts for the context). For instance, the translation [(p ⊃ q) ⊃ l]^{−} will be basically (([(q ⊃ l)]^{−}−∘[(p ⊃ q)]^{+})−∘b) ⊕ [l]^{−}. If Σ′ abbreviates l ⊃ r, (q ⊃ r) ⊃ q, the last step in the iil* proof displayed in Figure 3 will be translated basically as in Figure 4, where the middle branch will be provable.

The actual translation is more complicated; it also involves the “locksand-keys” technique from [25] in order to ensure faithfulness. We defer the discussion of details until Section 4.

1Grigori Mints directed our attention to iil*. Observe that after depth-reduction (see Section 6) iil* provides a direct proof-theoretic explanation for the membership in pspace of the decision problem for propositional intuitionistic logic. Cut-free proofs in iil* have a height that is linear in the number of connectives in the conclusion sequent. An alternating Turing machine can therefore generate and check the proof of a given sequent in a nondeterministic manner within polynomial time.

\begin{matrix}\underline{⋮ ; Σ′, (q ⊃ l) ⊢ (p ⊃ q)  ⋮ ; Σ′, l ⊢ r} & L ⊃ 2 \\ Σ′, ((p ⊃ q) ⊃ l) ⊢ r \end{matrix}
⇓

\underline{\begin{matrix}\underline{⋮ ; [Σ′]^{−}, [(q ⊃ l)]^{−} ⊢ [(p ⊃ q)]^{+} ; \overline{[Σ′]^{−} ⊢ [(q ⊃ l)]^{−}−∘[(p ⊃ q)]^{+}}^{−∘R} ⋮ ; b ⊢ [r]^{+}} & −∘L \\ [Σ′]^{−}, ([(q ⊃ l)]^{−}−∘[(p ⊃ q)]^{+})−∘b ⊢ [r]^{+} \end{matrix} ⋮ ; [Σ′]^{−}, [l]^{−} ⊢ [r]^{+}} _{⊕L}

[Σ′]^{−}, (([(q ⊃ l)]^{−}−∘[(p ⊃ q)]^{+})−∘b) ⊕ [l]^{−} ⊢ [r]^{+}

Figure 4: Toward imall translation of example.

In summary, we provide a transformation from iil sequents to imall sequents by transforming iil proofs. Our main result is:

Theorem 1.1 iil can be embedded into imall. The embedding preserves the structure of cut-free proofs in iil*.

iil proofs are transformed by eliminating any use of the cut and contraction rules, permuting the order of the inferences, and modifying the L ⊃ rule so as to eliminate the need for copying. The resulting iil* proofs can then be embedded in imall.

From the logic programming perspective given in [29], the main result of this paper addresses the issue of replacing copying and reuse in intuitionistic proofs by sharing. We believe that our results, together with [19, 4], contribute to the understanding of the role of linear logic as an expressive and natural framework for describing the control structure of logic programs.

2 Properties of iil

In this section, we present a series of lemmas about iil that eventually establish the eliminability of cut and contraction, the admissibility of weakening, and the redundancy of copying in iil proofs.

Proposition 2.1 shows that the rule of identity on atomic formulas can be extended to all formulas.

\overline{Γ, p_{i} ⊢ p_{i}}^{I}

\begin{matrix}Γ, A ⊢ B \\ \overline{Γ ⊢ (A ⊃ B)} & R ⊃ \end{matrix}

\begin{matrix}\underline{Γ, (A ⊃ B) ⊢ A Γ, (A ⊃ B), B ⊢ C} & L ⊃ \\ Γ, (A ⊃ B) ⊢ C \end{matrix}

\underline{Γ, A, A ⊢ B}_{Contraction}
Γ, A ⊢ B

\begin{matrix}\underline{Γ ⊢ C Γ, C ⊢ B} & Cut \\ Γ ⊢ B \end{matrix}

Figure 5: Rules for iil

Proposition 2.1 For all iil formulas A and iil sequents Γ, there exists a proof of Γ, A ⊢ A in iil.

Proof. We build the proof by induction on the structure of A. If A is an atomic proposition, the result is immediate. If A is an implication, that is A = A_{1} ⊃ A_{2}, then we may construct the following deduction:

\underline{⋮ ; Γ, (A_{1} ⊃ A_{2}), A_{1} ⊢ A_{1}  ⋮ ; Γ, (A_{1} ⊃ A_{2}), A_{1}, A_{2} ⊢ A_{2}} _{L ⊃}
Γ, (A_{1} ⊃ A_{2}), A_{1} ⊢ A_{2}
\overline{Γ, (A_{1} ⊃ A_{2}) ⊢ (A_{1} ⊃ A_{2})}^{R ⊃}

The two remaining sequents in the above deduction are provable by induction.

Proposition 2.2 For any sequent Σ ⊢ A appearing in any iil proof of Γ ⊢ B, the multiset Γ is a sub-multiset of Σ.

Proof. By induction on the size of proofs followed by a straightforward case analysis on the iil rules.

This conservation of antecedent formulas in iil proofs provides the key to the elimination of contraction as shown by Propositions 2.3 and 2.4. The size of a proof is taken to be the number of inferences in it.

Proposition 2.3 Given a proof of Γ, A, A ⊢ B of size n in iil, we can produce a proof of Γ, A ⊢ B of size n in iil.

Proof. From Proposition 2.2, we know that Γ, A, A appears as a submultiset of each sequent in the given proof tree. By case analysis on the rules in iil, we see that by replacing Γ, A, A with Γ, A everywhere in the derivation, we are left with a correctly formed iil proof of Γ, A ⊢ B.

Proposition 2.4 Given a proof of Γ ⊢ A of size n in iil, we can construct a proof of Γ ⊢ A in iil of size no greater than n that does not employ the contraction rule.

Proof. By induction on proof size. If the last rule in a derivation is contraction, then we simply apply proposition 2.3 to its premise to achieve a smaller proof of the desired sequent. In other cases we appeal to the induction hypothesis.

Proposition 2.5 If there is a proof of Γ ⊢ A in iil then there is a proof of Γ ⊢ A in iil that does not employ the cut rule.

Proof. The argument here is a straightforward adaptation of the cutelimination proof for G3 that appears in [22].

Proposition 2.6 Any proof of Γ ⊢ A in iil can be transformed into a proof of Γ ⊢ A in iil that does not employ the contraction or cut rules.

Proof. By application of Proposition 2.5 and then Proposition 2.4. Note that the latter lemma will never introduce cuts into a proof, and thus preserves the cut-free nature of proofs.

Proposition 2.7 Given a proof of Γ ⊢ B of size n in iil, we can produce a proof of Γ, A ⊢ B of size n in iil.

