Undecidability of the Lambek calculus with a relevant modality

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by  Max Kanovich

Undecidability of the Lambek Calculus with a
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Relevant Modality
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Max Kanovich1, Stepan Kuznetsov2, and Andre Scedrov3
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A UniversityCollege London;
National Research UniversityHigher School of Economics (Moscow)
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SteklovMathematical Institute(Moscow)

] 3University of Pennsylvania(Philadelphia);
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National Research UniversityHigher School of Economics (Moscow)
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h
t
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m Abstract
[ Morrill and Valent´ın in the paper “Computational coverage of TLG:

Nonlinearity”consideredanextensionoftheLambekcalculusenrichedby
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a so-called “exponential” modality. This modality behaves in the “rele-
v
vant”style,thatis,it allows contraction andpermutation,butnotweak-
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ening. Morrill and Valent´ın stated an open problem whether this system
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3 is decidable. Here we show its undecidability. Ourresult remains valid if
6 weconsiderthefragmentwherealldivisionoperationshaveonedirection.
0 Wealsoshowthatthederivabilityprobleminarestrictedcase,wherethe
. modality can be applied only to variables (primitive types), is decidable
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and belongs to theNP class.
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1 The Lambek Calculus Extended by a Relevant
:
v
Modality
i
X
r We start with the version of the Lambek calculus, L∗, that allows empty left-
a
handsidesofsequents(introducedin[10]). Wewillintroduce!L∗—anextension
of L∗ with one modality, denoted by !.
Formulae of !L∗ are built from a set of variables (Var = {p,q,r,...}) using
twobinary connectives, / (right division) and\ (left division), and additionally
one unary connective, !. Capital Latin letters denote formulae; capital Greek
letters denote finite (possibly empty) linearly ordered sequences of formulae.
Following the linguistic tradition, formulae of the Lambek calculus (and
its extensions) are also called types. In this terminology, variables are called
primitive types.
Thefinal publication (published inProc. FormalGrammar 2015/2016, LNCSvol.9804,
pp.240–256) isavailableatSpringerviahttp://dx.doi.org/10.1007/978-3-662-53042-9 14
1
We present !L∗ in the form of sequent calculus. Sequents of !L∗ are of
the form Π → A, where A is a formula and Π is a finite (possibly empty)
linearly ordered sequence of formulae. Π and A are called the antecedent and
the succedent respectively.
The axioms and rules of !L∗ are as follows:
A→A
Γ→A ∆ ,B,∆ →C Γ,A→B
1 2
(/→) (→/)
∆ ,B/A,Γ,∆ →C Γ→B/A
1 2
Γ→A ∆ ,B,∆ →C A,Γ→B
1 2
(\→) (→\)
∆ ,Γ,A\B,∆ →C Γ→A\B
1 2
Γ ,A,Γ →C !A ,...,!A →B
1 2 1 n
(!→) (→!)
Γ ,!A,Γ →C !A ,...,!A →!B
1 2 1 n
∆ ,!A,Γ,∆ →C ∆ ,Γ,!A,∆ →C
1 2 1 2
(perm ) (perm )
∆ ,Γ,!A,∆ →C 1 ∆ ,!A,Γ,∆ →C 2
1 2 1 2
∆ ,!A,!A,∆ →C
1 2
(contr)
∆ ,!A,∆ →C
1 2
We call ! the relevant modality, since it behaves in a relevant logic style,
allowing contraction and permutation, but not weakening. Recall that in the
original Lambek calculus there is neither contraction, nor permutation, nor
weakening. The modality is introduced to restore contractionand permutation
in a controlled way.
The cut rule is notofficially included in !L∗. Morrilland Valent´ın[12]claim
that it is admissible and that this fact can be proved using the standard pro-
cedure (cf. [11]). In this paper we consider the system without cut and don’t
need its admissibility.
We also consider fragments of !L∗. Since there is no cut rule in this system,
itenjoysthe subformulaproperty,andthereforeif werestrictthe setofconnec-
tives, we obtain conservative fragments of !L∗: !L∗ (where we have only / and
/
!), L∗ (this is the “pure” Lambek calculus without !), L∗.
/
Aswediscussinmoredetailin[7],L∗ canbeconsidered[1][18]asafragment
of non-commutative variantof Girard’s linear logic [5]. Our modality, !, follows
the spirit ofthe exponentialconnectivein linearlogic,allowingcontractionand
permutation. However, in contrast with the linear logic case, we don’t allow
weakening. On the other hand, as we discuss in Section 2, our ! is motivated
from the linguistical point of view.
