# A Unified Sequent Calculus for Focused Proofs.

**ABSTRACT** We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.

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**ABSTRACT:**We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative–additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut-elimination holds in such fragments. From cut-elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical–linear hybrid logics.Annals of Pure and Applied Logic 01/2011; · 0.45 Impact Factor - SourceAvailable from: diku.dk[Show abstract] [Hide abstract]

**ABSTRACT:**Existing focused proof systems for classical and intuitionistic logic allow contraction for exactly those formulas chosen for focus. For proof-search applications, contraction is undesirable, as we risk traversing the same path multiple times. We present here a contraction-free focused sequent calculus for classical propositional logic, called LKF CF , which is a modification of the recently developed proof system LKF. We prove that our system is sound and complete with respect to LKF, and therefore it is also sound and complete with respect to propositional classical logic. LKF can be justified with a compilation into focused proofs for linear logic; in this work we show how to do a similar compilation for LKF CF , but into focused proofs for linear logic with subexponentials instead. We use two subexponentials, neither allowing contraction but one allowing weakening. We show how the focused proofs for linear logic can then simulate proofs in LKF CF . Returning to proof-search, we end this work with a small experimental study showing that a proof-search implementation based on LKF CF performs well compared to implementations based on leanT A P and several variants and optimizations on LK and LKF. - SourceAvailable from: Dale Miller
##### Article: Finding unity in computational logic

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**ABSTRACT:**While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around the proof theory of the sequent calculus: recent results in the area of focused proof systems will play a central role in developing this unity.

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A Unified Sequent Calculus for Focused Proofs

Chuck Liang

Hofstra University

Department of Computer Science

Hempstead, NY, USA

Email: Chuck.C.Liang@hofstra.edu

Dale Miller

INRIA Saclay -ˆIle-de-France

LIX/Ecole Polytechnique

Palaiseau, France

Email: dale.miller@inria.fr

Abstract—We present a compact sequent calculus LKU for

classical logic organized around the concept of polarization. Fo-

cused sequent calculi for classical logic, intuitionistic logic, and

multiplicative-additive linear logic are derived as fragments

of LKU by increasing the sensitivity of specialized structural

rules to polarity information. We develop a unified, stream-

lined framework for proving cut-elimination in the various

fragments. Furthermore, each sublogic can interact with other

fragments through cut. We also consider the possibility of

introducing classical-linear hybrid logics.

Keywords-Proof theory; focused proof systems; linear logic

I. INTRODUCTION

While it is well-known how to describe proof systems for

intuitionistic and linear logics as restrictions on structural

rules and formula within a classical logic proof system, these

logics are usually studied separately. Girard merged these

three logics into a unified sequent calculus called LU [1] in

such a way that these three logics appear as fragments that

can interact via the cut rule. Central to LU is a classification

of formulas according to one of three polarities, which are

used to identify the formulas on which structural rules apply.

There are potentially many computer science applications

that could mix classical, intuitionistic, and linear logics. For

example, a logic program might be mostly classical logic

but in certain situations require the dynamics of linear logic

(e.g., multiset rewriting). Type systems for a programming

language can be intuitionistic when dealing with functions

but classical when dealing with control operators [2]. With

these kinds of computer science applications, one needs to

establish adequacy results that confirm that formal proofs

using the logic specification really capture what is intended

in the application. When proving such adequacy results over

sequent calculus proofs, one discovers that there can be

enormous numbers of sequent proofs, most of which differ

only in minor ways. For this reason, adequacy results are

usually based on the normal form of sequent calculus proofs

known as focused proofs. In the area of logic programming,

such proofs present backchaining as an atomic action [3]. In

type systems, focused proofs provide a canonical form for

terms.

Focused proof systems, such as Andreoli’s proof system

for linear logic (see [4] and Figure 1) organize proofs into

two phases: one phase contains the invertible inference rules

while the other phase contains the (possibly) non-invertible

rules. The two polarities of negative (invertible) and positive

(non-invertible) arise from these two phases. The polarity

notions used in LU can be seen as compatible with those

used in focusing proof systems. However, LU is not focused.

We have developed LKU as a focused proof system for a

unifying approach to logic. The differences between LKU

and LU, however, are not limited to focusing. In particular,

LU can be described by a translation to linear logic (except

at the level of atoms). The LKU proof system (see Figure 3)

is a classical logic and there is no translation of LKU

proofs into linear logic proofs: instead, each of its fragments

may require a different translation. As in Gentzen’s original

systems, intuitionistic (and linear) logic can be seen as

subsystems of LKU with restrictions on the use of structural

rules.

The proof system LKU contains a rich set of logical

connectives (a merging of the connectives in linear, intu-

itionistic, and classical logics) and each connective has one

inference rule. This stands in sharp contrast to LU where

several connectives have a large number of introduction

rules. On the other hand, LU provides a small and fixed set

of structural rules while LKU has a larger set of structural

rules (being a focusing proof system causes some growth in

these rules). In LKU, the meaning of a connective, such as ⊕

and

rule but also by the sensitivity of the structural rules to their

polarity. By adjusting this sensitivity we can use the various

symbols of LKU to derive focusing systems for classical

logic, intuitionistic logic, MALL (multiplicative additive

linear logic), and other interesting fragments of these logics.

Since these fragments are based on formulas containing the

same set of connectives, it is possible for these fragments

to interact through cut elimination. Placing more emphasis

on structural rules also seems to be valuable since much of

the effort in designing focusing proof systems is centered

on what structural rules they should include. For example,

one can have systems that focus on a unique formula or on

multiple formulas [5]. One can insist that an asynchronous

phase terminates when all asynchronous formulas are re-

moved or allow it to terminate before they are all removed.

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........, is determined not only by their (usual) introduction

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In applications of focusing proofs to tabled deduction [6],

once an atom is proved, there are good reasons to switch

its polarity from negative to positive, but in such a setting,

one should restrict the “reaction” rules to not be applicable

for positive atoms (in order to avoid reproving atoms). The

LKU system should be a framework for studying a range of

possible restrictions to the structural rules. While fragments

of LKU will be defined by imposing restrictions on the

structural rules, all fragments share the same set of nine

introduction rules. Such uniformity simplifies proving cut-

elimination for fragments.

We do not claim for LKU all that LU promises. In

particular, although never fully explained or further studied,

LU leaves open the possibility of allowing hybrid formulas

that use connectives from multiple logics without restriction,

e.g., (A⊗B)∧C. While such a possibility is not within the

scope of LKU, we consider limited classical-linear hybrid

logics in Section VII and as future work in Section VIII.