Proof. We simply add the iil formula A to each sequent in the entire iil derivation. That is, by case analysis on the rules in iil, we see that adding a formula A to the context (left hand side) of the hypotheses and conclusions of all the rules of iil leaves us with a correctly formed iil proof of Γ, A ⊢ B. Note that the formula A that has been weakened in, never occurs as the principal formula of any rule in the resulting proof.

Proposition 2.8 Given a proof of Γ, A ⊃ B, B ⊢ C of size n in iil, we can find a proof of Γ, B ⊢ C of size less than or equal to n in iil.

Proof. We prove this property by induction on the size of iil proof. At each step we perform case analysis on the last rule applied.

If the last rule is identity, which is restricted to atomic propositions, we may safely remove the formula A ⊃ B from the context.

If the last rule applied is R ⊃, Cut, or Contraction, then by induction we have our result.

In the final case of L ⊃, if some other implication in the context is analyzed, then by induction we have our result. If A ⊃ B is the formula analyzed, then we know the derivation is of the form:

\begin{matrix}\underline{⋮ ; Γ, A ⊃ B, B ⊢ A  ⋮ ; Γ, A ⊃ B, B, B ⊢ C} & L ⊃ \\ Γ, A ⊃ B, B ⊢ C \end{matrix}

Now applying Proposition 2.3 to the proof of the right hand hypothesis, we are able to obtain a proof of Γ, A ⊃ B, B ⊢ C which is smaller than the original proof, and thus by induction we may assume that A ⊃ B may be eliminated from the context, and we have a proof of Γ, B ⊢ C of size no more than n.

Proposition 2.9 For all iil formulas A, B, C the sequent (A ⊃ B) ⊃ C ⊢ B ⊃ C is provable in iil.

Proof. By the following iil proof:

\underline{\begin{matrix}\overline{(A ⊃ B) ⊃ C, B, A ⊢ B} & I \\ \overline{(A ⊃ B) ⊃ C, B ⊢ (A ⊃ B)} & R ⊃ \end{matrix}  \overline{(A ⊃ B) ⊃ C, B, C ⊢ C}^{I}}_{L ⊃}
(A ⊃ B) ⊃ C, B ⊢ C
\overline{(A ⊃ B) ⊃ C ⊢ B ⊃ C}^{R ⊃}

3 iil and iil*

We now introduce an interesting optimization of iil called iil*, and prove that cut-free, contraction-free iil proofs are easily transformed to proofs in

\begin{matrix}Γ, A ⊢ B \\ \overline{Γ ⊢ (A ⊃ B)} & R ⊃ \end{matrix}

\begin{matrix}\underline{Γ ⊢ p_{i} Γ, B ⊢ C} & L ⊃ 1 \\ Γ, (p_{i} ⊃ B) ⊢ C \end{matrix}

\begin{matrix}\underline{Γ, (B ⊃ C) ⊢ (A ⊃ B) Γ, C ⊢ D} & L ⊃ 2 \\ Γ, ((A ⊃ B) ⊃ C) ⊢ D \end{matrix}

Figure 6: Rules for iil*

iil*. The proof rules for iil* are given in Figure 6. Similar optimizations have been studied by others [34, 31, 20, 10].

Note that the identity rule is only applicable to atomic propositions, and that weakening is only allowed at the leaves of a proof, i.e., at an application of identity. Most important, however, is the property that the principal formula is not duplicated in the premises of any of the rules in iil*.

Proposition 3.1 Given a proof of Γ ⊢ B of size n in iil*, we can produce a proof of Γ, A ⊢ B of size n in iil*.

Proof. We simply add the iil* formula A to each sequent in the entire iil* derivation. That is, by case analysis on the rules in iil*, we see that adding a formula A to the context (left hand side) of the hypotheses and conclusions of all the rules of iil* leaves us with a correctly formed iil* proof of Γ, A ⊢ B.

Lemma 3.2 Given a proof of Γ ⊢ A in iil*, a proof of Γ ⊢ A can be constructed in iil.

Proof. By induction on iil* proofs using Propositions 3.1 and 2.9.

The other direction of the equivalence of iil and iil* is somewhat more complicated. Our original argument involved depth reduction (see Section 6.1). Here we adapt an argument due to Dyckhoff [10] by introducing Lemma 3.3 and modifying the definition of the weight of a sequent used to justify the induction in Lemma 3.4.

Consider an L ⊃ inference in an iil proof with a principal antecedent formula of the form p ⊃ A. Let Γ ⊢ C be the conclusion sequent of the inference. The inference is said to be backward if p does not occur in Γ. A forward proof is one with no backward inferences. These names are chosen to be reminiscent of forward and backward chaining.

Lemma 3.3 Any cut-free, contraction-free iil proof Π of size n can be transformed to a cut-free, contraction-free forward proof Θ of size no more than n with the same conclusion as Π.

Proof. The proof is by induction on the size of the cut-free, contractionfree proof Π.

If the final inference in Π is not a backward inference, then we have the result immediately by induction.

If the final step is a backward inference in Π, then we use the induction hypothesis to eliminate the backward inferences in the subproofs of the premises. This transforms the proof Π to the form below, where the only backward inference is the final one.

\begin{matrix}\underline{\begin{matrix}Θ_{1} & Θ_{2} \\ ⋮ & ⋮ \\ Γ, p ⊃ A ⊢ p & Γ, p ⊃ A, A ⊢ C \end{matrix}} & L ⊃ \\ Γ, p ⊃ A ⊢ C \end{matrix}

The premise Γ, p ⊃ A ⊢ p cannot be an axiom since p does not occur in Γ. The final inference in the proof Θ_{1} of Γ, p ⊃ A ⊢ p must therefore be an L ⊃ inference whose principal formula is either of the form (D ⊃ E) ⊃ F or of the form q ⊃ B where q occurs in Γ. In either case, these inferences can be permuted below the final inference in Π, as in Figure 7.

In Figure 7, the proof Θ′_{2} is obtained from Θ_{2} by Proposition 2.7 but has the same size as Θ_{2}. The backward inference with the subproofs Θ_{12} and Θ′_{2} is smaller than Π and we can therefore employ the induction hypothesis to eliminate the backward inference from it. The resulting proof is therefore free of backward inferences and has size no larger than Π.

