cyclic involutive fl

[10:51 22/3/2011 exq021.tex] LogCom: Journal of Logic and Computation Page: 231 231–252

Proof Theory Corner

Cyclic Involutive Distributive Full LambekCalculus is DecidableMICHAŁ KOZAK, Faculty of Mathematics and Computer Science, AdamMickiewicz University, Umultowska 87, 61-614 Poznan, Poland.E-mail: [email protected]

AbstractWe prove the decidability of the extension CyInDFL of DFL (Distributive Full Lambek Calculus), in which the involutivelaw is derivable under the assumption of cyclicity. So far, the decidability of this logic has been an open problem (Galatos et al.,2007, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Elsevier). We present a Gentzen-style consecutioncalculus which is complete with respect to the class of cyclic involutive distributive full Lambek algebras (CyInDFL). It is anextension of the system DFL introduced by the author (2009, Stud. Log., 91, 201–216). We prove the cut elimination theoremfor this system, and give a procedure of building a finite proof-search tree for any consecution.

Keywords: Full Lambek calculus, residuated lattice, distributive lattice, cyclicity, involution, cut elimination,decidability.

1 Introduction

Galatos et al. [15] pointed out (p. 219) that the decidability of the equational theory of cyclic involutivefull Lambek algebras (CyInFL) had been shown by Yetter [30] using proof-theoretic methods andalgebraically by Wille [29]. At the same book they indirectly raised the issue of the decidability ofits distributive cousin by indicating (p. 229) that its commutative variant CyInDFLe has a decidableequational theory.1 This result is due to Galatos and Raftery [16], who showed that the algebraicsemantics (in the sense of Blok and Pigozzi [5]) for RW (the contraction-free subsystem of the relevantlogic R) is the variety InDFLe (i.e. CyInDFLe, since cyclicity is a consequence of commutativity).2

The decidability of RW was earlier established by Brady [7] using Giambrone’s decidability argumentfor RW+ [17] (the positive fragment of RW). Both those results were based on Dunn’s Gentzenizationof R+ [1, 12].3 We should recall that independently of Dunn, Mints [23] introduced an analogousGentzen-style system for R+ supplemented with an additional S4 style necessity operator. These twoquite novel systems were a huge success, since seeking a Gentzen-style system in which distributioncan be proven had been a major problem for relevance logic (we refer to [1, 2] for history anddetails).

Both of these (Dunn and Mints’) innovations consist in allowing an antecedent of a sequent tobe a structure built from extensional and intensional structures inductively, with their own sets ofrules for these two kinds of structures. Due to this change, sequents were renamed consecutions.

1The naming convention corresponds to the one adopted for basic substructural logics, where subscripts stand for structuralrules determining properties of fusion [15]; e stands for exchange which just makes fusion commutative: x ·y=y ·x.

2Cyclicity (so named by Yetter [30] because of his rule of cyclic permutation) means that right and left negations coincide;commutativity makes that right and left residuals coincide, where from cyclicity follows (see Section 2 for details).

3To be accurate, mentioned relevant logics are often denoted in the literature with the addition of ◦t , since Dunn’sGentzenization covers R+ conservatively enriched by the logical connective ◦ and the logical constant t.

Vol. 21 No. 2, © The Author, 2010. Published by Oxford University Press. All rights reserved.For Permissions, please email: [email protected] online 24 September 2010 doi:10.1093/logcom/exq021


232 CyInDFL is Decidable

Providing a decision procedure for that consecution calculi in the case of the lack of the contractionrule for intensional structures was great Giambrone’s success. Brady cleverly extended this reasoningto calculi with signed formulas (formulas preceded by the sign T or F). Earlier such extension of asequent calculus by signed formulas for RW without distribution was done by Bull [8].4

Status of the decidability of Dunn and Mints’ systems, and consequently, the logic R+ is stillunknown. Only what we know is that the logic R is undecidable (by the famous result of Urquhart[28]). From this fact it follows immediately that the equational theory of the commutative and square-increasing variant of CyInDFL (i.e. CyInDFLec and InDFLec) is undecidable.5 This result is dueto Dunn’s algebraization of R extended with the mentioned constant t corresponding to the identityelement of de Morgan monoids [11] and the fact that the class of de Morgan monoids coincides withInDFLec [13, 15]. We should recall that the algebraization of R (without t) was done much later byBlok and Pigozzi [5].

In this article, we focus on cyclic variants of InDFL, but not necessarily with commutative fusion.We show that all basic subvarieties of CyInDFL, except these in which fusion is square-increasing,have decidable equational theories. The problem of the undecidability (as we conjecture) of theorieswith square-increasing fusion we leave to further research.

We introduce one negation satisfying the involutive law to the Gentzen–style system DFL [19].The technique employed to introduce it is similar to that of Brady [6, 7], however, unlike in thosepapers, we do not use signed formulas. We show completeness of the obtained system CyInDFL withrespect to CyInDFL by the construction of the Lindenbaum–Tarski algebra. It requires adding thecut rule to this system as well. Fortunately, as we prove by a syntactic argument, cut is eliminable.The proof is very long and tedious — it goes by triple induction involving mutual induction onthe structure of the cut formula and negation of the cut formula. It would be interesting to showcompleteness and cut admissibility by the method of nuclei and quasi-embedding as well as thefinite model property [3, 14, 18, 19, 24, 29], or even strong finite model property in the case of thenon-associative variants [9, 10].

The argument for the decidability of CyInDFL is similar to that of Giambrone. This line ofreasoning was also the inspiration for Restall [26] in his display systems in the style of Belnap [4] formany relevant logics. The heart of the matter is controlling the complexity of structures, and hencethe number of consecutions that can occur in a proof-search tree for a given consecution. Our proofresembles both that of Giambrone and that of Restall.

2 Preliminaries

A monoid is an algebra of the form M= (M,·,1), where · is a binary operation on M, called productor fusion, satisfying the associative law: (x ·y)·z=x ·(y ·z), for all x,y,z∈M, and 1 is the unit elementfor ·, i.e. for all x∈M, x ·1=1·x=x.

A lattice is an algebra of the form L= (L,∧,∨), where ∧ and ∨ are binary operations on L, called,respectively, meet and join, which are associative: (x∧y)∧z=x∧(y∧z), (x∨y)∨z=x∨(y∨z);commutative: x∧y=y∧x, x∨y=y∨x; and mutually absorptive: x∨(x∧y)=x, x∧(x∨y)=x.

A lattice L is distributive, if the distributive law of ∧ over ∨ holds: x∧(y∨z)≤ (x∧y)∨(x∧z);where as always a partial order relation ≤ on L is defined by: x≤y⇔x∨y=y.

4Bull refers to the book of Zaslavsky [31] on Nelson’s logics, in which this technique of dealing with negation was carriedout for the first time.

5Square-increasingness of fusion, i.e. x≤x ·x, is a consequence of the contraction rule, which is abbreviated by c.


CyInDFL is Decidable 233

A residuated lattice is an algebra of the form L= (L,∧,∨,·,→,←,1) such that (L,∧,∨) is a lattice,(L,·,1) is a monoid, and→ and← are, respectively, right and left residual for product, i.e. theysatisfy the following conditions: x ·y≤z⇔y≤x→z, and x ·y≤z⇔x≤z←y, for all x,y,z∈L.

A full Lambek algebra or a FL-algebra is a residuated lattice L with a constant 0 (which can denotearbitrary element of L). This constant 0 allows one to define two unary operations ∼ and − on L,called respectively right and left negation, by setting: ∼x=x→0 and −x=0←x.

