Focusing and Polarization in Intuitionistic Logic

15 Pages·2012·0.16 MB·English

Focusing and Polarization in Intuitionistic Logic
Chuck Liang1 and Dale Miller2
1 Departmentof Computer Science, Hofstra University,Hempstead, NY11550
chuck.liang at hofstra.edu
2 INRIA&LIX/Ecole Polytechnique,Palaiseau, France
dale.miller at inria.fr
Abstract. A focused proof system provides a normal form to cut-free
proofs that structures the application of invertible and non-invertible
inference rules. The focused proof system of Andreoli for linear logic
has been applied to both the proof search and the proof normalization
approaches to computation. Various proof systems in literature exhibit
characteristics of focusing to one degree or another. We present a new,
focused proof system for intuitionistic logic, called LJF, and show how
other proof systems can be mapped into the new system by inserting
logical connectives that prematurely stop focusing. We also use LJF to
design a focused proof system for classical logic. Our approach to the
design and analysis of these systems is based on the completeness of
focusing in linear logic and on the notion of polarity that appears in
Girard’s LC and LUproof systems.
1 Introduction
Cut-elimination providesanimportantnormalformforsequent calculusproofs.
But what normal forms can we uncover about the structure of cut-free proofs?
Since cut-free proofs play important roles in the foundations of computation,
suchnormalformsmight (cid:12)ndarangeof applicationsin the proof normalization
foundations for functional programming or in the proof search foundations of
logic programming.
1.1 About focusing
Andreoli’sfocusing proof system forlinear logic(the triadic proofsystemof [1])
provides a normal form for cut-free proofs in linear logic. Although we describe
this system, here called LLF, in more detail in Section 2, we highlight two
aspectof focusingproofshere.First,linear logicconnectivescanbedividedinto
the asynchronous connectives, whoseright-introductionrulesareinvertible, and
the synchronous connectives, whose right introduction rules are not (generally)
invertible. The searchfor a focused proof can capitalize on this classi(cid:12)cationby
applying(readinginferencerulesfromconclusion topremise)allinvertiblerules
in any order (without the need for backtracking) and by applying a chain of
non-invertible rules that focus on a given formula and its positive subformulas.
Such a chain of applications, usually called a focus, terminates when it reaches
an asynchronousformula. Proof search can then alternate between applications
ofasynchronousintroductionrulesandchainsofsynchronousintroductionrules.
A second aspect of focusing proofs is that the synchronous/asynchronous
classi(cid:12)cationof non-atomicformulasmust be extended to atomicformulas.The
arbitraryassignmentofpositive(synchronous)andnegative(asynchronous)bias
to atomic formulas can have a major impact on, not the existence of focused
proofs, but the shape of focused proofs. For example, consider the Horn clause
speci(cid:12)cation of the Fibonacci series:
(cid:12)b(0;0)^(cid:12)b(1;1)^8n8f8f0[(cid:12)b(n;f)^(cid:12)b(n+1;f0)(cid:27)(cid:12)b(n+2;f +f0)]:
If all atomic formulas are given negative bias, then the only focused proofs of
(cid:12)b(n;f )arethosethatcanbeclassi(cid:12)edas\backwardchaining"(thesizeofthe
n
smallestonebeingexponentialinn).Ontheotherhand,ifallatomicformulasare
given positive bias,then the only focused proofs arethose that can be classi(cid:12)ed
as \forwardchaining" (the size of the smallest one being linear in n).
1.2 Results
The contributions of this paper are the following. First, we introduce in Sec-
tion 5anew focusing proof systemLJF andshowthat it is sound andcomplete
for intuitionistic logic. Notable features of LJF are that it allows for atoms of
di(cid:11)erent bias and it contains two versions of conjunction: while these conjunc-
tions are logically equivalent, they are a(cid:11)ected by focusing di(cid:11)erently. Second,
in Section 6, we showhowseveralother focusingproof systemscan be captured
in LJF, in the sense of full completeness (one-to-one correspondence between
proofsindi(cid:11)erentsystems).Oneshouldnotethatwhiletherearemanyfocusing
proofsystemsforintuitionisticlogicintheliterature,weappeartobethe(cid:12)rstto
provide a single (intuitionistic) framework for capturing many of them. Third,
in Section 7, we use LJF to derive LKF, a focusing system for classical logic.
