Structural focalization - School of Computer Science - Carnegie

35 Pages·2014·0.43 MB·English

Structural focalization
Robert J. Simmons
Focusing,introducedbyJean-MarcAndreoliinthecontextofclassicallinearlogic[Andreoli1992],
definesanormalformforsequentcalculusderivationsthatcutsdownonthenumberofpossible
derivationsbyeagerlyapplyinginvertiblerulesandgroupingsequencesofnon-invertiblerules. A
focused sequent calculus is defined relative to some non-focused sequent calculus; focalization is
thepropertythateverynon-focusedderivationcanbetransformedintoafocusedderivation.
In this paper, we present a focused sequent calculus for propositional intuitionistic logic and
prove the focalization property relative to a standard presentation of propositional intuitionistic
logic. Comparedtoexistingapproaches,theproofisquiteconcise,dependingonlyontheinternal
soundness and completeness of the focused logic. In turn, both of these properties can be es-
tablished(andmechanicallyverified)bystructuralinductioninthestyleofPfenning’sstructural
cuteliminationwithouttheneedforanytediousandrepetitiousinvertibilitylemmas. Theproof
of cut admissibility for the focused system, which establishes internal soundness, is not particu-
larly novel. The proof of identity expansion, which establishes internal completeness, is a major
contributionofthiswork.
DraftasofJanuary22,2014
1. INTRODUCTION
The propositions of intuitionistic propositional logic are easily recognizable and
standard: we will consider a logic with atomic propositions, falsehood, disjunction,
truth, conjunction, and implication.
P,Q ::= p|⊥|P ∨P |(cid:62)|P ∧P |P ⊃P
1 2 1 2 1 2
Thesequentcalculuspresentationforintuitionisticlogicisalsostandard;thesystem
inFigure1ispreciselythepropositionalfragmentofKleene’ssequentsystemG as
3
presented in [Pfenning 2000]. Contexts Γ are, as usual, considered to be unordered
multisets of propositions P, and the structural properties of exchange, weakening,
and contraction are admissible (each left rule implicitly contains a contraction).
Sequent calculi are a nice way of presenting logics, and a logic’s sequent calculus
presentation is a convenient setting in which to establish the logic’s metatheory
in a way that is straightforwardly mechanizable in proof assistants (like Twelf or
Agda) that are organized around the idea of structural induction. There are two
key metatheoretic properties that we are interested in. The first, cut admissibility,
justifies the use of lemmas: if we know P (if we have a derivation of the sequent
Γ −→ P) and we know that Q follows from assuming P (if we have a derivation
of the sequent Γ,P −→ Q), then we can come to know Q without the additional
assumption of P (we can obtain a derivation of the sequent Γ −→ Q). 1 A
proof of the cut admissibility property establishes the internal soundness of a logic
1In common practice, the words proof and derivation are used interchangeably. In this article,
we will be careful to refer to the formal objects constructed using sequent calculus rules (such
as thosein Figure 1) asderivations. Except when discussingnatural deduction, the wordsproof
andtheoremwillrefertotheoremsprovedabouttheseformalobjects;thesearefrequentlycalled
metatheoremsintheliterature.
·
2 Robert J. Simmons
Γ−→P
Γ,p−→p init (no rule ⊥R) Γ,⊥−→Q ⊥L
Γ−→P1 Γ−→P2
∨R1 ∨R2
Γ−→P1∨P2 Γ−→P1∨P2
Γ,P1∨P2,P1−→Q Γ,P1∨P2,P2−→Q
∨L
Γ,P1∨P2−→Q
Γ−→P1 Γ−→P2
Γ−→(cid:62) (cid:62)R (no rule (cid:62)L) Γ−→P1∧P2 ∧R
Γ,P1∧P2,P1−→Q Γ,P1∧P2,P2−→Q
∧L1 ∧L2
Γ,P1∧P2−→Q Γ,P1∧P2−→Q
Γ,P1−→P2 Γ,P1⊃P2−→P1 Γ,P1⊃P2,P2−→Q
⊃R ⊃L
Γ−→P1⊃P2 Γ,P1⊃P2−→Q
Fig.1. Sequentcalculusforintuitionisticlogic.
