Three Topics in Non-decomposability of Generalized Multiplicative Connectives
aa r X i v : . [ m a t h . L O ] N ov Three Topics in Non-decomposability ofGeneralized Multiplicative Connectives
Yuki Nishimuta
Abstract
Danos and Regnier introduced generalized (non-binary) multiplicativeconnectives in (Danos and Regnier, 1989). They showed that there existthe generalized multiplicative connectives which cannot be defined by anycombinations of the tensor and par rules in the multiplicative fragment oflinear logic. These connectives are called non-decomposable generalizedmultiplicative connectives (ibid, p.192). In this short note, we investigatethe notion of Danos and Regnier’s non-decomposability and give three re-sults concerning (non-)decomposability of generalized multiplicative con-nectives.
Danos and Regnier introduced generalized (non-binary) multiplicative connec-tives in [2]. They showed that there exist the generalized connectives whichcannot be defined by any combinations of the tensor and par rules in the mul-tiplicative fragment of linear logic that satisfy the main reduction step of the cutelimination theorem. These connectives are called non-decomposable generalizedmultiplicative connectives . Danos and Regnier defined generalized multiplica-tive connectives only for classical multiplicative linear logic. In this short note,we investigate the (non-)decomposability of generalized multiplicative connec-tives in the other fragments of linear logic and give three results concerning (non-)decomposability of generalized multiplicative connectives. In the first part of thisnote, we define intuitionistic generalized multiplicative connectives by employingthe polarities which is inspired by [1]. We show that all of intuitionistic general-ized multiplicative connectives in Intuitionistic Multiplicative Linear Logic
IMLL are decomposable. Hence, there are no non-decomposable multiplicative connec-tives in
IMLL . Secondly, we generalize the definition of Danos and Regnier’s(non-)decomposability and show that all of non-decomposable generalized mul-tiplicative connectives are decomposable in Multiplicative Additive Linear Logic1
ALL and Multiplicative Exponential Linear Logic
MELL . Finally, we show thatelementary light linear logic
EMLL preserves non-decomposability of generalizedmultiplicative connectives.
In this section, we define an intuitionistic generalized connective by modifyingthe definition of the (classical) generalized multiplicative connective. The mainresult of this section is as follows; all of intuitionistic generalized connectivesin
IMLL are decomposable. This result says that there are no non-decomposablegeneralized connectives in
IMLL . We assume the knowledges of generalized mul-tiplicative connectives (see [2, 8])In the following part, we define intuitionistic generalized connectives.First, we define a polarized partition, then we define an intuitionistic polarizedpartition and an intuitionistic polarized meeting graph.
Definition 1.
A polarity of a natural number s is positive if it is not checkedand negative if it is checked. A partition p of a natural number n is a polarizedpartition if p takes a positive or negative polarity. In IMLL , a partition set P C of a generalized connective C corresponds to left-one-sided sequents. A formula A n which is in right hand side of a sequent cor-responds to the checked number ˇ n . Hence, a sequent of IMLL A , . . . , A n , ∼ A n ⊢ corresponds to the partition { , . . . , n − , ˇ n } . When an atomic formula (a meta-variable) A has positive (resp. negative) polarity, we often denote A as A + (resp. A − ). Definition 2.
A polarized partition p is an intuitionistic polarized partition if eachclass of p contains at most one checked element.
An intuitionistic polarized meeting graph G ( p , q ) is a polarized meeting graphwhich is obtained from two intuitionistic polarized partitions p and q in the fol-lowing way; we draw an edge between two nodes N ∈ p and N ′ ∈ q if two numbers n ∈ N and n ′ ∈ N ′ are the same and one of them is checked and the other is not. Definition 3.
A n-ary intuitionistic generalized connectives C is a pair of finiteintuitionistic partition set of a natural number n ( P L , P R ) such that ( P R ) ⊥ = P L and ( P L ) ⊥ = P R hold. Γ , A , . . . , A m − ⊢ A m · · · Γ k , A k , . . . , A km k − ⊢ A km k Γ ⊢ C ( A , . . . , A km k ) where Γ = S ki = Γ i and jm j ∈ { , . . . , n } .We show that all of intuitionistic generalized multiplicative connectives aredecomposable. Theorem 1.
Let C be an arbitrary intuitionistic multiplicative generalized con-nective. Then, C is IMLL -decomposable .Proof.