The other possibility is that the principal formula is of the form q ⊃ B where q occurs in Γ. In this case the inferences permute similarly, and the resulting proof may be seen to be forward by induction, and the fact that q occurs in Γ.

\begin{matrix}\underline{\begin{matrix}Θ_{11} & Θ_{12} \\ ⋮ & ⋮ \\ Γ, p ⊃ A ⊢ D ⊃ E & Γ, p ⊃ A, F ⊢ p \end{matrix}} & L ⊃ \\ Γ, p ⊃ A ⊢ p \end{matrix}  Θ_{2} ; ⋮ ; Γ, p ⊃ A, A ⊢ C L ⊃

Γ, p ⊃ A ⊢ C

becomes

Θ_{11} ; ⋮ ; Γ, p ⊃ A ⊢ D ⊃ E  \begin{matrix}\underline{Θ_{12} ; ⋮ ; Γ, p ⊃ A, F ⊢ p  Θ′_{2} ; ⋮ ; Γ, p ⊃ A, F, A ⊢ C} & L ⊃ \\ Γ, p ⊃ A, F ⊢ C \end{matrix} L ⊃

Γ, p ⊃ A ⊢ C

Figure 7: Permuting backward inferences

weight(A_{1}, …, A_{n} ⊢ C) = m(A_{1}, 1) + … + m(A_{n}, 1) + m(C, 1)
m(A ⊃ B, d) = m(A, d + 1) + d ∗ (m(B, d) + 1)
m(p, d) = d

Figure 8: Definition of weight

Lemma 3.4 Given a proof of Γ ⊢ C in iil, a proof of Γ ⊢ C can be constructed in iil*.

Proof. By Lemma 3.3, we can restrict our attention to forward proofs. We proceed by induction, not on the size of the given proof, but on weight(σ) for a sequent σ, as defined in Figure 8. There are four cases according to the final inference in the given proof.

It is easy to show by induction on the structure of A that if 0 < c < d, then 0 < m(A, c) < m(A, d).

If the given iil proof of Γ ⊢ C is an axiom, then the proof is also an iil* proof.

If the final inference in the given forward iil proof is R ⊃ applied to a conclusion of the form Γ ⊢ A ⊃ B to generate the premise Γ, A ⊢ B, weight(Σ, (D ⊃ E) ⊃ F ⊢ C) − weight(Σ, E ⊃ F, D ⊢ E)

= m((D ⊃ E) ⊃ F, 1) + m(C, 1) − m(E ⊃ F, 1) − m(D, 1) − m(E, 1)
= m(D, 3) + 2m(E, 2) + 2 + m(F, 1) + 1 +
m(C, 1) − m(E, 2) − m(F, 1) − 1 − m(D, 1) − m(E, 1)
> 0

Figure 9: Example calculation of weight

then this premise is of smaller weight. We can therefore apply the induction hypothesis to the premise to get an iil* proof of Γ, A ⊢ B from which the iil* proof of Γ ⊢ A ⊃ B can be completed by the R ⊃ rule of iil*.

If the final inference in the given forward iil rule is L ⊃ applied to a principal formula of the form p ⊃ B, then Γ has the form Σ, p ⊃ B and p must occur in Σ. Since p occurs in Σ, the sequent Σ ⊢ p is an iil* axiom. The nontrivial premise is then Σ, p ⊃ B, B ⊢ C. By Proposition 2.8, the sequent Σ, B ⊢ C must also have an iil proof and since it is of smaller weight than Σ, p ⊃ B ⊢ C, the induction hypothesis can be applied to it yielding an iil* proof of Σ, B ⊢ C. The required iil* proof of Σ, p ⊃ B ⊢ C can be constructed using the L ⊃ 1 rule with the premises Σ, B ⊢ C and Σ ⊢ p.

If the final inference in the given iil proof is L ⊃ applied to a principal formula of the form (D ⊃ E) ⊃ F, then Γ has the form Σ, (D ⊃ E) ⊃ F, and we have iil proofs for the two premises Σ, (D ⊃ E) ⊃ F ⊢ D ⊃ E and Σ, (D ⊃ E) ⊃ F, F ⊢ C. Proposition 2.8 applied to the second premise yields an iil proof of Σ, F ⊢ C to which the induction hypothesis can be applied yielding an iil* proof of Σ, F ⊢ C. Since in iil we can prove D, (E ⊃ F) ⊢ (D ⊃ E) ⊃ F and D, (D ⊃ E) ⊢ E, we can use the cut rule twice with the sequent Σ, (D ⊃ E) ⊃ F ⊢ D ⊃ E to get an iil proof of Σ, E ⊃ F, D ⊢ E. The difference in weight between this last sequent and the original conclusion sequent Σ, (D ⊃ E) ⊃ F ⊢ C is given in Figure 9.

So the induction hypothesis yields an iil* proof of Σ, E ⊃ F, D ⊢ E which by R ⊃ yields an iil* proof of Σ, E ⊃ F ⊢ D ⊃ E. This last sequent with Σ, F ⊢ C yield an iil* proof of Σ, (D ⊃ E) ⊃ F ⊢ C by the L ⊃ 2 rule of iil*.

The lack of contraction in iil* makes this formulation of the sequent rules for implicational intuitionistic propositional logic amenable to encoding into imall.

4 iil* to imall

An intuitionistic linear logic sequent is composed of two finite multisets of linear logic formulas separated by a ⊢ , where there is no more than one formula in the consequent (i.e., right-hand side) multiset. We assume a set of propositional atoms p_{i} to be given. Figure 10 gives the inference rules for the intuitionistic linear sequent calculus, with the slight restriction that the 0 rule is omitted.^{2} This omission does not pose problems for cut elimination.

We now define a pair of mutually recursive translation functions that transform any iil* formula into an imall formula. k and b are fresh propositional letters. The definitions of [ ]^{+} and [ ]^{−} given in Figure 11 can be seen to be well defined by induction on the size of the formulas.

For any iil* sequent Γ ⊢ C we define

θ(Γ ⊢ C) =^{Δ} [Γ]^{−}, k ⊢ [C]^{+}

Here [Γ]^{−} stands for the result of the application of [ ]^{−} to each element of Γ. Note that the “key” k is present in the context of the encoding of a sequent. We have chosen the notations [ ]^{+} and [ ]^{−} to suggest the interpretation of positive and negative polarity of occurrences.

Let us first demonstrate how parts of the example iil* proof given in Figure 3 are translated into imall. Consider the sequent Σ′, (p ⊃ q) ⊃ l ⊢ r, where Σ′ abbreviates l ⊃ r, (q ⊃ r) ⊃ q. This sequent has the θ-translation [Σ′]^{−}, [(p ⊃ q) ⊃ l]^{−}, k ⊢ [r]^{+}. By the above definition, [(p ⊃ q) ⊃ l]^{−} = k−∘((([(q ⊃ l)]^{−}−∘(k−∘[(p ⊃ q)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [l]^{−})). In the example iil* proof given in Figure 3, the proof of this sequent ends in an application of the L ⊃ 2 rule.