II. THE FOCUSING PROOF SYSTEMS LLF AND LKF

There are many examples of proof systems in literature

that exhibit characteristics of focusing to one degree or

another. These include, for example, uniform proofs [3],

“polarized” proof systems LJT/LJQ [7], [8] and LKη

well as the more recent “mixed polarization” proof system

λRCC [10]. Andreoli [4] identified focusing as arising from

a duality between invertible and non-invertible inference

rules and presented the “two-phase” proof system LLF

presented in Figure 1.

A literal is an atomic formula or the negation of an atomic

formula. Connectives of linear logic are either asynchronous,

negative (&,

Atoms are assigned arbitrary polarity. LLF uses two kinds

of sequents. In the sequent ? Γ:∆ ⇑ L, the “zones” Γ

and ∆ are multisets. In the original system L is a list, but

it is also valid to consider L as a multiset. This sequent

encodes the usual one-sided sequent − ?Γ,∆,L. The Γ to

the left of the : is the classical or unbounded context, while

the ∆ is the linear or bounded context. This sequent will

also satisfy the invariant that ∆ contains only literals and

synchronous formulas. In the sequent Γ:∆ ⇓ F, the zone

Γ is a multiset of formulas, ∆ is a multiset of literals and

synchronous formulas, and F is a single formula. The use

of these two zones replaces the need for explicit weakening

and contraction rules.

In the LLF proof system in Figure 1, inference rules

involving the ⇑ belong to the asynchronous phase and rules

involving the ⇓ belong to the synchronous phase. The I1

and I2rules are the initial rules. Of the structural rules, the

reaction rules R⇑ and R⇓ and the decision rules D1 and

D2stand out: they will collectively be called the structural

rules. Some formulations of focusing, e.g., [9], [11], avoid a

presentation with two arrows in favor of careful descriptions

of when a sequent proof is actually focused.

p[9], as

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........, ∀, ?) or synchronous, positive (⊕, ⊗, ∃, !).

Γ:∆ ⇑ L

Γ:∆ ⇑ ⊥,L

Γ:∆ ⇑ F,G,L

Γ:∆ ⇑ F

Γ:∆ ⇑ F,L

Γ:∆ ⇑ F & G,L

provided that F is not asynchronous

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

G,L

Γ,F:∆ ⇑ L

Γ:∆ ⇑ ?F,L

Γ:∆ ⇑ B[y/x],L

Γ:∆ ⇑ ∀x.B,LΓ:∆ ⇑ ?,L

Γ:∆,F ⇑ L

Γ:∆ ⇑ F,L[R ⇑]

Γ:∆ ⇑ G,L

Γ:· ⇓ 1

Γ:∆ ⇓ F1

Γ:∆ ⇓ F1⊕ F2

Γ:∆ ⇑ F

Γ:∆ ⇓ F

If K a positive literal:

Γ:∆1 ⇓ F

Γ:∆1,∆2 ⇓ F ⊗ G

Γ:∆ ⇓ F2

Γ:∆ ⇓ F1⊕ F2

provided that F is either asynchronous or a

negative literal

Γ:∆2 ⇓ G Γ:· ⇑ F

Γ:· ⇓ !F

Γ:∆ ⇓ B[t/x]

Γ:∆ ⇓ ∃x.B

[R ⇓]

Γ:K⊥⇓ K[I1]

Γ:∆ ⇓ F

Γ:∆,F ⇑ ·[D1]

The focused proof system LLF

Γ,K⊥:· ⇓ K[I2]

Γ,F:∆ ⇓ F

Γ,F:∆ ⇑ ·

If F is not a negative

literal:

[D2]

Figure 1.

In LLF, the structural rules and the initial rules I1and I2

are the rules that are directly sensitive to polarity informa-

tion: these rules show that it is polarity that drives focusing.

In fact, if the polarity-related side conditions for these rules

are removed, we are left with a rather convoluted version

of an unfocused sequent calculus for linear logic where one

would be able to switch between the ⇓ and the ⇑ states

without regard to change in polarity. Notice also that the

rules for the exponential operators ? and ! behave less like

other introduction rules and more like the reaction rules R⇑

and R⇓.

Structural rules in the style of LLF will play a critical

role in our unified sequent calculus. In fact, our project here

is first to present a rich set of LLF-like structural rules and

then to investigate different subsets of those structural rules

to see how they account for different proof systems (for, say,

intuitionistic or linear logics).

LLF-style focused systems have also been adapted to clas-

sical and intuitionistic logic. In [12], the authors presented

the focused intuitionistic sequent calculus LJF that can be

seen as an LU-inspired translation of intuitionistic logic into

linear logic. That paper also presented the focused classical

sequent calculus LKF that was inspired by a double-negation

translation into LJF (similar to Girard’s LC [13]). The

system LKF is given in Figure 2 (a one-side presentation of

LJF is given in Figure 5). Here, P is positive, N is negative,

C is a positive formula or a negative literal, Θ consists of

positive formulas and negative literals, and x is not free in Θ,

Γ. Focused and unfocused sequents have the form ?→ [Θ],A

and ? [Θ],Γ, respectively.

The additive and multiplicative versions of conjunction

and disjunction are available in both LKF and LU. In LU,

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? [Θ,C],Γ

? [Θ],Γ,C

[]

?→ [P,Θ],P

? [P,Θ]

ID+, atom P

Focus

? [Θ],N

?→ [Θ],N

Release

?→ [¬P,Θ],P

?→ [N,Θ],¬N

ID−, atom N

?→ [Θ],T

? [Θ],Γ,A

? [Θ],Γ,A ∧−B

? [Θ],Γ,A

? [Θ],Γ,∀xA

?→ [Θ],Ai

?→ [Θ],A1∨+A2

? [Θ],Γ,¬F

? [Θ],Γ,B

? [Θ],Γ

? [Θ],Γ,¬T

? [Θ],Γ,A,B

? [Θ],Γ,A ∨−B

?→ [Θ],A

?→ [Θ],A ∧+B

?→ [Θ],A[t/x]

?→ [Θ],∃xA

?→ [Θ],B

Figure 2.The Focused Proof System LKF

one must examine the polarity the connective’s arguments

to determine the additive/multiplicative nature of that con-

nective. In LKF, the top-level logical symbol provides this

information: ∧−and ∨+are additive while ∧+and ∨−are

multiplicative. The difference between the two conjunctions

and two disjunctions lies in the focused proofs that they

admit: they are, however, provably equivalent. In contrast,

the linear connectives ⊗ and & are not provably equivalent.