Let L , . . . , L k be an enumeration of left introduction rules of C and R , . . . , R m be that of right introduction rules. The main reduction step of the cut-elimination for a C -connective holds if and only if the two intuitionistic partition sets of C , ( P L , P R ) are orthogonal. This equivalence is proved as [2, Lemma 2]. By definitionof C , C satisfies the main reduction step of the cut-elimination. We prove thistheorem by case analysis on the number and the form of right introduction rules.Case P R = C is decomposed as follows. For each class x i ( i = , . . . , k ), weapply the left ⊗ -rules as much as possible. Then, we use ⊸ and obtain a formula α i . we connect these formulas by the tensor rules α = α ⊗ . . . ⊗ α k . In the fol-lowing, we call this form of formulas normal. By construction, P C = P α holds.Hence, C is decomposable.Case P R >
1: Consider two arbitrary right rules of C (say, R , R ∈ { R , . . . , R k } ). Case 1: For some formula A , a polarity of A is changed between R and R (say. A + ∈ R and A − ∈ R ). In this case, the cut-elimination does not hold. Forexample, let R = { ( A , B ⊢ E ) ( C , D ⊢ F ) } and R = { ( F , B ⊢ E ) ( C , D ⊢ A ) } .The dual of R should be L = { ( F ⊢ )( ⊢ A ) · · · } , where we omitted irrelevantformulas. The meeting graph between R and L is cyclic and the cut-eliminationdoes not hold.Case 2: For all formulas A i , a polarity of A i does not change in R k . In this case, ifwe apply the right ⊗ -rules as much as possible, we obtain one sequent. After that,we apply the left ⊗ -rules as much as possible and then apply one right ⊸ -rule.By this construction, we can obtain the same formula α from R k for each k . Weshow that α is the decomposition of C . We assume that P α > P R and show acontradiction. Consider arbitrary two elements p ∈ ( P α \ P R ) and q ∈ P ∼ α . Bythe relation P α ⊃ P R , P L ⊃ P ∼ α holds. Hence q ∈ P L holds. It contradicts with the3efinition of an intuitionistic generalized connective ( P L ) ⊥ = P R because p ⊥ q and p / ∈ P R hold. Therefore, P α = P R holds. Corollary 1.
There are no non-decomposable multiplicative connectives in
IMLL . MALL and
MELL
Danos and Regnier defined (non-)decomposability of a generalized connective [2]only in the it multiplicative framework .To investigate preservation of non-decomposability for various systems, weextend the definition of Danos and Regnier’s non-decomposability. After that, weshow that Multiplicative Additive Linear Logic
MALL and Multiplicative Expo-nential Linear Logic
MELL does not preserve non-decomposability.
Notations and terminologies ; The letter L represents some logical systemcontaining MALL as a subsystem. We call the formulas of the form ( A O . . . O A i ) ⊗ . . . ⊗ ( A m O . . . O A mi m ) normal (where 1 ≤ m , 0 ≤ i j and n is a natural number).We define the order relation L as follows; for any formulas P , Q ∈ L , P L Q if there is a proof of Q from that of P in the one-sided sequent calculus L usingthe rules of L . We omit the subscript L for readability. Definition 4.
Let I , . . . , I k (k is a natural number) be an enumeration of theintroduction rules of a generalized multiplicative connective C and S i , . . . , S im i (i = , . . . , k) be the premises of each introduction rule I i . A generalized connec-tive C is L -decomposable if either one of the following holds;1. there exists some formula α ∈ MLL such that P C = P α holds by using onlythe inference rules of MLL ,2. there exists some formula α ∈ L such that(1) for all i (i = , . . . , k), ⊢ α is derivable from S i , . . . , S im i by using at leastone inference rules of L other than those of MLL ,(2) S i , . . . , S im i is obtainable from ⊢ α by bottom-up construction and using atleast one inference rules of L other than those of MLL , We show that all of generalized multiplicative connectives are decomposablein
MALL and
MELL . Proposition 1.
Let C be an arbitrary MLL -non-decomposable generalized con-nective. C is MALL -decomposable. roof. Let P C = { p , . . . , p s } ( s ∈ N ) be the partition set of C ( A , . . . , A m ) . Foreach partition p j , the unique (up to commutativity and associativity) normal for-mula α j ( j ∈ { , . . . , s } ) which have the same assumptions as p j is determined.Put α = α ⊕ · · · ⊕ α s . ⊢ α ⊕ · · · ⊕ α s is derivable from p j ( j = , . . . , s ). For each j , p j is obtainable by bottom-up construction.The Figure 1 is an example of decomposition for the Danos and Regnier’snon-decomposable connective in MALL . ⊢ A , B , Γ ⊢ A O B , Γ ⊢ C , D , ∆ ⊢ C O D , ∆ ⊢ ( A O B ) ⊗ ( C O D ) , Γ , ∆ ⊢ (( A O B ) ⊗ ( C O D )) ⊕ (( A O C ) ⊗ ( B O D )) , Γ , ∆ ⊢ A , C , Γ ⊢ A O C , Γ ⊢ B , D , ∆ ⊢ B O D , ∆ ⊢ ( A O C ) ⊗ ( B O D ) , Γ , ∆ ⊢ (( A O B ) ⊗ ( C O D )) ⊕ (( A O C ) ⊗ ( B O D )) , Γ , ∆ Figure 1: Decomposition of a generalized connective in
MALL
Remark 1.
