for Intuitionisti Logi
Jens Otten ChristophKreitz
Fa hgebietIntellektik,Fa hberei h Informatik
Te hnis he Ho hs hule Darmstadt
Alexanderstr. 10, 64283 Darmstadt, Germany
fjeotten,kreitzgintellektik.i nfor matik .th- darm stadt .de
th
Abstra t. We present a proof method for intuitionisti logi based on
Wallen's matrix hara terization. Ourapproa h ombines the onne tion
al ulus and the sequent al ulus. The sear h te hnique is based on
no-tionsofpathsand onne tionsandthusavoidsredundan iesinthesear h
spa e. During the proof sear h the omputed rst-order and
intuition-isti substitutions are used to simultaneously onstru t a sequent proof
whi h is morehuman oriented thanthe matrixproof. This allowsto use
our method within intera tive proof environments. Furthermore we an
onsiderlo al substitutions instead ofglobal onesandtreatsubstitutions
o urring in di erent bran hes of the sequent proof independently. This
redu es thenumberofextra opies offormulae tobe onsidered.
1 Introdu tion
Intuitionisti logi (J),duetoits onstru tivenature,isoftenviewedasthelogi
of omputation.Ithasanessentialsigni an eforthederivationofveri ably
or-re tprogramssin e theoremsprovenwithinJ anbe onsideredasspe i ations
of algorithmswhi hare impli itly ontainedin the proof. Everyformula validin
intuitionisti logi isvalidin lassi allogi aswell.Theintuitionisti proof,
how-ever, ontains more informationthan a lassi aloneandmanyof thewell known
lassi alnormal formsand equivalen esare notvalid intuitionisti ally.As a
on-sequen e, it is onsiderablymore diÆ ult toproveatheorem in J than ndinga
lassi alproof.Reasoningin lassi allogi anbeautomated suÆ iently well(see
e.g. [5,12,19,2℄) but there isnot yet an eÆ ientintuitionisti proofpro edure.
Godelhas shownthat J anbe embedded into themodallogi S4[10℄:there
is a mapping M from J into S4 su h that a formula F is valid in J if M(F)
is valid in S4. In his investigations on non- lassi al logi s Wallen has used this
embedding to develop a matrix hara terization for the validity of intuitionisti
formulae[18℄whi h extendsBibel's hara terizationfor lassi alvalidity[3,4℄.
In propositional lassi al logi a formula F is valid if there is a spanning set
of onne tions for F. A onne tion is a pair of atomi formulae with di erent
polarities.Asetof onne tionsspans aformulaF ifeverypath through amatrix
representation of F ontains at least one onne tion. This hara terization also
appliesto predi ate logi ifall theterms ontainedin onne tedformulae an be
di eren e between lassi al and intuitionisti reasoning is expressed by ertain
restri tions on the intuitionisti rules. If rules are inverted for the purpose of
proof sear hthen these restri tions auseformulaeto be deleted from asequent.
Applyingarule(i.e.redu ing asub-formula)tooearly maythus deleteaformula
whi h later will be ne essary to omplete the proof. Therefore the order of rule
appli ationsmustbearranged appropriately.InWallen's matrix hara terization
this requirement is expressed by an intuitionisti substitution whi h makes the
pre xes of onne ted sub-formulae identi al where a pre x essentially des ribes
thepositionofasub-formulainthetreerepresentationoftheformulatobeproved.
Boththe rst-orderandtheintuitionisti substitutionhavetobe omputedby
uni ation algorithms. Forthe latter a spe ialized string uni ation is required.
Togetherwith the orderingofthe formulatree these substitutionsdeterminethe
orderinginwhi hagivenformulaF hasto beredu edbytherulesofthesequent
al ulus.Thisorderingmustbea y li sin eotherwisenoproofforF anbegiven.
During theproofpro essit maybe ome ne essaryto reatemultipleinstan esof
the same sub-formula. The number of opies generated to omplete the proof is
alledmultipli ity. Again,a multipli ity may bedueto aquanti erorspe i to
intuitionisti reasoning.
Developingan automated pro edure whi h onstru ts intuitionisti proofs on
thebasisofWallen'smatrix hara terizationmeansextending Bibel's onne tion
method[4℄ a ordingly.The advantageof su ha methodisthat the emphasison
onne tionsdrasti allyredu esthesear hspa e omparedto al ulianalyzingthe
outer stru ture of formulaesu has thesequent al ulus [9,7℄ ortableaux al uli
[1,16℄.Furthermoreitavoidsthenotationalredundan y ontainedinthese al uli
by avery ompa t representation.
The onne tion method is eÆ ient for nding proofs a ording to the
ma-trix hara terizationof validity. Itsresult, however, isalmostimpossible to read.
Therefore attempts have been made to onvert matrix proofs ba k into sequent
proofswhi haremu h loser to'natural'mathemati alreasoning.Thisis
ompa-rably easy for lassi alpropositional logi but be omes rather diÆ ult for
pred-i ate logi [11℄ or intuitionisti logi [15℄. In these ases the redu tion ordering
indu ed by thesubstitutions has tobe taken into a ount.
Althoughoriginallywewereinterestedonlyin ndingamatrixproofforagiven
formulatheabove onsiderationsledtothedevelopmentofaproofsear hmethod
whi h onstru tsthematrixproofandasequentproofalmostsimultaneously.The
partialsequentproof,however,ismorethan abyprodu tsin eit analsobeused
tosupporttheproofsear h.Itallows,forinstan e,to onsiderlo al substitutions
insteadofglobalones,i.e.substitutionswhi h anbeappliedindependentlywithin
sub-proofsof a sequentproof.Su ha lo al viewredu es the number of opies of
sub-formulaewhi hhavetobegeneratedto nda(global)substitutionandkeeps
thesear hspa eandthesizeof theproofsmaller.Thereforewehavedeveloped a
hybridmethod whi h ombines the onne tion methodand thesequent al ulus.
