Language Models for Some Extensions of the Lambek Calculus
aa r X i v : . [ m a t h . L O ] J u l Language Models for Some Extensions of the LambekCalculus
Max Kanovich a,d , Stepan Kuznetsov b,d , Andre Scedrov c,d a University College London b Steklov Mathematical Institute of RAS c University of Pennsylvania d National Research University Higher School of Economics
Abstract
We investigate language interpretations of two extensions of the Lambek calcu-lus: with additive conjunction and disjunction and with additive conjunctionand the unit constant. For extensions with additive connectives, we show thatconjunction and disjunction behave differently. Adding both of them leads toincompleteness due to the distributivity law. We show that with conjunctiononly no issues with distributivity arise. In contrast, there exists a corollary ofthe distributivity law in the language with disjunction only which is not deriv-able in the non-distributive system. Moreover, this difference keeps valid forsystems with permutation and/or weakening structural rules, that is, intuition-istic linear and affine logics and affine multiplicative-additive Lambek calculus.For the extension of the Lambek with the unit constant, we present a calculuswhich reflects natural algebraic properties of the empty word. We do not claimcompleteness for this calculus, but we prove undecidability for the whole rangeof systems extending this minimal calculus and sound w.r.t. language models.As a corollary, we show that in the language with the unit there exissts a se-quent that is true if all variables are interpreted by regular language, but nottrue in language models in general.
Keywords:
Lambek calculus, language models, relational models, distributivelaw, incompleteness, undecidability
Preprint submitted to Information and Computation August 4, 2020 . Introduction
The Lambek calculus was introduced by Joachim Lambek [1] for mathemat-ical modelling of natural language syntax. This suggests the natural interpre-tation of the Lambek calculus as the algebraic logic of operations on formallanguages. Such interpretations of the Lambek calculus are called languagemodels, or L-models for short.The Lambek calculus, as originally formulated by Lambek, includes threeoperations: · (product), \ (left division), and / (right division). A distinctivefeature of the Lambek calculus is the so-called Lambek’s non-emptiness restric-tion.
In terms of L-models, this means that the empty word is disallowed, andwe consider, for a given alphabet Σ, subsets of Σ + . Lambek operations onlanguages are defined as follows: A · B = { uv | u ∈ A, v ∈ B } ,A \ B = { u ∈ Σ + | ( ∀ v ∈ A ) vu ∈ B } ,B / A = { u ∈ Σ + | ( ∀ v ∈ A ) uv ∈ B } . The division operations, \ and / , are indeed residuals of the product w.r.t.the subset relation: B ⊆ A \ C ⇐⇒ A · B ⊆ C ⇐⇒ A ⊆ C / B.
These equivalences form the core of the Lambek calculus. Along with tran-sitivity ( A ⊆ B ⊆ C ⇒ A ⊆ C ), reflexivity ( A ⊆ A ), and associativity( A · ( B · C ) = ( A · B ) · C ), they form a complete axiomatization of all gen-erally true atomic statements about Lambek operations on formal languages.This axiomatization is the Lambek calculus in its non-sequential form.The sequential formulation of the Lambek calculus [1] is as follows. Formulaeare constructed from variables ( p, q, r, . . . ) using three connectives: · , \ , / . (Weuse capital Latin letters both for languages and for Lambek formulae.) Sequentsare expressions of the form Γ ⊢ C , where the antecedent Γ is a sequence offormulae and the succedent C is one formula (intuitionistic style). The calculus L includes axioms of the form A ⊢ A and the following rules of inference:2 ⊢ A Γ , B, ∆ ⊢ C Γ , Π , A \ B, ∆ ⊢ C \ L A, Π ⊢ B Π ⊢ A \ B \ R ; Π is non-emptyΠ ⊢ A Γ , B, ∆ ⊢ C Γ , B / A, Π , ∆ ⊢ C / L Π , A ⊢ B Π ⊢ B / A / R ; Π is non-emptyΓ , A, B, ∆ ⊢ C Γ , A · B, ∆ ⊢ C · L Γ ⊢ A ∆ ⊢ B Γ , ∆ ⊢ A · B · R Π ⊢ A Γ , A, ∆ ⊢ C Γ , Π , ∆ ⊢ C Cut
The cut rule is eliminable [1].An L-model, formally, is a mapping w of Lambek formulae to subsets of Σ + (languages without the empty word), which commutes with Lambek operations: w ( A · B ) = w ( A ) · w ( B ), w ( A \ B ) = w ( A ) \ w ( B ), and w ( B / A ) = w ( B ) / w ( A ).A sequent A , . . . , A n ⊢ B is true in this model, if w ( A ) · . . . · w ( A n ) ⊆ w ( B ).According to Lambek’s non-emptiness restriction, all sequents in derivationsare required to have non-empty antecedents. This constraint is motivated bylinguistic applications: without it, Lambek categorial grammars generate un-grammatical sentences [2, § \ R and / R —yields the Lambek calculus allowing empty antecedents, denoted by L ∗ [3]. Language models are easily adapted for the case of L ∗ :now we consider languages, which are subsets of Σ ∗ (that is, they are allowed toinclude the empty word ε ). The definition of division operations is also modified:for models of L ∗ , A \ B = { u ∈ Σ ∗ | ( ∀ v ∈ A ) vu ∈ B } ,B / A = { u ∈ Σ ∗ | ( ∀ v ∈ A ) uv ∈ B } . This modification can alter the values of A \ B and B / A even if A and B donot contain the empty word. For example, A \ A now always includes ε , andtherefore ( A \ A ) \ B is always a subset of B . Hence, L ∗ is not a conservativeextension of L : the sequent ( p \ p ) \ q ⊢ q has a non-empty antecedent, but isderivable only in L ∗ , not in L . For these modified L-models, let us use the term L ε -models.