1.3 Methodology and Related work
Thereareanumberof sequentcalculusproof systemsknownto be complete for
intuitionistic logic that exhibit characteristics of focusing. Some of these proof
systems are based on (cid:12)xing globally on either forward chaining or backward
chaining. The early work on uniform proofs [17] and the LJT proof system [11]
are both backward chaining calculi (all atoms have negative bias). The LJQ
calculus [11,7]similarlyselects the globalpreferenceto be forwardchaining (all
atoms have positive bias). Less has been published about systems that allow
for mixing bias on atoms. The (cid:21)RCC proof system of Jagadeesan, Nadathur,
and Saraswat [13] allows for both forward chaining and backward chaining in
a superset of the hereditary Harrop fragment of intuitionistic logic. Chaudhuri,
Pfenning, and Price in [3] observed that focusing proofs with mixed biases on
atoms can form a declarative basis for mixing forward and backward chaining
2
within Horn clauses. The PhD theses of Howe [12] and Chaudhuri [2] also ex-
plored various focusing proof systems for linear and intuitionistic logic.
We are interested in providing a (cid:13)exible and unifying framework that can
collect together important aspects of many of these proof systems. There are
severalwaystomotivateandvalidatethedesignofsuchasystem.Oneapproach
stays entirely within intuitionistic logic and works directly with invertibility
and permutability of inference rules. Such an approachhas been taken in many
papers, such as [17,18,7]. Our approach uses linear logic, with its exponential
operators ! and ?, as a unifying framework for looking at intuitionistic (and
classical) logic. The fact that Andreoli’s focused system was de(cid:12)ned for full
linearlogicprovidesuswithaconvenientplatformforexploringtheissuesaround
focusingandpolarity.Wetranslateintuitionisticlogicintolinearlogic,thenshow
thatproofsystemsforintuitionisticlogicmatchfocusedproofsof thetranslated
image (Section 3). A crucial aspect of understanding focusing in intuitionistic
logicisprovidedbyidentifyingthepreciserelationshipbetweenAndreoli’snotion
of polarity with Girard’s notion of polarity found in the LC [9] and LU [10]
systems (Section 4).
Another system concerning polarity and focusing is found in the work of
Danos,JoinetandSchellinx[5,6].Manytechniquesthattheydeveloped,suchas
inductive decorations, areusedthroughoutouranalysis.Ourworkdivergesfrom
theirs in the adaptation of Andreoli’s system (LLF) as our main instrument of
construction. The LK(cid:17) system of [6] describes focused proofs for classical logic.
p
Its connectionstopolarizationandfocusingwerefurther exploredandextended
by Laurent, Quatrini and de Falco [14] using polarized proof nets. It may be
tempting to speculate that the best way to arrive at a notion of intuitionistic
focusingisbysimple modi(cid:12)cationstothesesystems,suchasrestrictingthem to
single-conclusion sequents. Closer examination however, reveal intricate issues
concerning this approach. For example, the notion of classical polarity appears
to be distinct fromand contrary to intuitionistic polarity,especially atthe level
of atoms (see Sections 4 and 7). Resolving this issue would be central to (cid:12)nd-
ing systems that support combined forward and backward chaining. Although
the relationship between LK(cid:17) and our systems is interesting, we chose for this
p
workto deriveintuitionistic focusing from focusing in linear logic asopposed to
classical logic.
Much of the research into focusing systems has been motivated by their ap-
plication.Forexample,thepapers[13,17,12,2]aremotivatedbyfoundationalis-
suesinlogicprogrammingandautomateddeduction.Thepapers[11,5,6,14]are
motivatedbyfoundationalissuesinfunctionalprogrammingandthe(cid:21)-calculus.