In any FL-algebra, by the residuation laws, there holds: x≤ (y←x)→y and x≤y← (x→y). Hence,for y=0, we have x≤∼−x and x≤−∼x. A FL-algebra L is involutive, if the opposite inequalities∼−x≤x and −∼x≤x hold in L as well.

A FL-algebra L is cyclic, if negations coincide: ∼x=−x, for all x∈L.A FL-algebra L is distributive, if its lattice reduct (L,∧,∨) is distributive.An involutive distributive FL-algebra can be also defined in a different way. Namely, an algebra

of the form L= (L,∧,∨,→,←,∼,−,1) is an involutive distributive FL-algebra, if (L,∧,∨) is adistributive lattice, and the following six laws hold in L: Galois residuation x≤y→z⇔y≤z←x, involution ∼−x=x=−∼x, Galois negation x≤∼y⇔y≤−x, contraposition x→−y=∼x←y,associativity of residuals (x→y)←z=x→ (y←z) and residuation unit 1→x=x=x←1. As it wasshown in [15], in this approach, fusion and 0 are defined via:

x ·y def=−(y→∼x)=∼(−y←x), 0def=∼1=−1.

In the case of cyclicity, the equivalent definition is even simpler. Namely, an algebra of the formL= (L,∧,∨,→,←,∼,1) is an cyclic involutive distributive FL-algebra, if (L,∧,∨) is a distributivelattice, and the following five laws hold in L: Galois residuation, involution∼∼x=x, contrapositionx→∼y=∼x←y, associativity of residuals and residuation unit. Galois negation follows from theremaining laws:

x≤∼y ⇔ x≤1→∼y ⇔ 1≤∼y←x ⇔ 1≤y→∼x ⇔ y≤∼x←1 ⇔ y≤∼x.

3 CyInDFL

According to the Section 2, we can define the system CyInDFL without fusion and 0. So, thelanguage of CyInDFL consists of a denumerable infinite set of variables p, q, r, ..., the constant 1,one unary connective ∼, four binary connectives ∧, ∨,→ and←, and also the structure constant λ

(which corresponds to Dunn’s constant t) and two binary structure constructors� (which correspondsto Dunn’s intensional structures constructor — semicolon) and � (which corresponds to Dunn’sextensional structure constructor — comma). Formulas are formed out of variables and the constant 1by means of connectives. Structures are built from formulas and the structure constant λ by means ofstructure constructors. Consecutions are of the form X⇒A, where X is a structure and A is a formulaor the emptiness (they admit the empty succedent).

As above, we denote arbitrary formulas by A, B, C, ... (also the empty succedent, where it makessense) and structures by X , Y , Z and sometimes by V and W . By X[Y ] we denote a structure Xwith a designated substructure Y , and in this context by X[Z] we denote the substitution of Z forthat particular occurrence of Y in X . We can now specify axioms and inference rules of the systemCyInDFL.



Axioms:

(Id) A⇒A

(1R) λ⇒1

(∼1L) ∼1⇒

Structural rules:

(�A)X[(Y �Z)�V ]⇒A

X[Y � (Z �V )]⇒A

X[Y � (Z �V )]⇒A

X[(Y �Z)�V ]⇒A

(λW )X[Y ]⇒A

X[Y �λ]⇒A

X[Y ]⇒A

X[λ�Y ]⇒A

(λC)X[Y �λ]⇒A

X[Y ]⇒A

X[λ�Y ]⇒A

X[Y ]⇒A

(�A)X[(Y �Z)�V ]⇒A

X[Y � (Z �V )]⇒A

X[Y � (Z �V )]⇒A

X[(Y �Z)�V ]⇒A

(�W )X[Y ]⇒A

X[Y �Z]⇒A

(�E)X[Y �Z]⇒A

X[Z �Y ]⇒A

(�C)X[Y �Y ]⇒A

X[Y ]⇒A

Logical rules:

([∼]) X⇒A

X�∼A⇒X⇒∼A

X�A⇒X⇒A

∼A�X⇒X⇒∼A

A�X⇒

(1L)X[λ]⇒A

X[1]⇒A

(∼1R)X⇒

X⇒∼1

(∼∼L)X[A]⇒B

X[∼∼A]⇒B

(∼∼R)X⇒A

X⇒∼∼A

(∧L)X[A�B]⇒C

X[A∧B]⇒C



(∧R)X⇒A X⇒B

X⇒A∧B

(∼∧L)X[∼A]⇒C X[∼B]⇒C

X[∼(A∧B)]⇒C

(∼∧R)X⇒∼A

X⇒∼(A∧B)

X⇒∼B

X⇒∼(A∧B)

(∨L)X[A]⇒C X[B]⇒C

X[A∨B]⇒C

(∨R)X⇒A

X⇒A∨B

X⇒B

X⇒A∨B

(∼∨L)X[∼A]⇒C

X[∼(A∨B)]⇒C

X[∼B]⇒C

X[∼(A∨B)]⇒C

(∼∨R)X⇒∼A X⇒∼B

X⇒∼(A∨B)

(→L)X[B]⇒C Y⇒A

X[Y � (A→B)]⇒C

X[∼A]⇒C Y⇒∼B

X[(A→B)�Y ]⇒C

(→R)† A�X⇒B X�∼B⇒∼A

X⇒A→B

(∼→L)X[∼B�A]⇒C

X[∼(A→B)]⇒C

(∼→R)X⇒∼B Y⇒A

X�Y⇒∼(A→B)

(←L)X[B]⇒C Y⇒A

X[(B←A)�Y ]⇒C

X[∼A]⇒C Y⇒∼B

X[Y � (B←A)]⇒C

(←R)† X�A⇒B ∼B�X⇒∼A

X⇒B←A

(∼←L)X[A�∼B]⇒C

X[∼(B←A)]⇒C

(∼←R)X⇒A Y⇒∼B

X�Y⇒∼(B←A)

RemarkThe rules (→R)† and (←R)† are deliberately indicated with the symbol †, since we assume themonly to prove the completeness and cut elimination theorems. In Section 6, we replace these tworules with the corresponding rules from FL and give a simple proof of the equivalence of these twosystems.



In the standard way, the interpretation of consecutions in any cyclic involutive distributiveFL-algebra L is defined. Homomorphisms from the free algebra of formulas to L are calledassignments in L. Every assignment f is extended to structures and the empty succedent, by setting:

f (λ)=1, f ( )=∼1.

f (X�Y )=∼(f (Y )→∼f (X)),f (X�Y )= f (X)∧f (Y ),

A consecution X⇒A is true in a model 〈L,f 〉, if f (X)≤ f (A), and valid in L, if it is true in 〈L,f 〉 forany assignment f . We denote this fact by L |=X⇒A. By |=X⇒A, we denote the validity of X⇒Ain all algebras from CyInDFL (the validity in CyInDFL for short).

One can show that consecutions provable in this system (these which can be obtained from axiomsby applying inference rules repeatedly) are precisely those consecutions which are valid in CyInDFL.Soundness is straightforward — we leave it as an exercise to check that all axioms are valid and allrules preserve the validity. Completeness we will show using the construction of the Lindenbaum–Tarski algebra. It requires adding one more validity preserving rule to this system — the cut rule:

(CUT )X[A]⇒B Y⇒A

X[Y ]⇒B.