Afterresumingthesequent al ulus,thematrix hara terization,andaversion
of the onne tion method operating on non-normal forms in se tion 2 we shall
des ribethe relationbetween thesequent al ulus andthe onne tion methodin
few remarks on implementationissues and future investigations.
2 Preliminaries
2.1 Formula Trees, Types and Polarities
We assume the reader to be familiar with the language for rst-order logi . A
formula tree is arepresentation of a formula as tree whose nodes are marked by
positions denoted bya
0 ;a
1
;:::. Ea hposition orrespondstoa label onsisting of
the major onne tive or quanti er of a sub-formula or of the sub-formula itself
if it is atomi . Atomi positions are nodes labeledwith atomi formulae and are
leafs of thetree. Aformula treefor 8xPx)Pa^Pb isshown in gure 1. The
tree-ordering < isthe (partial) orderinggiven by the formula tree:a
i < a j if the position a i is below a j
in the formulatree.
y : 6 k 3 ) a 0 8x a 1 ^ a 3 Px a2 Pa a4 Pb a5 y : 6 k 3 ) 0 a0 8 1 x a 1 ( 0 ) ^ 0 a 3 ( 0 ) P 1 x a2( 0) P 0 a a4( 0) P 0 b a5( 0)
Fig.1.Formulatree without/withlabelsfor prin ipal/se ondary types andpolarities
Ea h sub-formula of a given formula F uniquely orresponds to a position in
the formula tree. A position is asso iated with a polarity, a prin ipal type, and
a se ondary type. The polarity (0 or 1) of a position is determined by the label
and polarity ofits parent. Theroot positionhaspolarity 0. Theprin ipal type of
a position is determined by its polarity and its label. Atomi positions have no
prin ipal type. The se ondary type of a position is determined by the prin ipal
type of its parents. The root position has no se ondary type. Polarity, prin ipal
type, andse ondarytypeofapositionarede ned intable1whose rstentry,for
instan e, means that a positionlabeledwith ^ and polarity1 has prin ipal type
andits su essornodeshavepolarity 1andse ondarytype
0 .
prin ipaltype se ondarytype
0 (A^B) 1 A 1 B 1 (A_B) 0 A 0 B 0 (A)B) 0 A 1 B 0 (:A) 1 A 0 (:A) 0 A 1
prin ipaltype se ondarytype
0 (8xA) 1 A 1 (9xA) 0 A 0
prin ipaltype se ondarytype
0 (A^B) 0 A 0 B 0 (A_B) 1 A 1 B 1 (A)B) 1 A 0 B 1
prin ipaltypeÆ se ondarytypeÆ
0 (8xA) 0 A 0 (9xA) 1 A 1
Table 1. Polarity, prin ipaltype, andse ondarytypeofpositions
Aformulatree for8xPx)Pa^Pb where the nodes additionally are labeled
with theirtypesandpolarityisalsogivenin gure1.Foragivenformulaweshall
use ; 0 ; ; 0 ; ; 0 ;, and 0
to denote the sets of positions of type ;
0 ; ; 0 ; ; 0 ;Æ, andÆ 0 respe tively.
A sequent has the form ` where (the ante edent) and (the su edent)
are sets of formulae. A proof of the sequent ` is a tree rooted with `
whosenodes aredetermined byrulesandwhose leafsare axioms.A formula F is
validi thereisaproofofthesequent`F.Table2shows theaxiomsand logi al
rulesof theintuitionisti sequent al ulus.
;A`A; axiom ;A` ;B` ;A_B` _left ;A;B` ;A^B` ^left ;:A`A; ;:A` :left ;A)B`A; ;B` ;A)B` )left ;8xA;A[xnt℄` ;8xA` 8left ;A[xna℄` ;9xA` 9left `A;B; `A_B; _right `A; `B; `A^B; ^right ;A` `:A; :right ;A`B `A)B; )right `A[xna℄ `8xA; 8right `A[xnt℄;9xA; `9xA; 9right
Table 2. A ut-freesequent al ulusfor intuitionisti logi
The parameter a of the rules 8right
and 9left
must not o ur free in the
on lusionof the rule(i.e. notin ;A, or). Similarly the term t in 9right and
8leftmustnot ontainvariableswhi ho urfree inthe on lusion. The al ulus
is omplete and orre t for intuitionisti logi [7℄. It di ers from the Gentzen's
al ulus LJ [9℄in thesensethat sets offormulaeare usedinsteadofsequen es {
whi h allows to omit stru tural rules like weakening and ontra tion { and that
more than one formula may o ur in the su edent of a sequent. It is, however,
possible to onvertproofsin the above al ulus into LJ-proofs(see e.g.[8℄).
The sequent al ulus forintuitionisti logi di ersfrom the lassi alone only
in the rules ) right,:right and 8right. Whereas in the intuitionisti ase the
su edent of the on lusion onsists of at most one formula, the orresponding
lassi alrulesmay ontainmultipleformulae.We alltheserulesspe ialrules.An
appli ationofan inverted rule(readfrom the on lusionto thepremise) is alled
aredu tion.Figure 2presents aproof ofthe formula8xPx)Pa^Pb.