The intuitive structure of the proof in Figure 12 is as follows. The leftmost application of I and the bottommost application of −∘L correspond to “unlocking” the formula of interest. The unlocked formula corresponding to (p ⊃ q) ⊃ l has ⊕ as its main connective. The proof tree therefore forks, and after a simple application of ⊗L, the rightmost branch can be seen to be the translation of the rightmost branch of the iil* proof.

The left main branch of the proof progresses by applying the −∘L rule. Here there is a choice to be made in the way we split the context Σ′ among

2Our arguments also apply to the sequent calculus given on p. 53 of [14] without the 0 rule.

I  \overline{p ⊢ p}  \frac{Σ ⊢ A  A, Γ ⊢ Δ}{Σ, Γ ⊢ Δ} Cut
⊗L  \frac{Σ, A, B ⊢ Δ}{Σ, (A ⊗ B) ⊢ Δ}  \frac{Σ ⊢ A  Γ ⊢ B}{Σ, Γ ⊢ (A ⊗ B)}  ⊗R
−∘L  \frac{Σ ⊢ A  Γ, B ⊢ Δ}{Σ, Γ, (A−∘B) ⊢ Δ}  \frac{Σ, A ⊢ B}{Σ ⊢ (A−∘B)}  −∘R
⊕L \frac{Σ, A ⊢ Δ  Σ, B ⊢ Δ}{Σ, (A ⊕ B) ⊢ Δ}  \frac{Σ ⊢ A  Σ ⊢ B}{Σ ⊢ (A&B)}  &R
&L1  \frac{Σ, A ⊢ Δ}{Σ, (A&B) ⊢ Δ}  \frac{Σ ⊢ A}{Σ ⊢ (A ⊕ B)}  ⊕R1
&L2  \frac{Σ, B ⊢ Δ}{Σ, (A&B) ⊢ Δ}  \frac{Σ ⊢ B}{Σ ⊢ (A ⊕ B)}  ⊕R2
⊥L  \overline{⊥ ⊢}  \frac{Σ ⊢}{Σ ⊢ ⊥}  ⊥R
1L  \frac{Σ ⊢ Δ}{Σ, 1 ⊢ Δ}  \overline{⊢ 1}  1R
\overline{Σ ⊢ ⊤}  ⊤R

Figure 10: Rules for imall

\begin{matrix}[A ⊃ B]^{+} & =^{Δ} & k ⊗ ([A]^{−}−∘(k−∘[B]^{+})) \\ [p_{i}]^{+} & =^{Δ} & k ⊗ ((p_{i} ⊕ b) ⊗ ⊤) \\ [p_{i}]^{−} & =^{Δ} & p_{i} \\ [p_{i} ⊃ A]^{−} & =^{Δ} & k−∘(((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−})) \\ [(A ⊃ B) ⊃ C]^{−} & =^{Δ} & k−∘((([(B ⊃C)]^{−}−∘(k−∘[(A⊃B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−})) \end{matrix}

Figure 11: Definition of translation

\begin{matrix}\underline{⋮ ; Σ′, (q ⊃ l) ⊢ (p ⊃ q)  ⋮ ; Σ′, l ⊢ r} & L ⊃ 2 \\ Σ′, ((p ⊃ q) ⊃ l) ⊢ r \end{matrix}
⇓

k ⊢ k I

\underline{⋮ ; [Σ′]^{−}, [(q ⊃ l)]^{−}, k ⊢ [(p ⊃ q)]^{+} ; \overline{[Σ′]^{−}, [(q ⊃ l)]^{−} ⊢ k−∘[(p ⊃ q)]^{+}}^{−∘R} ; \overline{[Σ′]^{−} ⊢ ([(q ⊃ l)]^{−}−∘(k−∘[(p ⊃ q)]^{+}))}^{−∘R} ⋮ ; k, b ⊢ [r]^{+} ; \overline{k ⊗ b ⊢ [r]^{+}}^{⊗L}}_{−∘L}
[Σ′]^{−}, ([(q ⊃ l)]^{−}−∘(k−∘[(p ⊃ q)]^{+}))−∘(k ⊗ b) ⊢ [r]^{+}

⋮ [Σ′]^{−}, k, [l]^{−} ⊢ [r]^{+}_{\overline{[Σ′]^{−}, k ⊗ [l]^{−} ⊢ [r]^{+}}^{⊗L}}

⊕L

[Σ′]^{−}, (([(q ⊃ l)]^{−}−∘(k−∘[(p ⊃ q)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [l]^{−}) ⊢ [r]^{+} −∘L

[Σ′]^{−}, k, k−∘((([(q ⊃ l)]^{−}−∘(k−∘[(p ⊃ q)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [l]^{−})) ⊢ [r]^{+}

Figure 12: iil* and imall proofs of example.

the branches of the proof. However, because of the form of our translation, we can without loss of generality choose to keep the entire context on the left branch. Lemma 4.1 implies that k, b ⊢ [r]^{+}, the upper right branch, is provable. Notice how [r]^{+} has been devised to ensure this. And finally, we see that after two applications of R ⊃ we are left with the translation of the right hand branch of the iil* proof.

In fact, the encoding is such that there are essentially no choices to be made in the proof of the imall translation that cannot be made in the proof of an iil* formula. For example, once a formula is unlocked with the “key” k, no other formula may be unlocked until the unlocked formula is reduced completely, at which point it provides another key k. This method of “locks and keys” was introduced in [25]. In the next section we show that an iil* formula is provable in iil* if and only if its translation is provable in imall.

Lemma 4.1 For any iil* formula C and any imall multiset Δ, the sequent Δ, k, b ⊢ [C]^{+} is provable in imall.

Proof. The proof is by induction on the right-hand depth of C. If C = p_{i} is a proposition, we can construct an imall proof as in Figure 13.

In the case that C = (A ⊃ B) is an implication, we know that B is of smaller depth than C, and we can construct the proof as in Figure 14.

\overline{k ⊢ k}^{I}  \underline{\begin{matrix}\overline{b ⊢ b} & I \\ \overline{b ⊢ p_{i} ⊕ b} & ⊕R \end{matrix} \overline{Δ ⊢ ⊤}^{⊤R}}_{⊗R} ; Δ, b ⊢ (p_{i} ⊕ b) ⊗ ⊤ ⊗R

Δ, k, b ⊢ k ⊗ ((p_{i} ⊕ b) ⊗ ⊤) Figure 13: Case 1 of Lemma 4.1.

\underline{\overline{k ⊢ k}^{I}  ⋮ ; Δ, [A]^{−}, k, b ⊢ [B]^{+} ; \overline{Δ, [A]^{−}, b ⊢ k−∘[B]^{+}}^{−∘R} ; \overline{Δ, b ⊢ [A]^{−}−∘(k−∘[B]^{+})}^{−∘R}}_{⊗R}

Δ, k, b ⊢ k ⊗ ([A]^{−}−∘(k−∘[B]^{+})) Figure 14: Case 2 of Lemma 4.1.