Theorem 1. The derivability problem for !L∗ is undecidable. Moreover, the
derivability problem is undecidable even for !L∗.
/
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Remark 1. !L∗ has been constructed as a conservative fragment of a larger
system Db!?, introduced in [12]. Thus Theorem 1 provides undecidability of
Db!? (solving an open question raised in [12]).
2 Linguistic Examples and Motivations
In this section we start from the standard examples of Lambek-style syntactic
analysis [9][4] and then follow [12].
InsyntacticformalismsbasedontheLambekcalculusanditsvariants,Lam-
bektypes(formulae)denotesyntacticcategories. Weusethefollowingstandard
primitivetypes: nstandsforcommonnoun(like“book”or“person”);npstands
for noun phrase (like “John” or “the book”); s stands for the whole sentence.
Actually,nandnprepresentnotonlyisolatednounsandnounphrases,butalso
syntactic groups with similar properties: e.g., “the red book” or “the person
whom John met yesterday,”from the lingustic point of view, should be treated
as a noun phrase (np) as well. The latter cannot be provedto be of type np by
means of L∗, but !L∗ can handle this.
For simplicity, in our examples we don’t distinguish singular and plural
forms.
Other parts ofspeechreceivecompound types: np\s stands for intransitive
verb (like “runs” or “sleeps”); (np\s)/np stands for transitive verb (“likes,”
“reads,”“met,”“admire”);(np\s)\(np\s)isthe type foradverbs like“yester-
day” (it takes an intransitive verb group from the left-hand side and yields a
compound intransitive verb group); np/n is the type for “the,” etc.
If the sequent A ,...,A → B is derivable in the Lambek calculus or its
1 n
extension, syntactic objects of type A ,...,A , taken together in the specified
1 n
linear order, are considered to form an object of type B. For example, since
(np\s)/np,np/n,n→np\s isderivable, “readsthe book”is anexpressionof
type np\s, or, in other words, acts as an intransitive verb.
Example 1.
“John met Pete.” “John met Pete yesterday.”
These two sentences receive type s, since the sequents np,(np\s)/np,np → s
and np,(np\s)/np,np,(np\s)\(np\s)→s are both derivable in L∗.
Example 2.
“the person whom John met”
Asmentionedabove,wewantthisphrasetoreceivetypenp. Thisisobtainedby
assigning type (n\n)/(s/np) to “whom.” “John met” has type s/np, which
means “a sentence that lacks a noun phrase on the right-hand side.” In other
terms, we have a gap after “met.” In Example 1 this gap is filled by “Pete,”
and here it is intentionally left blank.
Example 3.
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“the person whom John met yesterday”
Here the gap appears in the middle of the clause (between “met” and “yes-
terday”), therefore “John met yesterday” is neither of type s/np, nor of type
np\s. This situation is called medial extraction and is not handled by L∗.
To put np into the gap, we use ! and the (perm ) rule:
1
np,(np\s)/np,np,(np\s)/(np\s)→s
(!→)
np,(np\s)/np,!np,(np\s)/(np\s)→s
(perm )
np,(np\s)/np,(np\s)/(np\s),!np→s 1
np/n,n,n\n→np np,(np\s)/np,(np\s)/(np\s)→s/!np
np/n, n, (n\n)/(s/!np), np, (np\s)/np, (np\s)/(np\s) →np
the person whom John met yesterday
The sequent on top is the same schema as for “John met Pete yesterday”
(see Example 1).
Note that s/!np and !np\s are equivalent (due to the permutation rules).
Example 4.
“the paper that John signed without reading”
Finally, this is the case called parasitic extraction, with two np gaps (after
“signed”andafter “reading”). Ifthe that-clausewereanindependentsentence,
thegapswouldhavebeenfilledlikethis: “Johnsignedthepaperwithoutreading
the paper.” To fill both gaps with the same np, we use the (contr) rule:
np,(np\s)/np,np,((np\s)/(np\s))/np,np/np,np→s
np,(np\s)/np,!np,((np\s)/(np\s))/np,np/np,!np→s
np,(np\s)/np,((np\s)/(np\s))/np,np/np,!np,!np→s
np,(np\s)/np,((np\s)/(np\s))/np,np/np,!np→s
np/n,n,n\n→np np,(np\s)/np,((np\s)/(np\s))/np,np/np→s/!np
np/n,n, (n\n)/(s/!np), np, (np\s)/np, ((np\s)/(np\s))/np, np/np →np
thepaper that John signed without reading
Here“that”actsexactlyas“whom,”and“without”modifiestheverbgroup
like “yesterday” does, but also requires a noun phrase “reading the paper” on
the right side. The sequents on the top are easily derivable in L∗.