4 Completeness

Our resolution is to prove that a consecution X⇒A is provable in CyInDFL with cut, what we denoteby † X⇒A, if it is valid in CyInDFL. However, by the definition of the validity of consecutionsand the easy to prove (with the help of cut) Lemma 1 below, we can show a simpler form of thecompleteness theorem: |=A⇒B implies † A⇒B, for any formulas A,B.

Lemma 1For any structure X and formulas A,B,C there holds:

• if † X[1]⇒A, then † X[λ]⇒A,• if † X[∼(B→∼A)]⇒C, then † X[A�B]⇒C,• if † X[A∧B]⇒C, then † X[A�B]⇒C,• if † X⇒∼1, then † X⇒ .

In order to obtain this completeness theorem, we need some lemmas and definitions.

Lemma 2The following rules are derivable in CyInDFL:

(∼∼LR)A⇒B

∼∼A⇒∼∼B

(∧LR)A⇒B C⇒D

A∧C⇒B∧D

(∼∧LR)∼A⇒∼B ∼C⇒∼D

∼(A∧C)⇒∼(B∧D)

(∨LR)A⇒B C⇒D

A∨C⇒B∨D



(∼∨LR)∼A⇒∼B ∼C⇒∼D

∼(A∨C)⇒∼(B∨D)

(→LR)A⇒B C⇒D ∼B⇒∼A ∼D⇒∼C

B→C⇒A→D

(∼→LR)A⇒B ∼C⇒∼D

∼(A→C)⇒∼(B→D)

(←LR)A⇒B ∼B⇒∼A C⇒D ∼D⇒∼C

C←B⇒D←A

(∼←LR)A⇒B ∼C⇒∼D

∼(C←A)⇒∼(D←B)

Proof. We work out the proofs of only a couple of rules, leaving the rest as an exercise.

• (∼∨LR)∼A⇒∼B ∼C⇒∼D——————— (∼∨L) ———————– (∼∨L)∼(A∨C)⇒∼B ∼(A∨C)⇒∼D

————————————————————– (∼∨R)∼(A∨C)⇒∼(B∨D)

• (∼→LR)∼C⇒∼D A⇒B——————————— (∼→R)∼C�A⇒∼(B→D)—————————— (∼→L)∼(A→C)⇒∼(B→D)

• (←LR)C⇒D A⇒B ∼B⇒∼A ∼D⇒∼C

————————— (←L) ——————————— (←L)C←B�A⇒D ∼D�C←B⇒∼A

——————————————————————— (←R)†

C←B⇒D←A

�We define a binary relation ≡ on the set F of all formulas of CyInDFL as follows:

A≡B ⇔ † A⇒B and † B⇒A and †∼B⇒∼A and †∼A⇒∼B.

By (Id), (CUT ) and the rules from Lemma 2, it follows that ≡ is a congruence with respect to allconnectives. We construct the quotient algebra L≡F on the set F/≡, setting: |A|≡∧|B|≡=|A∧B|≡,|A|≡∨|B|≡=|A∨B|≡, |A|≡→|B|≡=|A→B|≡, |A|≡←|B|≡=|A←B|≡, ∼|A|≡=|∼A|≡ and 1=|1|≡. We should now verify that L≡F is an cyclic involutive distributive FL-algebra. In thisverification, the following lemma will be useful.

Lemma 3

|A|≡≤|B|≡ ⇔ † A⇒B and †∼B⇒∼A.



Proof. According to the above definitions we have: |A|≡≤|B|≡ if and only if |A∨B|≡=|B|≡,which is equivalent to † A∨B⇒B and †∼B⇒∼(A∨B), since B⇒A∨B and∼(A∨B)⇒∼B areprovable in CyInDFL.

Let us assume † A∨B⇒B and †∼B⇒∼(A∨B). We obtain † A⇒B and †∼B⇒∼A by theprovability of A⇒A∨B and ∼(A∨B)⇒∼A, and (CUT ).

Conversely, if we assume † A⇒B and †∼B⇒∼A, we will obtain † A∨B⇒B and †∼B⇒∼(A∨B) by (Id), and the rules (∨L) and (∼∨R). �

We verify only four axioms of cyclic involutive distributive FL-algebras, leaving the remainingaxioms to the reader.

• |A|≡≤|B|≡→|C|≡⇒|B|≡≤|C|≡←|A|≡We present proofs of the right implication of the Galois residuation law; proofs of the oppositeimplication are analogous.

∼C⇒∼C B⇒B—————————— (∼→R)∼(B→C)⇒∼A ∼C�B⇒∼(B→C)

——————————————————— (CUT )∼C�B⇒∼AC⇒C B⇒B——————— (→L)B�B→C⇒C A⇒B→C

——————————————— (CUT )B�A⇒C

———————————————————— (←R)†

B⇒C←A

� � � � � � � � � �

� � � � � � � � � � ��

��

∼B⇒∼B ∼C⇒∼C——————————— (→L)

B→C�∼C⇒∼B A⇒B→C————————————————— (CUT )

A�∼C⇒∼B——————— (∼←L)∼(C←A)⇒∼B

• |A|≡→∼|B|≡≤∼|A|≡←|B|≡We present proofs of the right inequality of the contraposition law; proofs of the oppositeinequality are analogous.

B⇒B A⇒A———— (∼∼R) ————– (∼∼L)

∼A⇒∼A B⇒∼∼B ∼B⇒∼B ∼∼A⇒A—————————— (→L) ——————————— (→L)

A→∼B�B⇒∼A ∼∼A�A→∼B⇒∼B———————————————————————— (←R)†

A→∼B⇒∼A←B

B⇒B A⇒A———— (∼∼R) ———— (∼∼L)B⇒∼∼B ∼∼A⇒A————————————— (∼→R)

B�∼∼A⇒∼(A→∼B)——————————— (∼←L)∼(∼A←B)⇒∼(A→∼B)

• (|A|≡→|B|≡)←|C|≡≤|A|≡→ (|B|≡←|C|≡)We present proofs of the right inequality of the associativity law; proofs of the opposite inequalityare analogous.



∼A⇒∼A ∼B⇒∼B——————————– (→L)

A→B�∼B⇒∼A C⇒C——————————————— (←L)

((A→B)←C�C)�∼B⇒∼A————————————— (�A)

(A→B)←C� (C�∼B)⇒∼A————————————— (∼←L)(A→B)←C�∼(B←C)⇒∼A

∼B⇒∼B A⇒A————————— (∼→R)∼C⇒∼C ∼B�A⇒∼(A→B)

——————————————— (←L)(∼B�A)� (A→B)←C⇒∼C————————————— (�A)∼B� (A� (A→B)←C)⇒∼C

B⇒B A⇒A———————– (→L)

A�A→B⇒B C⇒C————————————— (←L)

A� ((A→B)←C�C)⇒B———————————– (�A)(A� (A→B)←C)�C⇒B———————————————————— (←R)†

A� (A→B)←C⇒B←C—————————————————————————– (→R)†

(A→B)←C⇒A→ (B←C)

� � � � � � � � ��

��

� � � � � � � � ��

��

��

��

∼B⇒∼B A⇒A————————— (∼→R)

C⇒C ∼B�A⇒∼(A→B)———————————————– (∼←R)

C� (∼B�A)⇒∼((A→B)←C)—————————————— (�A)(C�∼B)�A⇒∼((A→B)←C)—————————————— (∼←L)∼(B←C)�A⇒∼((A→B)←C)

——————————————— (∼→L)∼(A→ (B←C))⇒∼((A→B)←C)

• |1|≡→|A|≡=|A|≡We present proofs of the first equality of the residuation unit law; proofs of the second equalityare analogous.

