Focus-preserving Embeddings of Substructural Logics in Intuitionistic Logic Jason Reed∗ Frank Pfenning University of Pennsylvania Carnegie Mellon University Philadelphia, Pennsylvania, USA Pittsburgh, Pennsylvania, USA Abstract a signature of function symbols suited to the substructural logic being embedded. We present a method of embedding substructural log- The embedding can be viewed as a constructive resource ics into ordinary first-order intuitionistic logic. This em- interpretation of substructural logics, where the first-order bedding is faithful in a very strong sense: not only does domain provides the notion of resources. Because the proof it preserve provability of sequents under translation, but it system of the target language is limited, and compatible also preserves sets of proofs — and the focusing structure with focusing, we are able to formulate and prove much of those proofs — up to isomorphism. Examples are given stronger claims about the faithfulness of our embedding for the cases of intuitionistic linear logic and ordered logic, than can usually be obtained for standard resource seman- and indeed we can use our method to derive a correct fo- tics into classical algebraic structures. Not only does prov- cusing system for ordered logic. Potential applications lie ability coincide with provability across the embedding, but in logic programming, theorem proving, and logical frame- proofs correspond bijectively to proofs, and focusing phases works for substructural logics where focusing is crucial for to focusing phases. the underlying proof theory. Focusing is deeply connected to notions of uniform proof for logic programming, to the analysis of canonical forms in dependently-typed logical frameworks, and to efficient au- 1 Introduction tomated proof search procedures. By showing how to relo- cate the problem of understanding focusing systems of sub- structural logics to the setting of a simpler and more easily Substructural logics can enforce restrictions on use of understood calculus, we open the door to more (and more hypotheses by having a structured context: hypotheses in convenient) application of the expressivity of substructural linear logic [Gir87], where the context is a multiset, must logics in all of these areas. be used exactly once, and in ordered logic [Lam58, Pol01] The rest of the paper is organized toward the aim of are used in a specified order, because the context is a list. In treating two examples of the embedding, intuitionistic lin- the same vein as display logic [Bel82] and work on graphi- ear logic (Section 3) and ordered logic (Section 4), but we cal representations of structured contexts [Lam07], we show must first describe the representation language into which how diverse substructural logics can be treated uniformly they will be embedded (Section 2). We finish with a discus- by isolating the reasoning about the algebraic properties of sion of related work (Section 5) and conclusions about our their context’s structure. Unlike these other approaches, contribution (Section 6). we do so without introducing a logic that itself has a so- phisticated notion of structured context, and instead use fo- cused proofs [And92] in a very simple nonsubstructural 2 The Logic of FF logic. This reduction of substructural to nonsubstructural has proved useful in understanding the design of substruc- Our representation language is FF, for Focused First- tural (especially dependent) type theories. We specifically order intuitionistic logic. We refer in the sequel to the sub- show how to embed substructural logics into in a fragment structural logic being embedded as the object language. of focused first-order intuitionistic logic with equality, over The notion of focusing, introduced by Andreoli [And92], is a way of narrowing eligible proofs down to those that de- ∗The work was supported by the Fundac¸ao˜ para a Cienciaˆ e Tecnolo- compose connectives in maximal contiguous runs of logical gia (FCT), Portugal, under a grant from the Information and Communica- connectives of the same polarity. Polarity is a trait of propo- tions Technology Institute (ICTI) at CMU. We would also like to thank the anonymous reviewers of a previous version of this paper, as well as Robert sitional connectives which, among other properties, charac- Simmons, for their helpful comments. terizes whether they can be eagerly decomposed as goals 1 (negative propositions) or eagerly decomposed as assump- We will also use the defined judgment of equivalence − − tions (positive propositions). Importantly, focused proof s1 ≡ s2 of negative atoms, which is defined to mean that − − search is complete: there is a focused proof of a proposition s1 and s2 have the same predicate symbol, and each of the iff there is an ordinary proof, but there are generally fewer corresponding pairs of term arguments are related by ≡. distinct focused proofs. It is by using