Remark 2. Our calculus !L∗, as well as Db!?, works well for pure complex
sentences and pure compound sentences. However, we meet with difficulties
in the mixed case, caused by sophisticated nature of “and” and the like. For
example, the fact that “John met Pete yesterday and Mary met Ann today”
hastypes,leadstoandunwantedclassificationof*“thepersonwhomJohnmet
yesterdayandMarymetAnntoday”asanounphrase(typenp),cf. Example3.
In order to address this issue, Morrilland Valent´ın[12] suggestanother variant
of the system, denoted by Db!?b. This variant includes brackets that disallow
gappingincertainsituations. MorrillandValent´ınposethedecidabilityquestion
both for Db!? and Db!?b. In this paper we solve the first question.
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Remark 3. The whole system !L∗ turns out to be undecidable (Theorem 1).
On the other hand, notice that in these examples and the like can be treated
using types of a very restricted form. Namely, ! is applied only to a primitive
type (for instance, !np). In Section 5 we show that this restricted fragment
of !L∗ is decidable. Moreover, it belongs to NP, i.e., can be resolved by a
nondeterministic polynomial algorithm.
3 L∗ with Buszkowski’s Rules
Inthis sectionwebuildanundecidable extensionofL∗ withafinite setofrules,
generally following the construction by W. Buszkowski from [2]. Buszkowski,
however,considersanotherversionoftheLambekcalculus,L,introducedin[9].
ThedifferencebetweenLandL∗ istheso-calledLambek’srestriction: inL, the
antecedents of all sequents are forced to be non-empty. In this paper, following
MorrillandValent´ın[12],weallowemptyantecedents,andLambek’srestriction
is not valid in L∗ (e.g., →p/p is derivable in L∗). The relationship between L
andL∗ isverysubtle. Forinstance,thesequentq/(p/p)→q is derivableinL∗,
but becomes underivable when Lambek’s restictionis imposed (despite the fact
that this sequent itself has a non-empty antecedent). Therefore one has to be
verycautiouswiththisissue,andforthisreasonhereweprovideamodification
of Buszkowski’s construction for L∗ rather than directly use results from [2].
Let L∗+R be L∗ extended with a finite set R of rules of two special forms:
Π →p Π →q Π,q →p
1 2
Π ,Π →r (B1) or Π→r (B2),
1 2
where p,q,r are fixed primitive types. We call these rules Buszkowski’s rules.
Theorem 2. The cut rule
Π→A ∆ ,A,∆ →C
1 2
(cut)
∆ ,Π,∆ →C
1 2
is admissible in L∗+R for an arbitrary set R of Buszkowski’s rules.
Proof. We proceed by double induction. We consider a number of cases, and
in each of them the cut either disappears, or is replaced by cuts with simpler
cut formulae (A), or is replaced by a cut for which the depth of at least one
derivation tree of a premise (Π → A or ∆ ,B,∆ → C) is less than for the
1 2
original cut, and the cut formula remains the same. Thus by double induction
(on the outer level—on the complexity of A, on the inner level—on the sum of
premise derivation tree depths) we get rid of the cut.
Case 1: A is not the type that is introduced by the lowermost rule in
the derivation of ∆ ,A,∆ → C. In this case (cut) can be interchanged with
1 2
that lowermost rule. Consider the situation when it was (B ) (other cases are
1
similar):
∆′1,A,∆′1′ →p ∆2 →q
′ ′′ (B1)
Π→A ∆1,A,∆1,∆2 →r
(cut)
′ ′′
∆1,Π,∆1,∆2 →r
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Π→A ∆′,A,∆′′→p
1 1
(cut)
′ ′′
∆1,Π,∆1 →p ∆2 →q
∆′1,Π,∆′1′,∆2 →r (B1)
Case 2: A = E/F, and it is introduced by the lowermost rules both into
Π→A and into ∆ ,A,∆ →C.
1 2
Γ,F →E Π→F ∆1,E,∆2 →C
(→/) (/→)
Γ→E/F ∆1,E/F,Π,∆2 →C
(cut)
∆1,Γ,Π,∆2→C

Π→F Γ,F →E
(cut)
Γ,Π→E ∆1,E,∆2 →C
(cut)
∆1,Γ,Π,∆2 →C
Case 2 for \ is handled symmetrically.