λ⇒1 A⇒A ∼A⇒∼A λ⇒1——————– (→L) —————————– (∼→R)λ�1→A⇒A ∼A�λ⇒∼(1→A)

——————– (λC) —————————– (λC)1→A⇒A ∼A⇒∼(1→A)

A⇒A A⇒A ∼A⇒∼A————— (λW ) —————– ([∼]) —————— (λW )λ�A⇒A A�∼A⇒ ∼A�λ⇒∼A

————— (1L) —————– (∼1R) —————— (1L)1�A⇒A A�∼A⇒∼1 ∼A�1⇒∼A

—————————————— (→R)† ——————— (∼→L)A⇒1→A ∼(1→A)⇒∼A

We can now prove the main theorem of this section.



Theorem 4For any formulas A, B, if |=A⇒B, then † A⇒B.

Proof. Just as for Lindenbaum–Tarski algebras, we define an assignment f : F→L≡F by f (A)=|A|≡. Let us now assume on the contrary that �† A⇒B. Then by Lemma 3, |A|≡� |B|≡. So, f (A)�f (B), i.e. A⇒B is not true in the model 〈L≡F ,f 〉.

�

5 Cut elimination

To achieve a syntactic proof of cut admissibility, it is expedient to prove a more general form of thecut rule, called the mix rule [1, 6, 7, 12, 17]. In our system, the mix rule has to have the followingtriple form:

(1)X[A]...[A]⇒B Y⇒A

X[Y ]...[Y ]⇒Bfor any B,

(MIX) (2)X[A]...[A]⇒B Y⇒A

X[Y ]...[Y ]�Y⇒ if ∼B=A or B=∼A,

(3)X[A]...[A]⇒B Y⇒A

Y �X[Y ]...[Y ]⇒ if ∼B=A or B=∼A.

We will prove the admissibility of all three forms of the (MIX) rule simultaneously by tripleinduction: (I) on the complexities of A and∼A (mutual), (II) on the length of proof of X[A]...[A]⇒Band (III) on the length of proof of Y⇒A. Clearly, from (MIX) (1) for one A we get the admissibilityof the (CUT ) rule.

• A=p.In the case when X[p]...[p]⇒B is the axiom (Id), the conclusion of (MIX) (1) coincides withthe premise Y⇒p. In most cases when X[p]...[p]⇒B is the conclusion of some rule, theconclusions of (MIX) (1)–(3) follow from the corresponding induction hypotheses of (II) for(MIX) (1)–(3), and this rule. Only in the cases when X[p]...[p]⇒B is the conclusion of ([∼]),and the introduced A on the left side of the consecution is one from the designated p from X,the conclusion of (MIX) (1) follows straightforwardly from the induction hypothesis of (II) for(MIX) (2) or (3).

• A=∼p.The proof does not stray much from the case above. Main differences occur in the cases whenX[∼p]...[∼p]⇒B is the conclusion of ([∼]) or (∼∼R). In the former case, we should alsoconsider the cases when the introduced ∼A on the left side of the consecution is one from thedesignated ∼p from X . In the latter, we should consider (MIX) (2)–(3) as well. In both thesecases, the conclusions of (MIX) (1) (for the former) and (MIX) (2)–(3) (for the latter) followstraightforwardly from the corresponding induction hypotheses of (II) for (MIX) (2) and (3).

• A=1.The proof differs from the case A=p only if X[1]...[1]⇒B is the conclusion of (1L) or (∼1R).In the case when the rule (1L) introduces one from the designated 1, we must switch on induction(III). The most special case of this induction is when Y⇒1 is an instance of the axiom (1R). Theconclusions of (MIX) (1)–(3) follow then straightforwardly from the corresponding induction



hypotheses of (II) for (MIX) (1)–(3). In the case of the rule (∼1R), we must switch on induction(III) to prove the admissibility of (MIX) (2)–(3). Again, the most special case is when Y⇒1 isan instance of the axiom (1R). Both conclusions of (MIX) (2)–(3) follow then from the inductionhypothesis of (II) for (MIX) (1) and the rule (λW ).

• A=∼1.The proof differs from the case A=∼p only if X[∼1]...[∼1]⇒B is an instance of the axiom(1R) or (∼1L). In both these cases we must switch on induction (III). In the case of the axiom(1R), the most special case is when Y⇒∼1 is the conclusion of (∼1R). The conclusions of(MIX) (2)–(3) follow then from the premise Y⇒ by the rule (λW ). In the case of the axiom(∼1L), the most special case is also when Y⇒∼1 is the conclusion of (∼1R). The conclusionof (MIX) (1) coincides with the premise Y⇒ .

• A=C∧D.The proof differs from the case A=p only if X[C∧D]...[C∧D]⇒B is the conclusion of (∧L)or (∼∧R). Analogously to the case A=1, two inductions (III) are required: when the rule (∧L)introduces one from the designated C∧D and in order to show the admissibility of (MIX) (2)–(3)in the case of (∼∧R). In both these inductions, the most special cases are when Y⇒C∧D isthe conclusion of (∧R). In the former induction (III), in this distinguished case, the conclusionsof (MIX) (1)–(3) follow from the corresponding induction hypotheses of (II) for (MIX) (1)–(3),the induction hypotheses of (I) for (MIX) (1), for C and D and the rule (�C). In the latter, inthis distinguished case, the conclusions of (MIX) (2)–(3) follow from the induction hypothesisof (II) for (MIX) (1) and the corresponding induction hypotheses of (I) for (MIX) (2)–(3), forC or D, depending on the version of the rule (∼∧R).

• A=∼(C∧D).The proof differs from the case A=∼p only if X[∼(C∧D)]...[∼(C∧D)]⇒B is the conclusionof (∼∧L) or (∧R). Analogously to the case A=∼1, two inductions (III) are required: when therule (∼∧L) introduces one from the designated∼(C∧D) and in order to show the admissibilityof (MIX) (2)–(3) in the case of (∧R). In both these inductions, the most special cases are whenY⇒∼(C∧D) is the conclusion of (∼∧R). In the former induction (III), in this distinguishedcase, the conclusions of (MIX) (1)–(3) follow from the corresponding induction hypotheses of(II) for (MIX) (1)–(3) and the induction hypothesis of (I) for (MIX) (1), for C or D, dependingon the version of the rule (∼∧R). In the latter, in this distinguished case, the conclusions of(MIX) (2)–(3) follow from the induction hypothesis of (II) for (MIX) (1) and the correspondinginduction hypotheses of (I) for (MIX) (2)–(3), for C or D, again depending on the version of therule (∼∧R).

• A=C∨D.The proof is quite analogous to the case A=C∧D. Two similar inductions (III) are required whenthe rule (∨L) introduces one from the designated C∨D and in order to show the admissibilityof (MIX) (2)–(3) in the case of (∼∨R).

• A=∼(C∨D).The proof is quite analogous to the case A=∼(C∧D). Two similar inductions (III) are requiredwhen the rule (∼∨L) introduces one from the designated ∼(C∨D) and in order to show theadmissibility of (MIX) (2)–(3) in the case of (∨R).