Pa`Pa 8xPx`Pa 8l eft Pb`Pb 8xPx`Pb 8l eft `8xPx)Pa^Pb )right
Fig.2.Sequentproof for8xPx)Pa^Pb
2.3 A Matrix Chara terization for Intuitionisti Logi
The matrix hara terization for intuitionisti logi developed by Wallen [17,18℄
is based on the notion of paths and onne tions pioneered by Bibel for lassi al
{ a multipli ity en oding the number of distin t instan es of sub-formulae to
be onsidered during the proof,
{ an admissible rst-order substitution
Q
assigning atermto everyvariable in
the formula,
{ a setof onne tions whi h are omplementary under
Q
su h that every path
through the formula F ontains a onne tion from this set.
For te hni al reasons we repla e the variables in atomi formulae by their
quanti er positions. Thus positions of type and Æ appear in atomi formulae
insteadofvariables.Consequentlya rst-ordersubstitution
Q
isamappingfrom
the set of positions of type to terms where again variables are repla ed by
positions. The substitution 1 Q indu es a relation < Q on in the following way:if Q (u)=tthenv < Q
uforallv2thataresub-termsoft.A onne tionis
apairofatomi positionslabeledwithatomi formulaehavingthesamepredi ate
symbol but di erent polarities. If they are identi al under
Q
the onne tion is
said to be omplementary under
Q
. A path through a formula F is a subset of
theatomi positionsofitsformulatree;itisahorizontalpath throughthematrix
representation ofF (see examplein gure 3and 4).
Sin ethequanti errules9leftand8rightare onstrainedbytheeigenvariable
onditionthe relationv <
Q
uexpressesthat the sub-formulalabeledbyv should
be redu ed before redu ing the one labeled by u. The transitive losure of the
union of<
Q
andthetree-ordering <is alledtheredu tion ordering , i.e.:=
(< [ < Q ) + . A rst-order substitution Q
isadmissible if the redu tion ordering
is irre exive. Inthis aseaproofin the sequent al ulus is onstru tible. This
te hnique was rst proposed by Bibel [4℄ as an alternative for skolemization in
lassi allogi .
The intuitionisti sequent al ulus ontains spe ial rules whi h, if used
ana-lyti ally, ause formulae to be deleted from a sequent. To ensure that formulae
ontainingtwo atomi formulae of a onne tion as sub-formulaeare not deleted
by spe ialrulesthe orrespondingatomi positions ofthis onne tion haveto be
made omplementaryunder an additionalintuitionisti substitution.
Toexplain thene essarymodi ationsofthe lassi almatrix hara terization
we extend the de nitions of types and positions. A spe ial position in a formula
treeisapositionlabeledwithanatomi formula,negation(:),impli ation()),or
auniversalquanti er(8x). Ifaspe ialpositionhaspolarity1it hasintuitionisti
type and otherwisetype .To denotethe set ofpositions of intuitionisti type
and weuse and respe tively.Withea hatomi positionuweasso iatea
sequen epre(u)ofpositions alledthepre xofuasfollows:ifu
1 <u 2 <:::<u n u
(1 n) are the elements of [ that dominate u in the formula tree then
pre(u)= u 1 u 2 u n
. Intuitionisti omplementarity of atomi positions requires
that theirpre xes an beuni ed 2 by an intuitionisti substitution. An intuitionisti substitution J is a mapping from to ([ ) . It indu es a relation< J on Pos([ ) 3
in the following way:if
J
(u)=p then v <
J u
1
Forte hni al reasons we onsiderasubstitution tobe idempotent (i.e. =).
2
Tounifytwopre xeswe needanalgorithmfor spe ial stringuni ation[14℄.
3
J
is the prede essor of u in the formula tree. As in the rst-order ase v <
J u
means that v should be `redu ed' before u. A ombined substitution onsist of a
rst-order substitution
Q
and an intuitionisti substitution
J
. It is admissible
if theredu tion ordering:=(<[ <
Q [ < J ) + isirre exive.
Theorem 2. A formula F is intuitionisti ally valid i there is
{ a multipli ity ,
{ an admissible ombined substitution :=(
Q ;
J ),
{ a set of onne tions whi h are omplementary under and su h that every
path through the formula F ontains a onne tion from this set.
Consider the formula8xPx) Pa^Pb. Its formula tree isshown in gure 3;
its matrix representation in gure 4 where we pla e omponents of -type
sub-formulae horizontallyand omponents of -type sub-formulaeverti ally.
y i 1 : 6 6 3 k ) 0 a 0 8 1 x a 1 1 P 1 a 1 1 a 1 2 8 1 x a 2 1 P 1 a 2 1 a 2 2 ^ 0 a 3 P 0 a a4 P 0 ba5
Fig.3. Formula treefor F
P 1 a 1 1 :a 0 a 1 1 a 1 2 |{z } pr efix P 1 a 2 1 :a 0 a 2 1 a 2 2 P 0 a:a 0 a 4 P 0 b:a 0 a 5
Fig. 4.Matrixrepresentationfor F
The two instan es of the formula 8xPx are to be onsidered omponents of
an impli it -typeposition(in the matrixthey stayside by side).In thepre xes
the positions of type are emphasized with an over-bar. There are two paths
throughthematrix,namelyfPa 1 1 ;Pa 2 1 ;PagandfPa 1 1 ;Pa 2 1
;Pbg.They ontainthe
onne tionsfPa 1 1 ;PagandfPa 2 1
;Pbgrespe tivelywhi hareboth omplementary
under the substitutions
Q = fa 1 1 na;a 2 1 nbg and J = f a 1 1 n;a 2 1 n;a 1 2 na 4 ;a 2 2 na 5 g.
Therefore theformulais intuitionisti ally valid.