Case 3: one of the premises of (cut) is the axiom (A→A). Then the goal
coincides with the other premise.
Note that since (B ) and (B ) introduce new primitive types only into the
1 2
succedent, the “bad” case, where both premises of the cut rule are derived
usingBuszkowski’srulesandthe cutformulais the formulaintroducedby both
of them, does not occur. This is the key trick that allows to formulate the
extended calculus in a cut-free way.
In the presence of (cut) Buszkowski’s rules (B ) and (B ) are equivalent to
1 2
axiomsp,q →randp/q →rrespectively,asshownbythefollowingderivations:
Π →q p,q→r Π,q →p
2 (cut) (→/)
Π →p p,Π →r Π→p/q p/q →r
1 2
(cut) (cut)
Π ,Π →r Π→r
1 2
and in the opposite direction:
q →q p→p
p→p q →q
(/→)
(B )
1 p/q,q →p
p,q→r (B )
2
p/q →r
From this perspective, L∗+R can be viewed as a finite axiomatic extension
of L∗ (with non-logical axioms of a special kind). However, for our purposes it
is more convenient to consider rules instead of axioms.
Theorem3. LetM bearecursivelyenumerablesetofwords over analphabet Σ
without the empty word. If Σ ⊂ Var, and Var also contains an infinite number
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ofvariables notbelonging toΣ, thenthereexistsafinitesetR ofBuszkowski’s
M
rules and s∈Var such that for any word a ...a over Σ
1 n
a ...a ∈M iff a ,...,a →s is derivable in L∗+R .
1 n 1 n M
We shall use the fact that any recursivelyenumerable language without the
empty word can be generated by a binary grammar [3]. A binary grammar is
a quadruple G = hN,Σ,P,si, where N and Σ are disjoint alphabets (Σ is the
originalalphabetofthe language),s∈N, andP is a finite setofproductions of
the form1
w⇒v v or v v ⇒w,
1 2 1 2
where v ,v ,w ∈ N ∪Σ. If (α ⇒ β) ∈ P and η and θ are arbitrary (possibly
1 2
empty) words over N∪Σ, then ηαθ ⇒ ηβθ. The relation⇒∗ is the reflexive-
G G
transitive closure of ⇒ . Finally, the language generated by G is the set of all
G
words a ...a over Σ such that s⇒∗ a ...a .
1 n G 1 n
Proof. Let M be an arbitrary recursively enumerable language without the
empty word and let G be a binary grammar that generates M. We construct
thecorrespondingextensionofL∗. LetN∪Σ⊂Var,andletVarcontainaninfi-
nite number of extra fresh variables that we’ll need later. For every production
(w ⇒v v )∈P we add one rule
1 2
∆ →v ∆ →v
1 1 2 2 (E)
∆ ,∆ →w
1 2
For productions of the form v v ⇒ w the construction is more complex.
1 2
Firstfor every pair p=h(v v ⇒w),xi, where (v v ⇒w)∈P andx∈N∪Σ,
1 2 1 2
we introduce new variables yp for every y ∈N ∪Σ and five extra variables ap,
bp, cp, ep, fp. Then for everyp we add the following rules. Some of these rules
e
are not in Buszkowski’s form. We’ll transform them into the correct format
below.
∆ →ep ∆ →x ∆ →yp ∆ ,y →ap
1 2 (1 ) 1 2 (2 )
∆1,∆2 →ap p ∆1e,∆2 →ap p
∆ →wp ∆ ,v ,v →ap ∆ →yp ∆ ,y →bp
1 2 1 2 1 2
(3 ) (4 )
∆e1,∆2 →bp p ∆1e,∆2 →bp p
∆ →fp ∆ ,ep →bp ∆ →y ∆ ,yp →cp
1 2 1 2
(5 ) (6 )
∆1,∆2 →cp p ∆1,∆2 →ecp p
∆,fp →cp
(7 )
p
∆→x
1Inthe definition from [3], P could alsoinclude productions ofthe form u⇒v foru,v∈
N∪Σ. Such a rulecan be equivalently replaced bytwo productions u⇒w1w2, w1w2 ⇒v,
wherew1andw2arenewelementsaddedtoN (differentfordifferentrules). Weencodethese
simple productions using more complex ones in order to reduce the number of cases to be
consideredintheproofs.