Also, Levy [15] presents focus-style proof systems for typing in the (cid:21)-calculus
andCurienandHerbelin[4](amongothers)havenotedtherelationshipbetween
forward chaining and call-by-value evaluation and between backward chaining
and call-by-name evaluation.
Our work can be extended to second order logic, although this paper is
concerned mainly with (cid:12)rst-order quanti(cid:12)cation.
Many details missing from this paper can be found in the report [16].
3
2 Focusing in Linear Logic
We summarize the key results from [1] on focusing proofs for linear logic.
Aliteral iseither anatomicformulaorthe negationof anatomicformula.A
linearlogicformulaisinnegation normal formifitdoesnotcontainoccurrences
of (cid:0)(cid:14) andif allnegationshaveatomicscope. IfK is literal, then K? denotesits
complement: in particular, if K is A? then K? is A.
Connectivesinlinearlogicareeitherasynchronous orsynchronous.Theasyn-
chronousconnectivesare?,...............................................................................................,?,>,&,and8whilethesynchronousconnectives
are their de Morgan dual, namely, 1, (cid:10), !, 0, (cid:8), and 9. Asynchronous connec-
tivesarethosewheretheright-introductionruleisalwaysinvertible.Formally,a
formulain negationnormalformisof threekinds:literal, asynchronous(i.e., its
top-level connective is asynchronous), and synchronous (i.e., its top-level con-
nective is synchronous).
(cid:9)(cid:9):(cid:1):(cid:1)**?L;L [?] (cid:9)(cid:9):(cid:1):(cid:1)**FF.....................;.................................................................G......... G;L;L [...............................................................................................] (cid:9)(cid:9):;(cid:1)F:*(cid:1)?*F;LL [?]
(cid:9):(cid:1)*F;L (cid:9):(cid:1)*G;L (cid:9):(cid:1)*B[y=x];L
[>] [&] [8]
(cid:9):(cid:1)*>;L (cid:9):(cid:1)*F &G;L (cid:9):(cid:1)*8x:B;L
(cid:9):(cid:1);F *L
[R*] providedthatF is not asynchronous
(cid:9):(cid:1)*F;L
(cid:9):(cid:1) +F (cid:9):(cid:1) +G (cid:9):(cid:1)*F
[1] 1 2 [(cid:10)] [!]
(cid:9):(cid:1)+1 (cid:9):(cid:1) ;(cid:1) +F (cid:10)G (cid:9):(cid:1)+!F
1 2
(cid:9):(cid:1)+F (cid:9):(cid:1)+F (cid:9):(cid:1)+B[t=x]
1 [(cid:8)] 2 [(cid:8) ] [9]
(cid:9):(cid:1)+F (cid:8)F l (cid:9):(cid:1)+F (cid:8)F r (cid:9):(cid:1)+9x:B
1 2 1 2
(cid:9):(cid:1)*F
[R+] providedthatF is either asynchronousor a negative literal
(cid:9):(cid:1)+F
If K a positive literal: [I ] [I ]
(cid:9):K? +K 1 (cid:9);K?:(cid:1)+K 2
(cid:9):(cid:1)+F (cid:9);F:(cid:1)+F
If F is not a negative literal: [D ] [D ]
(cid:9):(cid:1);F *(cid:1) 1 (cid:9);F:(cid:1)*(cid:1) 2
Fig.1.The focused proof systemLLF for linear logic
As mentioned in Section 1.1, the classi(cid:12)cation of non-atomic formulas as
asynchronous or synchronous is pushed to literals by assigning a (cid:12)xed but ar-
bitrary bias to atoms: an atom given a negative bias is linked to asynchronous
behaviorwhile anatomgivenpositive bias islinked tosynchronousbehavior.In
Andreoli’soriginalpresentationofLLF[1]allatomswereclassi(cid:12)edas\positive"
and theirnegations\negative."Girardmadea similarassignmentfor LC[9]. In
a classical setting, such a choice works (cid:12)ne since classical negation simply (cid:13)ips
bias. In intuitionistic systems, however, a more natural treatment is to assign
an arbitrary bias directly to atoms. This bias of atoms is extended to literals:
4
B B1 B0 (B0)?
atom Q Q Q Q?
true 1 > 0
false 0 0 >
P ^Q !(P1&Q1) !P0&!Q0 ?(P0)?(cid:8)?(Q0)?