3n an L ε -model w , a sequent of the form A , . . . , A n ⊢ B is true if w ( A ) · . . . · w ( A n ) ⊆ w ( B ), and a sequent of the form ⊢ B , with an empty antecedent,is true if ε ∈ w ( B ).Completeness theorems for L and L ∗ w.r.t. corresponding versions of L-models were proved by Pentus [4, 5]. Pentus’ proofs are highly non-trivial.If one considers the fragment without · (the product-free fragment), however,proving L-completeness becomes much easier. This was done by Buszkowski [6];Buszkowski’s proof applies both to L and L ∗ , w.r.t. L-models and L ε -models,respectively.Besides product and two divisions, natural operations on formal languagesinclude set-theoretic intersection and union. These operations correspond toso-called additive conjunction and disjunction. Additive operations are usuallyaxiomatized by the following inference rules (cf. [7]):Γ , A, ∆ ⊢ C Γ , B, ∆ ⊢ C Γ , A ∨ B, ∆ ⊢ C ∨ L Π ⊢ A Π ⊢ A ∨ B ∨ R l Π ⊢ B Π ⊢ A ∨ B ∨ R r Γ , A, ∆ ⊢ C Γ , A ∧ B, ∆ ⊢ C ∧ L l Γ , B, ∆ ⊢ C Γ , A ∧ B, ∆ ⊢ C ∧ L r Π ⊢ A Π ⊢ B Π ⊢ A ∧ B ∧ R The Lambek calculus L extended with these rules is denoted by MALC ( multi-plicative-additive Lambek calculus ); MALC ∗ is the variant of MALC withoutLambek’s restriction (that is, allowing empty antecedents). L-completeness,however, fails for
MALC in general. Further, in Section 2, we discuss this issuein detail.Following Abrusci [8], we put the Lambek calculus into a broader contextof linear logic. Namely,
MALC ∗ can be viewed as a fragment of intuitionistic non-commutative linear logic. (This fragment includes multiplicative and ad-ditive operations, but lacks the exponential and constants.) We also considercommutative systems: intuitionistic linear logic ILL and intuitionistic affinelogic
IAL .Calculi
ILL and
IAL can be obtained from
MALC ∗ by adding structuralrules: permutation for ILL and permutation and weakening for
IAL . In thelanguage of
MALC , the rules of permutation and weakening are formulated as4ollows: Γ , B, A, ∆ ⊢ C Γ , A, B, ∆ ⊢ C P Γ , ∆ ⊢ C Γ , A, ∆ ⊢ C W
Adding only weakening yields non-commutative intuitionistic affine logic, oraffine (monotone) multiplicative-additive Lambek calculus. We denote this sys-tem by
AMALC ∗ (in the presence of extra structural rules, we do not imposeLambek’s restriction).We shall also use alternative calculi for the commutative systems ILL and
IAL , in which structural rules are hidden in axioms and in the format of se-quents. First, we change the language of formulae, introducing one connective A ⊸ B instead of A \ B and B / A (these are equivalent in
ILL and
IAL ). Wealso write A ⊗ B instead of A · B , following Girard’s [9] linear logic notations.Sequents are now going to be expressions of the form Γ ⊢ C , where Γ is a multiset of formulae. Further Γ , A means Γ ⊎ { A } , and Γ , Π means Γ ⊎ Π, where ⊎ is multiset union.Axioms are of the form p ⊢ p , for each variable p , in the case of ILL , and ofthe form Γ , p ⊢ p for IAL . Inference rules for both systems are as follows:Π ⊢ A Γ , B ⊢ C Γ , Π , A ⊸ B ⊢ C ⊸ L Π , A ⊢ B Π ⊢ A ⊸ B ⊸ R Γ , A, B ⊢ C Γ , A ⊗ B ⊢ C ⊗ L Γ ⊢ A ∆ ⊢ B Γ , ∆ ⊢ A ⊗ B ⊗ R Γ , A ⊢ C Γ , B ⊢ C Γ , A ∨ B ⊢ C ∨ L Π ⊢ A Π ⊢ A ∨ B ∨ R l Π ⊢ B Π ⊢ A ∨ B ∨ R r Γ , A ⊢ C Γ , A ∨ B ⊢ C ∧ L l Γ , B ⊢ C Γ , A ∨ B ⊢ C ∧ L r Π ∧ A Π ∧ B Π ⊢ A ∧ B ∧ R For
IAL , the weakening rule is not officially included in the system, but isadmissible: Γ ⊢ C Γ , A ⊢ C W (it is hidden in axioms).The cut rule of the following form is admissible both in
ILL and
IAL :Π ⊢ A Γ , A ⊢ C Γ , Π ⊢ C Cut multiplicative unit constant, . The unit con-stant is added to systems without Lambek’s restriction extending L ∗ ( i.e., L ∗ itself, MALC ∗ , AMALC ∗ , ILL , IAL ). The
Lambek calculus with the unit, L [10], is obtained from L ∗ by adding one axiom, ⊢ (its antecedent is empty),and one inference rule, Γ , ∆ ⊢ C Γ , , ∆ ⊢ C L L-completeness, however, does not hold for L . Indeed, since should be theunit w.r.t. · , that is A · = A = · A for any A , in L ε -models it should be inter-preted as { ε } . The following sequent is a counter-example for L-completeness: / p, / p ⊢ / p . This sequent is true in all models for any interpretation of p ,but is not derivable in L .Throughout this paper, we shall frequently consider fragments of the calculidefined above in languages with restricted sets of connectives. Such a fragmentwill be denoted by the name of the calculus, followed by the list of connectivesin parentheses: e.g., MALC ( \ , /, ∧ ).
2. Distributivity Law in Fragments with One Additive
It is well known, that
MALC is incomplete w.r.t. L-models. The reason isthe distributivity principle,( A ∨ C ) ∧ ( B ∨ C ) ⊢ ( A ∧ B ) ∨ C. ( D )On one hand, this principle is obviously true in all L-models. On the other hand,as noticed by Ono and Komori [11], one needs the structural rules of contractionand weakening to derive it. In particular, the distributivity principle is notderivable in MALC , MALC ∗ , AMALC ∗ , ILL , and
ILL .The distributivity principle, as formulated above, includes both additive con-nectives, ∧ and ∨ . We investigate fragments of MALC with only one additive, ∧ or ∨ . The result of our study is that with respect to distributivity ∧ and ∨ behave in opposite ways. 6et MALC + D denote MALC with the distributivity principle added asan extra axiom scheme. In the presence of this axiom scheme, we have to keepcut as an official rule of the system (it is now not eliminable). A hypersequentialsystem for
MALC + D was developed by Kozak [12].Let us restrict ourselves to the product-free language (with product, provingL-completeness is hard even without extra connections [4, 5]). We also con-sider calculi without the unit constant: issues connected with are discussedin Section 3. Thus, we consider two fragments of the multiplicative-additiveLambek calculus: MALC ( \ , /, ∧ ) and MALC ( \ , /, ∨ ), and the correspondingfragments of bigger system up to IAL . (For commutative calculi, we have onlyone implication, that is, consider fragments in the language of ⊸ , ∧ and ⊸ , ∨ .)As shown by Buszkowski [6], MALC ( \ , /, ∧ ) is complete w.r.t. L-models.This yields the following corollary: MALC ( \ , /, ∧ ) is a conservative fragmentof both MALC and
MALC + D . Indeed, any sequent provable in MALC + D is true in all L-models; if it is in the language of \ , /, ∧ , it is derivable in MALC ( \ , /, ∧ ) by L-completeness. In other words, the distributivity principlehas no non-trivial corollaries in the language of \ , /, ∧ .The situation with MALC ( \ , /, ∨ ) is opposite. Namely, we present a corol-lary of the distributivity principle in the language of \ , /, ∨ , which is not prov-able in MALC ( \ , /, ∨ ). Thus, MALC ( \ , /, ∨ ) is not a conservative fragmentof MALC + D , and is therefore incomplete w.r.t. L-models. Moreover, we showthat this effect is of a more general nature. Namely, the same holds for the cor-responding fragments of MALC ∗ , AMALC ∗ , ILL , and
IAL : distributivityhas no new corollaries in the language with ∧ , but has such in the languagewith ∨ . For the first series of results, concerning ∧ , we give a semantic proof. Foreach system, we consider a specific version of L-semantics. For MALC ( \ , /, ∧ )and MALC ∗ ( \ , /, ∧ ), these are L-models and L ε -models respectively. For othersystems, let us first give some definitions and prove correctness statements for7hem. Definition 1.
A language A is called monotone, if for any word u u ∈ A andan arbitrary word w the word u wu also belongs to A . Proposition 1. If A and B are both monotone, then so are A \ B , B / A , and A ∧ B .Proof. Let u = u u ∈ A \ B . Then for any v ∈ A we have vu u ∈ B . Now take u ′ = u wu for an arbitrary w . By monotonicity of B , the word vu ′ = vu wu is also in B . Since this holds for any v ∈ A , we get u ′ ∈ A \ B . The reasoningfor B / A is symmetric. The case of A ∧ B is trivial. Definition 2.
A language A is called commutative, if for any word u = a . . . a n belonging to A and an arbitrary transposition σ ∈ S n on { , . . . , n } the word a σ (1) . . . a σ ( n ) also belongs to A .Commutative languages are in one-to-one correspondence with multisets ofletters from Σ. Thus, we can define the operation of multiset union, A ⊎ B , fortwo commutative languages A and B , which can be expressed as follows: A ⊎ B = { a σ (1) . . . a σ ( n ) | σ ∈ S n and a . . . a n ∈ A · B } . If A is a commutative language, then vu ∈ A if and only if uv ∈ A . Therefore,for commutative A and B , we have A \ B = B / A ; we denote this language by A ⊸ B . Proposition 2. If A and B are commutative, then so is A ⊸ B and A ∧ B .Proof. Commutativity of A ∧ B is obvious. For A ⊸ B , take any u = a . . . a n ∈ A ⊸ B = B / A and let u ′ = u σ (1) . . . u σ ( n ) . Now for any v = a n +1 . . . a m ∈ A .By definiton of B / A , we have uv ∈ B . Now by commutativity of B , theword u ′ v also belongs to B . Indeed, it is obtained from uv by the followingtransposition: ˜ σ = . . . n n + 1 . . . mσ (1) σ (2) . . . σ ( n ) n + 1 . . . m . Since v ∈ A was taken arbitrarily, we conclude that u ′ ∈ B / A = A ⊸ B .8aving the class of monotone languages and the class of commutative lan-guages closed under our operations ( \ , / , ∧ ), we can define the classes of re-stricted L ε -models for all our systems. Definition 3.