While LKF inherits the structural rules of LLF, the

reaction rules Release and [] of LKF correspond not to R⇓

and R⇑ but to the ! and ? introduction rules of LLF. The

decision rule D2 in LLF corresponds directly to the LKF

rule Focus: both embody an explicit contraction. There is,

however, an important difference between these two proof

systems regarding the formulas that are contracted. In LKF

(and LJF), formulas selected for focus (and thus subjected

to contraction) are always positive. In LLF, however, the

? introduction rule stops asynchronous decomposition and

so asynchronous formulas are also subject to contraction.

The restriction of contraction to only positive formulas is

an important characteristic of LKF and prompts us to adopt

this feature to our unified system. In fact, we exchange

the ability to represent full linear logic for the benefits of

a system that is better behaved with respect to focusing,

and which can still accommodate classical, intuitionistic,

and multiplicative-additive linear logic. This simplification

of LKF also leads to a more direct proof of cut-elimination

(without the need for Gentzen’s mix rule [14]).

III. LKU

Central to the LKU proof system, found in Figure 3,

are the four polarities +2 (positive classical), +1 (positive

linear), −1 (negative linear), and −2 (negative classical).

Atomic formulas are assigned polarities from this set. Other

formulas derive their polarity from their top-level connective

as follows: ∧+, ∨+, ∃, 1, 0 are given polarity +2; ⊗, ⊕, Σ

are given polarity +1;

∧−, ∨−, ∀, ?, ⊥ are given polarity −2. Negation (A⊥) is

defined by the following De Morgan dualities: ⊗/

∧+/∨−, ∨+/∧−, Σ/Π, ∃/∀, 1/⊥, ?/0, A/A⊥for literals

A. The dual polarity of +1 is −1 and the dual of +2 is −2.

Formulas are assumed to be in negation normal form.

A large number of connectives are included in LKU.

The polarity scheme of LKU is superficially similar to

that of LU. The “neutral” polarity of LU is here divided

into the +1 and −1 polarities. Unlike LU, the distinctions

between the connectives is not fixed within the unified logic.

The introduction rules make no distinction between the

linear and classical versions of each connective: the notation

[⊗|∧+] means that the rule is applicable to both connectives.

LKU can be divided into two principal components: the

introduction rules, which are invariant for all fragments,

and the collection of initial, decision, and reaction rules

(collectively called “structural” rules here), which can be

restricted to define sublogics. The structural rules are further

divided between the “level-1” and “level-2” rules.

LKU sequents of the form ? Γ : ∆ ⇓ B and ? Γ : ∆ ⇑ Θ

and will always satisfy the invariant that Γ and ∆ contain

only positive formulas or negative literals. As a consequence,

the only possible instances of the two initial rules I1and I2

(Figure 3) will be such that P is a positive literal.

As given, LKU can only be called classical logic. The

four connectives for conjunction, ∧+, ∧−, ⊗, and & are all

provably equivalent, as are the four for disjunction and the

pairs of quantifiers and units. The structural rules are only

sensitive to the positive/negative distinction and not to the

linear/classical distinction. If we removed even this basic

level of sensitivity to polarity, then we are left with a verbose

version of the unfocused LK. Clearly, the inference rules of

LKU are sound with respect to classical logic. The classical

completeness of LKU follows from the completeness of LKF

[12], which it contains.

A focusing phase of a proof ends in either a reaction rule,

R1⇓ or R2⇓, or in an initial rule I1or I2. The ? and ! rules of

LLF reappear in LKU as the level-2 reaction rules R2⇑ and

R2⇓. As given, R2⇓ is subsumed by R1⇓: their distinction

will become clear when we consider sublogics. Both the

R1⇑ and R2⇑ rules exclude asynchronous formulas, as does

the D2 rule. This divergence from LLF means that we

will not be able to represent full-linear logic for reasons

explained in the previous section. These restrictions are

similar to those of polarized linear logic [15]. In LKU the

role of the exponential operators is replaced entirely by

polarity information. If we relaxed these restrictions and

allowed R2⇑ and D2 to be applicable for asynchronous

formulas, then clearly every LLF proof can be mimicked.

Although a unified logic that accommodates full linear

logic is certainly an interesting topic (see Section VIII), the

restriction that we adopt is also worthy of separate study.

Fragments of LKU are defined by restricting the structural

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........, &, Π are given polarity −1; and

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........, ⊕/&,

Page 4

? Γ : P⊥⇓ P

I1

? Γ : ∆,C ⇑ Θ

? Γ : ∆ ⇑ C,ΘR1⇑

? C,Γ : ∆ ⇑ Θ

? Γ : ∆ ⇑ C,ΘR2⇑

? Γ : ∆ ⇑ N

? Γ : ∆ ⇓ N

? Γ :⇑ N

? Γ :⇓ N

R1⇓

? Γ : ∆ ⇓ P

? Γ : ∆,P ⇑D1

? P,Γ : ∆ ⇓ P

? P,Γ : ∆ ⇑? Γ,P⊥:⇓ P

I2

R2⇓

D2

? Γ : ∆ ⇑ A,B,Θ

? Γ : ∆ ⇑ A [

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........|∨−] B,Θ

? Γ : ∆1 ⇓ A

? Γ : ∆1∆2 ⇓ A [⊗|∧+] B

[

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........|∨−]

? Γ : ∆ ⇑ Θ

? Γ : ∆ ⇑ ⊥,Θ⊥

? Γ : ∆2 ⇓ B

? Γ : ∆ ⇑ A,Θ

? Γ : ∆ ⇑ A [&|∧−] B,Θ

? Γ : ∆ ⇑ B,Θ

[&|∧−]

? Γ : ∆ ⇑ ?,Θ?

[⊗|∧+]

? Γ :⇓ 11

? Γ : ∆ ⇓ Ai

? Γ : ∆ ⇓ A1 [⊕|∨+] A2

? Γ : ∆ ⇑ A,Θ

? Γ : ∆ ⇑ [Π|∀]x.A,Θ

[⊕|∨+]

? Γ : ∆ ⇓ A[t/y]

? Γ : ∆ ⇓ [Σ|∃]y.A

[Σ|∃][Π|∀]

The rules for Initial, Reaction, and Decision as well as three introduction rules for the units, eight rules for the binary connectives, and

four rules for the quantifiers. Here, P is positive (+2 or +1), N is negative (−2 or −1), and C is a positive formula or negative literal.

Figure 3.The Unified Focusing Sequent Calculus LKU

rules and possibly also the forms of formulas used. Not all

fragments, however, can be called “logics” (see Section VII).

Assume that end-sequents of LKU all have the form ?:⇑ Γ.

The following fragments are immediate.