2.4 The Conne tion Method
Aproofmethodfor lassi al rst-orderlogi basedontheorem1isthe onne tion
methoddevelopedby Bibel[4℄.Theproofsear hisdrivenby onne tions instead
of onne tives as in the sequent al ulus. On e a onne tion has been identi ed
all paths ontainingthis onne tion are eliminated. If every paths is deleted the
formulaisvalid.Inthefollowingwepresentaproofmethodsimilartotheoriginal
onne tion methodwhi hdeals with formulaein non-normal formbe auseofthe
absen eof su ha formin the intuitionisti logi .
De nition 1. Twoatomi formulaeP andQare -/ -related ifthe rst ommon
node in the formula tree - going from the nodes labeled with P and Q down to
theroot-isapositionoftype / .Noatomi formulaP is -/ -relatedto itself.
If two atoms (atomi formulae) are -related they appear side by side in a
Q are -/ -related for all formulae Q2 S. Every atom P is -/ -relatedto the
empty set ;.
Let A be the set of all atoms 4
in a given rst-order formula F. Then the
following pro edure returnstrue i F isintuitionisti ally valid.
Main-pro edure
repeat
:=(;;;); valid:=Proof(;;;);
ifvalid= false thenin reasethe multipli ityof thegiven formulaF
until valid= true
Sub-pro edureProof(P;C) (P Aisthea tive path.C Aareproven subgoals.)
ifnoatomA2Ais -relatedtoP and -relatedto C then returntrue
E :=;; 0
:=
repeat
sele tanatom A2Awhi his -relatedto P[E and -relatedtoC
ifthere isnosu h atomAthen return false
E :=E [fAg; D:=;; valid:= false; no onne t:= false
repeat sele t an atom A 2 A where A 62 D and either A 2 P or A is -related
to P [fAg and (A;
A) is a onne tion whi h is omplementary under an
admissible ombined substitution omputedusing 0
ifthere isnosu hatom
A
thenno onne t :=true
elseD:=D[f
Ag; valid :=Proof (P[fAg;f
Ag)
ifvalid =true thenvalid :=Proof (P;C[fAg)
untilvalid = trueorno onne t= true
until valid= true
returntrue
Note that in Proofall variablesex ept forthe set Aandthe substitution are
lo al.Anexampleproofusingthe onne tionmethodisgiveninthenextse tion.
3 Relating Sequent Cal ulus and Conne tion Method
In this se tion we point out the relationship between a proof with the
onne -tion method and the orresponding sequent proof.Firstly we deal with lassi al
propositional logi . After that we onsider the intuitionisti propositional ase.
3.1 Classi al (Propositional) Logi
ConsiderF (S^(:(T )R ))P)) )(:((P )Q)^(T )R )))(S^::P)).
The formula tree (skeleton) of this formula is shown in gure 5, its matrix
rep-resentation in gure 6. In the skeleton only the positions of prin ipal type ,
i.e. 1 ; 2 ; 3 and 4 , are marked. 5
Additionally ea h bran h rooted at su h a
-positionis marked with a letter, namely a,b,...,h. Sin e wedeal with formulae
4
Di erentatomshavingthesamepredi atesymbolare onsidered distin t.
5
Positionsoftype playtheessentialroleduringtheproofpro ess.Inourpresentation
amatrix anitselfbeamatrix.Components ofsub-formulaeoftype are pla ed
one upon the other. Atoms are marked with their polarities, whereas polarity 0
indi ates that the atom o urspositivelywithin thenegationalnormal form and
polarity 1 means that it o urs negatively. A redu tion of a position means the
sub-formula rooted at thisposition hasto be redu ed in thesequent al ulus.
}> }> y : Y * o7 I Y 1 * i y : P 1 Q 0 R 0 T 1 P 0 S 0 R 1 S 1 T 0 P 1 1 2 3 4 a b d g h e f
Fig.5. Skeletonofthe formulatree forF
P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1
Fig. 6.Matrixof theformulaF
We begin by proving the lassi al validity of the formula F. After ea h
on-ne tion step we show the stru ture of the orresponding sequent proof. In the
rststep {shown in gure 7{we onne t atom P 1
whi his in bran h'a' ofthe
formula tree with P 0
in bran h ' '. If these atoms shall form an axiom in the
sequentproofwehaveto redu epositions
1 and
2
.Wheneverweredu ea
posi-tionofprin ipaltype thesequentproof willsplit intotwobran hes.Thusafter
redu ing
1
there is a split into two bran hes 'a' and 'b'. Now we redu e
2 in
the'a'-bran hof thesequentproofwhi hresults in the bran hes ' ' and'd'. The
' '-bran hnow ontains an axiom of the form ;P 1
`P 0
;. Thisbran his said
to be losed. 6
Note that we do not perform redu tionsof positions whi hdo not
havetype (i.e.areoftype , ,orÆ)expli itly.Sin eredu ingpositionsoftype
, , orÆ do notsplit thesequentproofthey anbe redu ed straightforwardly.
P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 a d b P 1 `P 0 ? ? 2 1
Fig.7.The rststep inthe onne tion/sequentproofofF
In the sequent proof there are twobran hes 'b' and 'd' whi h do not ontain
an axiom. They are said to be open. We rst want to lose bran h 'd'. In the
formula tree this bran h only ontains the atom S 0
. Conne ting it with atom
S 1
(obtained without redu ing any -position) leads to an axiom of the form
;S 1
`S 0
; whi h loses thisbran hasshown in gure 8.