An L ε -model is monotone, if all languages in it are monotone.Truth of sequents is defined as in ordinary L ε -models. Definition 4.
A commutative L ε -model is an L ε -model, where all languagesare commutative.In commutative models ⊎ actually plays the role of product (while we donot have product as a connective, we still have the sequential comma, which isa hidden product), due to the following fact. Proposition 3.
In a commutative L ε -model w , a sequent A , . . . , A n ⊢ B istrue if and only if w ( A ) ⊎ . . . ⊎ w ( A n ) ⊆ w ( B ) .Proof. The “if” part is due to the fact that w ( A ) · . . . · w ( A n ) ⊆ w ( A ) ⊎ . . . ⊎ w ( A n ). The “only if” part holds since w ( B ) is closed under transpositions.Now we prove an extension of Buszkowski’s completeness result Theorem 4.
Each of
MALC ( \ , /, ∧ ) , MALC ∗ ( \ , /, ∧ ) , AMALC ∗ ( \ , /, ∧ ) , ILL ( ⊸ , ∧ ) , IAL ( ⊸ , ∧ ) is sound and complete w.r.t. the corresponding class ofmodels, according to the following table:Calculus Models MALC ( \ , /, ∧ ) L-models MALC ∗ ( \ , /, ∧ ) L ε -models AMALC ∗ ( \ , /, ∧ ) monotone L ε -models ILL ( ⊸ , ∧ ) commutative L ε -models IAL ( ⊸ , ∧ ) L ε -models, which are both monotone and commutative Proof.
The cases of
MALC ( \ , /, ∧ ) and MALC ∗ ( \ , /, ∧ ) are due to Buszkow-ski [6]. Let us consider the remaining three systems.9he soundness part is easy: our conditions on models were specifically de-signed to reflect structural rules. In a monotone model, if w ( A ) · . . . · w ( A k ) · w ( A k +1 ) · . . . · w ( A n ) ⊆ w ( B ), then also w ( A ) · . . . · w ( A k ) · w ( A ) · w ( A k +1 ) · . . . · w ( A n ) ⊆ w ( B ), thus the weakening rule is valid. If we have a commutative L ε -model, then the permutation rule is valid. This is obvious from Proposition 3:unlike · , ⊎ is just commutative. All other rules and axioms are valid in arbitraryL ε -models.Completeness is proved by Buszkowski’s canonical model argument. We doit uniformly for all systems. In the canonical model, the alphabet Σ is the setof all formulae of the given calculus, and for any formula A let w ( A ) = { Γ | Γ ⊢ A is derivable in the given system } . First we show that w is indeed an L ε -model: w ( A \ B ) = w ( A ) \ w ( B ); w ( B / A ) = w ( B ) / w ( A ); w ( A ∧ B ) = w ( A ) ∧ w ( B ) . This is performed exactly as in Buszkowski’s proof. Indeed, if Γ ∈ w ( A \ B ),then for an arbitrary ∆ ∈ w ( A ) we have Γ ⊢ A \ B and ∆ ⊢ A . Applying cutwith A, A \ B ⊢ B , we get ∆ , Γ ⊢ A derivable in our system. Thus, ∆Γ ∈ w ( B ),therefore Γ ∈ w ( A ) \ w ( B ). Notice that cut is available in all systems weconsider. Dually, if Γ ∈ w ( A ) \ w ( B ), then, since A ∈ w ( A ) by the axiom, A Γ ∈ w ( B ). This means derivability A, Γ ⊢ B , thus Γ ⊢ A \ B . Hence,Γ ∈ w ( A \ B ).The / case is symmetric. For ∧ , we use the equivalence Γ ⊢ A ∧ B if andonly if Γ ⊢ A and Γ ⊢ B . Here the “if” part is an application of ∧ R , and the“only if” part is by cut with A ∧ B ⊢ A and A ∧ B ⊢ B .Next, is easy to see that the canonical model w belongs to the correspondingclass of models: monotone for AMALC ∗ ( \ , /, ∧ ), commutative for ILL ( \ , /, ∧ ),commutative and monotone for IAL ( \ , /, ∧ ).10inally, suppose a sequent Π ⊢ B is not derivable. Consider two cases. IfΠ = A , . . . , A n is non-empty, then, since each A i belongs to w ( A i ), we haveΓ ∈ w ( A ) · . . . · w ( A n ). On the other hand, Γ / ∈ w ( B ). This falsifies Π ⊢ B under interpretation w . If Π is empty, then we have ε / ∈ w ( B ), which againfalsifies Π ⊢ B . This finishes the completeness proof.It is easy to see that soundness actually extends to the language with ∨ (interpreted as set-theoretic union). Unions of monotone languages are alsomonotone, the same for commutative languages. The situation with productis a bit more complicated for commutative systems, since A · B is usually notcommutative, even for commutative A and B . Thus, we have to alter thedefinition of language models in the commutative case, requiring w ( A · B ) = w ( A ) ⊎ w ( B ) instead of w ( A · B ) = w ( A ) · w ( B ). Under this modification,soundness holds for product also. Finally, notice that in all models we consider ∨ and ∧ are interpreted set-theoretically, thus, obey the distributivity law. Theseconsiderations yield the following soundness result: Proposition 5.
Each of
MALC + D , MALC ∗ + D , AMALC ∗ + D , ILL + D , IAL + D is sound w.r.t. the corresponding class of models, according to the tablein Theorem 4; for ILL and
IAL in the models we use ⊎ to interpret · . Now we are ready to state and prove our conservativity result.
Theorem 6.
The systems in the restricted language without ∨ , MALC ( \ , /, ∧ ) , MALC ∗ ( \ , /, ∧ ) , AMALC ∗ ( \ , /, ∧ ) , ILL ( ⊸ , ∧ ) , and IAL ( ⊸ , ∧ ) are conser-vative fragments of MALC + D , MALC ∗ + D , AMALC ∗ + D , ILL + D , and IAL + D respectively.Proof. Let Π ⊢ B be a sequent in the language of \ , /, ∧ (in the commuta-tive case, ⊸ , ∧ ). Suppose it is derivable in one of the distributive systems, MALC + D , . . . , IAL + D . Then by Proposition 5 it is true in all mod-els of the corresponding class. By Theorem 4 it is derivable in, respectively, MALC ( \ , /, ∧ ), . . . , IAL ( ⊸ , ∧ ). 11 .2. Incompleteness with Additive Disjunction Only If we take ∨ instead of ∧ , however, no analog of the conservativity result likeTheorem 6 is possible, due to the following counter-example. Theorem 7.