Focused MALL. If we forbid all uses of the level-2 struc-

tural rules and only allow I1, D1, R1⇑, and R1⇓, then the

resulting system is essentially the same as LLF restricted

to the MALL fragment (but with quantifiers). We shall call

this fragment MALLF. Note that “forbidding level-2 rules”

is not the same as forbidding the +2/−2 polarities: the units

0, 1, ⊥ and ? are all still accounted for in MALLF. In fact,

we still retain all the connectives of LKU, but symbols such

as ∧+and ⊗ will both be interpreted as linear connectives.

LKF. If we forbid all the level-1 rules and only allowed the

level-2 structural rules then we arrive at a more conventional

sequent calculus for classical logic, one that’s similar to

LKF. Symbols such as ⊗ and

have the same meaning as their classical counterparts.

Retaining seemingly redundant symbols facilitates the

communication between different fragments of LKU through

cut, which is difficult to formalize if the fragments use

disjoint sets of connectives.

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

are retained but they will

IV. THE INTUITIONISTIC FRAGMENT

Intuitionistic logic appears as a linear-classical hybrid

fragment within LKU and, as a result, that fragment provides

a focused proof system for intuitionistic logic. We illustrate

this by showing how the LJF proof system of [12] can be

seen as a fragment of LKU. Since LJF is itself a framework

for describing a range of focused proof systems (e.g., LJT

[7], LJQ’ [8], and λRCC [10]) and unfocused proof systems

(e.g., LJ [14]) for intuitionistic logic, describing LJF is a

good test of LKU’s expressiveness.

Formulas in LJF are “annotated” intuitionistic formulas:

that is, atomic formulas are assigned an arbitrary but fixed

polarity (either positive or negative) and conjunctions are

annotated as being either additive ∧−or multiplicative ∧+.

The LJF proof system is a two-sided sequent calculus

using sequents of the following styles. The premises and

conclusion of invertible inference rules use sequents of the

form [Γ],Θ −→ R: such sequents lack focus. Dually, the

premises and conclusion of non-invertible inference rules use

sequents such as [Γ]

−→ [R], which provides a “left-focus”

formula L or sequents such as [Γ] −R→, which provides

a “right-focus” formula R. A set of “structural rules” are

provided in LJF that mix sequents of both kinds.

LJF formulas are mapped into LKU formulas using the

two functions [·]Rand [·]Ldefined in Figure 4. This is

a shallow, syntactic mapping of intuitionistic connectives

(whose proof rules are described using two-sided sequents)

to classical connectives (whose proof rules are described

using one-sided sequents). That is, a left-occurrence of the

intuitionistic ⊃ is exactly the same as (a right-occurrences

of) the LKU connective ⊗. Positive LJF atoms are assigned

polarity +2 in LKU while negative LJF atoms are assigned

polarity −1 in LKU. Formulas in the range of [·]Rare called

essentially right intuitionistic formulas (they have polarity

+2 or −1) and formulas in the range of [·]Lare called

essentially left intuitionistic formulas (they have polarity −2

or +1). Notice that R is an essentially right intuitionistic

formula if and only if the negation normal form of R⊥is

an essentially left intuitionistic formula.

The usual symbols of linear logic are used to define

the negative intuitionistic connectives. The left-hand side

of an essentially right implication is essentially left (and

vice versa) and is given a classical treatment by the reaction

rules, thus mimicking the usual linear-logic interpretation of

intuitionistic implication as !A−◦B. As with LKF, the LJF

fragment contains both positive and negative connectives for

conjunction: ∧+and & respectively on the right (∨−and ⊕

on the left). However, there is only the positive disjunction

∨+(∧−on the left), with

L

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

only used in the representation

Page 5

[B ∧−C]R= [B]R& [C]R

[B ⊃ C]R= [B]L. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

[∀x.B]R=

[B ∧−C]L= [B]L⊕ [C]L

[B ⊃ C]L= [B]R⊗ [C]L

[∀x.B]L=

For atomic A, [A]R= A and [A]L= A⊥.

[B ∧+C]R=[B]R∧+[C]R

[B ∨ C]R=[B]R∨+[C]R

[∃x.B]R=

[B ∧+C]L= [B]L∨−[C]L

[B ∨ C]L= [B]L∧−[C]L

[∃x.B]L=

[C]R

Πx.[B]R

∃x.[B]R

Σx.[B]L

∀x.[B]L

Figure 4.Mapping LJF formulas into LKU.

of intuitionistic implication. Intuitionistic negation ∼A is

defined as A⊥

0 when appearing essentially right. (For

a minimal logic treatment of negation, replace 0 in the

language with some designated +2 atom.)

While we do not present the full LJF proof system (see

[12]) to save space, it can be seen as a simple fragment

within LKU. In particular, if one restricts the structural

rules of LKU to be those given in Figure 5, we are left

with a one-sided sequent presentation of LJF. To illustrate

how two sided inference rules for intuitionistic logic can

be represented in the one-sided, focused setting of LKU,

consider the additive and multiplicative versions of the

conjunction-left rule in (unfocused) LJ:

Ai,Γ ? C

A1∧ A2,Γ ? C

These inference rules correspond to the focused LKU rules

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

and

A1,A2,Γ ? C

A1∧ A2,Γ ? C.

? Γ⊥: C ⇓ A⊥

? Γ⊥: C ⇓ A⊥

For the reader familiar with LJF [12], the two-sided

LJF sequents correspond to the one-sided LKU sequents as

follows:

i

1⊕ A⊥

2

and

? Γ⊥: C ⇑ A⊥

? Γ⊥: C ⇑ A⊥

1,A⊥

1∨−A⊥

2

2

.

[Γ],Θ −→ R

[Γ],Θ −→ [R]

[Γ] −R→

[Γ]

−→ [R]

The original structural rules of LJF and those in Figure 5

correspond as follow: Lf ↔ D2, Rf ↔ D1, Rl↔ R1⇓,

Rr↔ R2⇓, []l↔ R2⇑, []r↔ R1⇑.

In the following we only consider LJF in its form as a

fragment of LKU. Of the structural rules of LJF, I1, R1⇓,

R2⇑, and D2can be called “left rules” while I2, R2⇓, R1⇑,

and D1are the right rules.

Observe that the R1⇑ and R1⇓ rules allow only one

essentially right formula inside the linear context of an LJF

sequent. If we are interested only in mapping complete

LJF proofs to intuitionistic proofs, then this restriction is

not necessary: the single-conclusion condition is essentially

already enforced by other rules such as R2⇓, I1, and I2.