P _Q !P1(cid:8)!Q1 !P0(cid:8)!Q0 ?(P0)?&?(Q0)?
P (cid:27)Q !(?(P0)? ............................................................................................... Q1) !P1(cid:0)(cid:14)!Q0 !P1(cid:10)?(Q0)?
:P !(0............................................................................................... ?(P0)?) !P1(cid:0)(cid:14)0 !P1(cid:10)>
9xP 9x!P1 9x!P0 8x?(P0)?
8xP !8xP1 8x!P0 9x?(P0)?
Table 1.The 0/1 translation usedto encode LJ proofs into linear logic.
negating a negative atom yields a positive literal and negating a positive atom
yields a negative literal.
ThefocusingproofsystemLLFforlinearlogic,presentedinFigure1,contains
twokindsofsequents.Inthesequent(cid:9):(cid:1)*L,the\zones"(cid:9) and(cid:1)aremultisets
and L is a list. This sequent encodes the usual one-sided sequent (cid:0) ?(cid:9);(cid:1);L
(here, we assume the natural coercion of lists into multisets). This sequent will
alsosatisfytheinvariantthatrequires(cid:1)tocontainonlyliteralsandsynchronous
formulas.Inthe sequent(cid:9):(cid:1)+F,the zone(cid:9) isamultiset offormulasand(cid:1)is
amultisetofliteralsandsynchronousformulas,andF isasingleformula.Notice
that the bias of literals is explicitly referred to in the [R *] and initial rules: in
particular, in the initial rules, the literal on the right of the + must be positive.
Changes to the bias assigned to atoms does not a(cid:11)ect provability of a linear
logic formula: instead it a(cid:11)ects the structure of focused proofs.
3 Translating Intuitionistic Logic
Table1containsatranslationof intuitionistic logicintolinear logic.Thistrans-
lation induces a bijection between arbitrary LJ proofs and LLF proofs of the
translated image in the following sense. First notice that this translation is
asymmetric: the intuitionistic formula A is translated using A1 if it occurs on
the right-side of an LJ sequent and as A0 if it occurs on the left-side. Since this
translationisusedtocapturecut-freeproofs,suchdistinctionsarenotproblem-
atic. Since the left-hand side of a sequentin LJ will be negatedwhen translated
to a one-sided linear logic sequent, (B0)? is also shown. The following Proposi-
tion essentially says that, via this translation, linear logic focusing can capture
arbitrary proofs in LJ. Here, ‘ denotes an intuitionistic logic sequent.
I
Proposition 1. Let((cid:0)0)? bethemultisetf(D0)? jD2(cid:0)g.Thefocused proofs
of ‘((cid:0)0)? :*R1 are in bijective correspondence with the LJ proofs of (cid:0) ‘ R.
I
5
For a detailed proof, see [16]. We simply illustrate one case in constructing
the mapping from a collection of LLF rules to an LJ inference rule.
‘((cid:0)0)?;(D0)? :R1 *
i [?]
‘((cid:0)0)? :R1 *?(D0)? (cid:0);D ‘ R
i [R+] 7(cid:0)! i I [^L]
‘((cid:0)0)? :R1 +?(Di0)? [(cid:8)] (cid:0);D1^D2 ‘I R
‘((cid:0)0)? :R1 +?(D0)?(cid:8)?(D0)?
1 2
The liberal use of ! in this translation throttles focusing. This translation is
reminiscent of the earliestembedding of classicalinto intuitionistic logicof Kol-
mogorov, which uses the double negation in a similarly liberal fashion.