The sequent (( x / y ) ∨ w ) / (( x / y ) ∨ ( x / z ) ∨ w ) , ( x / y ) ∨ w, (( x / y ) ∨ w ) \ (( x / z ) ∨ w ) ⊢ ( x / ( y ∨ z )) ∨ w is derivable in MALC + D but this sequent is not derivable in IAL . This sequent is in the language of \ , /, ∨ . The theorem states that it isderivable in MALC + D , and therefore in all its extensions up to IAL + D ,but not in the corresponding ( \ , /, ∨ ) fragments without the distributivity lawadded. Thus, this is a non-trivial corollary of D in the language without ∧ .In particular, Theorem 7 implies that MALC ( \ , /, ∨ ) is incomplete w.r.t. L-models, as well as MALC ∗ ( \ , /, ∨ ), AMALC ∗ ( \ , /, ∨ ), ILL ( ⊸ , ∨ ), IAL ( ⊸ , ∨ ) are incomplete w.r.t. the corresponding modifications of L-models (comparewith Theorem 4).Before proving Theorem 7, let us make some remarks. First, let us noticethat the sequent in this theorem is slightly different from the one in our WoLLIC2019 paper [13], where one variable is used for x and w . The reason is that theold example happens to be derivable in IAL (but still not in
ILL and weakersystems).Second, the hard part of Theorem 7 is, of course, the second one (non-derivability). Fortunately, the derivability problem in
MALC is algorithmicallydecidable (belongs to PSPACE), thus, it is possible to establish non-derivabilityby exhaustive proof search. This proof search was first performed, as a pre-verification of the result, automatically using an affine modification of llprover by Tamura [14]. (For the WoLLIC 2019 paper, we used a
MALC prover byJipsen [15], based on the algorithm by Okada and Terui [16].) In order to makethis article self-contained and independent from proof-search software, here wepresent a complete manual proof search.12ne of the WoLLIC 2019 reviewers suggested a shorter method of prov-ing non-derivability of the given sequent in
MALC , via an algebraic counter-model. This counter-model is a commutative residuated lattice on the set R = { , a, b, c, } . The order is defined as follows: 0 ≺ a, b, c ≺ a, b, c areincomparable. Product and residual are defined as follows: · a b c
10 0 0 0 0 0 a a b c b b a c c c c c c ⊸ a b c
10 1 1 1 1 1 a a b c b b a c c c c c c ⊸ .) Variables are interpreted as follows: y as b , z as c , x and w both as a . Thisalgebraic model falsifies the sequent in Theorem 7. However, is insufficient forour new purposes. The reason is that in this model a · b = b a , while in thepresence of weakening A · B ⊢ A should be true. Thus, in order to establishnon-derivability of our sequent not only in MALC , but also in
IAL , we use thegood old syntactic method.
Proof of Theorem 7.
The first statement is proved using the joining (diamond)construction, the idea of which goes back to Lambek [1] and Pentus [17]. Indeed,let A = ( x / y ) ∨ w and B = ( x / z ) ∨ w . Then A ∨ B is equivalent to ( x / y ) ∨ ( x / z ) ∨ w . One can easily check derivability of A / ( A ∨ B ) , A, A \ B ⊢ A and A / ( A ∨ B ) , A, A \ B ⊢ B in MALC . Notice that the antecedent of this sequentis exactly the one in the sequent of our theorem. Next, we derive
A / ( A ∨ B ) , A, A \ B ⊢ A ∧ B , and further by distributivity A ∧ B ≡ (( x / y ) ∧ ( x / z )) ∨ w ≡ ( x / ( y ∨ z )) ∧ w .The second statement is proved by an exhaustive proof search for the sequent(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , ( y ⊸ x ) ∨ w, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w IAL .In order to facilitate proof search, we take into account the following con-siderations.First, the rules ∨ L and ⊸ R are invertible. Thus, we can suppose theyare applied immediately. Moreover, ∨ L has two premises, and when disprovingderivability we have the right to choose one and establish non-derivability there.Second, we can suppose that in our (hypothetic) derivation instances of ∨ L r of the form Γ ⊢ w Γ ⊢ F ∨ w are directly preceded by axioms. Indeed, such instancesare interchangeable upwards with ⊸ L and ∨ L , and ⊸ R cannot appear beforethis ∨ L r , since w is a variable. Other rules are impossible by the polarizedsubformula property.Third, we establish non-derivability of several sequents, which will appearfrequently in our proof search: ( y ⊸ x ) ∨ w (1) z ( y ⊸ x ) ∨ w (2) y ( y ⊸ x ) ∨ w (3) z, y ( y ⊸ x ) ∨ w (4) z, z ( y ⊸ x ) ∨ w (5) z ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w (6) ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w (7) z, y ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w (8)14ow we are ready to start proof search. First we invert ∨ L introducing( y ⊸ x ) ∨ w and choose y ⊸ x :(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , y ⊸ x, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w Now we have a choice of 4 principal connectives (denoted by numbers in thesequent) to be decomposed first.
Case 1.
In this case, we use ∨ R l , thanks to our consideration that ∨ R r with w should be applied immediately after an axiom.(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , y ⊸ x, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ ( y ∨ z ) ⊸ x Invert ⊸ R and ∨ L , choosing z out of y ∨ z :(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , y ⊸ x, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z ⊢ x Now we can decompose (by ⊸ L ) one of the implications 2–4, and for eachwe have a choice of 8 = 2 ways of splitting the rest of the antecedent into Πand Γ. Making use of the weakening rule, however, we can reduce the numberof cases. Subcase 1–2.
If Π includes y ⊸ x , then the right premise is Γ , ( y ⊸ x ) ∨ w ⊢ x , where Γ is a subset of z, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ). Noticethat if Γ ′ ⊆ Γ and the sequent is not derivable with Γ, it is also not derivablewith Γ ′ (otherwise we could derive it with Γ using the weakening rule). However,the sequent is not derivable even with the maximal Γ: z, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , ( y ⊸ x ) ∨ w x. Indeed, invert ∨ L and choose w : z, w, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) x. ⊸ L , but then in its right premise we can again invert ∨ L choosing w , which yields one of: w ⊢ x z, w ⊢ x w, w ⊢ x z, w, w ⊢ x. None of these is derivable.If Π does not include y ⊸ x , then Π is a subset of (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z , and we again take the maximal Π in the left premise:(( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w (9)Decomposing ⊸ yields either ⊢ ( y ⊸ x ) ∨ w or z ⊢ ( y ⊸ x ) ∨ w , both notderivable by (1) and (2). Thus, we have to decompose ∨ on the right.Taking y ⊸ x (and inverting ⊸ R ) yields(( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z, y ⊢ x. Now we again have to use ⊸ L . The new cases are y ⊢ ( y ⊸ x ) ∨ w and z, y ⊢ ( y ⊸ x ) ∨ w , both not derivable (3)(4).Taking z ⊸ x and inverting ⊸ R gives(( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z, z ⊢ x. Decomposing ⊸ fails due to (1)(2)(5). Subcase 1–3.
Apply ⊸ L for ⊸ and consider its left premise with themaximal possible Π:(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z ⊢ y. (10) Subsubcase 1–3–2.
Decompose ⊸ . If the big formula with ⊸ goes to the newΓ, then the new Π is either z or empty. However, neither z ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w nor ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w is derivable (6)(7). If the formula with ⊸ goes to the new Π, then the new Γ is either z or empty. This gives, at maximum, z, ( y ⊸ x ) ∨ w ⊢ y , which is falsified by choosing w in the inverted ∨ L : z, w y . Subsubcase 1–3–4.
Decompose ⊸ . Again, if the big formula (now with ⊸ )goes to the new Γ, we falsify the left premise by (1) or (2). Otherwise, the right16remise is, at maximum, z, ( z ⊸ x ) ∨ w ⊢ y , which is again falsified by choosing w . Subcase 1–4.
If Π includes y ⊸ x , then the right premise is, at maximum,(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , z, ( z ⊸ x ) ∨ w ⊢ x Invert ∨ L and choose w :(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , z, w ⊢ x Now we have to use ⊸ L . Its right premise is, at maximum, z, w, ( y ⊸ x ) ∨ w ⊢ x . Choosing w falsifies it.If y ⊸ x is in Γ, then the maximal version of the left premise is(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , z ⊢ ( y ⊸ x ) ∨ w. (11)Applying ⊸ right now is impossible: its left premise gets falsified by (7) or (6).Apply ∨ R l (recall that ∨ R r is used only directly below axiom) and invert ⊸ R :(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , z, y ⊢ x. Here the left premise of ⊸ is also falsified by (7), (6), or (8). Case 2.