When building a proof from the bottom-up, malformed

←→

←→

←→

←→

? [Γ]L:⇑ [Θ]L,[R]R

? [Γ]L: [R]R⇑ [Θ]L

? [Γ]L: · ⇓ [R]R

? [Γ]L: [R]R⇓ [L]L

L

? Γ : Q⊥⇓ Q

? Γ : ∆ ⇓ P

? Γ : ∆,P ⇑D1

I1

? Γ : C ⇑ Θ

? Γ :⇑ C,Θ

R1⇑

? Γ : C ⇑ N

? Γ : C ⇓ N

? D,Γ : ∆ ⇑ Θ

? Γ : ∆ ⇑ D,ΘR2⇑

R1⇓

? Γ,Q⊥:⇓ Q

I2

? Γ :⇑ N

? Γ :⇓ N

R2⇓

? P,Γ : ∆ ⇓ P

? P,Γ : ∆ ⇑

D2

Q: +1 atom, C: +2 formula or −1 atom, N: −2 formula,

P: +2 formula, Q: +2 atom, D: +1 formula or −2 literal,

N: −1 formula, P: +1 formula

Figure 5.LJF as a Fragment of LKU

sequents, i.e., those with multiple essentially right formulas,

will be rejected by the initial rules if not sooner.

The stronger restrictions for the R1⇑ and R1⇓ rules allow

us to establish the stronger correspondence between open

proofs as well. In LJF, malformed sequents could appear as

a consequence of splitting the context when applying the ⊗

rule. The essentially-left occurrence of an implication A ⊃

B has the form A⊗B⊥where A is essentially right and B⊥

essentially left. The implication-left rule of LJ thus appears

in the form

? Γ⊥:⇓ A

? Γ⊥: C ⇓ A ⊗ B⊥

But it is also possible to split the context so as to have

? Γ⊥: C ⇓ A, which is a sequent with two essentially right

formulas. The reaction rules of LJF are designed, however,

to reject such a malformed sequent at the end of a focusing

(⇓) phase. Such a phase must end in either a reaction or an

initial rule. In an incomplete proof structure, there could be

occurrences of malformed sequents inside the synchronous

phases of proofs, but we shall only consider completed

phases as marking the boundary of inference rules: what

defines a focused proof is not what happens in the details of

each synchronous or asynchronous phase but what happens

at the borders of such phases. Each synchronous or asyn-

chronous phase can be thought of as the introduction of a

synthesized connective; that is to say a single introduction

rule. A border sequent of LJF will be either an axiom or have

the form ? Γ⊥: C ⇑, which corresponds to a well-formed

intuitionistic sequent. Without the explicit restriction to one

formula in the level-1 reaction rules, malformed sequents

may survive across focusing phases.

Thus if we strictly use only polarity information in

restricting the structural rules, we can achieve a weak form

of full-completeness. With the stronger forms of the rules

as presented, the local structure of even partial intuitionistic

proofs are preserved.

There is, however, one scenario in which a malformed

sequent may also appear as part of a complete LJF proof.

When considering full intuitionistic logic, as opposed to

? Γ⊥: C ⇓ B⊥

⊗

Page 6

? Γ : Q⊥⇓ Q

I1

? Γ : C ⇑ Θ

? Γ :⇑ C,Θ

R1⇑

? P,Γ : ∆ ⇑ Θ

? Γ : ∆ ⇑ P,ΘR2⇑

? Γ :⇑ N

? Γ :⇓ N

R2⇓

? P,Γ : ∆ ⇓ P

? P,Γ : ∆ ⇑

D2

Q: +1 literal, C: −1 atom, P: +1 formula, N: −1 formula

Figure 6.The nLJF fragment of LKU.

minimal logic, the intuitionistic context may be inconsistent.

That is to say the ? rule (0 on the left) may appear in a proof.

This problem is likewise encountered by LU and several

other works that encodes intuitionistic logic into linear logic

(including LJF). To resolve this problem we must show that

even in such situations there is a LJF proof that corresponds

to a well-formed LJ proof. Such an argument relies on cut-

elimination (see Section V).

This treatment of intuitionistic logic in LKU is similar

to that in LU with two differences. First, LU is a two-

sided sequent calculus in order to accommodate intuitionistic

logic. The richness of polarity information in LKU replaces

the need for a two-sided system: the polarity of a formula

unambiguously determines its essentially left or right status.

(Of course, one may still prefer a two-sided system for

readability.) Second, it is equally valid in LKU to use the

−2/+1 polarities for essentially right formulas and +2/−1

for the left ones by altering the restrictions on the structural

rules.

The Negative Intuitionistic Fragment: There is a sig-

nificant fragment of LJF where the problem with context

splitting in the ⊗ rule does not appear. We shall call this

fragment the negative intuitionistic fragment nLJF and it

corresponds to the neutral intuitionistic fragment of LU. The

structural rules that correspond to nLJF are found in Figure

6. In this fragment, essentially right formulas have only

polarity −1 and essentially left formulas have only polarity

+1. In an essentially left implication A⊗B⊥, A will have -1

polarity, which means that the appearance of a malformed

sequent ? Γ⊥: C ⇓ A will immediately invoke the R2⇓

rule, which fails because the linear context is not empty.

V. CUT ELIMINATION

In order to claim that a fragment of LKU is, in fact, a

logic, one needs to show that the result of eliminating a cut

between two proofs in the given fragment yields a proof

still in that fragment. In this section, we illustrate how to

prove such a result. We later examine what kinds of cut-

free proofs can be derived when different fragments of LKU

cross-cut. The LKU framework provides a uniform structure

to cut-elimination arguments. Since the introduction rules

are shared by all the fragments, the permutation of cut above

introductions can be demonstrated just once. Furthermore,

instead of considering individual rules, we can define the

following relations to characterize the structure of complete

synchronous and asynchronous phases. For convenience, we

write ΓΓ?to denote the multiset union of Γ and Γ?.

Definition 1: Let ↑ and ↓ represent relations between

formulas and multisets of formulas defined as follows:

◦ A ↑ {A} if A is a negative literal or positive.

◦ ⊥ ↑ {}.

◦ (A [

◦ (A [&|∧−] B) ↑ Φ if A ↑ Φ.

◦ (A [&|∧−] B) ↑ Φ?if B ↑ Φ?.

◦ A ↓ {A} if A is a positive literal or negative.

◦ 1 ↓ {}.

◦ (A [⊗|∧+] B) ↓ ΨΨ?if A ↓ Ψ and B ↓ Ψ?.

◦ (A [⊕|∨+] B) ↓ Ψ if A ↓ Ψ.

◦ (A [⊕|∨+] B) ↓ Ψ?if B ↓ Ψ?.

(First-order quantification [Π | ∀], [Σ | ∃] can be treated

similarly.)