– it implies that there are no closed proofs of contradiction, even by circuitous
reasoning using lemmas. The identity property asserts that assuming P is always
sufficienttoconcludeP,thatis,thatthesequentΓ,P −→P isalwaysderivable. A
proof of the identity property establishes the internal completeness of a logic. We
call these properties internal, following Pfenning [2010], to emphasize that these
are properties of the deductive system itself and not a comment on the system’s
relationship to any external semantics.
Thereisatraditioninlogic,datingbacktoGentzen[1935],thatviewsthesequent
calculus as a convenient formalism for proving a logic’s metatheoretic properties
while viewing natural deduction proofs as the “true proof objects.”2 One reason
for this bias towards natural deduction is that natural deduction proofs have nice
normalization properties. A natural deduction proof is normal if if there are no
instances of an introduction rule immediately followed by an elimination rule of
the same connective; such detours give rise to local reductions which eliminate the
detour, such as this one:
D D
1 2
P true P true
1 2 ∧
P1∧P2true ∧ I D1
P true E1 =⇒ P true
1 R 1
The normalization property says that every natural deduction proof can be trans-
formed into a normal natural deduction proof.
We are frequently interested in the set of normal natural deduction proofs of a
given proposition. As an example, there is exactly one normal natural deduction
prooffor(p∧q)⊃(r∧s)⊃(p∧r). Presentedasaderivation,thatnaturaldeduction
2Thisdiscussionassumesabasicfamiliaritywithnaturaldeduction. Werefertheinterestedreader
to Girard, Taylor, and Lafont’s Proofs and Types [Girard et al. 1989]; the aforementioned quote
comesfromSection5.4ofthatwork.
·
Structural focalization 3
proof looks like this:
hyp hyp
u v
p∧qtrue r∧strue
∧ ∧
E1 E1
ptrue rtrue
∧
I
p∧rtrue
⊃v
(r∧s)⊃(p∧r)true I
⊃u
(p∧q)⊃(r∧s)⊃(p∧r)true I
Underthestandardprooftermassignmentfornaturaldeduction,this(normal)nat-
ural deduction proof corresponds to the (irreducible) proof term λx.λy.(cid:104)π x,π y(cid:105).
1 1
Incontrast, therearemanysequentcalculusderivationsofthesameproposition.
Here’s one of them:
init init
p∧q,r∧s,p−→p p∧q,r∧s,r −→r
∧ ∧
L1 L1
p∧q,r∧s−→p p∧q,r∧s−→r
∧
R
p∧q,r∧s−→p∧r
⊃
R
p∧q −→(r∧s)⊃(p∧r)
⊃
R
·−→(p∧q)⊃(r∧s)⊃(p∧r)
Reading from bottom to top, this derivation decomposes (p∧q)⊃(r∧s)⊃(p∧r)
ontheright,thendecomposes(r∧s)⊃(p∧r)ontheright,thendecomposesp∧ron
theright, andthen(inonebranch)decomposesp∧q ontheleftwhile(intheother
branch)decomposingr∧sontheleft. Otherpossibilitiesincludedecomposingp∧q
on the left before decomposing (r∧s)⊃(p∧r) on the right and decomposing r∧s
on the left before p∧r on the right; there are at least six different derivations even
if you don’t count derivations that do useless decompositions on the left.
These different derivations are particularly problematic if our goal is to do proof
searchforsequentcalculusderivations,asinessentialdifferencesbetweenderivations
correspond to unnecessary choice points that a proof search procedure will need to
backtrack over. It was in this context that Andreoli originally introduced the idea
of focusing. Some connectives, such as implication A⊃B, are called asynchronous
becausetheirrightrulescanalwaysbeappliedeagerly,withoutbacktracking,during
bottom-up proof search. Other connectives, such as disjunction A∨B, are called
synchronous because their right rules cannot be applied eagerly. For instance, the
∨ rule cannot be applied eagerly if we are looking for a proof of p −→ ⊥∨p.
R1
This asynchronous or synchronous character is the connective’s polarity.3
3Andreolidealtwithaone-sidedclassicalsequentcalculus;inintuitionisticlogic,itiscommonto
call asynchronous connectives right-asynchronous and left-synchronous. Similarly, it is common
tocallsynchronousconnectivesright-synchronousandleft-asynchronous.
