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 multiplicativeadditive 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 cutelimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classicallinear hybrid logics.

Conference Paper: Superdeduction in LambdaBarMuMuTilde
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ABSTRACT: Superdeduction is a method specially designed to ease the use of firstorder theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proofterm language and a cutelimination reduction already exist for superdeduction, both based on Christian Urban's work on classical sequent calculus. However the computational content of Christian Urban's calculus is not directly related to the (lambdacalculus based) CurryHoward correspondence. In contrast the Lambda bar mu mu tilde calculus is a lambdacalculus for classical sequent calculus. This short paper is a first step towards a further exploration of the computational content of superdeduction proofs, for we extend the Lambda bar mu mu tilde calculus in order to obtain a proofterm langage together with a cutelimination reduction for superdeduction. We also prove strong normalisation for this extension of the Lambda bar mu mu tilde calculus.Proceedings Third International Workshop on Classical Logic and Computation; 01/2010  SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: It is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by doublenegation, while the other direction has no truthpreserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics. Comment: 15 pages, to appear in 19th EACSL Annual Conference on Computer Science Logic (CSL 2010)06/2010;  SourceAvailable from: Dale Miller[Show abstract] [Hide abstract]
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 cutelimination holds in such fragments. From cutelimination 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;
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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 ˆIledeFrance
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
multiplicativeadditive 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 cutelimination in the various
fragments. Furthermore, each sublogic can interact with other
fragments through cut. We also consider the possibility of
introducing classicallinear hybrid logics.
KeywordsProof theory; focused proof systems; linear logic
I. INTRODUCTION
While it is wellknown 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) noninvertible
rules. The two polarities of negative (invertible) and positive
(noninvertible) 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 classicallinear 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 noninvertible inference
rules and presented the “twophase” 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 onesided 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 polarityrelated 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 LLFlike 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).
LLFstyle 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 LUinspired translation of intuitionistic logic into
linear logic. That paper also presented the focused classical
sequent calculus LKF that was inspired by a doublenegation
translation into LJF (similar to Girard’s LC [13]). The
system LKF is given in Figure 2 (a oneside 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,
Page 3
? [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 toplevel 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 multiplicativeadditive linear logic. This simplification
of LKF also leads to a more direct proof of cutelimination
(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 toplevel 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 “level1” and “level2” 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 level2 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 fulllinear 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 endsequents of LKU all have the form ?:⇑ Γ.
The following fragments are immediate.
Focused MALL. If we forbid all uses of the level2 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 level2 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 level1 rules and only allowed the
level2 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 linearclassical 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 twosided 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 noninvertible inference rules use
sequents such as [Γ]
−→ [R], which provides a “leftfocus”
formula L or sequents such as [Γ] −R→, which provides
a “rightfocus” 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 twosided sequents)
to classical connectives (whose proof rules are described
using onesided sequents). That is, a leftoccurrence of the
intuitionistic ⊃ is exactly the same as (a rightoccurrences
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 lefthand 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 linearlogic 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 onesided sequent presentation of LJF. To illustrate
how two sided inference rules for intuitionistic logic can
be represented in the onesided, focused setting of LKU,
consider the additive and multiplicative versions of the
conjunctionleft 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 twosided
LJF sequents correspond to the onesided 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 singleconclusion condition is essentially
already enforced by other rules such as R2⇓, I1, and I2.
When building a proof from the bottomup, 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 essentiallyleft occurrence of an implication A ⊃
B has the form A⊗B⊥where A is essentially right and B⊥
essentially left. The implicationleft 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 wellformed
intuitionistic sequent. Without the explicit restriction to one
formula in the level1 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 fullcompleteness. 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 wellformed 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 twosided system: the polarity of a formula
unambiguously determines its essentially left or right status.
(Of course, one may still prefer a twosided 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
crosscut. The LKU framework provides a uniform structure
to cutelimination 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?.
(Firstorder 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 cutfree 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 cutfree 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 cutfree provable. Then
? Γ : ∆1...∆n⇓ R is also cutfree provable. Furthermore,
every cutfree proof of ? Γ : ∆1...∆n⇓ R is of the form
? Γ : ∆1⇓ a1
...
? Γ : ∆1...∆n⇓ R
where R ↓ {a1,...,an}.
Cutelimination follows from the following lemma.
Lemma 4: R ↑ {a1,...,an} if and only if R⊥
{a⊥
The inductive measure for the cutelimination 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 asynchronoussynchronous 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. Keycase
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
cutfree, since we can apply the procedure to the lowest
height cuts first. The cutelimination 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 leftside 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 keycase 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 keycase 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 rightside subproof will continue with the
form ?: ∆?,A⊥⇑, which will then also require a formula
to be selected for focus. This is a critical choicepoint in
cutelimination (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 keycase 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 proofheight
measure. Again by lemma 3, we can then synthesize the
conclusion ?: B,∆∆?.
The difference between cutelimination 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 cutfree proof, then
? A,Γ : ∆ ⇑ Θ has a cutfree proof of the same height. If
? Γ : ∆ ⇑ Θ has a cutfree proof, then ? A,Γ : ∆ ⇑ Θ has
a cutfree proof of the same height.
It is important to note that the explicit contraction in D2is
restricted to positive formulas. The keycase 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
keycase cut while the “copy” is erased by parametric cuts.
The parametric cuts have lower proofheight 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 cutelimination 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, cutelimination 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 cutfree classical proofs, then all
cuts between fragments collapse to classical cuts. In certain
circumstances, cutelimination 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 cutfree proof in LJF.
The proof of this theorem follows the same format as
other cutelimination 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 endsequent (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 cutelimination in
LJF, we also have the following admissible crosscut:
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 cutfree proof in LJF.
VII. A LIMITED CLASSICALLINEAR 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 level1 structural rules to +1/−1
formulas and the level2 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 “HighLow 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 cutreduction 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 keycase cut will be
reduced to Cut2rules for the classical subformulas of the
cut formula and Cut1rules for the linear subformulas.
While HLL exhibits reasonable cutelimination 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 linearclassical context,
∨−is equivalent to
contradict cutelimination 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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