When we consider only cut-free proofs, we can regard a general-ized connective C ( A , . . . , A m ) as the formula α ⊕ · · · ⊕ α s as the above proof.However, when we consider a proof containing the cut-rules, we cannot identify C ( A , . . . , A m ) with α ⊕ · · · ⊕ α s . The reason is as follows; the dual connective C ∗ ( ∼ A , . . . , ∼ A m ) of C ( A , . . . , A m ) can be decomposed as β ⊕ · · · ⊕ β s by thesame method. However, α ⊕ · · · ⊕ α s and β ⊕ · · · ⊕ β s are not de Morgan dualand the cut is not defined between these formulas (Figure 2). If we consider only cut-free proofs, the introduction of non-decomposableconnectives amount to the introduction of additive disjunction in the restrictedform.
In this section, we show that Elementary Multiplicative Linear Logic
EMLL sat-isfies non-decomposability preservation. Namely, if a generalized multiplicative5 Γ , A , B ⊢ Γ , C , D ⊢ Γ , Γ , ( A O B ) ⊗ ( C O D ) ⊕ ( A O C ) ⊗ ( B O D ) ⊢ ∆ , ∼ A , ∼ D ⊢ ∆ , ∼ B ⊢ ∆ , ∼ C ⊢ ∆ , ∆ , ∆ , (( ∼ A O ∼ D ) ⊗∼ B ⊗∼ C ) ⊕ (( ∼ B O ∼ C ) ⊗∼ A ⊗∼ D ) ⊢ Γ , Γ , ∆ , ∆ , ∆ ⊢ Γ , A , B ⊢ ∆ , ∼ A , ∼ D ⊢ Γ , ∆ , B , ∼ D ⊢ ∆ , ∼ B ⊢ Γ , ∆ , ∆ , ∼ D ⊢ Γ , C , D ⊢ Γ , Γ , ∆ , ∆ , C ⊢ ∆ , ∼ C ⊢ Γ , Γ , ∆ , ∆ , ∆ Figure 2: Undefined cut between ⊕ and ⊕ connective C is non-decomposable in MLL , then C is also non-decomposable in EMLL .The multiplicative fragment of Elementary Linear Logic
EMLL is defined asfollows.
Definition 5.
Formulas of
EMLL are defined as follows;A : = P |∼ P | A ⊗ A | A O A | ! A | ? ANegation is inductively defined as follows; ∼∼ P : = P , ∼ ( A ⊗ B ) : = ∼ A O ∼ B , ∼ ( A O B ) : = ∼ A ⊗∼ B , ∼ ( ! A ) : = ? ∼ A , ∼ ( ? A ) = ! ∼ A(where P is an atomic formula ) The inference rules of
EMLL is as Figure 3.The following is the general form of an introduction rule for a generalizedmultiplicative connective. Note that each premise of a generalized connectivecontains at least one principal formula. ⊢ Γ , A , . . . , A i · · · ⊢ Γ m , A m , . . . , A mi m ⊢ Γ , . . . , Γ m , C ( A , . . . , A n ) where ji j ∈ { , . . . , n } and i k = ( ≤ k ≤ m ) . Theorem 2.
Let C be an arbitrary MLL -non-decomposable connective. Then C is EMLL -non-decomposable. ⊢ P , ∼ P ⊢ Γ , A ⊢ ∆ , ∼ A (Cut) ⊢ Γ , ∆ ⊢ Γ , A ⊢ ∆ , B ( ⊗ ) ⊢ Γ , ∆ , A ⊗ B ⊢ Γ , A , B ( O ) ⊢ Γ , A O B ⊢ Γ (?-weakening) ⊢ Γ , ? A ⊢ Γ , ? A , ? A (?-contraction) ⊢ Γ , ? A ⊢ Γ , A ( K ) ⊢ ? Γ , ! A Figure 3: Inference rules of
EMLL
Proof.
We assume that C is EMLL -decomposable and show that it implies a con-tradiction. By definition, there is a
EMLL -formula α such that ⊢ Γ , α (where Γ = S ms = Γ s and “ ∪ ” is a multiset union) is derivable from the same assumptionsfor C in EMLL . The formula α must contain at least one modal operator. Other-wise, it contradicts with the assumption that C is MLL -non-decomposable. Themodal operator contained in α cannot be a bang connective. If that is a case, Γ must have the form ? ∆ . It contradicts the assumption that Γ is arbitrary. If α contains a why-not connective, an introduction of a why-not connective is by ei-ther K -rule or ?-weakening rule. It cannot be the K -rule as the above explanation.Hence, some formula ? A is introduced by the weakening rule. If an operand of atensor in the formula α has the form ? A , then there is some decomposition of theformula α such that some sequent has only one formula ( · · · ⊢ ? A · · · ) . We apply?-weakening rule to this sequent and obtain the empty sequent ⊢ . This contradicts with the definition of C -introduction rule. If any operands of all tensor con-nectives in the formula α have the form ? A , then we can construct the formula α ′ from α by deleting the formulas having the form ? A and the superfluous parconnectives. This formula α ′ is a decomposition of the connective C in MLL ,contradiction.Preservation of non-decomposability in Multiplicative Light Linear Logic
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