• A=C→D.The proof is analogous to the case A=C∧D, as regards the need of two inductions (III). However,procedures for the most special cases of these inductions are slightly different. In the casewhen X[C→D]...[C→D]⇒B is the conclusion of (→L) and Y⇒C→D is the conclusion of(→R)†, the conclusions of (MIX) (1)–(3) follow from the corresponding induction hypotheses



of (II) for (MIX) (1)–(3) applied to the left premise of (→L), the induction hypothesis of (II)for (MIX) (1) applied to the right premise of (→L) and the induction hypotheses of (I) for(MIX) (1), for D and C, or ∼C and ∼D, depending on the version of the rule (→L). In the casewhen X[C→D]...[C→D]⇒∼(C→D) is the conclusion of (∼→R) and Y⇒C→D is theconclusion of (→R)†, the conclusion of (MIX) (2) follows from the induction hypothesis of (II)for (MIX) (1) applied to the both premises of (∼→R) and the induction hypotheses of (I), firstfor (MIX) (2) and D, and next for (MIX) (1) and C, whereas in order to get the conclusion of(MIX) (3) we should rather apply the induction hypotheses of (I) for (MIX) (3) and∼C, and for(MIX) (1) and ∼D.

• A=∼(C→D).The proof is analogous to the case A=∼(C∧D), however induction (III) in the case whenX[∼(C→D)]...[∼(C→D)]⇒C→D is the conclusion of (→R)† and Y⇒∼(C→D) is theconclusion of (∼→R) is slightly different. The conclusion of (MIX) (2), in this distinguishedcase, follows from the induction hypothesis of (II) for (MIX) (1) applied to the right premise of(→R)† and the induction hypotheses of (I), first for (MIX) (1) and ∼D, and next for (MIX) (2)and C. Whereas the conclusion of (MIX) (3) follows from the induction hypothesis of (II) for(MIX) (1) applied to the left premise of (→R)† and the induction hypotheses of (I), for (MIX)(1) and C, and for (MIX) (3) and ∼D.

• A=C←D.The proof is quite similar to the case A=C→D.

• A=∼(C←D).The proof is quite similar to the case A=∼(C→D).

• A=∼C.This case follows straightforwardly from the mutual induction (I) on ∼A.

• A=∼∼C.The proof differs from the case A=∼(C∧D) only in the cases when X[∼∼C]...[∼∼C]⇒B isthe conclusion of the rules introducing negations on the right side of consecution. For all theserules we need induction (III) if introduced formula ∼E=B in the consequent is identical with∼C. In all these inductions, the most special cases are when Y⇒∼∼C is the conclusion of(∼∼R). The conclusions of (MIX) (2)–(3) follow then from the induction hypothesis of (II) for(MIX) (1), the corresponding rule that introduced B and the corresponding induction hypothesesof (I) for (MIX) (2)–(3), for C.

6 Simplification

On first sight, the rules (→R)† and (←R)† are a little bit strange. It seems that premises are redundant.Indeed they are, as we will show using the completeness theorem. So, let us replace these two ruleswith the corresponding simpler rules (→R) and (←R) from FL:

(→R)A�X⇒B

X⇒A→B(←R)

X�A⇒B

X⇒B←A

We will denote the provability relation of this new system CyInDFL by . The following theoremjustifies naming this new system as the old one.



Theorem 5For any structure X , and any formula A there holds:

† X⇒A ⇔ X⇒A.

Proof. Let † X⇒A. From the Section 6, we have that there exists a cut-free proof of X⇒A. Weremove from this proof all branches leading down to consecutions which are the right premise of anyinstance of (→R)† or (←R)†. We then get a (simpler) proof of X⇒A in the new system CyInDFL.

Let X⇒A. Clearly X⇒A is valid in CyInDFL, since the rules (→R) and (←R) also preservethe validity. Therefore † X⇒A follows directly from Section 4. �

7 Decidability

For FL and its all basic variants, except these with contraction, it is enough to prove the cut eliminationtheorem to get the decidability [15, 21, 25]. The property of the finite proof-search tree of this calculusfollows immediately from the subformula property of all remaining rules: any premise contains onlysubformulas of formulas appearing in the conclusion. The contraction rule requires a special treatment— showing that any proof-search needs only to consider a bounded number of applications of thisrule. In the presence of exchange, it is possible by Kripke’s Lemma [2, 13, 20, 22, 27]. In the case oftwo structure constructors and contraction (with weakening) for only one of these constructors, suchrestriction can be obtained by the method of Giambrone [2, 13, 17, 27].6 We adapt this method in amanner similar to how Restall adapted it [26].

First, let us notice that all rules of the system CyInDFL have the following form of subformulaproperty: any premise contains only λ and subformulas and negations of subformulas of formulasappearing in the conclusion. Second, for all these rules the following complexity measure ofconsecutions decreases from the conclusion to any of the premise, or stay the same:

m(A)= the total number of occurrences of variables, constants and connectives in Am(λ)=0, m( )=1 (for the empty succedent)m(X�Y )=m(X)+m(Y ),m(X�Y )=max{m(X),m(Y )},m(X⇒A)=m(X)+m(A).

Thus, indeed, it suffices to show that any proof-search needs only to consider a bounded numberof applications of (�C) and (λC). In order to show this, let us first notice that for all structures X, Y ,Z , V and any formula A the following equivalence holds:

X[(Y �Z)�V ]⇒A ⇔ X[Y � (Z �V )]⇒A. (&)

This equivalence allows us to consider all structures that can be transformed into each other bymeans of the following two transformations

X[(Y �Z)�V ]X[Y � (Z �V )]

X[Y � (Z �V )]X[(Y �Z)�V ]

6The associativity of this constructor is also crucial, as well as the associativity of comma in the case of FLec.



as inferentially equivalent. As a consequence, we assume the convention of omitting bracketswhich separate constituents of such structures. Therefore, the rule (�A) is not needed anymore, sowe drop it from the system CyInDFL.7 We can now define the needed notions.

In any structure X , the substructures Y and V are said to be near if they occur in a substructure ofthe form Y �Z1�Z2� ...�Zk �V or V �Z1�Z2� ...�Zk �Y , for some structures Z1,Z2,...,Zk .8

In any structure X , λ is said to be superfluous if λ occurs in a substructure of the form λ�Y orY �λ, for some structure Y .

A structure X is said to be reduced if X does not contain any superfluous λ and X does not containany identical substructures near one another.

It is easy to see that for any structure X, there exists a reduced structure Xr , which is obtained fromX by the following term rewriting procedure:

• for any superfluous λ, delete it, together with the adjacent �,• for any substructure Y near one another Y , delete one, together with the adjacent �.

Moreover, it is easy to see that for every such obtained reduced structure Xr from X, there holds:

X⇒A ⇔ Xr⇒A.

It follows directly from the structural rules of the system CyInDFL. This fact allows one to searchfor a proof of any consecution Xr⇒A instead of X⇒A. It remains to show that this proof-search isalways finite. We need only next four lemmas.

Lemma 6For any consecution X⇒A, if m(X⇒A)=n, then there exist at most finitely many reduced structuresof complexity not greater than n, built from subformulas and negations of subformulas of formulasoccurring in X⇒A, and λ.

Proof. The proof is similar to the counterparts from [17, 19, 26] — it goes by induction on n. Thebase step is trivial, since there is only one reduced structure of complexity 0.