Consider again two cases, depending on whether y ⊸ x goes to Πor to Γ. If it goes to Π, then the right premise is, at maximum,( y ⊸ x ) ∨ w, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w. Invert ∨ L and choose y ⊸ x : y ⊸ x, (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w. (12)For reusal of our reasoning in further cases, we shall falsify a stronger sequent y ⊸ x, (( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w. (13)Indeed, ( y ⊸ x ) ∨ w ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w , and therefore (( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) is derivable in IAL .Thus, if (12) happens to be derivable then, by cut, so will be (13).17ow we decompose one of ∨ , ⊸ , ⊸ in (13). Subcase 2– Π –1. Recall that we never choose w in ∨ R , and invert ⊸ R : y ⊸ x, (( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , y ∨ z ⊢ x. Invert ∨ L and choose z : y ⊸ x, (( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z ⊢ x. Subsubcase 2– Π –1–5. The left premise is, at maximum,(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , z ⊢ y. Applying ⊸ L is impossible, since its right premise is falsified by choosing w : w, z y and w y . Subsubcase 2– Π –1–4. Again, if y ⊸ x goes to the new Π, then the rightpremise is, at maximum, z, ( z ⊸ x ) ∨ w ⊢ x, which is falsified by choosing w .If it goes to the new Γ, then the new left premise is, at maximum, z ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w , which is not derivable by (6). Subcase 2– Π –4. If the new Π is empty, then the left premise is falsifiedby (7). Otherwise, the right premise is( z ⊸ x ) ∨ w ⊢ (( y ∨ z ) ⊸ x ) ∨ w. Invert ∨ L and choose z ⊸ x : z ⊸ x ⊢ (( y ∨ z ) ⊸ x ) ∨ w. Applying ⊸ L is impossible ( z ); also z ⊸ x w . Thus, we have to use ∨ R l ,and we can immediately apply ⊸ R afterwards: z ⊸ x, y ∨ z ⊢ x. Inverting ∨ L and choosing y falsifies this sequent: z ⊸ x, y x . Subcase 2– Π –5. The left premise is, at maximum,(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ y. This is not derivable. 18ow let, in Case 2, y ⊸ x go to Γ. Then the left premise is, at maximum,(( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ ( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w. This sequent is stronger than (9)—that is, (9) can be obtained from it by weak-ening. Therefore, it cannot be derivable, since we’ve already falsified (9) inCase 1.
Case 3.
Take the maximal possible Π and consider the left premise:(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , (( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) ⊢ y. This sequent is stronger than (10), and therefore not derivable: (10) was falsifiedin Case 1.
Case 4. If y ⊸ x goes to Π, then the maximal version of the right premiseof ⊸ L is(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , ( z ⊸ x ) ∨ w ⊢ (( y ∨ z ) ⊸ x ) ∨ w. Invert ∨ L and choose z ⊸ x :(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) , z ⊸ x ⊢ (( y ∨ z ) ⊸ x ) ∨ w. Suppose this sequent is derivable. Then it will also be derivable after swappingvariables y and z :(( z ⊸ x ) ∨ ( y ⊸ x ) ∨ w ) ⊸ (( z ⊸ x ) ∨ w ) , y ⊸ x ⊢ (( z ∨ y ) ⊸ x ) ∨ w. This sequent, however, is exactly (13), up to commutativity; (13) was falsifiedin Case 2.Finally, if y ⊸ x , in Case 4, goes to Γ, then the maximal version of the leftpremise of ⊸ L is(( y ⊸ x ) ∨ ( z ⊸ x ) ∨ w ) ⊸ (( y ⊸ x ) ∨ w ) ⊢ (( y ∨ z ) ⊸ x ) ∨ w. This sequent is stronger than (11) and therefore cannot be derivable.19 . Undecidability with \ , ∧ , and 1 L + ε ( \ , ∧ , ) and Its Undecidability In this section we consider the extension of the Lambek calculus with themultiplicative unit constant. The language of our fragment will be as follows: \ , ∧ , . As shown by Buszkowski [6], in the fragment of \ and ∧ the Lambekcalculus with empty antecedents is complete w.r.t. L ε -models. As noticed inthe Introduction, however, this is not the case if we add . In L ε -models,because of the principle A · ⊢ , the unit constant is necessarily interpretedas the singleton set { ε } , where ε is the empty word. (In the presence of theunit constant, we allow the empty word to belong to our languages and abolishLambek’s non-emptiness restriction.) This particular interpretation of satisfiescertain principles, including A ·{ ε } = { ε }· A and { ε }·{ ε } = { ε } . Moreover, theseprinciples keep valid for languages of the form { ε } ∩ B (for any B ). Indeed, thislanguage is either { ε } or ∅ , and for the empty set ∅ we also have A · ∅ = ∅ · A and ∅ · ∅ = ∅ .Below we present a calculus denoted by L + ε ( \ , ∧ , ), which reflects theseprinciples as sequential rules: A ⊢ A Id A, ⊢ A Π ⊢ A Γ , B, ∆ ⊢ C Γ , Π , A \ B, ∆ ⊢ C \ L A, Π ⊢ B Π ⊢ A \ B \ R Γ , A ⊢ C Γ , A ∨ B ⊢ C ∧ L l Γ , B ⊢ C Γ , A ∨ B ⊢ C ∧ L r Π ∧ A Π ∧ B Π ⊢ A ∧ B ∧ R Γ , A, ∧ G, ∆ ⊢ C Γ , ∧ G, A, ∆ ⊢ C Lε Γ , ∧ G, A, ∆ ⊢ C Γ , A, ∧ G, ∆ ⊢ C Rε Γ , ∧ G, ∧ G, ∆ ⊢ C Γ , ∧ G, ∆ ⊢ C Dε
The rules Lε and Rε are called “commuting” rules; they reflect the fact that,for any set X , X · { ε } = { ε } · X and X · ∅ = ∅ · X . The “doubling” rule Dε is caused by { ε } · { ε } = { ε } and ∅ · ∅ = ∅ . Thus, these rules express naturalalgebraic properties of the interpretation of as ∅ . However, we emphasize thatthey are not admissible in the standard calculus L , introduced by Lambek [10],that is, non-commutative intuitionistic multiplicative-additive linear logic.20he rules Lε , Rε , and Dε are not new. Their underlying principles, namely,( ∧ G ) · A ≡ A · ( ∧ G ) and ( ∧ G ) · ( ∧ G ) ≡ ∧ G appear in works ofthe Hungarian school (Andr´eka, Mikul´as, N´emeti). Namely, in [18] one canfind the first of these equivalences (denoted there as formula 3.2), as one ofthe principles which is true in language algebras, but not in algebras of binaryrelations. The second equivalence is true for binary relations also; formula (CbI)in [19] is actually its stronger version, ( ∧ G ) · ( ∧ F ) ≡ ∧ G ∧ F . We get our( ∧ G ) · ( ∧ G ) ≡ ∧ G by taking F = G .Andr´eka, Mikul´as, and Sain [20] also sketch an undecidability proof for asystem related to the one considered here. Their proof is based on the techniqueof Kurucz et al. [21]. The system considered in [20] is the logic of residuateddistributive lattices over monoids. Unlike the case we consider in this section,their system requires product, the unit and also the zero constant (the minimalelement of the lattice) to be present in the language. Here we require onlydivision, additive conjunction, and the unit. The trade-off is that we considera narrower class of models. Namely, we consider only L ε -models, and thesemodels, as shown above, allow extra principles for .We do not claim that L + ε ( \ , ∧ , ) is an L ε -complete system. Indeed, theL ε -complete extension of L happens to be quite involved (cf. [22]). In partic-ular, it is still an open problem whether such a complete system is recursivelyenumerable. The cut rule is not included in L + ε ( \ , ∧ , ), so all our derivationswill be cut-free. We do not claim that cut is admissible in this system.We prove undecidability for the whole range of systems between L + ε ( \ , ∧ , )and the L ε -complete system in the language of \ , ∧ , . Theorem 8.