Synchronicity or polarity, a property of connectives, is closely connected to (and sometimes
conflatedwith)apropertyofrulescalledinvertibility;aruleisinvertibleiftheconclusionofthe
ruleimplieseachofthepremises. So⊃Risinvertible(Γ−→P1⊃P2impliesΓ,P1−→P2)but⊃L
is not (Γ,P1⊃P2 −→C does not imply Γ,P1⊃P2 −→P1). Rules that can be applied eagerly
need to be invertible, so asynchronous connectives have invertible right rules and synchronous
connectives have invertible left rules. Therefore, another synonym for asynchronous is right-
invertible, and another synonym for synchronous is left-invertible. This terminology would be
misleadinginourcase,asboththeleftandrightrulesforconjunctioninFigure1areinvertible.
·
4 Robert J. Simmons
Andreoli’s key observation was that proof search only needs to consider deriva-
tions that have two alternating phases. In inversion phases, we eagerly apply
(invertible) right rules to asynchronous connectives and (invertible) left rules to
synchronous ones. When this is no longer possible, we begin a focusing phase by
putting a single remaining proposition in focus, repeatedly decomposing it (and
only it) by applying right rules to synchronous connectives and left rules to asyn-
chronous ones. Andreoli described this restricted form of proof search as a regular
proof search procedure in a restricted sequent calculus; such sequent calculi, and
derivations in them, are called focused as opposed to unfocused [Andreoli 1992].
In order to adopt such a proof search strategy, it is important to know that the
strategy is both sound (i.e., the proof search strategy will only say “the sequent
hasaderivation”ifthatisthecase)andcomplete(i.e., theproofsearchstrategyis
capable of finding a proof if one exists). Soundness proofs for focusing are usually
easy: focused derivations are essentially a syntactic refinement of the unfocused
derivations. Completeness, the nontrivial direction, involves turning unfocused
derivations into focused ones. This process is focalization.4 Thus, an effective
procedureforfocalizationisaconstructivewitnesstothecompletenessoffocusing.
The techniques described in this article are general and can be straightforwardly
transferred to other modal and substructural logics, as explored in the author’s
dissertation [Simmons 2012]. Our approach has three key components, which we
will now discuss in turn.
Focalization via cut and identity. Existingfocalizationproofsalmostallfallprey
totheneedtoprovemultipletediousinvertibilitylemmasdescribingtheinteraction
of each rule with every other rule; this results in proofs that are unrealistic to
write out, difficult to check, and exhausting to contemplate mechanizing. The way
forwardwasfirstsuggestedbyChaudhuri[2006]. Inhisdissertation,heestablished
the focalization property for linear logic as the consequence of the focused logic’s
internal soundness (the cut admissibility property) and completeness (the identity
property). Statingandprovingtheidentitypropertyforafocusedsequentcalculus
has remained a challenge, however. A primary contribution of this work is identity
expansion,ageneralizationoftheidentitypropertythatisamenabletomechanized
proof by structural induction on propositions. This identity property is, in turn,
part of our larger development, a proof of the focalization property that entirely
avoidsthetediousinvertibilitylemmasthatplagueexistingapproaches. (Wereview
existing techniques used to prove the focalization property in Section 6.)
Refining the focused calculus. The focused logic presented in this article is es-
sentially equivalent to the presentation of LJF given by Liang and Miller [2009], a
point we will return to in Section 2.1. A reader familiar with LJF will note three
non-cosmetic differences. The first two, our use of a polarized variant of intuition-
4Theusageoffocus,focusing,focussing,andfocalizationisnotstandardintheliterature. Weuse
the words focus and focusing to describe a logic (e.g. the focused sequent calculus) and aspects
of that logic (e.g. focused derivations, propositions in focus, left- or right-focused sequents, and
focusing phases). Focalization, derived from the French focalisation, is reserved exclusively for
theactofproducingafocusedderivationgivenanunfocusedderivation;thefocalizationproperty
establishesthatfocalizationisalwayspossible.
·
Structural focalization 5
istic logic and our novel treatment of atomic and suspended propositions, will be
discussed further in Section 2. A third change is that LJF does not force any par-
ticular ordering for the application of rules during a inversion phase. A seemingly
inevitable consequence of this choice is that the proof of focalization must estab-
lish the equivalence of all permutations of these invertible rules; this is one of the
aforementioned tedious invertibility lemmas that plague proofs of the focalization
property. Ourfocusedlogic,likemanyothers(includingAndreoli’soriginalsystem)
fixes a particular inversion order.