Let us assume by the inductive hypothesis that the thesis holds for all m<n. Then, it suffices toshow that there exist at most finitely many reduced structures of complexity equal to n, built fromsubformulas and negations of subformulas of formulas occurring in X⇒A, and λ. But any suchreduced structure is one of three forms:

• A formula.Clearly, there are only finitely many formulas of complexity n, built from subformulas andnegations of subformulas of formulas occurring in X⇒A.

• A structure of the form: Y1�Y2.From the definition of m, Y1 and Y2 are structures of complexity less than n (none of them can beλ, lest Y1�Y2 is not reduced). So, by the induction hypothesis, there are at most finitely manyreduced structures which might serve as left and right constituents. So, we can build from thesereduced structures at most finitely many reduced structures of this form of complexity n.

• A structure of the form: Y1�Y2� ...�Yk .

7Clearly, these two approaches of the system CyInDFL, with explicit (�A) and with (�A) in the metatheory, are decidablyequivalent.

8For k=0 these structures are near in the literal meaning of this word; for k≥1 they are near in the literal meaning, in atleast one of inferentially equivalent structure X — taking into account the rule (�E) this time.



According to the assumed convention followed from (&), each of these constituents Y1, Y2, ...,Yk must be either a formula, or λ, or of the form (V �W ) for some V and W , and must havecomplexity less than or equal to n (by the definition of m). So by the induction hypothesis andtwo reasonings in the above, there are at most finitely many reduced structures which might serveas such constituents. So, by the constraint saying that none of two identical substructures canbe near one another, we can build from these reduced structures at most finitely many reducedstructures of this form of complexity n. �

Like in [17, 26], we need the notion of semi-reduced structures and we can show the analogouslemma for structures of this kind.

A structure X is said to be semi-reduced if either X is reduced, or X has at most one superfluous λ,or X has at most two identical substructures near one another.

Lemma 7For any consecution X⇒A, if m(X⇒A)=n, then there exist at most finitely many semi-reducedstructures of complexity not greater than n, built from subformulas and negations of subformulas offormulas occurring in X⇒A, and λ.

Proof. By analogous induction on n. �As a corollary, we have that there exist at most finitely many consecutions with a semi-reduced

antecedent (semi-reduced consecutions for short), of complexity not greater than n, built fromsubformulas and negations of subformulas of formulas occurring in X⇒A, and λ. Finally, we canshow that proof-search within this finite set of consecutions is sufficient, where from we obtain thedecidability of CyInDFL.

However, we first need to prove some technical lemma about semi-reduced consecutions andsemi–reduced proofs (proofs which consist only of semi-reduced consecutions).

Lemma 8Let Xs

1, Xs2 be any semi-reduced structures of X and let A be any formula. Then, there exists a

semi-reduced proof of Xs2⇒A, which employs only structural rules and Xs

1⇒A as the only initialconsecution (shortly: a semi-reduced structural proof of Xs

2⇒A from Xs1⇒A).

Proof. We prove this lemma by induction on the build-up of the structure X. The base steps aretrivial, since for any atomic structure X (λ or a formula), the only semi-reduced structure of X is X.

The inductive steps are much more complicated; especially the second one.

• X=Y1�Y2.By the inductive hypothesis we have that the thesis holds for Y1 and Y2. We can divide allsemi-reduced structures of X for the following two groups:• Group I.

The group consisting of all structures of the forms Ys1 �Yr

2 and Yr1 �Ys

2, where Ysi is any

semi-reduced structure of Yi and Yri is any reduced structure of Yi, for i=1,2.

• Group II.The group consisting of all structures of the forms Ys

1 and Ys2, where Ys

i is any semi-reducedstructure of Yi, for i=1,2. This is the case when the reduced structure of the second constituentis λ, and this λ is deleted by the procedure as superfluous.

We should consider four cases, depending on the groups which the structures Xs1 and Xs

2 belong to.We work out only the proof of the case when Xs

1 and Xs2 belong to the Group I, since all others

are very similar (it can even be said that they are particular cases of this one).



Let Xs1=Y11 �Y21 and Xs

2=Y12 �Y22 . We restrict our attention to the case when Y11 and Y12

are reduced. The remaining cases are analogous.First, what we want to have is a semi-reduced structural proof of Y11 �Yr

21⇒A from Y11 �

Y21⇒A, where Yr21

is some reduced structure of Y21 , unless Y21 is already reduced. From theinduction hypothesis for Y2, we know that there exists a semi-reduced structural proof of Yr

21⇒A

from Y21⇒A. So, we repeat the same structural rules which are used in this proof and we obtainthe semi-reduced structural proof of Y11 �Yr

21⇒A from Y11 �Y21⇒A.

Next, we want to have a semi-reduced structural proof of Y12 �Yr21⇒A from Y11 �Yr

21⇒A.

From the induction hypothesis for Y1, we know that there exists a semi-reduced structural proofof Y12⇒A from Y11⇒A. So, again we repeat the same structural rules which are used in thisproof, and we obtain the semi–reduced structural proof of Y12 �Yr

21⇒A from Y11 �Yr

21⇒A. In

this case we must be careful when Yr21

is λ. If Yr21=λ, then we should apply (λC) at the beginning

and (λW ) at the end of this semi-reduced structural proof of Y12 �λ⇒A from Y11 �λ⇒A.Lastly, what we want to have is a semi-reduced structural proof of Y12 �Y22⇒A from

Y12 �Yr21⇒A. Again, we use the induction hypothesis for Y2, and by repeating the structural

rules from a semi-reduced structural proof of Y22⇒A from Yr21⇒A, we obtain the semi-reduced

structural proof of Y12 �Y22⇒A from Y12 �Yr21⇒A, as desired. This time we must be careful

when Y12 is λ. Analogously, the rules (λC) and (λW ) are essential in this case.

• X=Y1�Y2� ...�Yk .By the inductive hypothesis, we have that the thesis holds for Y1, Y2, ..., Yk . We can divide allsemi-reduced structures of X for the following similar two groups.• Group I.

The group consisting of all structures of the following forms:

Ys1 �Yr

2 � ...�Yrk ,

Yr1 �Ys

2 � ...�Yrk ,

...,

Yr1 �Yr

2 � ...�Ysk ,

where Ysi is any semi-reduced structure of Yi and Yr

i is any reduced structure of Yi, fori=1,2,...,k.

• Group II.The group consisting of all structures of the following forms:

Ysm1

�Yrm2

� ...�Yrml

,

Yrm1

�Ysm2

� ...�Yrml

,

...,

Yrm1

�Yrm2

� ...�Ysml

,

where m1,m2,...,ml∈{1,2,...,k} and mi �=mj for i �= j and Ysmi

is any semi-reduced structureof Ymi and Yr

miis any reduced structure of Ymi (i,j=1,2,...,l). This is the case when some

reduced or semi-reduced structures of the constituents Y1,Y2,...,Yk are identical, and they aredeleted by the procedure.

We should consider four cases, depending on the groups which the structures Xs1 and Xs

2 belong to.Similarly, we work out only the proof of the case when Xs

1 and Xs2 belong to the Group I, since

all others are very similar (also it can be said that they are particular cases of this one).



Let Xs1=Y11 �Y21 � ...�Yk1 and Xs

2=Y12 �Y22 � ...�Yk2 . We restrict our attention to thecase when Y11 ,Y21 ,...,Yk−11 and Y12 ,Y22 ,...,Yk−12 are reduced. We also assume that Yi1 �=Yj1and Yi2 �=Yj2 for i �= j (i,j=1,2,...,k−1). The remaining cases are analogous.