The 0/1 translation can be used as a starting point in establishing the com-
pleteness of other proof systems. These systems can be seen as induced from
alternative translations of intuitionistic logic.
Consider,forexample,the LJQ0 proof
system presented in [7]. The translation
for this system is given here using the F Fq (right) Fj (left)
\q=j" mapping. The \(cid:10)1" device is an-
atom C C C
other way to control focusing, or the lack
thereof. All atoms must be given positive false 0 0
bias for this translation. A^B Aq (cid:10)Bq !Aj (cid:10)!Bj
With minor changes,Girard’soriginal A_B Aq (cid:8)Bq !Aj (cid:8)!Bj
(non-polarized) translation of intuitionis-
A(cid:27)B (!Aj(cid:0)(cid:14)Bq)(cid:10)1 Aq(cid:0)(cid:14)!Bj
tic logic [8] induces the complement to
LJQ0 called LJT [11], which is itself de-
rivedfromLKT[5](wherethisconnection
wasnotedimplicitly.) All atoms must be given negative bias forthistranslation.
Given a translation such as that of LJQ0, one can give a completeness proof
for the system using a \grand tour" through linear logic as follows:
1. Show that a proof under the 0/1 translation can be converted into a proof
under the new translation. This usually follows from cut-elimination.
2. De(cid:12)neamappingbetweenproofsinthe newsystem(suchasLJQ)andLLF
proofs of its translation.
3. ShowsoundnessofthenewsystemwithrespecttoLJ.Thisisusuallytrivial.
The\tour"isnowcomplete,sinceproofsin LJmaptoproofsunderthe0/1
translation.
An intuitionistic system that contains atoms of both positive and negative
bias is (cid:21)RCC [13]. Two special cases of the (cid:27)L rule are distinguished involving
E (cid:27)D for positive atom E and G(cid:27)A for negative atom A. Each rule requires
that the complementary atom (E on the left, A on the right) is present when
applied,thusterminatingonebranchoftheproof.Onecantranslatethesespecial
casesusingformsE(cid:0)(cid:14)!D0 and!G0(cid:0)(cid:14)A,respectively,inlinearlogic.Thestrategy
outlined above can then be used to not only prove its completeness but also
extend it with more aggressivefocusing features.
6
Ourinteresthereisnottheconstructionof individualsystemsbutthe build-
ing of a unifying framework for focusing in intuitionistic logic. Such a task re-
quires a closer examination of polarity and its connection to focusing.
4 Permeable Formulas and their Polarity
Focused proofs in linear logic are characterized by two di(cid:11)erent phases: the in-
vertible(asynchronous)phaseandthenon-invertible(synchronous)phase.These
two phases are characterized by introduction rules for dual sets of formulas. In
order to construct a general focusing scheme for intuitionistic logic, the non-
linear (exponential) aspects of proofs need special attention, especially in light
ofthefactthatthe[!]rulestopsabottom-upconstructionoffocusedapplication
of synchronousrules (the arrow + in the conclusion (cid:13)ips to * in the premise).
For our purposes here, a particularly (cid:13)exible way to deal with the exponen-
tials in the translations of intuitionistic formulas is via the notion of perme-
ation that is used in LU [10]. In particular, there are essentially three grades
of permeation. The formula B is left-permeable if B (cid:17) !B, is right-permeable if
B (cid:17) ?B, and neutral otherwise. Within sequent calculus proofs, a formula is
left-permeable if it admits structural rules on the left and right-permeable if it
admits structural rules on the right. An example of a left-permeable formula
is 9x!A. All left-permeable formulas are synchronous and all right-permeables
asynchronous.IntheLUsystem,boththeleftandrightsidesofsequentscontain
two zones | one that treats formulas linearly and one that permits structural
rules. A left-permeable (resp., right-permeable) formula is allowed to move be-
tween both zones on the left (right). In addition, LU introduces atoms that are
inherently left or right-permeable or neutral. Although they appear to properly
extend linear logic, one can simulate LU in \regular"linear logic by translating
left-permeable atoms A as !A and right-permeable ones as ?A.