We introduce a new calculus rather than reusing an existing one in order to
present a focused logic and focalization proof that is computationally clean and
straightforward to mechanize. Our desire to mechanize proofs of the focalization
property also informed our decision to use propositional intuitionistic logic. All
proofsinthisarticlearemechanizedinbothTwelf[PfenningandSchu¨rmann1999]
and Agda [Norell 2007], though we will only mention the Twelf development. The
concretebasisforourclaimofcomputationalcleanlinessisthatourmechanizations
are complete artifacts capturing the constructive content of the proofs we present,
and the size of this artifact scales linearly relative to the number of connectives;
the approaches we call “tedious” tend to scale quadratically.
Proof terms. SinceAndreoli’soriginalwork,focusedsequentcalculusderivations
havebeenshowntobeisomorphictonormalnaturaldeductionproofsforrestricted
fragments of logic [Cervesato and Pfenning 2003] and variations on the usual fo-
cusing discipline [Howe 2001]. Such results challenge the position that natural
deductionproofsaresomehowmorefundamentalthansequentcalculusderivations
andalsoindicatethatfocalizationisafundamentalpropertyoflogic. InSection2.4,
wepresentaprooftermlanguageforpolarizedintuitionisticlogicthatdirectlycap-
tures the branching and binding structure of focused derivations. The result is a
term language generalizing the spine form of Cervesato and Pfenning [2003].
By presenting our cut admissibility and identity expansion proofs at the level of
proofterms,weemphasizethattheconstructivecontentofacutadmissibilityproof
is a substitution function on proof terms generalizing the hereditary substitution of
Watkins et al. [2002] in a spine form setting. Similarly, the constructive content of
our identity expansion proof is a novel η-expansion property on proof terms.
1.1 Outline
This article is dealing with three “soundness” properties and three “completeness”
properties, so it is important to carefully explain what we’re doing and when; the
following discussion is represented graphically in Figure 2.
Wepresentanewproofofthecompletenessoffocusing(thefocalizationproperty,
Theorem 4) for intuitionistic logic; the proof of the focalization property follows
from the internal soundness and completeness of the focused sequent calculus (cut
admissibility, Theorem 2, and identity expansion, Theorem 3). We will start, in
Section2,bymotivatingapolarizedpresentationoflogicthatsyntacticallydifferen-
tiates the synchronous and asynchronous propositions. We then present a focused
sequent calculus for polarized propositional intuitionistic logic and formally state
the soundness and completeness of focusing. We also prove the soundness of fo-
cusing (the de-focalization property, Theorem 1) in this section, but it’s pretty
·
6 Robert J. Simmons
“Soundness” “Completeness”
Cut admissibility Identity expansion
Theorem 2, Theorem 3,
Section 3 Section 4
Focused,
polarized
Unfocused
admissibility lemmas
Section 5.1
De-focalization Focalization
(cid:108) Theorem 1, Theorem 4,
Section 2.5 Section 5.2
Cut admissibility Identity
Unfocused,
Corollary 1, Corollary 2,
unpolarized
Section 5.3 Section 5.3
Fig.2. Theoremsandtheirdependencies.
boring and independent of the proofs of cut admissibility, identity expansion, and
the completeness of focusing.
Internal soundness for the focused sequent calculus is established by the cut ad-
missibility theorem in Section 3, and internal completeness for the focused sequent
calculus is established in Section 4 using a generalization of the identity expansion
theorem first developed in [Simmons and Pfenning 2011b]. In Section 5 we prove
the focalization property by showing unfocused admissibility, a group of lemmas
establishing that the focused sequent calculus can act like an unfocused sequent
calculus. Finally, rather than proving the internal soundness and completeness
(cut and identity) for the unfocused system directly, we show that these properties
canbeestablishedascorollariesofthefirstfourtheorems. InSection6weconclude
with an overview of existing proofs of the focalization property.
We will henceforth avoid using the words soundness and completeness as much
aspossible. Instead,wewillrefertothecutadmissibilityandidentitytheoremsfor
the focused and unfocused sequent calculi by name, and will refer to the soundness
and completeness of focusing as de-focalization and focalization, respectively.