Let L be an L ε -sound logic which includes L + ε ( \ , ∧ , ) . Then thederivability problem for L is undecidable. Our undecidability proof is based on encoding computations of 2-counterMinsky machines [23]. In the forward encoding, from Minsky computations toderivations in our calculus, we present explicit derivations in L + ε ( \ , ∧ , ). Forthe backwards direction, from derivations to computations, we use a semantic21pproach using L-models (cf. [24, 25, 16], where phase semantics was used forsimilar purposes). Thus, we get undecidability not only for L + ε ( \ , ∧ , ) itself,but for the whole range of its L ε -sound extensions.Before going further, let us introduce the relative double negation construc-tion. We fix a variable (atomic proposition) b and define relative negation A b as A b = A \ b. The term “negation” here is motivated as follows. In linear logic with thefalsity constant ⊥ , negation is expressed as A ⊥ = A ⊸ ⊥ . Here we do the samenon-commutatively, but due to lack of the ⊥ constant we replace it by a fixedvariable. This is the minimal logic approach: variable b can be read as “false,”but no specific axioms like b ⊢ A ( ex falso ) are imposed for b .The relative double negation now is A bb = ( A \ b ) \ b. Notice the difference from the more usual in the Lambek calculus “type raising”version of something like double negation: b A b = b / ( A \ b ). In our setting, wehave neither A bb ⊢ A (due to the intuitionistic nature of the Lambek calculus),nor A ⊢ A bb (due to non-commutativity; in contrast, A ⊢ b A b is derivable).Nevertheless, A bb will be useful for our construction.Given a sequence of formulae Φ = A , A , . . . , A m − , A m and a formula C ,we introduce the notationΦ \ C = A m \ ( A m − \ . . . \ ( A \ ( A \ C )) . . . ) . In particular, Φ b = A m \ ( A m − \ . . . \ ( A \ ( A \ b )) . . . )and Φ bb = (cid:0) A m \ ( A m − \ . . . \ ( A \ ( A \ b )) . . . ) (cid:1) \ b. In what follows, we suppose that the bb operation has a higher priority thanordinary division \ . 22onsider a non-deterministic Minsky machine M with a finite set of states { L , L , . . . , L n } . A configuration of M is a triple ( L i , k , k ), where L i is thecurrent state and k and k are the current values of M ’s two counters. Thecounters themselves are denoted by c and c . The configuration ( L , ,
0) isconsidered the final one; the initial configuration can be taken arbitrarily.Configurations of Minsky machines are encoded as follows. We introduce dis-tinct variables e , e , p , p , l , l , . . . , l n and represent configuration ( L i , k , k )as e , p , . . . , p | {z } k times , l i , p , . . . , p | {z } k times , e . In particular, the final configuration ( L , ,
0) is represented as e , l , e .Minsky instructions are encoded according to the following table: Instruction I Formula A I inc ( L i , , L j ) l i \ ( p , l j ) bb inc ( L i , , L j ) l i \ ( l j , p ) bb dec ( L i , , L j ) ( p , l i ) \ l bbj dec ( L i , , L j ) ( l i , p ) \ l bbj jz ( L i , , L j ) ( e , l i ) \ ( e , l j ) bb jz ( L i , , L j ) ( l i , e ) \ ( l j , e ) bb Here instruction inc ( L i , r, L j ) (increment) means “at state L i , increase c r by 1 and go to L j ” ( r = 1 , dec ( L i , r, L j ) (decrement) means “atstate L i , decrease c r by 1 and go to L j .” If k r = 0, then this instruction cannotbe applied. Finally, jz ( L i , r, L j ) (zero-test) means “at state L i , if k r = 0, go to L j .” Now if k r = 0, then the instruction cannot be applied.Notice that our version of zero-test and decrement instructions are veryrestrictive. Once the counter has a wrong value (zero for decreasing or non-zero for zero-test), the machine just fails to proceed. Usually, in such cases themachine is allowed to perform conditional branching (e.g., zero-test jumps to L j if the counter is zero and safely stays at L i if not). These restrictions, however,are compensated by the allowed non-determinism of M . Indeed, the compound jzdec ( L i , r, L j , L j ) instruction from Minsky’s original formalism [26], meaning23at state L i , if k r = 0, decrease c r by one and go to L j , and if k r = 0, go to L j ,” is modelled by adding simultaneously two instructions: dec ( L i , r, L j )and jz ( L i , r, L j ). This non-deterministically branches computation; however,exactly one branch (depending on whether k r = 0) could be successful, the otherone immediately fails.Let us denote the set of our variables, except b , by V : V = { e , e , p , p , l , l , . . . , l n } . Finally, the Minsky machine M is represented by the following formula G = (( e , l , e ) \ b ) ∧ ^ I A I ∧ ^ q ∈V ( q \ q bb ) . Here in the first big conjunction I ranges among all instructions of M .Now we are ready to state our main encoding theorems. Theorem 9. If M can reach the final configuration ( L , , , starting from ( L i , k , k ) , then the following sequent is derivable in L + ε ( \ , ∧ , ) : ∧ G, e , p , . . . , p | {z } k times , l i , p , . . . , p | {z } k times , e ⊢ b. ( ∗ ) Theorem 10.
If the sequent ( ∗ ) is true in all L ε -models, then M can reach ( L , , from ( L i , k , k ) . Notice that our encodings are in a sense “upside-down”: the starting con-figuration corresponds to the goal sequent in our derivation, and the sequentencoding the final configuration ( L , ,
0) is on the top of the derivation, veryclose to axioms (see proof of Theorem 9 below). The right intuition here isto consider the derivation in the direction of proof search, developing from thegoal up to axioms. This direction correctly reflects the direction of Minskycomputation.Theorem 8 (our undecidability result) immediately follows from Theorems 9and 10. Indeed, if L is a logic which is L ε -sound and includes L + ε ( \ , ∧ , ),then ( ∗ ) is provable in L if and only if M can reach ( L , ,
0) from ( L , k , k ).24ndeed, the “if” direction is by Theorem 9, and the “only if” direciton is byTheorem 10. Since reachability in Minsky computations is undecidable, we getundecidability of L .Before proving Theorems 9 and 10, we establish several technical results.Notice that each formula in the big conjunction G , except the first one, isof the form G Φ , Ψ = Ψ \ Φ bb . The key lemma for such formulae, in the view ofTheorem 9, is as follows. Lemma 11.
If the big conjunction G includes G Φ , Ψ and ∧ G, Φ , ∆ ⊢ b isderivable in L + ε ( \ , ∧ , ) , then so is ∧ G, ∆ , Ψ ⊢ b .Proof. The derivation is as follows:Ψ ⊢ Ψ ∧ G, Φ , ∆ ⊢ b Φ , ∧ G, ∆ ⊢ b Rε several times b ⊢ b ∧ G, ∆ ⊢ Φ \ b \ R ∧ G, ∆ , (Φ \ b ) \ b ⊢ B \ L ∧ G, ∆ , Ψ , Ψ \ Φ bb ⊢ b \ L ∧ G, ∆ , Ψ , ∧ (Ψ \ Φ bb ) ⊢ b ∧ L r ∧ G, ∧ (Ψ \ Φ bb ) , ∆ , Ψ ⊢ b Lε several times ∧ G, ∧ G, ∆ , Ψ , ⊢ b ∧ L several times ∧ G, ∆ , Ψ ⊢ b Dε Corollary 12 (“Post-ish productions”) . Let ∆ and ∆ be sequences of vari-ables from V (no complex formulae). Then, provided that G includes q \ q bb for any q ∈ V , the sequent ∧ G, ∆ , ∆ ⊢ b is derivable in L + ε ( \ , ∧ , ) from ∧ G, ∆ , ∆ ⊢ b .Proof. It is sufficient to consider the case of ∆ = q ; then we proceed by induc-tion on the length of ∆ . For ∆ = q , we apply Lemma 11 with Φ = Ψ = q . Corollary 13 (One step of Minsky computation) . Suppose the Minsky machine M can make a computation step from configuration ( L i , k , k ) to configuration ( L i ′ , k ′ , k ′ ) , and let ( ∗ ′ ) be the instance of ( ∗ ) for ( L i ′ , k ′ , k ′ ) . Then ( ∗ ) isderivable from ( ∗ ′ ) in L + ε ( \ , ∧ , ) . roof. The proof is performed uniformly for all Minsky instructions. For anyinstruction I , the corresponding formula A I is of the form G Φ , Ψ = Ψ \ Φ bb .On the other hand, ( ∗ ′ ) is obtained from ( ∗ ) by replacing Ψ with Φ in theantecedent.For example, for the instruction inc ( L i , , L j ) in the center of ( ∗ ) we have l i = Ψ, which is replaced with p , l j = Φ in ( ∗ ′ ). This exactly corresponds to thecomputation step: the number of p ’s (that is, the value of c ) gets increased by1, and the state is changed to l j . For jz , the replacement happens at the edgeof the antecedent, involving e or e .Thus, ( ∗ ) is of the form ∧ G, ∆ , Ψ , ∆ ⊢ b and ( ∗ ′ ) is ∧ G, ∆ , Φ , ∆ ⊢ b .Now we derive ( ∗ ) from ( ∗ ′ ) in the following way: ∧ G, ∆ , Φ , ∆ ⊢ b ∧ G, Φ , ∆ , ∆ ⊢ b Corollary 12 ∧ G, ∆ , ∆ , Ψ ⊢ b Lemma 11 ∧ G, ∆ , Ψ , ∆ ⊢ b Corollary 12
Now we are ready to prove Theorem 9.