Using these dual relations, we can study how cuts permute

only where it matters the most: at the borders between posi-

tive and negative focusing phases where the rules of reaction

and decision come into play. In MALL, the distributive laws

can be used to provide the following normal forms for all

positive and negative synthetic connectives:

⊕i∈I(⊗j∈JiNij)

where I and Ji(for i ∈ I) are finite set of indices and Nij

denotes a negative formula or a literal and Pij denotes a

positive formula or a literal. Using the notation above, the

following are satisfied:

⊕i∈I(⊗j∈JiNij) ↓ {Nij| j ∈ Ji}

&i∈I(

Thus, the ↓ selects the premises for a possible introduction

rule of a positive synthetic connective while the ↑ selects

a possible premise for the introduction rule of a negative

synthetic connective. While normal forms for synthetic

connectives are equivalent to using the ↓ and ↑ within

MALL, one does not expect that similar distributive laws

hold for all fragments of LKU and, as a consequence,

normal forms for synthetic connectives might be hard to

write down. For this reason, we employ the notation using

arrows since they provide natural and immediate descriptions

of the introduction rules for synthetic connectives in all of

LKU.

Lemmas 2 through 4 below are all proved by induction

on the structure of formulas.

Lemma 2: Given a formula R, let Φ1,...,Φm be mul-

tisets such that R ↑ Φ1,...,R ↑ Φm and if R ↑ Φ then

Φ = Φifor some unique 1 ≤ i ≤ m. Every cut-free proof

of ? Γ : ∆ ⇑ R,Θ is of the form

? ΓΦ1

...

? Γ : ∆ ⇑ R,Θ

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........|∨−] B) ↑ ΦΦ?if A ↑ Φ and B ↑ Φ?.

&i∈I(

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

j∈JiPij),

(i ∈ I)

(i ∈ I)

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

j∈JiPij) ↑ {Pij| j ∈ Ji}

1: ∆Φ2

1⇑ Θ

···

? ΓΦ1

m: ∆Φ2

...

m⇑ Θ

Page 7

such that Φ1

such that Φi= Φ1

iΦ2

i= Φi for each 1 ≤ i ≤ m. For every Φi

iΦ2

i, if

? ΓΦ1

1: ∆Φ2

1⇑ Θ,...,? ΓΦ1

m: ∆Φ2

m⇑ Θ

are all cut-free provable, then ? Γ : ∆ ⇑ R,Θ is also cut-

free provable.

The splitting of Φiinto Φ1

between R1⇑ and R2⇑. The above lemma does not specify

how Φ1

a fragment of LKU, we can specialize this lemma by

being more specific as to how the multiset is split between

the linear and classical contexts. That is, we can deduce

specialized versions of this lemma for each fragment from

the general lemma above. In the MALLF fragment, Φ1

be empty. In the intuitionistic LJF fragment, Φ1

of essentially left formulas and Φ2

essentially right formula (see lemma 7 below).

The dual lemma for ↓ is the following.

Lemma 3: Let R ↓ {a1,...,an} and assume that ? Γ :

∆1⇓ a1,...,? Γ : ∆n⇓ anare all cut-free provable. Then

? Γ : ∆1...∆n⇓ R is also cut-free provable. Furthermore,

every cut-free proof of ? Γ : ∆1...∆n⇓ R is of the form

? Γ : ∆1⇓ a1

...

? Γ : ∆1...∆n⇓ R

where R ↓ {a1,...,an}.

Cut-elimination follows from the following lemma.

Lemma 4: R ↑ {a1,...,an} if and only if R⊥

{a⊥

The inductive measure for the cut-elimination proof is

the usual lexicographical ordering on the size of the cut

formula and the heights of subproofs. In a focused proof,

the height of a proof can be taken as the maximum number

of alternating asynchronous-synchronous phases (i.e., the

number of D1and D2rules) along a path to a leaf.

Theorem 5: The cut rule

? Γ :⇑ A,Θ

? ΓΓ?:⇑ ΘΘ?

can be eliminated in LKF. Similarly, the cut rule

iand Φ2

irepresents the choice

iand Φ2

iare split: e.g., Φ2

imay be empty. Given

imust

iconsists

iconsists of at most one

···

? Γ : ∆n⇓ an

...

↓

1,...,a⊥

n}.

? Γ?:⇑ A⊥,Θ?

cut

?: ∆ ⇑ A,Θ

?: ∆?⇑ A⊥,Θ?

?: ∆∆?⇑ ΘΘ?

cut

can be eliminated in MALLF. Finally, the cut rule

? Γ⊥:⇑ A

? ∆⊥: Ω ⇑ A⊥,Θ

? Γ⊥∆⊥: Ω ⇑ Θ

cut

can be eliminated in LJF and nLJF. In this latter case, Ω

consists of at most one essentially right formula.

Instances of cut are divided into two categories. Key-case

cuts are cuts where both cut formulas are principal, i.e., when

the positive cut formula is under focus and the negative one

is being decomposed. Parametric cuts refer to cuts when,

in at least one subproof, the cut formula is not principal.

The parametric formula can be a synchronous formula under

focus or an asynchronous formula.

We first illustrate the proof in the MALLF case. As usual,

we can assume that the two subproofs involved in a cut are

cut-free, since we can apply the procedure to the lowest-

height cuts first. The cut-elimination procedure permutes

the cut above the introduction of parametric formulas until

a key case is reached. By lemma 2, this holds easily for

asynchronous parametric cases. But when reduction reaches

the following state:

?: ∆,A ⇑?: ∆?⇑ A⊥

?: ∆∆?⇑

cut

the left-side subproof must end in a decision rule (D1),

which selects a formula for focus. If the formula selected

for focus is the cut formula A, then we have a key-case cut.

If D1selects some other formula in ∆ for focus, then we

have a parametric case with a positive parametric formula.

By lemmas 2 and 3, the key-case cut will have the form

?: ∆1 ⇓ a1

...

···

?: ∆n ⇓ an

...

?: ∆ ⇓ A

?: ∆,A ⇑D1

?: ∆ ⇑ A

R1⇑

?: ∆?Φ1 ⇑

...

···

?: ∆?Φm ⇑

...

?: ∆?⇑ A⊥

?: ∆∆?

cut

where A ↓ {a1,...,an}, A⊥↑ Φifor each 1 ≤ i ≤ m, and

∆ = ∆1...∆n.

The positive parametric cut will have the form

? ∆1⇓ b1

...

···

? ∆n⇓ bn

...

? ∆,A ⇓ B

? ∆,A,B ⇑D1

?: ∆?⇑ A⊥

?: B,∆∆?

cut

where B ↓ {b1,...,bn} and ∆,A = ∆1...∆n.