2. POLARIZED LOGIC
ThereisasignificantlineofworkonpolarityinlogicdatingbacktoAndreoli[1992]
and Girard [1993]. That line of work holds that polarity is a refinement property
of propositions, that the asynchronous and synchronous propositions are syntactic
refinements of the set of all propositions. We can determine the synchronous or
·
Structural focalization 7
(p+)•=p+ (p−)•=p−
(↓A−)•=(A−)• (↑A+)•=(A+)•
(⊥)•=⊥
(A+∨B+)•=(A+)•∨(B+)• (A+⊃B−)•=(A+)•⊃(B−)•
((cid:62)+)•=(cid:62) ((cid:62)−)•=(cid:62)
(A+∧+B+)•=(A+)•∧(B+)• (A−∧−B−)•=(A−)•∧(B−)•
Fig.3. Erasureofpolarizedpropositions.
asynchronous character of a proposition by inspecting its outermost connective.5
Ina1991notepublishedtotheLINEARlist[Girard1991],Girardintroducedthe
idea of syntactically differentiating the positive propositions (those Andreoli called
synchronous) from the negative propositions (those Andreoli called asynchronous)
while mediating between the two with shifts: the upshift ↑A+ includes positive
propositions in the negative ones, and the downshift ↓A− includes negative propo-
sitions in the positive ones. This polarization6 of logic was developed further by
Girard in Ludics [Girard 2001] and treated extensively in Laurent and Zeilberger’s
Ph.D.theses[Laurent2002; Zeilberger2009b]. Thesearethepropositions, positive
and negative, for polarized intuitionistic logic:
A+,B+,C+ ::= p+ |↓A− |⊥|A+∨B+ |(cid:62)+ |A+∧+B+
A−,B−,C− ::= p− |↑A+ |A+ ⊃B− |(cid:62)− |A−∧−B−
Linearlogicisabletounambiguouslyassignallconnectivestoonecategoryorthe
other,butinintuitionisticlogic,truth(cid:62)andconjunctionP ∧P canbeunderstood
1 2
as having either a positive character (corresponding to 1 and A+ ⊗B+ in linear
logic) or a negative character (corresponding to (cid:62) and A−&B− in linear logic).
We take the maximally general approach and allow both versions of truth and
conjunction, which are decorated to emphasize their polarity.
TheshiftsintroducedbyGirardweremodalitiesthatmightchangetheprovability
of a proposition. We adopt the later stance of Zeilberger [2009b], McLaughlin and
Pfenning [2009], and others: shifts influence the structure of proofs, but not the
provability of propositions. Therefore, we expect there to be a focused proof of A+
or A− if and only if there is an unfocused proof of (A+)• or (A−)•, where (−)• is
the erasure function given in Figure 3. The polarity of an atomic proposition can
be arbitrary as long as it is consistent, as if each individual atomic proposition p is
really intrinsically positive or negative, but the calculus in Figure 1 didn’t notice.
Shifts and polarization are computationally interesting phenomena. Our view of
polarization lines up with the call-by-push-value system independently developed
5Linearlogicnaturallyhastwopolarities. Othersystems,likeGirard’sLUandLiangandMiller’s
LKU, use more than these two polarities [Girard 1993; Liang and Miller 2011]. Furthermore, in
LUthepolarityofapropositionisdeterminedbymorethanjusttheoutermostconnective.
6For the purposes of this article we are making somewhat artificial distinction between polarity,
Andreoli’s classification of propositions as asynchronous and synchronous, and polarization, the
segregationofthesetwoclassesaspositiveornegativepropositionsbyusingshifts. Thisdistinction
isnotastandardone: “synchronous”and“positive”areelsewhereusedinterchangeably,“polarity”
isusedtodescribewhywecall“polarization,”andsoon.
·
8 Robert J. Simmons
by Levy: positive propositions correspond to value types and negative propositions
correspond to computation types [Levy 2004]. Shifts are also useful in theorem
proving. By employing different polarization strategies, the name for partial in-
verses of erasure, the Imogen theorem prover can simulate fully-focused LJF proof
search, proof search in an unfocused logic like Kleene’s G , and proof search in
3
many other partially-focused systems in between [McLaughlin and Pfenning 2009].