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As already said, some of these rules are not actually Buszkowski’s rules.
However,any rule of the form
∆ →p ∆ ,q →r
1 2
∆ ,∆ →t
1 2
(these are rules (2 ), (4 ), (5 ), and (6 )) can be equivalently replaced by two
p p p p
rules
∆,q →r ∆ →p ∆ →u
1 2
∆→u (B2) and ∆ ,∆ →t (B1)
1 2
where u is a fresh variable.
Similarly, (3 ) is a shortcut for three rules:
p
∆,v →ap ∆,v →u ∆ →wp ∆ →u
2 (B ) 1 1 (B ) 1 2 2 (B )
∆→u1 2 ∆→u2 2 ∆1,e∆2 →bp 1
Rules (1 ), (7 ), and (E) are already in the correct format. Thus we’ve
p p
actually constructed a calculus of the form L∗+R. Denote it by L∗+R .
M
Now to achieve our goal it is sufficient to prove that for any x,z ,...,z ∈
1 m
N ∪Σ
x⇒∗ z ...z iff z ,...,z →x is derivable in L∗+R .
G 1 m 1 m M
The proof consists of two directions.
⇐ All types in the sequent z ,...,z → x are primitive, therefore its
1 m
derivation includes only axioms and Buszkowski’s rules, but not original rules
of L∗ ((→/), (→\), (/→), (\→)).
Since ep, fp, and yp (for all y ∈ N ∪Σ, including w) do not appear in the
succedents of goal sequents in Buszkowski’s rules from R , the only possible
e M
situation when ep, fp, or yp actually appears in the succedent is the axiom.
Hence rules(1 )–(5 )canbe rewritteninasimplerway(rules(6 )and(7 )are
p p e p p
not affected by this simplification):
Φ→x (1′) Φ,y →ap (2′) Φ,v1,v2 →ap (3′)
ep,Φ→ap p yp,Φ→ap p wp,Φ→bp p
Φ,y →bp Φe,ep →bp ∆ →ey ∆ ,yp →cp
(4′) (5′) 1 2 (6 )
yp,Φ→bp p fp,Φ→cp p ∆1,∆2 →ecp p
eΦ,fp →cp
(7 )
p
Φ→x
Proceedbyinductiononthecut-freederivation. Thesequentz ,...,z →x
1 m
could either be an axiom (and then n=1, z =x, and trivially x⇒∗ x) or be
1 G
derived by one of the Buszkowski’s rules. Since x ∈ N ∪Σ, the only possible
rules are (E) and (7 ).
p
If z ,...,z →x is derived using (E):
1 m
z ,...,z →v z ,...,z →v
1 k 1 k+1 m 2
(E),
z ,...,z ,z ,...z →x
1 k k+1 m
8
then we have z ,...,z → v , z ,...,z → v , and (x ⇒ v v ) ∈ P. By
1 k 1 k+1 m 2 1 2
induction hypothesis, v ⇒∗ z ...z and v ⇒∗ z ...z , therefore we get
1 G 1 k 2 G k n
x⇒ v v ⇒∗ z ...z z ...z .
G 1 2 G 1 k k+1 m
If the last rule in the derivation is (7 ), then we get z ,...,z ,fp → cp.
p 1 m
Tracethetypeinthesuccedent. Sincetheantecedentdoesn’tcontaincp,bp,or
ap, sequents with these types in the succedentcould notappear as axioms,and
the only ways to derive such sequents are represented by the following schema
(the arrows go from goal to premises):
(6p) (4′p) (2′p)
(cid:7)(cid:7) (cid:7)(cid:7) (cid:7)(cid:7)
// cp //bp // ap //x
(5′) (3′) (1′)
p p p
Therefore, the sequent z ,...,z ,fp → cp is derived in the following way:
1 m
several (possibly zero) applications of (6 ), then (5′), then several (4′), then
p p p
several(2′),then(1′). Finally,ontopofthislast(1′)ruleweagaingetasequent
p p p
with x in the succedent. The whole derivation has the following form. Here ∗
means several consecutive applications of the same rule, and ∆ ,...,∆ =
1 n
z ,...,z .