To preserve the focusing characteristics of permeable atoms as positively or
negativelybiasedatoms,weusethefollowingLU-inspiredasymmetricaltransla-
tion. The superscript (cid:0)1 indicates the left-side translation and +1 indicates the
right-side translation:
P(cid:0)1 =!P and P+1 =P, for left-permeable (positive) atom P.
N(cid:0)1 =N and N+1 =?N, for right-permeable (negative) atom N.
B(cid:0)1 =B+1 =B, for neutral atom B.
The ! rule of LLF causes a loss of focus in all circumstances, and is the main
reasonwhyweuseanasymmetricaltranslation.Thetranslationofpositiveatoms
abovepreservespermeation ontheleft whileallowingforfocus ontheright.That
is,left-permeableatomscannowbeinterpretedmeaningfullyaspositivelybiased
atomsinfocusedproofs,andduallyforright-permeableatoms.Furthermore,the
permeationofpositiveatomsis\one-wayonly:" theycannotbeselectedforfocus
again once they enter the non-linear context.
Intuitionistic logic uses the left-permeable and neutral formulas and atoms.
LU de(cid:12)nes atranslationfor intuitionistic logicsothat all synchronous formulas
7
areleft-permeable.Forexample,_istranslatedasfollows(here,P,Qarepositive
and N, M are negative): (P _Q)(cid:0)1 = P(cid:0)1 (cid:8)Q(cid:0)1, (P _N)(cid:0)1 = P(cid:0)1 (cid:8)!N(cid:0)1,
(N _P)(cid:0)1 = !N(cid:0)1 (cid:8)Q(cid:0)1, and (N _M)(cid:0)1 = !N(cid:0)1(cid:8)!M(cid:0)1. The (cid:12)nal element
of intuitionistic polarity is that neutral atoms should be assigned negative bias
in focused proofs. Neutral atoms that are introduced into the left context (e.g.
by a (cid:27)L rule) must immediately end that branch of the proof in an identity
rule. Otherwise, the unique stoup is lost when multiple non-permeable atoms
accumulate in the linear context.
TheLUandLLFsystemsserveasaconvenientplatformfortheuni(cid:12)edchar-
acterizationof polarity and focusing in all three logics. We can now understand
the terminology of \positive" and \negative" formulas in each logic as follows:
Linear logic: Positive formulasaresynchronousformulasandpositivelybiased
neutral atoms. Negative formulas areasynchronousformulas and negatively
biased neutral atoms.
Intuitionistic logic: Positive formulas are left-permeable formulas and nega-
tive formulas are asynchronousneutral formulas and negatively biased neu-
tral atoms.
Classical logic: Positive formulasareleft-permeableformulas.Negative formu-
las are right-permeable formulas.
5 The LJF Sequent Calculus
Sincethe polaritiesof intuitionistic logicobservestrongerinvariances,intuition-
istic focused proofs are more well-structured than LLF proofs. The non-linear
context of LLF containsboth synchronousand asynchronousformulas,whereas
in intuitionistic logic sequents can be clearly divided into zones respecting po-
larity. That is, when translating an intuitionistic sequent into a LLF sequent,
synchronous formulas on the left are placed in the linear context.
We also make an adjustment on the LU translation of intuitionistic logic.
Insteadof using& or(cid:10)depending onthe polaritiesof the subformulas,we con-
structtwoversionsofintuitionisticconjunction,whichhasthefollowingmeaning
in linear logic (P, Q for positives, N, M for negatives, A, B arbitrary):
(P ^+Q)(cid:0)1 =P(cid:0)1(cid:10)Q(cid:0)1 (A^+B)+1 =A+1(cid:10)B+1
(P ^+N)(cid:0)1 =P(cid:0)1(cid:10)!N(cid:0)1
(N ^+P)(cid:0)1 =!N(cid:0)1(cid:10)P(cid:0)1 (A^(cid:0)B)(cid:0)1 =A(cid:0)1&B(cid:0)1
(N ^+M)(cid:0)1 =!N(cid:0)1(cid:10)!M(cid:0)1 (A^(cid:0)B)+1 =A+1&B+1
The connectives ^(cid:0) and ^+ are equivalent in intuitionistic logic in terms of
provabilitybutdi(cid:11)erintheirimpactonthe structure offocusedproofs.Theuse
of two conjunctions means that the top-level structure of formulas completely
determines their polarity. Polarity in intuitionistic logic is de(cid:12)ned as follows.