Furthermore, forward-chaining versus backward-chaining logic programming can
be seen as arising from particular polarization strategies in uniform proof search
[Chaudhurietal.2008]. However,inthisarticlepolarizationisprimarilyatechnical
device. Shifts makes it clearer that certain proofs are structurally inductive over
propositions, especially identity expansion (Theorem 3). Additionally, the focus-
interrupting nature of the shift is a critical part of our unfocused admissibility
lemmas.7
2.1 Sequent calculus
We will develop our focused sequent calculus in two stages; in the first stage we do
not consider atomic propositions. We can present sequents for our polarized logic
in two equivalent ways. In the one-sequent view, we say that all sequents have the
formΓ;L(cid:96)U. Thecomponentsofasequentaredefinedbythefollowing(currently
incomplete) grammar:
Hypothetical contexts Γ::=·|Γ,A− |...
Inversion contexts Ω::=·|A+,Ω
Antecedents L::=Ω|[A−]
Succedents U ::=[A+]|A+ |A− |...
This first view requires us to further restrict the form of sequents for two reasons.
First, we only want to focus on one proposition at a time, so only one right focus
[A+] or left focus [A−] should be present in a sequent. Second, focus and inversion
phases should not overlap, so it must be the case that L = · when U = [A+] and
that U (cid:54)= A− when L = [A−]. Given this restriction, the first view of sequents is
equivalent to a three-sequent view in which there are three different sequent forms:
—Right focus: Γ(cid:96)[A+], where L=· and is therefore omitted,
—Inversion: Γ;Ω(cid:96)U, where U (cid:54)=[A+], and
—Left focus: Γ;[A−](cid:96)U, where U is stable (more about this shortly).
The sequent calculus for polarized intuitionistic logic in Figure 4 is presented in
termsofthisthree-sequentview. Theright focussequentΓ(cid:96)[A+]describesastate
in which non-invertible right rules are being applied to positive propositions, the
left focus sequent Γ;[A−] (cid:96) U describes a state in which non-invertible left rules
7SomethinglikeashiftisnecessaryforLiangandMiller’sfocalizationproofsaswell. Becausetheir
logic has polarity but no shifts, they construct delays δ+(A)=(cid:62)+∧+A and δ−(A)=(cid:62)+ ⊃A
to force asynchronous or synchronous connectives to behave (respectively) like synchronous or
asynchronous ones [Liang and Miller 2009]. This gets in the way of defining logical fragments
byaddingorremovingconnectives,asthepresenceoftheconnective(cid:62)+ isfundamentaltotheir
completenessproof.
·
Structural focalization 9
Γ(cid:96)[A+]–right focus
Γ;·(cid:96)A−
↓
Γ(cid:96)[↓A−] R
Γ(cid:96)[A+] Γ(cid:96)[B+]
(no rule ⊥R) Γ(cid:96)[A+∨B+] ∨R1 Γ(cid:96)[A+∨B+] ∨R2
Γ(cid:96)[A+] Γ(cid:96)[B+]
(cid:62)+ ∧+
Γ(cid:96)[(cid:62)+] R Γ(cid:96)[A+∧+B+] R
Γ;Ω(cid:96)U –inversion, U (cid:54)=[A+]
Γ(cid:96)[A+] Ustable Γ,A−;[A−](cid:96)U
foc foc
Γ;·(cid:96)A+ R Γ,A−;·(cid:96)U L A+stable
Γ,A−;Ω(cid:96)U
Γ;↓A−,Ω(cid:96)U ↓L
Γ;A+,Ω(cid:96)U Γ;B+,Ω(cid:96)U
Γ;⊥,Ω(cid:96)U ⊥L Γ;A+∨B+,Ω(cid:96)U ∨L
Γ;Ω(cid:96)U Γ;A+,B+,Ω(cid:96)U
(cid:62)+ ∧+
Γ;(cid:62)+,Ω(cid:96)U L Γ;A+∧+B+,Ω(cid:96)U L
Γ;·(cid:96)A+ Γ;A+(cid:96)B−
Γ;·(cid:96)↑A+ ↑R Γ;·(cid:96)A+⊃B− ⊃R
Γ;·(cid:96)A− Γ;·(cid:96)B−
(cid:62)− ∧−
Γ;·(cid:96)(cid:62)− R Γ;·(cid:96)A−∧−B− R
Γ;[A−](cid:96)U –left focus, U must be stable
Γ;A+(cid:96)U Γ(cid:96)[A+] Γ;[B−](cid:96)U
Γ;[↑A+](cid:96)U ↑L Γ;[A+⊃B−](cid:96)U ⊃L
Γ;[A−](cid:96)U Γ;[B−](cid:96)U
∧− ∧−
(no rule (cid:62)−) Γ;[A−∧−B−](cid:96)U L1 Γ;[A−∧−B−](cid:96)U L2
L
Fig.4. Focusedsequentcalculusforpolarizedintuitionisticlogic(sanssuspendedpropositions).