Proof of Theorem 9.
Using Corollary 13 and induction on the number of stepsin Minsky computation from ( L i , k , k ) to ( L , , ∗ ) from ∧ G, e , l , e ⊢ b ( ∗ )This sequent ( ∗ ) is derived as follows: e ⊢ e l ⊢ l e ⊢ e b ⊢ be , e \ b ⊢ b \ Le , l , l \ ( e \ b ) ⊢ b \ Le , l , e , e \ ( l \ ( e \ b )) ⊢ b \ Le , l , e , ∧ G ⊢ b ∧ L several times ∧ G, e , l , e ⊢ b Lε ε -model. Let Σ = V and fefine B M as the set of “terminating words” for M ,defined as follows: B M = { e p . . . p | {z } k times l i p . . . p | {z } k times e | M can reach ( L , ,
0) from ( L i , k , k ) } . Now define the interpreting function w on variables as follows: w ( q ) = { q } for q ∈ V ; w ( b ) = { ΞΥ | Ξ and Υ are words over Σ such that ΥΞ ∈ B M } . Lemma 14. w ( ∧ G ) = { ε } .Proof. It is sufficient to show that ε ∈ w ( G ), that is, ε belongs to interpretationof all formulae in the big conjunction G .First, ε ∈ w (( e , l , e ) \ b ). Indeed, w (( e , l , e ) \ b ) = { e l e } \ w ( b ), thus,we have to show that e l e ε = e l e ∈ w ( b ). This is indeed so by the definitionof B M , since ( L , ,
0) is reachable from itself in zero steps.Second, we prove that ε ∈ w ( A I ) for each instruction I of M . Recall that A I = Ψ \ Φ bb = Ψ \ ((Φ \ b ) \ b ), and if instruction I changes the configurationfrom ( L i , k , k ) to ( L i ′ , k ′ , k ′ ), then the code of the second configuration isobtained from the code of the first one by replacing Ψ with Φ. In other words,the code of ( L i , k , k ) is ∆ Ψ∆ and the code of ( L i ′ , k ′ , k ′ ) is ∆ Φ∆ . Wehave to prove that ε ∈ w (Ψ) \ w ((Φ \ b ) \ b ). Since w (Ψ) = { Ψ } (Ψ containsonly letters from V ), this means that Ψ should belong to w ((Φ \ b ) \ b ).In turn, Ψ ∈ w ((Φ \ b ) \ b ) means that for any word ∆ ∈ w (Φ \ b ) we have∆Ψ ∈ w ( b ). The fact that ∆ ∈ w (Φ \ b ), since w (Φ) = { Φ } , actually means thatΦ∆ ∈ w ( b ). Thus, we have to prove, for an arbitrary ∆, that if Φ∆ ∈ w ( b ),then ∆Ψ ∈ w ( b ).If Φ∆ ∈ w ( b ), then we have ∆ = ∆ ∆ , and ∆ Φ∆ ∈ B M . Here Φ cannotbe split between Ξ and Υ, because any word in B M should begin with e and endon e . This means that ∆ Φ∆ is a code of some configuration ( L i ′ , k ′ , k ′ ), fromwhich M can reach the final configuration. As noticed above, this means that27 Ψ∆ encodes a configuration ( L i , k , k ), which transforms into ( L i ′ , k ′ , k ′ )by applying instruction I . Therefore, from ( L i , k , k ) our Minsky machine canalso reach the final state, hence ∆ Ψ∆ ∈ B M . This yields ∆Ψ = ∆ ∆ Ψ ∈ w ( b ), which is our goal.Third, consider q \ q bb , where q ∈ V . We have to show that ε ∈ w ( q ) \ w ( q bb ),that is, q ∈ w ( q bb ). The latter means that for any ∆ ∈ w ( q \ b ) the word ∆ q should belong to w ( b ). This is indeed so: if ∆ ∈ w ( q \ b ), then q ∆ ∈ w ( b ), andsince w ( b ) is closed under cyclic transpositions, also ∆ q ∈ w ( b ).Now we are ready to prove Theorem 10. Proof of Theorem 10.
If ( ∗ ) is true in all L ε -models, it is true in the specificmodel defined above. By Lemma 14, w ( ∧ G ) = { ε } ; w ( q ) = { q } for any q ∈ V .Thus, we have e p . . . p | {z } k times l i p . . . p | {z } k times e ∈ w ( b ) , and therefore e p . . . p | {z } k times l i p . . . p | {z } k times e ∈ B M . (No cyclic transposition is possible, since e and e should start and end theword.)By definition of B M , this means that M can reach the final state ( L , , L i , k , k ). Let Th(L ε -models; \ , ∧ , ) denote the set of all sequents in the language of \ , ∧ , which are true in all L ε -models, that is, the complete theory of this classof models.As noticed above, the question of axiomatizing this theory is quite involved.We know that this theory includes L + ε ( \ , ∧ , ), introduced in the previous sec-tion, but it is probably much more complicated. For example, as shown in [22],Soboci´nski’s 3-valued logic RM can be embedded into Th(L ε -models; \ , ∧ , ).28t follows from Theorem 8 that Th(L ε -models; \ , ∧ , ) is undecidable. Moreprecisely, it is Σ -hard (hard w.r.t. the class of recursively enumerable sets). Theupper complexity bound, however, is not known: this theory could possibly beeven not recursively enumerable. Having the algorithmic complexity questionfor Th(L ε -models; \ , ∧ , ) open, we can still obtain an interesting corollary ofour complexity estimations.Recall the standard notion of regular expression. Regular expressions areconstructed from constants 0 and 1 using two binary operations, · and +, and oneunary operation, ∗ . The language L ( R ) described by a given regular expression R is defined recursively: L (0) = ∅ ; L (1) = { ε } ; L ( A · B ) = L ( A ) · L ( B ); L ( A + B ) = L ( A ) ∪ L ( B ); L ( A ∗ ) = (cid:0) L ( A ) (cid:1) ∗ = { u . . . u n | n ≥ , u i ∈ L ( A ) } . Languages described by regular expressions are called regular languages.By
Lreg ε -models let us denote a subclass of L ε -models in which every vari-able as interpreted as a regular language, that is, a set of words described by aregular expression. It is well-known that the class of regular languages is closedunder intersection (see, for example, [27, Theorem 2.8]). Moreover, it is alsoclosed under division: Proposition 15. If A and B are regular languages, then so are A \ B and B / A .Proof.