De(cid:12)nition 1. Atoms in LJF are arbitrarily positive or negative. Positive for-
mulas are among positive atoms, true, false, A^+B, A_B and 9xA. Negative
formulas are among negative atoms, A^(cid:0)B, A(cid:27)B and 8xA.
8
[N;(cid:0)](cid:0)N![R] [(cid:0)](cid:0) ! [(cid:0)];P (cid:0)![R] [(cid:0)](cid:0)!N
P R
Lf Rf l R
[N;(cid:0)](cid:0)![R] [(cid:0)](cid:0)![P] [(cid:0)](cid:0)P![R] [(cid:0)](cid:0) ! r
N
[C;(cid:0)];(cid:2)(cid:0)!R [(cid:0)];(cid:2)(cid:0)![D]
[] []
[(cid:0)];(cid:2);C (cid:0)!R l [(cid:0)];(cid:2)(cid:0)!D r
I; atomic N
I ; atomic P l
[P;(cid:0)](cid:0) ! r [(cid:0)](cid:0)N![N]
P
[(cid:0)];(cid:2)(cid:0)!R
falseL trueL trueR
[(cid:0)];(cid:2);false(cid:0)!R [(cid:0)];(cid:2);true(cid:0)!R [(cid:0)](cid:0) !
true
[(cid:0)](cid:0)A!i [R]
^(cid:0)L [(cid:0)];(cid:2);A;B(cid:0)!R
[(cid:0)]A1(cid:0)^!(cid:0)A2[R] [(cid:0)];(cid:2);A ^+B(cid:0)!R ^+L
[(cid:0)];(cid:2)(cid:0)!A [(cid:0)];(cid:2)(cid:0)!B [(cid:0)](cid:0) ! [(cid:0)](cid:0) !
[(cid:0)];(cid:2)(cid:0)!A ^(cid:0)B ^(cid:0)R [(cid:0)A](cid:0)A^+B!B ^+R
[(cid:0)];(cid:2);A(cid:0)!R [(cid:0)];(cid:2);B (cid:0)!R [(cid:0)](cid:0) !
Ai
_L _R
[(cid:0)];(cid:2);A_B(cid:0)!R [(cid:0)](cid:0)A1_A2!
[(cid:0)](cid:0) ! [(cid:0)](cid:0)B![R]
A [(cid:0)];(cid:2);A(cid:0)!B
(cid:27)L
[(cid:0)]A(cid:0)(cid:27)!B[R] [(cid:0)];(cid:2)(cid:0)!A(cid:27)B (cid:27)R
A[t=x]
[(cid:0)];(cid:2);A(cid:0)!R [(cid:0)](cid:0)A[t=x]! [(cid:0)] (cid:0)! [R] 8L [(cid:0)];(cid:2)(cid:0)!A
[(cid:0)];(cid:2);9yA(cid:0)!R 9L [(cid:0)](cid:0)9xA! 9R [(cid:0)](cid:0)8x!A[R] [(cid:0)];(cid:2)(cid:0)!8yA 8R
Fig.2. The Intuitionistic Sequent Calculus LJF. Here, P is positive, N is negative,
C is a negative formula or positive atom, and D a positive formula or negative atom.
Otherformulas are arbitrary. Also, y is not free in (cid:0),(cid:2), or R.
The above translation induces the sequent calculus LJF for intuitionistic
logic, shown in Figure 2. Sequents in LJF can be interpreted as follows:
1. [(cid:0)];(cid:2) (cid:0)!R (end sequent): this is an unfocused sequent. (cid:0) contains nega-
tive formulas and positive atoms. R represents either a formula R or [R].