First, what we want to have is a semi-reduced structural proof of Y11 �Y21 � ...�Yrk1⇒

A from Y11 �Y21 � ...�Yk1⇒A, where Yrk1

is some reduced structure of Yk1 , unless Yk1 isalready reduced. From the induction hypothesis for Yk , we know that there exists a semi-reducedstructural proof of Yr

k1⇒A from Yk1⇒A. So, we repeat the same structural rules which are used

in this proof, and we obtain the semi-reduced structural proof of Y11 �Y21 � ...�Yrk1⇒A from

Y11 �Y21 � ...�Yk1⇒A. We expect Y11 �Y21 � ...�Yrk1⇒A to be a reduced consecution. If

not (since Yrk1

is identical to some of Y11 ,Y21 ,...,Yk−11 ), we can use (�C) (together with(�E), if necessary) to eliminate this identical structure and obtain the reduced consecutionYm11 �Ym21 � ...�Yml1 �Yr

k1⇒A, where m1,m2,...,ml∈{1,2,...,k−1} and mi �=mj for i �= j.

Using the induction hypotheses for Yml ,...,Ym2 ,Ym1 and reasoning as above (including theelimination of the identical structure after every usage of the induction hypothesis), we obtaina semi-reduced structural proof of Yn12 �Yn22 � ...�Ynh2 �Yr

k1⇒A from Ym11 �Ym21 � ...�

Yml1 �Yrk1⇒A, where n1,n2,...,nh∈{m1,m2,...,ml} and ni �=nj for i �= j.

Next, we can return to the constituent Yrk1

. We obtain a semi-reduced structural proof of Yn12 �

Yn22 � ...�Ynh2 �Yk2⇒A from Yn12 �Yn22 � ...�Ynh2 �Yrk1⇒A, by the induction hypothesis

for Yk and the same procedure.Finally, we can obtain a semi-reduced structural proof of Y12 �Y22 � ...�Yk−12 �Yk2⇒A

from Yn12 �Yn22 � ...�Ynh2 �Yk2⇒A, by applying the rule (�W ) as many times as necessary(possibly zero times), in order to add the missing reduced structures. �

With the help of this lemma, we can prove the main lemma of this section.

Lemma 9Any provable reduced consecution has a semi-reduced proof.

Proof. Like in [26], we take a proof and apply the reduction procedure to the antecedent of everynode of this proof. The result is almost a proof. Clearly all its leaves are axioms. All we need to do inorder to obtain any semi-reduced proof is to add some semi-reduced consecutions between obtainedpremises and conclusions. The procedure uses Lemma 8 in an essential way, and it is analogous forevery rule. So, we present only one case, the rest being treated in a similar manner (with three smallexceptions).

• X[∼(A∧B)]⇒C is obtained from X[∼A]⇒C and X[∼B]⇒C by applying (∼∧L).Let Xr[∼(A∧B)]⇒C, Xr[∼A]⇒C and Xr[∼B]⇒C be results of applying the reduction pro-cedure on X[∼(A∧B)]⇒C, X[∼A]⇒C and X[∼B]⇒C, respectively. Obtained consecutionsmay not match the schema of the rule (∼∧L), for two reasons. First, the reduction procedurecould delete some of the designated formulas∼(A∧B),∼A,∼B or some superstructures of them.Second, since the reduction procedure is ambiguous, it could delete corresponding substructuresin X[∼(A∧B)], X[∼A] and X[∼B] in a completely different way.

Therefore, let us impose certain restrictions on the reduction procedure, so that the rule (∼∧L)could be applied to:• treat the designated formula as unique in the entire structure,• treat the substructures from the antecedents of the premises in the same way as the

corresponding substructures from the antecedent of the conclusion.



ExampleLet the input instance of the rule (∼∧L) be as follows:

((∼A�∼B�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

((∼A�∼A�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

((∼A�∼(A∧B)�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

��

In order to treat the designated formulas as unique, we mark them with the sign ′. So we getthe scheme as follows:

((∼A� (∼B)′�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

((∼A� (∼A)′�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

((∼A� (∼(A∧B))′�D�∼A)�E)� ((∼A�∼(A∧B)�D�∼A)�E)⇒C

��

After applying the reduction procedure to the antecedent of the conclusion (and removingsign ′), we get:

((∼(A∧B)�D�∼A)�E)� ((∼(A∧B)�D�∼A)�E)⇒C.

Treating in the same way substructures from the antecedents of the premises, we finally get:

for the left premise: ((∼A�D�∼A)�E)� ((∼(A∧B)�D�∼A)�E)⇒C,

for the right premise: ((∼B�D�∼A)�E)� ((∼(A∧B)�D�∼A)�E)⇒C.

Let Xs[∼(A∧B)]⇒C, Xs[∼A]⇒C and Xs[∼B]⇒C be results of applying the reductionprocedure with the above restrictions.

It is easy to see that such restrictions give not only consecutions matching the scheme of therule (∼∧L) but also consecutions which are semi-reduced. So, in order to finish this case, weonly need to use Lemma 8 and add some semi-reduced structural proofs of Xs[∼A]⇒C fromXr[∼A]⇒C, Xs[∼B]⇒C from Xr[∼B]⇒C and Xr[∼(A∧B)]⇒C from Xs[∼(A∧B)]⇒Cbetween the premises Xr[∼A]⇒C, Xr[∼B]⇒C and the conclusion Xr[∼(A∧B)]⇒C.

• Exceptions.For the structural rules (�W ) and (�E), the first restriction (treating the designated structuresas unique in the entire structure) can lead to structures which are not semi-reduced — in caseswhen both designated structures Y and Z are near other instances of Y and Z . In such cases, itis very important to delete all these identical instances from the antecedents of the premise andthe conclusion, first. The same applies to the rule (�C), if there are other instances of Y nearthe designated structures Y .

Similar situation occurs for the rule (∧L), when both designated A and B are near otherinstances of A and B. We should delete all these identical instances of A and B from the antecedentof the premise in order to get a semi-reduced structure after applying the reduction procedure. Weshould also delete all corresponding instances of A and B from the antecedent of the conclusionin order to get a consecution matching the conclusion of the rule (∼∧L). This deletion of A andB from the antecedent of the conclusion causes that the obtained consecution after applying thereduction procedure is not a semi-reduced consecution of X[A∧B]⇒C. Fortunately, we canrestore the missing A and B by the rule (�W ).



Example 2Let the input instance of the rule (∧L) be as follows:

((B�A�D�A∧B)�E)� ((B�A�D�A�B)�E)� ((D�A�B)�E)⇒C

((B�A�D�A∧B)�E)� ((B�A�D�A∧B)�E)� ((D�A�B)�E)⇒C

After the deletion of all instances of A and B which are near the designated A�B in theantecedent of the premise, and all corresponding instances of A and B in the antecedent of theconclusion, we get:

((B�A�D�A∧B)�E)� ((D�A�B)�E)� ((D�A�B)�E)⇒C

((B�A�D�A∧B)�E)� ((D�A∧B)�E)� ((D�A�B)�E)⇒C

The reduction procedure with the restrictions changes nothing in this case. Finally, by restoringthe missing A and B, we get the following semi-reduced form of the conclusion:

((B�A�D�A∧B)�E)� ((B�A�D�A∧B)�E)� ((D�A�B)�E)⇒C

The last exception is very simple: do not delete the designated λ in the premise of the rule (1L).

�As a corollary we get the following result.

Theorem 10CyInDFL is decidable.