the tight control over The focus judgments are used when we have selected a proof search and proof identity that focusing affords that proposition and have committed to continue decomposing we are able to faithfully mimic not only of which proposi- it until we reach a polarity shift. Inversion takes place when tions are provable in the object language, but how they are we are trying to prove a negative proposition, and we ap- proved. ply right rules eagerly, because all right rules for negative FF is a multi-sorted first-order logic, and is parametrized propositions are characteristically invertible. over the structure of its first-order domain: we leave it open For uniformity, we write Γ ` J to stand for either for each particular embedding to choose a collection of Γ;[A] ` s− or Γ ` A. On occasion, when we need to con- sorts, which function symbols exist to build terms of those trast the judgment of FF with that of the object language, sorts, and how an equivalence relation ≡ on those terms is decorate the turnstile as `FF. axiomatized. In the embedding, first-order terms serve to describe the 2.2 Proof Theory shape of sequents with substructural contexts. The relation ≡ is used to express that two sequent shapes are considered The valid deductions of this judgment are defined by the equivalent. For instance, in the case linear logic, it captures inference rules in Figure 1. They are mostly standard, but the property that the order of hypotheses does not matter. we note some consequences of focusing discipline: when we are focused on a negative atomic proposition s−, the cur- − − 2.1 Syntax rent conclusion s0 must be already equivalent to s ; when focused on a positive atom s+, that same atom must already The basic syntax of FF is as follows. be found in the current context. Encountering ↓A on the right blurs focus, and begins inversion of A. Decomposing ↓ on the left begins a focus phase, which is only allowed Negative Props A ::= B ⇒ A | A ∧ A | > | ∀x:σ.A | s− once the conclusion has finished inversion, and arrived at a Positive Props B ::= s+ | ↓A negative atomic proposition s−. Contexts Γ ::= · | Γ, B The right rule for the quantifier is understood to have Sorts σ ::= ··· the usual side conditions about the freshness of variable it Terms t ::= x | · · · introduces. We write {t/x} for substitution of a first-order term for the variable x. By ` t : σ we mean that t is a well- The bulk of the propositions are built of negative log- formed term of sort σ; for space reasons we avoid giving a ical connectives: implication, conjunction, truth, univer- complete explanation of this rather standard notion. We will − sal quantification, and negative atomic propositions s . write f :(σ1, . , σn) → σ to indicate that f is a function Atomic propositions, both positive and negative (hereafter symbol taking n arguments of sorts σ1, . , σn and yielding sometimes ‘atoms’), are generally predicates on first-order a term of sort σ. terms. We leave it again to the particular embedding to de- cide which atoms (i.e., which predicates) exist in the lan- 2.3 Metatheory guage, and of which polarity. The first argument to implica- tion is, as usual for focusing systems, a positive proposition. This calculus satisfies the usual pair of properties that es- Ordinarily the positives might include existential quantifi- tablish its internal soundness (cut admissibility) and internal cation and disjunction, but for our purposes we only need completeness (identity expansion). Because of the equiva- positive atoms s+, and an inclusion of negative propositions lence relation allowed at negative atoms, we must first show back into positives, via the shift connective ↓ that interrupts a congruence lemma with respect to the relation. focus phases. − − Contexts Γ are built out of positive propositions, and are Lemma 2.1 (Congruence) Suppose s ≡ s0 . subject to tacit weakening, contraction, and exchange. − − The judgments of the system are: 1. If Γ ` s , then Γ ` s0 − − 2. If Γ;[A] ` s , then Γ;[A] ` s0 Right Focus Γ ` [B] Left Focus Γ;[A] ` s− Proof By induction on the derivation. Use transitivity for Right Inversion Γ ` A the case of s−L. 2 − − − − s1 ≡ s2 Γ ` A Γ, ↓A;[A] ` s Γ ` A1 Γ ` A2 Γ;[Ai] ` s + − s R s L ↓R ↓L ∧R ∧Li + + − − − − Γ, s ` [s ] Γ;[s1 ] ` s2 Γ ` [↓A] Γ, ↓A ` s Γ ` A1 ∧ A2 Γ;[A1 ∧ A2] ` s Γ, B ` A Γ ` [B] Γ;[A] ` s− Γ ` A ` t : σ Γ;[{t/x}A] ` s− >R ⇒R ⇒L ∀Rx ∀L Γ ` > Γ ` B ⇒ A Γ;[B ⇒ A] ` s− Γ ` ∀x:σ.A Γ;[∀x:σ.A] ` s− Figure 1. FF Inference Rules The admissibility of cut now follows. polarity shift connectives ↑ and ↓ passing between them. Their syntax is as follows. Theorem 2.2 (Cut Admissibility) The following rules are − admissible: Negative N ::= ↑P | N & N | > | P ( N | a Positive P ::= ↓N | P ⊗P | 1 | P ⊕ P | 0 | a+ | !N Γ ` [B] Γ, B ` J Γ ` A Γ, ↓A ` J Now to instantiate FF, we first choose a set of sorts, and Γ ` J Γ ` J function symbols to inhabit them.