P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 a d b P 1 `P 0 ? S 1 `S 0 2 1
Fig.8.The se ondstep inthe onne tion/sequentproofofF
6
Open bran hes ( orresponding to open subgoals) are marked with a '?' whereas ''
stepwe onne tR 0
withR 1
(see gure9).Sin eR 1
o ursinthe'e'/'g'-bran hof
theformulatreewe rsthavetoredu eposition
3 and
4
su essively.Therefore
the 'b'-bran h in the sequent proof is split twi e. Whereas
3
is responsible for
splitting into the bran hes'e' and'f',
4
splits the 'e'-bran hinto 'g' and'h'. As
the'g'-bran his losedbyan axiomtheonlyopenbran hesare 'h'and'f'.Inthe
nextstepwe onne tfromT 0
to T 1
losingthe'h'-bran hin thesequent al ulus
asshown in gure 9. Sin e the 'b'-bran halready ontains T 1 wedo nothaveto redu e a -position. P 1 : R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 a d b g e h f P 1 `P 0 S 1 `S 0 R 1 `R 0 T 1 `T 0 ? 2 1 4 3
Fig.9. Thethird/fourth step inthe onne tion/sequentproofofF
Thereisonlyoneopenbran hleft,namelythe'f'-bran h.Conne tingfromP 1
toP 0
splitsitintothebran hes' 'and'd',astheatomP 0
o ursinthe' '-bran h
of the formula tree (see gure 10). Closing this ' '-bran h with an axiom in the
sequentproof,we nallyhaveto losethe'd'-bran h. Inthislast stepwe onne t
from S 0
to S 1
,whi hdoes notleadto anyopen bran hes,sin e theatom S 1
an
be rea hedwithout aredu tion of -positions.
P 1 : R 0 P 0 S 0 Q 0 T 1 R 1 : T 0 : P 1 S 1 a d b g e h f d P 1 `P 0 S 1 `S 0 R 1 `R 0 T 1 `T 0 P 1 `P 0 S 1 `S 0 2 1 4 3 2
Fig.10. The fth/sixthstep in the onne tion/sequent proofofF
Wesu essfully ompleted the onne tion proof and every leaf in the sequent
proofisan axiom. Therefore theformulaF is lassi allyvalid.
3.2 Intuitionisti (Propositional) Logi
In intuitionisti logi we additionally have to unify the pre xes of the atomi
formulaeinevery onne tion.Thisleadstoanintuitionisti substitution
J whi h
indu esarelation<
J
onthepositionsoftheformulatreeasde nedinse tion2.3.
Togetherwith the treeordering< it determines the redu tionordering where
vumeans that position v should beredu ed before positionu. Performing all
these stepsw.r.t. theformulaF aboveeventually leadsto thefollowing redu tion
orderingon the positions ofprin ipal type (i.e.
1 ; 2 ; 3 and 4 ): 2 3 1 4 :
2
redu e
3
andsoon.Thereforetheintuitionisti sequentproofshownin gure11
di ersfrom the lassi alone in orderofrule appli ation. 7 (P 1 `P 0 ) (R 1 `R 0 ) (T 1 `T 0 ) (P 1 `P 0 ) (S 1 `S 0 ) 1 4 3 2 a b e f d g h
Fig.11.The stru tureofan intuitionisti sequentproofofF
Thesequentproofin intuitionisti logi annot bederived aseasilyasin
las-si al propositional logi . In the latter ase ea h onne tion in a matrix proof
orrespondsto exa tlyone axiom in thesequent al ulus. Forintuitionisti logi
(even in the propositional part) this property does not hold anymore. The
sit-uation is similar for lassi al predi ate logi be ause the eigenvariable ondition
restri ts the order in whi h positions an be redu ed (en oded in the relation
<
Q
de ned in se tion 2.3). To avoid these problems our approa h will take the
redu tionordering into a ount duringthe onstru tion ofthe proof.
4 A Conne tion Based Proof Method
Before we present our proof pro edure we shall investigate the intuitionisti
va-lidity ofthe previousse tion's example alittle more detailed.
4.1 An Introdu tory Example
We have seenthat it ismore eÆ ient to onsider the redu tion ordering
(par-ti ularly <
J
) during the pro ess of onstru ting a matrix proof and a sequent
proof simultaneously.Due to the importan eof -positions within the redu tion
orderingweslightlymodifythede nitionofa tivepathsandde neopensubgoals.
De nition 3. The -pre xof an atomi position u, denoted by -pre(u), isthe
set of all elements v
1 ;:::;v n 2 0 (positions of type 0
) that dominate u in the
formula tree,i.e. -pre(u) := fv2
0
jv <ug.
In the previous se tion as well as in the example below we have marked
bran hes in the sequent proof with letters (e.g. a, b,...) to keep the notation
simple. For the following de nitions we have to point out that ea h letter
or-responds to exa tly one position of type
0
. If, for instan e, the redu tion of a
-position
1
leads to the bran hes'a' and'b' in the sequent proof, they will be
identi ed bythe twosu essorpositions of
1
in theformulatreewhi hareboth
oftype
0 .
7
0
0
allthelabels(positionsoftype
0
)obtainedbygoingfromtherootofthesequent
proof to the node marked with u while olle ting the label of every bran h. An
a tive -path P
indu es an a tive path P for the position u where P = fv j v
atomi position and -pre(v)P
g.
Thea tive path P for uis thus theset of all the atomswhi h anbe rea hed
from the u-bran hin the sequent proof (i.e. the bran h leadingfrom the root to
thepositionu)without passingthrough a -position.Inotherwords,itistheset
of atoms whi h an be obtained by redu ing the orrespondingsequent without
redu ing positionsof type .
De nition 5. ThesetofopensubgoalsC
0
isthesetofthepositionsoftype
0
labeling theopenbran hesin thesequentproof.Ea hopenbran hisassigned
its a tive( -)path.