2. [(cid:0)] (cid:0)! [R]: this represents a sequent in which all asynchronous formulas
have been decomposed, and is ready for a formula to be selected for focus.
A
3. [(cid:0)] (cid:0)! [R]: this is a left-focusing sequent, with focus on formula A. The
meaning of this sequent remains (cid:0);A‘ R.
I
4. [(cid:0)](cid:0) !: this is a right-focusing sequent on formula A, with the meaning
A
(cid:0) ‘ A.
I
Theorem 1. LJF is sound and complete with respect to intuitionistic logic.
Proof. Using the \grand tour" strategy. See [16, Section 6] for details.
9
Giventhedi(cid:11)erentformsofsequents,thecutruleforLJFtakesmanyforms:
[(cid:0)];(cid:2)(cid:0)!P [(cid:0)0];(cid:2)0;P (cid:0)!R [(cid:0)];(cid:2)(cid:0)!C [C;(cid:0)0];(cid:2)0 (cid:0)!R
[(cid:0)(cid:0)0];(cid:2)(cid:2)0 (cid:0)!R Cut+ [(cid:0)(cid:0)0];(cid:2)(cid:2)0 (cid:0)!R Cut(cid:0)
[(cid:0)](cid:0)B![P] [(cid:0)0];P (cid:0)![R] [(cid:0)](cid:0)!N [N;(cid:0)0](cid:0)B![R]
Cut Cut
[(cid:0)(cid:0)0](cid:0)B![R] 1 [(cid:0)(cid:0)0](cid:0)B![R] 2
[(cid:0)](cid:0) ! [C;(cid:0)0](cid:0) !
C R Cut!
[(cid:0)(cid:0)0](cid:0) !
R
Notice that the last three cut rules retain focus in the conclusion. These rules
extend those of LJQ’ [7], which were shown to be useful for studying term-
reduction systems. See [16] for a proof of the admissibility of these rules.
LikeLLF, a keycharacteristicof LJF is the assignmentof arbitrarypolarity
toatoms.Toillustratethee(cid:11)ectoftheseassignmentsonthestructureoffocused
proofs, consider the sequent a;a(cid:27)b;b(cid:27)c‘c where a, b and c are atoms. This
sequent can be proved either by forward chaining through the clause a (cid:27) b, or
backward chaining through the clause b(cid:27)c. Assume that atoms a and b areas-
signedpositivepolarityandthatcisassignednegativepolarity.Thisassignment
e(cid:11)ectively adopts the forward chaining strategy, re(cid:13)ected in the following LJF
proof segment (here, (cid:0) is the set fa;a(cid:27)b;b(cid:27)cg):
Ir c Il
[b;(cid:0)](cid:0) ! [b;(cid:0)](cid:0)![c]
b
(cid:27)L
b(cid:27)c
[b;(cid:0)](cid:0)![c]
Lf
[b;(cid:0)](cid:0)![c]
[]
l
[(cid:0)];b(cid:0)![c]
R
I l
r b
[(cid:0)](cid:0) ! [(cid:0)](cid:0)![c]
a
(cid:27)L
a(cid:27)b
[(cid:0)](cid:0)![c]
The polarities of a and c do not fundamentally a(cid:11)ect the structure of the proof
in this example. However, assigning negative polarity to atom b would restrict
the proof to use the backwardchaining strategy:
I
I l
r b
[(cid:0)](cid:0) ! [(cid:0)](cid:0)![b]
a
(cid:27)L
a(cid:27)b
[(cid:0)](cid:0)![b]
Lf
[(cid:0)](cid:0)![b]
[]
r
[(cid:0)](cid:0)!b
Rr c Il
[(cid:0)](cid:0) ! [(cid:0)](cid:0)![c]
b
(cid:27)L
b(cid:27)c
[(cid:0)](cid:0)![c]
10