are being applied to negative propositions, and the inversion sequent Γ;Ω (cid:96) U
describes everything else. While we treat the hypothetical context Γ informally as
a multiset, the inversion context Ω is not a multiset. Instead, it should be thought
of as an ordered sequence of positive propositions: the empty sequence is written
as “·” and “,” is an associative append operator. Whenever the context Ω is non-
empty, the only applicable rule is the one that decomposes the left-most positive
connective in the context.
The picture in Figure 4 is quite uniform: every rule except for foc and foc
R L
breaks down a single connective, while the shift rules regulate the focus and in-
version phases. Read from bottom up, they put an end to the process of breaking
·
10 Robert J. Simmons
down a proposition under focus (↓ , ↑ ) and to the process of breaking down a
R L
proposition with inversion (↓ , ↑ ). All other rules maintain focus or inversion on
L R
the subformulas of a proposition.
The conclusions of foc and foc are inversion sequents with empty inversion
R L
contexts and succedents U that are stable, meaning that there is no possibility of
applying an invertible rule. These stable sequents (sometimes called neutral) have
animportantplaceinfocusedsequentcalculi. Intheintroduction,weclaimedthat
there was an essential equivalence between LJF and our focused presentation, but
this is only true if we ignore the internal structure of focus and inversion phases
and work with synthetic rules, the derivation fragments comprised of one or more
inversion phases stacked on top of a single focused phase. In the synthetic view
of focusing, we abstract away from the internal structure of focusing phases to
emphasize the stable sequents that lie between them [Andreoli 2001]. LJF and our
presentation of polarized intuitionistic logic have different internal structure, but
we claim that the systems give rise to the same synthetic rules.8 Aside from our
use of shifts, we depart from Liang and Miller in ways that are technically relevant
but that are invisible at the level of synthetic connectives.
The stability requirement for U in left focus sequents Γ;[A−] (cid:96) U can equiva-
lently be stated as a extra premise Ustable for the rule ↑ , which is done in the
L
accompanying Twelf development.
2.2 Suspended propositions
The pleasant picture of focusing given above is complicated when we consider
atomicpropositions. Atomicpropositionsarebestunderstoodasstand-insforarbi-
trarypropositions,andsoourpolarizedlogichasbothpositiveatomicpropositions
(stand-ins for arbitrary positive propositions) and negative atomic propositions
(stand-ins for arbitrary negative propositions).
Whenweareperforminginversionandwereachanatomicproposition,wedonot
have enough information to break down that proposition any further, but we have
not reached a shift. We have to do something different. What we do is suspend
that atomic proposition, either in the hypothetical context or in the succedent.
We represent a suspended atomic proposition as (cid:104)p+(cid:105) or (cid:104)p−(cid:105). If we wanted to
closely follow existing focused sequent calculi, we would introduce two more rules
forprovingatomicpropositionsinfocususingasuspendedatomicproposition. The
resulting extension of Figure 4 would look something like this:
Γ,(cid:104)p+(cid:105);Ω(cid:96)U Γ;·(cid:96)(cid:104)p−(cid:105)
Γ;p+,Ω(cid:96)U Γ;·(cid:96)p− Γ,(cid:104)p+(cid:105)(cid:96)[p+] Γ;[p−](cid:96)(cid:104)p−(cid:105) (cid:104)p−(cid:105)stable
This treatment is not incorrect and is obviously analogous to the init rule from
the unfocused system in Figure 1. Nevertheless, we contend that this is a design
error and a large part of why it has historically been difficult to prove the identity
theorem for focused systems. We generalize these rules by allowing the hypothet-
ical context to contain arbitrary suspended positive propositions (not just atomic
8Ourpresentationdoesnothavethefirst-orderquantifierspresentinLJF,andLJFlacksanegative
unit(cid:62)−,butquantifierscanbeaddedtooursystemeasily,andthesameistruefor(cid:62)− inLJF.