ConsideragainF (S^(:(T)R ))P)))(:((P)Q)^(T)R )))(S^::P))
and its formula treegiven below
}> }> y : Y * o7 I Y 1 * i y : P 1 Q 0 R 0 T 1 P 0 S 0 R 1 S 1 T 0 P 1 1 2 3 4 a b d g h e f
ToproveF we rstsele t anatom 8
,say P 1
,in bran h'a'of theformulatree
and onne t it with the atom P 0
in the ' '-bran h. For that we have to redu e
two -positions, namely
1
and
2
. Unifying the pre xes ofthe twoatoms leads
to an intuitionisti substitution. Together with the tree ordering it indu es the
redu tion ordering
2
1
. Thus we have to split into the bran hes ' ' and 'd'
( orrespondingto
2
)beforewesplitthe' '-bran hinto'a'and'b'( orresponding
to
1
). This loses the 'a'-bran h in the sequent proof as shown in gure 12. In
the next step we hoose the 'd'-bran h from the set of open subgoals C
= fb,
dg.Thea tive -pathP
=fdgfor'd' indu esan a tivepathP = fS 1 ;S 0 g. The only atom S 0
in the'd'-bran hofthe formulatree an thereforebe onne tedto
S 1
in the a tivepath whi h loses thisbran h.
P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 (P 1 `P 0 ) a b 1 2 d ? (S 1 `S 0 )
Fig.12. The rstandse ondproofstep
The only open bran h is now the 'b'-bran h (C
= fbg). In the formula tree
this bran h ontains twoatoms R 0
andT 1
from whi hwesele t R 0
and onne t
it with R 1
whi h is not in luded in the a tive path P = fS 1 ;P 0 ;R 0 ;T 1 g for 'b' (P = f ,bg). To make R 0
form an axiom with R 1
we have to redu e
3
8
4
bran hinto 'g' and 'h'. The uni ation of the pre xes of these two atoms yields
an intuitionisti substitution whi h { together with the tree ordering { indu es
theredu tionordering( on erningthe -positions)
2 3 1 4 .Thatmeans
we have to insert the split into 'e' and 'f' between the redu tion of
2
and
1
(leaving the rest of the partial sequent proof remains un hanged) and split into
thebran hes 'g' and'h' after redu ing
1 ,asshownin gure 13. P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 (R 1 `R 0 ) ? ? 1 4 3 2 a b e f d g h
Fig.13.The thirdproofstep
After losingbran h'g' weget twoadditionalopen bran hes'f'and'h' (C
=
ff, hg). The a tive -paths for 'f' (P
= f ;fg) and for 'h' (P = f ;e;b;hg) indu e P = fS 1 ;P 0 ;P 1 g and P = fS 1 ;P 0 ;R 0 ;T 1 ;T 0 g respe tively. To lose
these bran hes we onne t P 1
in the 'f'-bran h of the formula tree to P 1
in the
a tivepath for'f'andT 0
in the'h'-bran htoT 1
inthe a tivepathfor'h'.These
steps on lude the intuitionisti proof for F, sin e C
= ; and therefore ea h
bran hin thesequent proofis losed.
P 1 R 0 P 0 S 0 Q 0 T 1 R 1 T 0 P 1 S 1 (T 1 `T 0 ) (P 1 `P 0 ) 1 4 3 2 a b e f d g h
Fig.14.The fourthand fthproofstep
4.2 The Proof Pro edure
The explanations given in the aboveexample should be suÆ ient to understand
the pro edure arrying out our proof method. In prin iple it is similar to the
version ofthe onne tion methodintrodu ed in se tion 2.4.There is, however, a
di eren e in the handling of subgoals and a tive paths. The original onne tion
method fo uses on onne ting new atoms whi h are sele ted a ording to the
urrent a tive path P and the set C of already proven subgoals. P and C are
parametersof thepro edure.Themethod whi hweshall des ribebelow aimsat
losingopensubgoalsoftype
0
(asetwhi hmaygroworde reaseinthepro ess)
and uses onne tions related to their a tive -paths forthis purpose. Thea tive
path dependson the sele ted subgoaland willbe omputed withinthe pro ess.
Let A be the set of all atoms in a given rst-order formula F. The following
pro edure returnstrue i F isintuitionisti ally valid.
Main-pro edure
repeat
:=(;;;); initialize
; valid :=Proof(f;g)
ifvalid = false then in reasethemultipli ityofthe givenformula F
Sub-pro edure Proof(C ) (C B 0
aresubgoals whi h still haveto beproven)
ifC =;then returntrue
E :=;; 0 :=; 0 := ; sele tanelementa 2C repeat
sele tanatom A2Awherea
-pre(A)whi his -relatedto E
ifthere isnosu h atomA then returnfalse
E :=E [fAg; D:=; ; valid :=false; no onne t:=false
omputethe a tive -pathP fora and itsa tive pathP
using the -redu tionordering 0 repeat sele t an atom A 2 A where A 62 D and either A 2 P or A is -related
to P [fAg and (A;
A) is a onne tion whi h is omplementary under an
admissible ombinedsubstitution andanadmissible -redu tionordering
omputedusing 0
and 0
ompute theset C
expanded bythe new opensubgoals
ifthere isnosu hatom
A
thenno onne t :=true
elseD:=D[f
Ag; valid :=Proof (C
nfa
g)
untilvalid = trueorno onne t= true
until valid= true
returntrue
Note that all variablesin Proof {ex ept forA, ,and
{ arelo al.
Theabovealgorithmusesafewnew on eptswhi hdeserveexplanation.Sin e
itispossibletoredu ethesameformulaindi erentbran hesofthesequentproof
wehaveto distinguish thesebran hes(identi edwith positionsoftype
0
)byan
index. B i
0
isa setof indexed positionsoftype
0
in luded in the sequent proof.