Lemma 4.2 For any iil* proposition p_{i} and any imall multiset Δ, the sequent Δ, [p_{i}]^{−}, k ⊢ [p_{i}]^{+} is provable in imall. Proof. The proof follows from expanding the definition of [p_{i}]^{+}, as seen in Figure 15.

\overline{k ⊢ k}^{I}  \underline{\begin{matrix}\overline{p_{i} ⊢ p_{i}} & I \\ \overline{p_{i} ⊢ p_{i} ⊕ b} & ⊕R \end{matrix} \overline{Δ ⊢ ⊤}^{⊤R}}_{⊗R} ; Δ, p_{i} ⊢ (p_{i} ⊕ b) ⊗ ⊤ ⊗R

Δ, p_{i}, k ⊢ k ⊗ ((p_{i} ⊕ b) ⊗ ⊤) Figure 15: Proof of Lemma 4.2.

⋮ ; Γ, A ⊢ B ; \overline{Γ ⊢ (A ⊃ B)}^{R ⊃}

⇒

\underline{\overline{k ⊢ k}^{I}  ⋮ ; [Γ]^{−}, [A]^{−}, k ⊢ [B]^{+} ; \overline{[Γ]^{−}, [A]^{−} ⊢ k−∘[B]^{+}}^{−∘R} ; \overline{[Γ]^{−} ⊢ [A]^{−}−∘(k−∘[B]^{+})}^{−∘R}}_{⊗R}
[Γ]^{−}, k ⊢ k ⊗ ([A]^{−}−∘(k−∘[B]^{+}))

Figure 16: Case Right ⊃.

5 Completeness of Translation

In order to prove Theorem 1.1, we have to show that the translation is correct and faithful, i.e., there exists a cut-free proof of Γ ⊢ C in iil* if and only if there is a cut-free proof of θ(Γ ⊢ C) in imall. This will be established in two lemmas below.

Lemma 5.1 If there is a cut-free proof of Γ ⊢ C in iil*, then there is a cut-free proof of θ(Γ ⊢ C) in imall.

Proof. One proceeds by induction on the depth of proof in iil*.

In the case that the proof of Γ ⊢ C is simply one application of identity, C is actually a proposition p_{i} (identity is only applicable to atomic propositions in iil*), and therefore Γ must contain p_{i} as an element. Thus one can use Lemma 4.2.

In the case that the proof of Γ ⊢ C ends in an application of the Right ⊃ rule of iil*, then one may simply unlock the conclusion formula and then apply −∘R to the imall translation. Note that by definition, the translation of [A ⊃ B]^{+} is k⊗([A]^{−} −∘ (k−∘[B]^{+})). This case is given in Figure 16, where the required imall proof of [Γ]^{−}, [A]^{−}, k ⊢ [B]^{+} is given by the induction hypothesis.

Suppose that the iil* proof ends in an application of Left ⊃ 1. Consider the proof given in Figure 17. Lemma 4.1 implies that k, b ⊢ [C]^{+} is provable, and by induction hypothesis there exists imall proofs of the other two branches.

In the final case, suppose that the iil* proof ends in an application of Left ⊃ 2. Consider the proof given in Figure 18. As in the previous case, Lemma 4.1 implies that k, b ⊢ [D]^{+} is provable, and by induction hypothesis there exists imall proofs of the other two branches.

\begin{matrix}\underline{⋮ ; Γ ⊢ p_{i}  ⋮ ; Γ, A ⊢ C} & L ⊃ 1 \\ Γ, (p_{i} ⊃ A) ⊢ C \end{matrix}

⇓

k ⊢ k I

\underline{⋮ ; [Γ]^{−}, k ⊢ [p_{i}]^{+} ; \overline{[Γ]^{−} ⊢ k−∘[p_{i}]^{+}}^{−∘R} ⋮ ; k, b ⊢ [C]^{+} ; \overline{k ⊗ b ⊢ [C]^{+}}^{⊗L}}_{−∘L} ; [Γ]^{−}, (k−∘[p_{i}]^{+})−∘(k ⊗ b) ⊢ [C]^{+} ⋮ ; [Γ]^{−}, k, [A]^{−} ⊢ [C]^{+} ; \overline{[Γ]^{−}, k ⊗ [A]^{−} ⊢ [C]^{+}}^{⊗L} ⊕L

[Γ]^{−}, ((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−}) ⊢ [C]^{+} −∘L

[Γ]^{−}, k, k−∘(((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−})) ⊢ [C]^{+} Figure 17: Case Left ⊃ 1.

\begin{matrix}\underline{⋮ ; Γ, (B ⊃ C) ⊢ (A ⊃ B)  ⋮ ; Γ, C ⊢ D} & L ⊃ 2 \\ Γ, ((A ⊃ B) ⊃ C) ⊢ D \end{matrix}

⇓

k ⊢ k I

\underline{⋮ ; [Γ]^{−}, [(B ⊃ C)]^{−}, k ⊢ [(A ⊃ B)]^{+} ; \overline{[Γ]^{−}, [(B ⊃ C)]^{−} ⊢ k−∘[(A ⊃ B)]^{+}}^{−∘R} ; \overline{[Γ]^{−} ⊢ ([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))}^{−∘R} ⋮ ; k, b ⊢ [D]^{+} ; \overline{k ⊗ b ⊢ [D]^{+}}^{⊗L}}_{−∘L}

[Γ]^{−}, ([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b) ⊢ [D]^{+}

⋮ [Γ]^{−}, k, [C]^{−} ⊢ [D]^{+}_{\overline{[Γ]^{−}, k ⊗ [C]^{−} ⊢ [D]^{+}}^{⊗L}}

[Γ]^{−}, (([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−}) ⊢ [D]^{+} −∘L

[Γ]^{−}, k, k−∘((([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−})) ⊢ [D]^{+}

Figure 18: Case Left ⊃ 2.

⊕L

We now introduce three propositions that simplify the other direction of Theorem 1.1. These propositions are mild alterations of lemmas used to establish the pspace-completeness of imall [25]. Proposition 5.2 is only used to prove Propositions 5.3 and 5.4, and the latter two Propositions formally state that in a cut-free imall proof, no lock can be opened before there is a key available at the top level.

Proposition 5.2 For any atomic proposition p, and multiset Δ not containing the constant 1 or the constant 0, if the sequent Δ ⊢ p is provable in imall, then Δ is identically p, or contains a positive subformula of the form p&A, A&p, p ⊕ A, A ⊕ p, or A−∘p for some formula A.

Note that the clause about the constant 0 is not actually needed in our formulation of imall. However, this property could be of interest outside the scope of this paper, and thus we state it exactly for full intuitionistic two-sided multiplicative additive linear logic.