A more well-known fact (see, for example, [27, Exercise 2.3.17a]) is thatthe class of regular languages is closed under the following modified divisionoperation with the existential quantifier instead of the universal one: A ∼ B = { u ∈ Σ ∗ | ( ∃ v ∈ A ) vu ∈ B } . Our “normal” division \ can be reduced to ∼ by the complement operation: A \ B = A ∼ B , where X = Σ ∗ − X . Since the29lass of regular languages is closed under ∼ and complement (again, see [27,Theorem 2.8]), it is also closed under \ . The / case is symmetric.Thus, in Lreg ε -models interpretations of all formulae are regular languages.In the language without the unit constant, namely, \ , / , ∧ , the theory of Lreg ε -models coincides with the theory of all L ε -models: Proposition 16.
Th(
Lreg ε -models ; \ , /, ∧ ) = Th( L ε -models ; \ , /, ∧ ) . Proof.
On the one hand, the calculus
MALC ∗ ( \ , /, ∧ ) is sound w.r.t. all L ε -models. On the other hand, as shown by Buszkowski [28], it is complete w.r.t.a class of models which is even narrower than the class of Lreg ε -models.Namely, MALC ∗ ( \ , /, ∧ ) is complete w.r.t. the class of L ε -models in whichvariables are interpreted by cofinite languages. (A cofinite language is a lan-guage which includes all words over a given alphabet, except for a finite set.) Inthis case, formulae are interpreted by cofinite or finite languages, and any finiteor cofinite language is regular. Therefore, both Th( Lreg ε -models; \ , /, ∧ ) andTh(L ε -models; \ , /, ∧ ) are axiomatized by the same calculus MALC ∗ ( \ , /, ∧ ).The unit changes things dramatically. With the unit, there is no complete-ness result, like Theorem 4, but also no equivalence between theories of allL ε -models and Lreg ε -models. Theorem 17.
Th(
Lreg ε -models ; \ , ∧ , ) = Th( L ε -models ; \ , ∧ , ) . Proof.
As follows from Theorem 8, Th(L ε -models; \ , ∧ , ) is Σ -hard. On theother hand, following Buszkowski [29], we can show that Th(L ε -models; \ , ∧ , )belongs to the Π class. Indeed, a sequent belongs to this theory if and only ifit is true in all Lreg ε -models. A concrete sequent includes only a finite numberof variables, p , . . . , p n . Thus, a model for this sequent is defined by a finitenumber of regular expressions R , . . . , R n , which describe the languages w ( p ),. . . , w ( p n ). This means that the general truth condition for this sequent can be30ritten down as the following formula: ∀ R . . . ∀ R n (cid:0) the sequent is true under interpretationwhere w ( p i ) is the language of R i (cid:1) . Quantifiers ∀ R , . . . , ∀ R n can be encoded as quantifiers over natural numbersrepresenting the regular expressions. The quantifier-free part of the formula(truth condition under a concrete w ) is decidable, because all necessary opera-tions on regular expressions are computable. Thus, we get a Π representationof the set of sequents which are true in all Lreg ε -models.It is well known that a set cannot belong to Π and be Σ -hard at the sametime. (Otherwise, for any set in Σ there would be a computable reduction toa set in Π , which would yield Σ ⊆ Π , which is not the case.) Therefore,Th( Lreg ε -models; \ , ∧ , ) = Th(L ε -models; \ , ∧ , ) . Notice that our proof of Theorem 8 does not apply to Th(
Lreg ε -models; \ , ∧ , ), because the language w ( b ) there is non-regular (in fact, it is undecid-able).Since the class of Lreg ε -models is narrower than the class of all L ε -models,we have (by Galois connection) an inverted inclusion of theories:Th( Lreg ε -models; \ , ∧ , ) ⊃ Th(L ε -models; \ , ∧ , ) . By our Theorem 17, this inclusion is strict. Thus, the other inclusion shouldfail: Th(
Lreg ε -models; \ , ∧ , ) Th(L ε -models; \ , ∧ , ) . In other words, there exists a sequent which is true in all
Lreg ε -models, but notin all L ε -models. Our proof, however, is non-constructive, and we do not presenta concrete example of such sequent. Constructing such a concrete example isleft for further research. 31otice that if we apply the reasoning establishing the upper Π bound ofTh( Lreg ε -models; \ , ∧ , ) to Th(L ε -models; \ , ∧ , ), we shall have to quantifyover arbitrary formal languages w ( p ), . . . , w ( p n ). This results in hyperarith-metical quantifiers, and yields only a very high, Π complexity upper bound forTh(L ε -models; \ , ∧ , ).
4. Concluding Remarks
In this article, we have investigated language interpretations of natural ex-tensions of the Lambek calculus: with additive operations ( ∨ and ∧ ) and withadditive conjunction ( ∧ ) and the unit constant ( ).For extensions with additive connectives (Section 2), we have shown thatconjunction and disjunction show significantly different behaviour. It is knownthat adding both conjunction and disjunction leads to incompleteness due tothe distributivity law D . This law is true in all language models, but notderivable in the multiplicative-additive Lambek calculus ( MALC ). Adding onlyconjunction, however, still provides completeness. Any sequent in the languageof \ , / , ∧ (but not ∨ ) that is derivable with the help of D , is also derivablewithout it. For disjunction the situation is opposite: there exists a sequent inthe language of \ , / , ∨ , which is derivable using D , but not derivable withoutit. Moreover, this difference between ∧ and ∨ keeps valid for systems with per-mutation and/or weakening structural rules, that is, intuitionistic linear ( ILL ),and affine (
IAL ) logics and affine
MALC .For the extension of the Lambek calculus with the unit, , it is well-knownthat its standard axiomatization in the style of linear logic does not give an L ε -complete system. In Section 3, we present a system in the language \ , ∧ , ,where rules for reflect natural algebraic properties of the empty word inthe algebra of formal languages. This system is denoted by L + ε ( \ , ∧ , ). Wedo not claim L ε -completeness of L + ε ( \ , ∧ , ). Instead, we consider the wholerange of logics between L + ε ( \ , ∧ , ) and the L ε -complete system denoted by32h(L ε -models; \ , ∧ , ). For any logic within this range, we show that it is unde-cidable; more precisely, Σ -complete. As a corollary, we also show that, in thelanguage of \ , ∧ , , the complete theory of all L ε -models is different from thatof Lreg ε -models, where formulae are interpreted by regular languages.A preliminary version of this article was presented at WoLLIC 2019 andpublished in its lecture notes [13]. Let us briefly list the results which are newin this article, compared to the WoLLIC paper. • In the language without additive conjunction, we show incompleteness notonly for
MALC , but also for its extensions:
MALC ∗ , AMALC ∗ , ILL ,and
IAL . • We prove that
MALC ( \ , /, ∧ ) is a conservative fragment of MALC ex-tended with the distributity law D . Moreover, we prove similar results for MALC ∗ , AMALC ∗ , ILL , and
IAL . • We prove that, in the language including , the theory of all L ε -modelsis different from the theory of L Reg ε -models, in which formulae areinterpreted by regular languages. In the language of \ , /, ∧ (without ), the corresponding theories coincide due to a completeness result byBuszkowski [28].While in Section 2 we have presented a quite completed study, Section 3leaves many questions open for further investigations. Among these, we wouldlike to emphasize the following ones. • The question of axiomatization, or even recursive enumerability for com-plete theories Th(L ε -models; \ , ∧ , ) and Th( Lreg ε -models; \ , ∧ , ) is stillopen, and potentially very hard. Notice that these theories are different(Theorem 17) and that for Th( Lreg ε -models; \ , ∧ , ) enumerability willimmediately mean decidability. • A possibly easier question would be to construct a concrete formula dis-tinguishing Th(L ε -models; \ , ∧ , ) and Th( Lreg ε -models; \ , ∧ , ). Thatis, we are looking for an explicit example for Theorem 17.33 Without the unit, we know that