Previouslywehadrequiredthat theredu tionorderingde nesade nite
rela-tion between all -positions. Thisis notstri tly ne essary.If a substitutiondoes
not lead to an ordering between two bran hes in the sequent proof we have to
en ode the permutability between these bran hes. This is done by an extended
de nitionofpathstogetherwith aso- alled -redu tionordering
.
onsists
oftworelations,namely
0 0 and6 0 0
.Therelationuv (u;v 2
0 )
means that there isa sequentproof wherethe bran hes u andv are in the same
-path (that is a way from the root to a leaf), whereas u 6 v (u;v 2
0
) means
that there is no su h a sequentproof.These tworelations indu e an a tivepath
P :=(P n ;P p ). Then-pathP n fora 0
-positionu ontainsall
0
-positionswhi h
are ne essarily in the a tive -path for u in all sequent proofs under
onsidera-tion. The p-path P p
for u denotes the set of
0
-positions whi h are possibly in
the a tive -path ofu. 9
Our method always attempts to sele t a redu tion ordering whi h allows to
onne t to the a tivepath. Thisshortensproofssubstantially sin e a onne tion
to the a tive path does not lead to any new open subgoal. If we ignore the
re-du tion ordering during the sear h for onne tions we will get a version of the
onne tion method. Therefore ourmethod isageneralizationof theoriginal
on-ne tion method.
9
Thesequentproofmakes it possibleto use so- alledlo al substitutionsinsteadof
global ones. We present an approa h to treat rst-order as well as intuitionisti
substitutionslo ally.
The onne tionmethodandour proofmethodpresentedaboveuseglobal
sub-stitutions. If we substitute a term t for a variable x then every o uren e of x
in the orrespondingsequent proof has to berepla ed by t. Thisis not very
rea-sonable, sin ein asequentproofweareallowedto repla edi erenttermsforthe
same variableif it o urs in di erentbran hes ofthe proof.
Let us onsider the formula 8xPx)Pa^Pb fromse tion 2. In the matrix
proofinse tion2.3weneededa opyofthesubformula8xPx(eveninthe lassi al
ase) sin e wehad to assign twoterms a andb tothe variable x. Howeverin the
sequent proof (see gure 15) a dupli ation does not (expli tly) o ur. We ould
avoid this dupli ation if we treat the substitutions of the two bran hes of the
sequent proof independently. Therefore we take two substitutions into a ount,
namely
1
= fxnag and
2
= fxnbg, whi h are related to the two di erent
bran hesin the sequentproof shownin gure 15.
Pa`Pa 8xPx`Pa Pb`Pb 8xPx`Pb `8xPx)Pa^Pb 1 =fxnag 2=fxnbg =fg Pa`Pa Pb`Pb
Fig.15. Asequentprooffor8xPx)Pa^Pbanditsstru ture
Toperformsu hastepitisne essarythatthe -positiona
(a
3
inourexample)
responsibleforthesplit isredu ed beforethe -positiona
(a 1 labeledwith 8xin our example). 10
That is, either the redu tion-orderingyields a
a
or wehave
to introdu e this ordering and look if it is admissible. This te hnique is similar
to Bibel's splitting te hnique [4℄. Our approa h, however, is simpler and an be
applied more rigorously sin e we are able to exploit the sequent proof. When
omputing the substitution whi h has to make a onne tion omplementary we
onlyhaveto onsidersubstitutionsrelatedtobran hesofthea tive -path.After
thatwehaveto dividethe omputedsubstitutionsu hthatitspartsrelatetothe
orrespondingbran hes.
Inthefollowing examplewedealwith theintuitionisti substitution.Consider
theformula F ::P )::P ^::P .Itsmatrix representationtogether with
thepre xes of theatoms isgiven in gure 16.
P 0 :a 0 a 1 a 2 a 3 P 1 :a0a4a6a7 P 1 :a 0 a 4 a 8 a 9
Fig.16.MatrixforF
1 =f a 2 na 6 ;a 7 na 3 g 2 =f a 2 na 8 ;a 9 na 3 g 0=f a4na1g P 1 `P 0 P 1 `P 0
Fig. 17. Stru tureofthesequent proofforF
There are two paths through the matrix ea h of them ontaining a
onne -tion. To make the rst onne tion omplementary we have to unify the
pre-10
Otherwise we have to repla e the variable in the ommon bran h before the split
xes a 0 a 1 a 2 a 3 and a 0 a 4 a 6 a 7
whi h results in the intuitionisti substitution
J = f a 4 na 1 b; a 2 n ba 6 ; a 7 n a 3 g 12 where
b and are new variables.
Apply-ingthis substitutiontothepre xes ofthese ond onne tionleads tothe pre xes
a 0 a 1 ba 6 a 3 anda 0 a 1 ba 8 a 9
respe tivelywhi hdonotunify.Itwouldbene essary
to dupli ate thesubformula:P 0
althoughthis opy does notappear(expli itly)
in the sequentproof. To avoidthis dupli ation weagain onsider lo al
substitu-tions. Sin e the position labeled with ^ is redu ed before :P 0
(indu ed by the
substitution) thesubformula:P 0
withthepre xa
2 a
3
o ursinbothbran hesof
thesequentproof.Thereforealsothe variablea
2
ansubstituted bytwodi erent
stringswhi hmakethese ond onne tion omplementary.Thelo alsubstitutions
1
and
2
and the substitution
0
whi h is ommon to both bran hes together
withthestru tureofthesequentproofareshownin gure17. 13
Both onne tions
are now omplementary under thesubstitution
0 [ 1 and 0 [ 2 respe tively.