It is also possible that both A and A⊥are literals, which

means that the right-side subproof will continue with the

form ?: ∆?,A⊥⇑, which will then also require a formula

to be selected for focus. This is a critical choice-point in

cut-elimination (see [9]). We must permute the cut above the

subproof that contains the positive cut formula. The positive

cut formula is “attractive” in the terminology of [9].

The key-case cut can be permuted into zero or more cuts

involving formulas of smaller size. By lemma 4, one of the

Φjwill have the form {a⊥

are exhaustive). For each ai, if it is a positive literal then

∆i = {a⊥

have a subproof ending in ?: ∆i⇑ ai, to which we can apply

1,...,a⊥

n} (assuming Φ1,...,Φm

i}. If ai is negative, then by reaction (R1⇓) we

Page 8

an inductive hypothesis, i.e., a cut between ?: ∆i⇑ aiand

the subproof containing Φj. In either case we obtain a proof

of ? ∆i∆?,a⊥

(each involving a cut formula of smaller size), we get the

conclusion ? ∆∆?⇑.

For the parametric cut, exactly one of the ∆iwill contain

the cut formula A. If bi is a positive literal, it cannot be

the case that bi= A⊥because A is assumed positive. This

critical fact relies on the choice to always permute the cut

above the subproof with the positive cut formula. Thus bi

must be negative and by R1⇓ we have a subproof of ?:

∆i⇑ bi. The original cut is permuted to a cut between

?: ∆i⇑ bi and ?: ∆?⇑ A⊥with a lower proof-height

measure. Again by lemma 3, we can then synthesize the

conclusion ?: B,∆∆?.

The difference between cut-elimination for MALLF and

for the classical LKF involves contraction (via D2) and

weakening (via I1and I2). The following lemma shows that

weakening and contraction are admissible in LKU.

Lemma 6: If ? A,A,Γ : ∆ ⇑ Θ has a cut-free proof, then

? A,Γ : ∆ ⇑ Θ has a cut-free proof of the same height. If

? Γ : ∆ ⇑ Θ has a cut-free proof, then ? A,Γ : ∆ ⇑ Θ has

a cut-free proof of the same height.

It is important to note that the explicit contraction in D2is

restricted to positive formulas. The key-case cut is preceded

above by several parametric cuts. That is, for the sequent

? A,Γ :⇓ A, the occurrence of A under focus is erased by a

key-case cut while the “copy” is erased by parametric cuts.

The parametric cuts have lower proof-height measures while

the key cut reduces to smaller cut formulas. This argument

would fail if we cannot assume that the A is positive: if A

is negative then there would be no key case.

The LKU framework allows us to identify elements of cut-

elimination proofs that are “reusable.” This is exemplified

in the intuitionistic case. A proof of cut-elimination for

LJF as originally presented is found in [12]. It involves a

simultaneous induction on seven versions of cut. A much

simpler proof is possible in the unified context. The principal

additional work needed for the intricacies of intuitionistic

polarity is captured by the following lemma, which is

provable by simultaneous induction on formulas.

Lemma 7: Let A be an essentially right formula and B an

essentially left formula. Let ΦA,ΦB,ΨA,ΨB, be multisets

such that A ↑ ΦA, B ↑ ΦB, A ↓ ΨA, and B ↓ ΨB. Then:

ΦAcontains exactly one essentially right formula;

ΨBcontains exactly one essentially left formula;

ΦBconsists of only essentially left formulas; and

ΨAconsists of only essentially right formulas.

With this lemma, cut-elimination for LJF can be proved

in essentially the same way as it is proved for other logics:

by applying lemmas 2, 3, and 4. The technical argument

involves a few more cases than for MALLF or LKF since

both D1and D2are used.

1...a⊥

i−1,a⊥

i+1,...a⊥

n⇑, and by repeated cuts

VI. COMMUNICATION BETWEEN FRAGMENTS

Since all the fragments of LKU share the same connec-

tives and atoms, different fragments can interact using cuts.

If we are only interested in cut-free classical proofs, then all

cuts between fragments collapse to classical cuts. In certain

circumstances, cut-elimination can preserve more structure.

We give two such examples. A formula is pure with respect

to a polarity if all of its subformulas have the same polarity.

Focusing on purely positive formulas leads to constructive

proofs.

Theorem 8: Let A be a purely +2 formula and let ∆

consist of purely −2 formulas. Given an LKF proof of ?:⇑

A,∆ and a LJF proof of ? Γ⊥: Ω ⇑ A⊥,Θ, the following

cut:

?:⇑ A,∆

Γ⊥: Ω ⇑ ∆Θ

can be replaced by a cut-free proof in LJF.

The proof of this theorem follows the same format as

other cut-elimination proofs and uses the observation that A

must be selected for focus in the LKF proof.

Note that formulas such as P ∨+P⊥are excluded from

the scope of the theorem because they cannot be purely of

one polarity. The scope of the theorem is expanded when

one considers that, except for the quantifiers, every classical

connective has an equivalent one of the opposite polarity.

Now consider cutting between a MALLF proof and an

LJF proof. It is not immediate that a MALLF proof of an

intuitionistic end-sequent (all formulas on the right side of

⇑) can be transformed into a intuitionistic proof. MALLF

proofs may “split the context” differently from an intuition-

istic proof.

The following lemma generalizes a key property of intu-

itionistic sequents that is valid in any fragment of LKU.

Lemma 9: Let Γ,∆,Θ consist of only essentially left in-

tuitionistic formulas. There is no LKU proof of ? Γ : ∆ ⇑ Θ

that does not include an instance of the ? rule.

This lemma is proved by contradiction: there cannot be

such a proof of minimum height. With this lemma we can

show that a MALLF proof of an intuitionistic sequent can

be transformed into an LJF proof. From cut-elimination in

LJF, we also have the following admissible cross-cut:

Theorem 10: Given an LJF proof of ? Γ⊥:⇑ A and a

MALLF proof of ?:⇑ A⊥,B where B is an essentially right

intuitionistic formula, the following cut

? Γ⊥: Ω ⇑ A⊥,Θ

cut

? Γ⊥:⇑ A

?:⇑ A⊥,B

? Γ⊥:⇑ B

cut

can be replaced by a cut-free proof in LJF.