Proof. The argument is by induction on the size of the cut-free imall proof.

For each inductive step, one considers case analysis on the rules of imall. If identity is the last rule applied, then Δ ≡ p. ⊗R, −∘R, ⊕R, and &R do not apply because the right hand side is an

atomic propositional literal.

The ⊥R, ⊥L, and ⊤R rules do not apply because the right hand side of the sequent is an atomic propositional literal.

The 1L and 1R rules do not apply because Δ does not contain 1 as a subformula.

If ⊗L is the last rule applied, then Δ cannot be p, so the induction hypothesis implies that the premise contains a positive subformula of the form p&A, A&p, p ⊕ A, A ⊕ p, or A−∘p for some formula A. Because all positive subformulas present in the premise of ⊗L are also positive subformulas of the conclusion, the result follows.

If −∘L is the last rule applied, the induction hypothesis implies that the context of the right hand premise is identically p, or contains a positive subformula of the form p&A, A&p, p ⊕ A, A ⊕ p, or A−∘p for some formula A. If it is identically p, then the conclusion contains the formula A−∘p for some A, and the result follows. Otherwise, because all positive subformulas present in the context of the right hand premise of −∘L are also positive subformulas of the context of the conclusion, the result follows.

If ⊕L is the last rule applied, the induction hypothesis implies that in each premise the context is identically p, or contains a positive subformula of the form p&A, A&p, p ⊕ A, A ⊕ p, or A−∘p for some formula A. If the context of at least one premise is identically p, then the conclusion contains the formula A ⊕ p or p ⊕ A for some A, and the result follows. Otherwise, because all positive subformulas present in the premises of ⊕L are also positive subformulas of the conclusion, the result follows.

If &L is the last rule applied, the induction hypothesis implies that the context of the premise is identically p, or contains a positive subformula of the form p&A, A&p, p ⊕ A, A ⊕ p, or A−∘p for some formula A. If it is identically p, then the conclusion contains the formula A&p or p&A for some A, and the result follows. Otherwise, because all positive subformulas present in the premise of &L are also positive subformulas of the conclusion, the result follows.

Proposition 5.3 In any cut-free imall proof of θ(Γ ⊢ C) the last rule applied must be left implication or right tensor. In either case the rule must have k ⊢ k as the left hand premise.

Proof. Given a cut-free imall proof of a sequent θ(Γ ⊢ C), in other words [Γ]^{−}, k ⊢ [C]^{+}, we need to show that the context [Γ]^{−} cannot be split in the last rule applied. Let us consider which imall proof rule can be applied last in the given cut-free proof. Investigating the forms of imall formulas that can appear in a θ-translation, one sees that the last proof rule applied must be either −∘L, ⊗R, or identity. However, even identity cannot apply, because k always appears on the left in any θ-translation, and k never appears at top level on the right in such a translation. Thus there are only two cases to consider, −∘L and ⊗R. In either case, if any part of [Γ]^{−} were to be included in the left premise, there could be no imall proof of that premise, as stated in Proposition 5.2.

Proposition 5.4 If formula F is a proper subformula of an encoding [ ]^{−} or [ ]^{+} , respectively, and is not identically k, then F must be reduced below any other formula in any imall proof of [Γ]^{−}, F ⊢ [C]^{+} or [Γ]^{−} ⊢ F , respectively.

Proof. The proof of this property is almost immediate from Proposition 5.2, because our encoding functions [ ]^{−} and [ ]^{+} have the requisite properties.

Lemma 5.5 If there is a proof of θ(Γ ⊢ C) in imall, then there is a proof of Γ ⊢ C in iil*.

Proof. In order to prove Lemma 5.5, one performs cut-elimination on the given imall proof, and then observes that the resulting proof must be of a very special form. In fact, an iil* proof can be directly read from any such cut-free imall proof. The action of the “locks and keys” encoded by the positive and negative occurrences of k in the imall translations forces any cut-free imall proof of a sequent to have a very specific form. Propositions 5.3 and 5.4 state this formally. It is exactly this sort of control over the shape of a proof that one can encode in linear logic sequents, but that is impossible to encode in intuitionistic and classical logic. The proof of this lemma proceeds by induction on the size of cut-free imall proof.

Given a cut-free imall proof of a sequent θ(Γ ⊢ C), by Proposition 5.3 the last rule applied must be either −∘L or ⊗R. In each case one first applies Proposition 5.3 to establish that the context [Γ]^{−} must be entirely contained in the right hand premise of the last rule. Then, because there is no k at top level in the right hand premise of the last rule, Proposition 5.4 implies that reducing any formulas in [Γ]^{−} could not lead to a proof. In each case Proposition 5.4 is applied in this way several times, forcing the proof to take a specific form.

First, let us consider the case when ⊗R is the last rule applied in a proof. There are two possible forms this formula can take in any θ-translation: k ⊗ ([A]^{−}−∘(k−∘[B]^{+})), or k ⊗ ((p_{i} ⊕ b) ⊗ ⊤).

The first possibility would imply that the assumed imall proof must take the form given in Figure 16. This proof may be mimicked in iil*as simply the application of Right ⊃, and the hypothesis, which is itself a translation, may be mimicked by induction.

The second possibility would imply that the assumed imall proof has the form:

\underline{\overline{k ⊢ k}^{I}  \begin{matrix}\underline{⋮ ; [Δ]^{−} ⊢ p_{i} ⊕ b  ⋮ ; [Σ]^{−} ⊢ ⊤} & ⊗R \\ [Γ]^{−} ⊢ (p_{i} ⊕ b) ⊗ ⊤ \end{matrix}}_{⊗R} ; [Γ]^{−}, k ⊢ k ⊗ ((p_{i} ⊕ b) ⊗ ⊤)

for some Δ and Σ which together make up Γ. Investigating the left unfinished branch, one sees by Proposition 5.4 that p_{i} ⊕ b must be reduced. Furthermore, it can be seen that this p_{i} ⊕ b must be reduced to p_{i} . Proposition 5.2 implies that [Δ]^{−} ≡ p_{i} , and thus [Γ]^{−} ≡ [Σ]^{−}, [p_{i}]^{−}. On the other hand, the right unfinished branch could be completed by one application

k ⊢ k I

\underline{⋮ ; [Γ_{1}]^{−}, k ⊢ [p_{i}]^{+} ; \overline{[Γ_{1}]^{−} ⊢ k−∘[p_{i}]^{+}}^{−∘R} ⋮ ; [Γ_{2}]^{−}, k, b ⊢ [C]^{+} ; \overline{[Γ_{2}]^{−}, k ⊗ b ⊢ [C]^{+}}^{⊗L}}_{−∘L}
[Γ]^{−}, (k−∘[p_{i}]^{+})−∘(k ⊗ b) ⊢ [C]^{+}

⋮ [Γ]^{−}, k, [A]^{−} ⊢ [C]^{+}_{\overline{[Γ]^{−}, k ⊗ [A]^{−} ⊢ [C]^{+}}^{⊗L}}

⊕L

[Γ]^{−}, ((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−}) ⊢ [C]^{+} −∘L

[Γ]^{−}, k, k−∘(((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−})) ⊢ [C]^{+}

Figure 19: First possibility of −∘L.

of ⊤R. Whatever its form, one may mimick this entire proof in iil* by an application of identity.