8 Variants

The method of this article can be extended to several variants of CyInDFL and, by analogy, can bealso applied to the corresponding variants of DFL (these which have the finite model property [19]).

The first example is CyInDFL⊥ which is complete with respect to the class of Bounded CyInDFL,i.e. the class that satisfies identities:⊥≤x and x≤�, for all elements x. Since�=∼⊥, CyInDFL⊥can be defined as CyInDFL with one constant ⊥ and two new axioms:

(⊥L) X[⊥]⇒A,

(∼⊥R) X⇒∼⊥.

It is not difficult to extend the cut elimination theorem with these two axioms as well as the decisionprocedure.

It is also not difficult to extend them if we add exchange to CyInDFL or CyInDFL⊥:

(�E)X[Y �Z]⇒A

X[Z �Y ]⇒A.

It is also possible to add (independently to the previous extensions) weakening (�W ) to CyInDFL.But, in order to get the system complete with respect to the class of Integral CyInDRL, i.e. the class



in which 1 is the upper bound, we must add right weakening (�O) as well:

(�W )X[λ]⇒A

X[Y ]⇒A,

(�O)X⇒X⇒A

.

In any of above systems it is also possible to omit the rule (�A). This rule is not significant inthe proof of the decidability of CyInDFL, so the non-associative variants of CyInDFL are alsodecidable.

As we mention at the beginning, by omitting the connective ∼ we get the system DFL. Clearly,the decidability of corresponding variants of DFL can be obtained in an analogous way.

In the end, let us notice that we can also omit the constructor � from CyInDFL and replace therule (∧L) with the two corresponding rules from FL:

X[A]⇒C

X[A∧B]⇒C

X[B]⇒C

X[A∧B]⇒C

Then we get the decidable system CyInFL with ‘intuitionistic’ sequents.9

Acknowledgments

This research was supported with funds granted for research projects by the Polish Ministry of Scienceand Higher Education in the year 2009. Thanks are also due to Professors Wojciech Buszkowski andKazimierz Swirydowicz for their valuable comments helping me to improve this article.

References[1] A. R. Anderson and N. D. Belnap. Entailment: The Logic of Relevance and Necessity, vol. I.

Princeton University Press, 1975.[2] A. R. Anderson, N. D. Belnap, and J. M. Dunn. Entailment: The Logic of Relevance and

Necessity, vol. II. Princeton University Press, 1992.[3] F. Belardinelli, P. Jipsen, and H. Ono. Algebraic aspects of cut elimination. Studia Logica, 77,

209–240, 2004.[4] N. D. Belnap. Display logic. Journal of Philosophical Logic, 11, 375–417, 1982.[5] W. J. Blok and D. Pigozzi.Algebraizable logics. Memoirs of the American Mathematical Society,

77, 1989.[6] R. T. Brady. Gentzenization and decidability of some contraction-less relevant logics. Journal

of Philosophical Logic, 20, 97–117, 1991.[7] R. T. Brady. The Gentzenization and decidability of RW. Journal of Philosophical Logic, 19,

35–73, 1990.

9The Gentzen-style calculi complete with respect to CyInFL developed by Yetter [30] and Wille [29] are one-sided sequentsystems. They have also counterparts built from two-sided ‘classical’ sequents, similar to the cyclic variant of the systemInFL introduced by Galatos and Jipsen [14].



[8] R. A. Bull. Survey of generalizations of Urquhart semantics. Notre Dame Journal of FormalLogic, 28, 220–237, 1987.

[9] W. Buszkowski. Interpolation and FEP for logics of residuated algebras. Logic Journal of theIGPL, Forthcoming.

[10] W. Buszkowski and M. Farulewski. Nonassociative Lambek calculus with additives and context-free languages. In Languages: From Formal to Natural, Vol. 5533 of Lecture Notes in ComputerScience, O. Grumberg, M. Kaminski, S. Katz, and S. Wintner, eds, pp. 45–58. Springer, 2009.

[11] J. M. Dunn. The Algebra of Intensional Logics. PhD Thesis, University of Pittsburgh,1966.

[12] J. M. Dunn. A Gentzen system for positive relevant implication. Journal of Symbolic Logic, 38,356–357, 1973.

[13] J. M. Dunn and G. Restall. Relevance logic. Handbook of Philosophical Logic, vol. 6, 2nd edn.,D. M. Gabbay and F. Guenthner, eds, pp. 1–128. Springer, 2002.

[14] N. Galatos and P. Jipsen. Residuated frames with applications to decidability. Transactions ofthe American Mathematical Society, Forthcoming.

[15] N. Galatos, P. Jipsen, T. Kowalski, and H. Ono. Residuated Lattices: An Algebraic Glimpseat Substructural Logics, Vol. 151 of Studies in Logic and the Foundations of Mathematics.Elsevier, 2007.

[16] N. Galatos and J. G. Raftery. Adding involution to residuated structures. Studia Logica, 77,181–207, 2004.

[17] S. Giambrone. TW+ and RW+ are decidable. Journal of Philosophical Logic, 14, 235–254,1985.

[18] P. Jipsen and C. Tsinakis. A survey of residuated lattices. In Ordered Algebraic Structures,J. Martinez, ed., pp. 19–56. Kluwer Academic Publishers, 2002.

[19] M. Kozak. Distributive full Lambek calculus has the finite model property. Studia Logica, 91,201–216, 2009.

[20] S. A. Kripke. The problem of entailment. Journal of Symbolic Logic, 24, 324, 1959.[21] J. Lambek. The mathematics of sentence structure. American Mathematical Monthly, 65,

154–170, 1958.[22] R. K. Meyer. Topics in Modal and Many-valued Logic. PhD Thesis, University of Pittsburgh,

1966.[23] G. Mints. Cut elimination theorem for relevant logics. Journal of Mathematical Sciences, 6,

422–428, 1976. Translated from Issledovanija po konstructivnoj mathematike i matematiceskojlogike V, Izdatelstvo Nauka, 1972.

[24] M. Okada and K. Terui. The finite model property for various fragments of intuitionistic linearlogic. Journal of Symbolic Logic, 64, 790–802, 1999.

[25] H. Ono. Decidability and finite model property of substructural logics. In the Tbilisi Symposiumon Logic, Language and Computation (Studies in Logic, Language and Information),J. Ginzburg, Z. Khasidashvili, C. Vogel, J.–J. Lévy, and E. Vallduví, eds, pp. 263–274. CSLIPublications, 1998.

[26] G. Restall. Displaying and deciding substructural logics 1: logics with contraposition. Journalof Philosophical Logic, 27, 179–216, 1998.

[27] G. Restall. Relevant and substructural logic. In Logic and the Modalities in the TwentiethCentury, D. M. Gabbay and J. Woods, eds, Vol. 7 of Handbook of the History of Logic, pp.289–398. Elsevier, 2006.

[28] A. Urquhart. The undecidability of entailment and relevant implication. Journal of SymbolicLogic, 49, 1059–1073, 1984.



[29] A. M. Wille. A Gentzen system for involutive residuated lattices. Algebra Universalis, 54,449–463, 2005.

[30] D. N. Yetter. Quantales and (noncommutative) linear logic. Journal of Symbolic Logic, 55,41–64, 1990.

[31] I. D. Zaslavsky. Symmetric Constructive Logic. Publishing House of Academy of Sciences ofArmenia SSR, 1978, (in Russian).

Received June 29, 2009

cyclic involutive fl

Documents