Employing lo al substitutionsredu es the number of opies of formulae to be
onsideredinaproofandthus themultipli ity.A opywillberequiredifandonly
ifthis opyalsoappearsexpli itlyin thesequentproof.Sin edupli ated formulae
an beverylargethis redu esthesear hspa e for aproofaswell asits size.
We on ludethisse tionbypresentingamatrix hara terisationfor
intuition-isti logi using lo al substitutions.
De nition 6. Alo al onne tion((A;a); ( A;a)),wherea;a2B i 0 ,a= ` -pre(A) 14 and a = ` -pre(
A), is lo ally omplementary under
and if the onne -tion (A;
A) is omplementary under the admissible ombined substitution :=
L ( (u)j u 2 P for a or u 2 P for a) where L is the ombination of
substitutions (for detailssee [14℄).
Theorem 3. A formula F is intuitionisti ally valid i there is
{ a multipli ity ,
{ an admissible -redu tion ordering
(en oding the sequentproof stru ture),
{ a lo al substitution
whi h assigns ea h indexed
0 -position u2 B i 0 a om-bined substitution :=( Q ; J ),
{ a set of lo al onne tions whi h are lo ally omplementary under
and
su h that every path through F ontains a onne tion from this set.
6 Con lusion
In this paper he havepresented a proof methodfor intuitionisti logi whi h
de-velops a matrix proof and a sequent proof simultaneously. Our method extends
Bibel's onne tionmethod[4℄ a ordingWallen's matrix hara terizationof
intu-itionisti validity[18℄ butit does notrequire anormal form.Due to an emphasis
on onne tions instead of the outer stru ture of formulae the sear h spa e an
be kept omparably small. Developing the sequent proof during the proof
pro- ess leadsto anaturalrepresentationof aformal proofwhi h an beusedwithin
11
Weemphasizethe positionsoftype whi h play thepartofvariablesbyanoverbar.
12
This (andonlythis!)is infa tthe mostgeneral uni er.
13
Wehaveomitted theextra variables
band .
14
`
-pre(A)isthe lastpositionof type
0
instead of global ones whi h redu es the sear h spa e even more than a purely
matrix-orientedproofmethod would do.
TheeÆ ien yofourproofpro edurealsodependsontheuni ationalgorithm
omputing the so- alled intuitionisti substitutions. In [14℄ we have developed
a spe ialized string uni ation algorithm whi h is more eÆ ient than the one
presented in [13℄ sin e it omputes only the most general substitutions whi h
makethepre xes equal.
The sequentproofgeneratedbyour pro edure an easilybe transformedinto
a Gentzen-stylesequent proof (see [15℄for details). Thus we an realize our
pro- edureas ata ti of the NuPRLsystem [6℄ in orderto support the development
ofproofsand veri ed routineprograms withinari h onstru tivetheory.
Referen es
1. E. W. Beth. The foundations ofmathemati s. North{Holland,1959.
2. W. Bibel, S. Br
uning, U. Egly, T. Rath. Komet. In Pro eedings of the 12 th
CADE,LNAI814, p.783{787. SpringerVerlag,1994.
3. W. Bibel. On matri eswith onne tions. Jour. ofthe ACM,28,p. 633{645, 1981.
4. W. Bibel. Automated Theorem Proving. ViewegVerlag,1987.
5. K. Bl
asius, N.Eisinger, J. Siekmann, G. Smolka, A.Herold, C. Walther.
The MarkgrafKarlrefutation pro edure. InIJCAI-81,p. 511{518, 1981.
6. R. L.Constableet. al. Implementing Mathemati swiththe NuPRLproof
devel-opmentsystem. Prenti eHall, 1986.
7. M. C. Fitting. Intuitionisti logi , model theory and for ing. Studiesinlogi and
thefoundationsofmathemati s. North{Holland,1969.
8. J. Gallier. Constru tive logi s. Part I: A tutorial on proof systems and typed
- al uli. Te hni al Report8, Digital EquipmentCorporation,1991.
9. G. Gentzen. Untersu hungen uber das logis he S hlieen. Mathematis he
Zeits hrift,39:176{210, 405{431, 1935.
10. K. G
odel.Aninterpretationoftheintuitionisti sententiallogi .InThePhilosophy
ofMathemati s, p. 128{129. OxfordUniversity Press,1969.
11. D. S. Korn. KonSequenz { Ein Konnektionsmethoden-gesteuertes
Sequen-zenbeweis-Verfahren. Master'sthesis, TH Darmstadt,FG Intellektik,1993.
12. R. Letz, J. S humann, S. Bayerl, W. Bibel. Setheo: A high-performan e
theoremprover. Journalof Automated Reasoning, 8:183{212,1992.
13. H.J.Ohlba h. Aresolution al ulusformodallogi s. Ph.D.Thesis(SEKIReport
SR-88-08), FBInformatik,Universitat Kaiserslautern, 1988.
14. J.Otten. EinkonnektionenorientiertesBeweisverfahrenfurintuitionistis heLogik.
Master'sthesis, TH Darmstadt,FG Intellektik,1995.
15. S. S hmitt, C. Kreitz. On transforming intuitionisti matrix proofs into
stan-dard-sequentproofs. InPro eedings Tableaux Workshop1995, thisvolume.
16. R. M. Smullyan. First-Order Logi , Ergebnisse der Mathematik 43. 1968.
17. L.Wallen. Matrixproof methods formodallogi s. IJCAI-87, p.917{923. 1987.
18. L.Wallen. Automated dedu tion in non lassi al logi . MITPress,1990.
19. L. Wos et. al. Automated reasoning ontributes to mathemati s and logi . In
Pro eedings ofthe 10 th