This completes the analysis in the case that the last proof rule applied is right tensor. In the case that the last rule applied is left implication, there are two possible forms this formula can take in any θ-translation: k−∘(((k−∘[p_{i}]^{+})−∘(k ⊗ b)) ⊕ (k ⊗ [A]^{−})) and k−∘((([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−})).

The first possibility would imply that the assumed imall proof must take the form displayed in Figure 19, where [Γ_{1}]^{−} and [Γ_{2}]^{−} form a partition of [Γ]^{−}. (Note that this form is almost the same as that in Figure 17.) Applying the induction hypothesis to the leftmost branch in Figure 19 yields an iil* proof of Γ_{1} ⊢ p_{i}. By Proposition 3.1 one obtains a iil* proof of Γ ⊢ p_{i}. On the other hand, applying the induction hypothesis to the rightmost branch in Figure 19 yields an iil* proof of Γ, A ⊢ C. Using Left ⊃ 1, one then constructs an iil* proof of Γ, (p_{i} ⊃ A) ⊢ C. The middle unfinished branch in Figure 19 figure is irrelevant to the translation, but happens to always be provable by Lemma 4.1.

The second possibility would imply that the assumed imall proof must take the form displayed in Figure 20, where [Γ_{1}]^{−} and [Γ_{2}]^{−} form a partition of [Γ]^{−}. (Note that this form is almost the same as that in Figure 18.) Applying the induction hypothesis to the leftmost branch in Figure 20 yields an iil* proof of Γ_{1}, B ⊃C ⊢ A⊃B. By Proposition 3.1 one obtains an iil* proof of Γ, B ⊃ C ⊢ A ⊃ B. Moreover, applying the induction hypothesis to the rightmost branch in Figure 20 yields an iil* proof of Γ, C ⊢ D. By Left ⊃ 2, one then obtains an iil* proof of Γ, (A ⊃ B) ⊃ C ⊢ D. The middle unfinished branch in Figure 20 figure is irrelevant to the translation, but is always provable in imall by Lemma 4.1.

k ⊢ k I

⋮ ; [Γ_{1}]^{−}, [(B ⊃ C)]^{−}, k ⊢ [(A ⊃ B)]^{+} ; \overline{[Γ_{1}]^{−}, [(B ⊃ C)]^{−} ⊢ k−∘[(A ⊃ B)]^{+}}^{−∘R} ; \overline{[Γ_{1}]^{−} ⊢ ([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))}^{−∘R} ⋮ ; [Γ_{2}]^{−}, k, b ⊢ [D]^{+} ; \overline{[Γ_{2}]^{−}, k ⊗ b ⊢ [D]^{+}}^{⊗L} −∘L
[Γ]^{−}, ([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b) ⊢ [D]^{+}

⋮ [Γ]^{−}, k, [C]^{−} ⊢ [D]^{+}_{\overline{[Γ]^{−}, k ⊗ [C]^{−} ⊢ [D]^{+}}^{⊗L}}

⊕L

[Γ]^{−}, (([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−}) ⊢ [D]^{+} −∘L

[Γ]^{−}, k, k−∘((([(B ⊃ C)]^{−}−∘(k−∘[(A ⊃ B)]^{+}))−∘(k ⊗ b)) ⊕ (k ⊗ [C]^{−})) ⊢ [D]^{+}

Figure 20: Second possibility of −∘L.

Remark. The reader will observe that in order to prove Lemma 5.5, in the course of the argument given above it sometimes suffices to rely on the general permutability properties of imall rules instead on Proposition 5.4. However, using Proposition 5.4 throughout the argument yields a stronger result than Lemma 5.5, namely that any cut-free imall proof of a θ-translation must take a specific form given above.

6 Efficiency of Transformation

For any iil sequent ρ we have provided an equiprovable imall sequent θ(ρ). This encoding into imall could be exponential in the size of ρ, but if ρ is of depth two or less, then θ(ρ) is linear in the size of ρ. Below we give a depth-reduction procedure that takes polynomial time and that produces a sequent Ξ(ρ) of depth at most two, which is only linearly larger than ρ. The transformation θ(Ξ(ρ)) therefore provides an argument for the pspacehardness of the decision problem for imall. The argument for membership of this problem in pspace is immediate and appears in [25].

The transformation from iil* to imall is efficient in another stronger manner. It preserves the structure of iil* proofs. The imall translation of an iil* proof is linear in the size of the given iil* proof. Note that our transformation from iil to iil* does not necessarily preserve the structure of cut-free proofs in iil due to the permutations that are needed to make copying redundant. Neither of our transformations preserves the structure of proofs with cut.

Γ, (A ⊃ B) ⊃ (C ⊃ D) ⊢ Z ⇒ x ⊃ (C ⊃ D), Γ, (A ⊃ B) ⊃ x ⊢ Z
Γ, p_{i} ⊃ ((A ⊃ B) ⊃ C) ⊢ Z ⇒ (A ⊃ B) ⊃ x, Γ, p_{i} ⊃ (x ⊃ C) ⊢ Z
Γ, p_{i} ⊃ (A ⊃ (B ⊃ C)) ⊢ Z ⇒ x ⊃ (B ⊃ C), Γ, p_{i} ⊃ (A ⊃ x) ⊢ Z
Γ, ((A ⊃ B) ⊃ C) ⊃ p_{i} ⊢ Z ⇒ x ⊃ (A ⊃ B), Γ, (x ⊃ C) ⊃ p_{i} ⊢ Z
Γ, (A ⊃ (B ⊃ C)) ⊃ p_{i} ⊢ Z ⇒ (B ⊃ C) ⊃ x, Γ, (A ⊃ x) ⊃ p_{i} ⊢ Z
Γ ⊢ (A ⊃ B) ⊃ (C ⊃ D) ⇒ (C ⊃ D) ⊃ x, Γ ⊢ (A ⊃ B) ⊃ x
Γ ⊢ p_{i} ⊃ (A ⊃ (B ⊃ C)) ⇒ (B ⊃ C) ⊃ x, Γ ⊢ p_{i} ⊃ (A ⊃ x)
Γ ⊢ p_{i} ⊃ ((A ⊃ B) ⊃ C)) ⇒ x ⊃ (A ⊃ B), Γ ⊢ p_{i} ⊃ (x ⊃ C)
Γ ⊢ (A ⊃ (B ⊃ C)) ⊃ p_{i} ⇒ x ⊃ (B ⊃ C), Γ ⊢ (A ⊃ x) ⊃ p_{i}
Γ ⊢ ((A ⊃ B) ⊃ C)) ⊃ p_{i} ⇒ (A ⊃ B) ⊃ x, Γ ⊢ (x ⊃ C) ⊃ p_{i}