1 m
y1,...,yk−1,v1,v2,yk+1,...,yn →x ′
(1 )
ep,y1,...,yk−1,v1,v2,yk+1,...,yn→ap p′ ∗
(2 )
yekp+1,...,yenp,ep,y1,...,yk−1,v1,v2 →ap (3′p)
wep,yekp+1,...,yenp,ep,y1,...,yk−1 →bp ′p∗
(4 )
ye1p,...,yekp−1,wep,yekp+1,...,yenp,ep→bp p′
(5 )
∆1 →y1 ... ∆k →w ... ∆∆1n,.→..,y∆nn,fpfp→,yec1pp,...,yekp−1,wep,yekp+1,...,yenp →cp (6pp)∗
∆1,...,∆n→x (7p)
Here rule (1′) introduces ep, (2′) moves ep to the left and marks y as yp,
p p i i
(3′) actually applies the production (v v ⇒ w), which is possible, since v ,v
p 1 2 1e2
is now on the edge of the antecedent, (4′) continues the movement, and finally
p
(5′), (6 ), and (7 ) move the letters backwards, unmark them and return the
p p p
antecedent to x.
By induction hypothesis, y ⇒∗ ∆ , ..., y ⇒∗ ∆ , w ⇒∗ ∆ ,
1 G 1 k−1 G k−1 G k
y ⇒∗ ∆ , ..., y ⇒∗ ∆ , and x ⇒∗ y ...y v v y ...y . By
k+1 G k+1 n G n G 1 k−1 1 2 k+1 n
application of v v ⇒w we get x⇒∗ ∆ ...∆ .
1 2 G 1 n
Wenoticethatthefirsttypeofproductionsofthebinarygrammarishandled
much easier than the second one. This is due to the fact that in the first
case we simulate standard context-free derivation, while in the second case the
production is not context-free and even not context-sensitive.
⇒ Proceed by induction on ⇒∗. For the base case (x ⇒∗ x) the corre-
G G
sponding sequent (x→x) is an axiom.
9
If the last production is w ⇒v v :
1 2
x⇒∗ z ...z wz ...z ⇒ z ...z v v z ...z ,
G 1 k−1 k+1 m G 1 k−1 1 2 k+1 m
then by induction hypothesis z ,...,z ,w,z ,...,z → x is derivable in
1 k−1 k+1 m
L∗ + R . Also v ,v → w is derivable by (B ), and by (cut) we obtain
M 1 2 1
z ,...,z ,
1 k−1
v ,v ,z ,...,z →x. The cut rule is admissible in L∗+R by Theorem 2.
1 2 k+1 m
For the v v ⇒w case, i.e., the last production is applied like this:
1 2
x⇒∗ z ...z v v z ...z ⇒ z ...z wz ...z ,
G 1 k−1 1 2 k+1 m G 1 k−1 k+1 m
the derivation is as follows (here p=h(v v ⇒w),xi):
1 2
z1,...,zk−1,v1,v2,zk+1,...,zm →x ′
(1 )
ep,z1,...,zk−1,v1,v2,zk+1,...,zm →ap ′p ∗
(2 )
zekp+1,...,zemp,ep,z1,...,zk−1,v1,v2 →ap (3p′)
wep,zekp+1,...,zemp ,ep,z1,...,zk−1→bp ′p∗
(4 )
ze1p,...,zekp−1,wep,zekp+1,...,zemp,ep→bp ′p
(5 )
z1→z1 ... w→w ... zm →zm fp,ze1p,...,zekp−1,wep,zekp+1,...,zemp →cp p ∗
z1,...,zk−1,w,zk+1,...,zm,fp →cp (6p)
(7p)
z1,...,zk−1,w,zk+1,...,zm →x
The sequent on the top is derivable by inductive hypothesis.
Since there exist undecidable recursively enumerable languages, Theorem 3
now yields the following result:
Theorem 4. There exists a finite set of Buszkowski’s rules R such that the
0
derivability problem for L∗+R is undecidable.
0
Note that the \ connectiveis notusedin the construction,so we’veactually
obtained undecidability for L∗ +R .
/ 0
4 Undecidability of !L∗
We prove undecidability of !L∗ by encoding L∗+R derivations in this calculus.
In order to do that, we first prove a technical proposition.
If R is a set of Buszkowski’s rules, let
G = (r/q)/p| Π1→p Π2→q ∈R ∪ r/(p/q)| Π,q→p ∈R .
R n Π1,Π2→r o n Π→r o
If B ={B ,...,B } is a finite set of formulae, let !Γ =!B ,...,!B . (The
1 n B 1 n
order of the elements in B doesn’t matter, since !Γ will appear in left-hand
B
sides of !L∗ sequents, and in !L∗ we have the (perm ) rules.)
1,2
10