VII. A LIMITED CLASSICAL-LINEAR HYBRID LOGIC

The existence of intuitionistic logic as a hybrid logic with

both linear and classical characteristics suggests that other

such hybrids may also exist. It is tempting to define such a

Page 9

? Γ : Q⊥⇓ Q

? Γ : ∆ ⇓ P

? Γ : ∆,P ⇑D1

I1

? Γ : ∆,C ⇑ Θ

? Γ : ∆ ⇑ C,ΘR1⇑

? Γ : ∆ ⇑ N

? Γ : ∆ ⇓ N

? D,Γ : ∆ ⇑ Θ

? Γ : ∆ ⇑ D,ΘR2⇑

? P,Γ :⇓ P

? P,Γ :⇑

R1⇓

? Γ,Q⊥:⇓ Q

? Γ :⇑ N

? Γ :⇓ N

I2

R2⇓

D2

Q: +1 atom, C: +1 formula or −1 atom, N: −1 formula, P:

+1 formula. Q: +2 atom, D: +2 formula or −2 literal, N: −2

formula, P: +2 formula

Figure 7.The Hybrid Logic HLL

logic by restricting the level-1 structural rules to +1/−1

formulas and the level-2 rules to +2/−2 formulas. Cut-

elimination fails, however, for such a system since it is possi-

ble to arbitrary interleaving linear and classical connectives.

To illustrate this issue, consider translating formulas such as

(A ⊗ B) ∧+C into linear logic, say, into !(A ⊗ B)⊗!C (as

suggested by the LU tables). Focusing cannot continue past

the !. It would be valid to transfer from a linear focusing

state to a classical one, but not vice versa. One can require

that classical formulas contain no linear subformulas. Let

us call this logic “High-Low Logic”, or HLL. The structural

rules of HLL are found in Figure 7.

Clearly, both classical logic and MALL are found as sub-

fragments of HLL. The sample hybrid formula A

A⊥) is provable if A is classical but not if A is linear.

It is also possible to understand HLL by a translation to

linear logic. We preserve the linear connectives and translate

the classical connectives as suggested by LU. For example,

if A is +2 and B is −2 then A∧+B is translated as A⊗!B

and A ∧−B is translated as ?A&B.

The following cuts can be eliminated within HLL.

? Γ : ∆ ⇑ A,Θ

? ΓΓ?: ∆∆?⇑ ΘΘ?

? A,Γ : ∆ ⇑ Θ

? ΓΓ?: ∆ ⇑ Θ

Here, A is a +1 or −1 formula and A is a classical formula.

To illustrate cut-reduction in HLL, assume that N is a −2

formula and that M is a −1 formula. The proof

? Γ :⇑ N

? Γ :⇓ NR2⇓

? Γ : ∆ ⇓ N ⊗ M

? Γ : ∆,N ⊗ M ⇑D1

? Γ : ∆ ⇑ N ⊗ M

? ΓΓ?: ∆∆?⇑

is reduced to

? Γ?,N⊥: ∆?⇑ M⊥

? ΓΓ?: ∆?⇑ M⊥

? ΓΓΓ?: ∆∆?⇑

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

(A⊥⊗

? Γ?: ∆?⇑ A⊥,Θ?

Cut1

? Γ?:⇑ A⊥

Cut2

? Γ : ∆ ⇑ M

? Γ : ∆ ⇓ MR1⇓

⊗

R1⇑

? Γ?,N⊥: ∆?⇑ M⊥

? Γ?: ∆?⇑ N⊥,M⊥R2⇑

? Γ?: ∆?⇑ N⊥. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

M⊥

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........

Cut1

? Γ :⇑ N

Cut2

? Γ : ∆ ⇑ M

Cut1

This example can be generalized into the core of the cut-

elimination proof, which shows that a key-case cut will be

reduced to Cut2rules for the classical subformulas of the

cut formula and Cut1rules for the linear subformulas.

While HLL exhibits reasonable cut-elimination properties,

it also diverges from the other major fragments of LKU

in significant ways. In the other fragments, provability is

invariant under the assignment of different polarities to

atomic formulas. This invariance fails in HLL. Switching

between linear and classical atoms may affect provability,

as with switching between −2 and +2 atoms. Moreover,

the classical equivalences between the positive and negative

versions of connectives, such as ∨−and ∨+, hold only in a

purely classical context. In a mixed linear-classical context,

∨−is equivalent to

contradict cut-elimination as stated above. Observe that one

cannot replace a ∨−with a ∨+through cut except in a purely

classical context. No admissible cut can be applied on the

sequents

. . . ...... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... This apparent anomaly does not

? A⊥∧+A⊥:⇑ A ∨+A

because the linear context in the right sequent is not empty.

and

?: A⊥⊗ A⊥⇑ A ∨−A

VIII. FUTURE WORK: UNRESTRICTED HYBRID LOGIC

Girard’s LU system leaves open the possibility of mixing

logics without restriction. However, designing a focused

system that is entirely faithful to LU faces difficulties.

For example, the De Morgan dualities fail when “neutral”

formulas are mixed with classical ones.

In this paper, we have employed the different approach

of carefully restricting the structure of formulas and se-

quents using polarity information. Extracting a focused proof

system for intuitionistic logic is a powerful validation of

this approach. Extracting the logic HLL is another example.

One may consider another hybrid system in this same style

in which the only classical connectives allowed are the

additives ∨+and ∧−. In this system A ∨+B is equivalent

to ?(A ⊕ B) in linear logic, and unlike HLL, it would be

valid to transfer from classical to linear focusing. But linear

connectives may not join classical subformulas without a

change in positive/negative polarity1.

Still another approach to developing hybrid logic is to

extend LKU with new polarities and structural rules. Restric-

tions on formulas are replaced by even greater sensitivity to

polarity information. Focusing can be separated into distinct

levels. For example, classical focusing can be represented by

1The fact that LU’s notion of classical polarity is compatible with

focusing can be explained by equivalences such as !(!A⊗!B) ≡!A⊗!B:

the ! can be dropped on subformulas of the same polarity. But we also note

the equivalences ?(?A⊕?B) ≡?(A ⊕ B) and ?∃x.?A ≡?∃x.A, as well

as their duals by negation. Unlike the LU equivalences, these apply only

to the additives. Structural rules on a formula need only be applied at the

outset, not to subformulas, thus enabling hybrid focusing.

Page 10

⇓2and linear focusing by ⇓1. Transition between focusing

modes can be formulated by lateral reaction rules such as

? Γ : ∆ ⇓2A

? Γ : ∆ ⇓1A

where A is a classical formula. More flexible variants of

R⇑ are also needed, including those that insert asynchronous

formulas into the classical context, as one would expect from

a system with the full power of linear logic. We are, in fact,

currently studying a system with three distinct types of ⇓

and three of ⇑.

ACKNOWLEDGMENT

This work has been supported by INRIA through the

“Equipes Associ´ ees” Slimmer.

L⇓

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