Figure 21: Definition of Ξ

6.1 Depth Reduction in IIL

An iil formula of depth one is either an atom p or has the form (p_{i} ⊃ p_{j}). A formula of depth two is one of the form (p_{i} ⊃ (p_{j} ⊃ p_{k})), or the form ((p_{i} ⊃ p_{j}) ⊃ p_{k}). Given a sequent Γ ⊢ D, we define Ξ(Γ ⊢ D) to be the result of repeatedly applying any of the the set of transformations given in Figure 21 until none of them apply.

These transformations each reduce the depth of implications, at the expense of building a new implication (which is also shallower than the original). Thus this sequence of reductions always terminates. Notice that the only kinds of formulas left after the Ξ transformation are of the form: p_{i}, p_{i} ⊃ p_{j}, p_{i} ⊃ (p_{j} ⊃ p_{k}), or (p_{i} ⊃ p_{j}) ⊃ p_{k}, where p_{i}, p_{j}, and p_{k} are atomic propositions. Although all the formulas appearing are very small, there may be many more of them. This technique goes back to [35], see also [30].

We define a positive contextual formula, written F^{+}[C], to be a formula with a specific occurrence of a subformula C identified, which has positive polarity in the formula F. Similarly, a negative contextual formula, written F^{−}[C], is one where the specific occurrence of a subformula C has negative polarity in the formula F. Note that the occurrence specified is unique. That is, even if the formula C occurs multiple times as a subformula of F, the occurrence indicated by F^{+}[C] or F^{−}[C] is unique.

Proposition 2.1 readily yields:

Lemma 6.1 For any iil formula A, if a sequent involving a proposition p_{i} is provable in iil, then that sequent with p_{i} replaced with A is also provable in iil.

The main lemma regarding the soundness of depth reduction relies on the following two lemmas, which are easily shown by simultaneous induction on the structure of F:

Lemma 6.2 For all iil formulas A and B, all iil multisets Γ, and positive contexts F^{+}[ ], the sequent Γ, A ⊃ B, F^{+}[A] ⊢ F^{+}[B] is provable in iil.

Lemma 6.3 For all iil formulas A and B, all iil multisets Γ, and negative contexts F^{−}[ ], the sequent Γ, B ⊃ A, F^{−}[A] ⊢ F^{−}[B] is provable in iil.

The soundness of depth reduction follows:

Lemma 6.4 A sequent Γ ⊢ A is provable in iil if and only if Ξ(Γ ⊢ A) is provable in iil.

Proof. The argument is by induction on the steps of transformation Ξ applied to Γ ⊢ A. Each of the individual transformations may be written in one of four forms:

Γ, F^{−}[A] ⊢ B ⇒ Γ, (A ⊃ x), F^{−}[x] ⊢ B
Γ, F^{+}[A] ⊢ B ⇒ Γ, (x ⊃ A), F^{+}[x] ⊢ B
Γ ⊢ F^{+}[A] ⇒ Γ, (A ⊃ x) ⊢ F^{+}[x]
Γ ⊢ F^{−}[A] ⇒ Γ, (x ⊃ A) ⊢ F^{−}[x]

In the if direction, assuming we have a proof of the transformed sequent, we simply apply lemma 6.1, and we have a proof of the desired sequent with the unpleasant addition of the formula (A ⊃ A) in the context. Since ⊢ (A ⊃ A) is provable we may cut against this to achieve the desired proof.

In the only if direction, there are four cases, although they are all very similar.

Assuming that there was a proof of Γ, F^{−}[A] ⊢ B, one simply cuts this against the proof of (A ⊃ x), F^{−}[x] ⊢ F^{−}[A] guaranteed by Lemma 6.3, and thus obtains a proof of Γ, (A ⊃ x), F^{−}[x] ⊢ B.

Assuming that there was a proof of Γ, F^{+}[A] ⊢ B, one simply cuts this against the proof of (x ⊃ A), F^{+}[x] ⊢ F^{+}[A] guaranteed by Lemma 6.2, and thus obtains a proof of Γ, (x ⊃ A), F^{+}[x] ⊢ B.

Assuming that there was a proof of Γ ⊢ F^{+}[A], one simply cuts this against the proof of (A ⊃ x), F^{+}[A] ⊢ F^{+}[x] guaranteed by Lemma 6.2, and thus obtains a proof of Γ, (A ⊃ x) ⊢ F^{+}[x].

Assuming that there was a proof of Γ ⊢ F^{−}[A], one simply cuts this against the proof of (x ⊃ A), F^{−}[A] ⊢ F^{−}[x] guaranteed by Lemma 6.3, and thus obtains a proof of Γ, (x ⊃ A) ⊢ F^{−}[x].

For completeness, we state:

Lemma 6.5 For any intuitionistic sequent ρ , Ξ(ρ) is computable in polynomial time and its size is linear in ρ.

7 Conclusion

Linear logic is a refinement of both classical and intuitionistic logic. It admits a cut-elimination theorem. An interesting aspect of cut elimination is that in linear logic it is possible to encode constraints on the form of a cutfree proof in the conclusion sequent. Linear logic is therefore expressive in a manner that intuitionistic and classical logic are not. The classification of the complexity and decidability of fragments of linear logic highlights some of this expressiveness [25, 21].

Our embedding of the implicational fragment of propositional intuitionistic logic in the imall fragment of linear logic provides an alternative proof for the pspace-hardness of imall. More importantly, it provides insight into the use and elimination of the structural rules from iil through the embedding of iil into iil*. The system iil* is an interesting optimization of intuitionistic logic that could be useful in theorem proving and logic programming applications [10, 19, 28].

A number of questions remain open. An extension of our techniques to all intuitionistic propositional connectives should be investigated. On the other hand, it would be interesting to know whether there is an embedding of intuitionistic implication in imall that preserves the structure of all cutfree proofs. We would also like to know the complexity of cut-elimination for the system iil* with a cut rule. It is worth examining what transformations such as depth reduction mean at the level of proof terms given by the Curry-Howard isomorphism, and whether there are some useful optimizations in the evaluation of proof terms arising from such a study.

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