### Table 1. The sequent calculus of permutative logic.

"... In PAGE 4: ... If = f(a; b); (c); (d; e)g; 2, then its genus is given by the couple (2; 3) and rk( ) = 6. The multiplicative permutative calculus is recalled in Table1 ; moreover, the involutive duality is given by De Morgan rules: (A O B)? = B? A? ([A)? = #A? }? = h ?? = 1 (A B)? = B? O A? (#A)? = [A? h? = } 1? = ?: By the fact that basic commutations are not provable keeping the lowest topo- logical complexity, PL turns out to be an inference system able to deal with logical noncommutativity. As suggested by some of the next propositions, basic commutations can be recovered throughout the two permutative modalities [ and #.... ..."

### Table 2. Sequent calculi and their conditions depending on the selected logic

1999

"... In PAGE 7: ...Table 2. Sequent calculi and their conditions depending on the selected logic Table2 uniformly describes the rules of all sequent calculi. The rules are arranged according to the tableau classi cation and directly usable for cumulative domains.... ..."

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### Table 2: Sequent rules corresponding to circuit links

"... In PAGE 6: ... In valid (or sequential) nets C will havetobevalid as well;; this will be checked using the sequentialization process of Appendix B, or, equivalently,by showing that the circuit can be built inductively. Since some of these connectives will not be familiar (and because we use a di erent notation from just about anyone else!|Lambek uses n ;;=for ; ;; ;,and : ; ;; ; : for 5;; 4), the sequent rules that correspond to these links are given in Table2 . In commutative logics the reader can add the exchange rule for himself.... In PAGE 12: ...n the noncommutative logic. In Appendix B we presentavalid sequentialization process. An example of a planar non-sequential circuit which satis es the net criterion is given in Figure 16. The sequent rules given in Table2 are all valid in the noncommutative logics;; for the commutative logics, where the circuits need not be planar, one must add the exchange rules in the obvious way. In Figure 3 are some (valid) circuits.... In PAGE 15: ... Logical theories and categorical doctrines We shall deal with several logical theories (and the corresponding categorical structures) in this paper. The full system using all the binary connectives ;; ;; ; ;; ;;; 4;; 5 and the constants gt;;; ? and using the sequent rules of Table2 (or equivalently the circuit links of Table 1) is Lambek apos;s bilinear logic BILL.We also consider the fragment of bilinear logic which omits the connectives 4;; 5;;we call this noncommutative logic GILL.... In PAGE 15: ....1. Remark. (Cut elimination and FILL) Neither the commutative nor noncommuta- tiveversions of FILL, if presented as a sequent calculus (as in Table2 , with the restriction of Remark 1.1) satis es cut elimination.... In PAGE 18: ...102 in Table2 ) corresponds categorically to having an inverse (costrength) to this family of maps: A ; (B C) ;! (A ; B) C.Wecancheck that in the category of circuits with the more general \boxed quot; links we do indeed havesuch an isomorphism;; half of this exercise is illustrated in Figure 4.... ..."

### Table 1: Sequent rules for LF

2002

"... In PAGE 6: ... However, it turns out that once we establish the appropriate left and right introduction rules for the two way rules above, in particular for the typing rules, these mixed cut rules will be derivable, and so need not be added to the sequent calculus. In Table1 we list the sequent rules for this logic LF, using the convention that when a rule applies to either sequent calculus, we use a variable such as x to represent either of S or T. We have given the left and right introduction rules corresponding to the two way rules discussed above.... In PAGE 6: ...ules. We shall illustrate this with the following pair of lemmas. Lemma 1.2 In LF (as presented in Table1 ), the mixed cut rules are derivable rules. Proof.... In PAGE 8: ... 2 Lemma 1.3 In LF (as presented in Table1 ), the two way tensor and the two way typing rules are derivable rules. Proof.... ..."

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### Table 1: Inference rules for sequents.

2000

"... In PAGE 12: ...2 (Sequents) Let R = h ; E; L; Ri be a rewrite theory. We say that R entails a sequent [s] ) [t], written R ` [s] ) [t], if and only if [s] ) [t] can be obtained by a nite number of applications of the inference rules in Table1 , where t(~ w=~x) denotes the simultaneous substitution of wi for xi in t. A rewrite theory is just a static description of `what a system can do apos;; the behaviour of the theory is instead given by the rewrite relation induced by the rules of deduction.... In PAGE 12: ... A rewrite theory is just a static description of `what a system can do apos;; the behaviour of the theory is instead given by the rewrite relation induced by the rules of deduction. The deduction system in Table1 was introduced in [54], and it is only one of the possible, equivalent ways to entail the same class of sequents. It has, however, the advantage of being rather intuitive.... In PAGE 17: ...Extending the paradigm to non-cartesian structures The deduction rules presented in Table1 make clear that the underlying idea of the rewriting logic paradigm is that the rewrite relation has to be built in- ductively, lifting to computations the structure of terms. Such an intuition can be exploited to describe suitable notions of computation also over structures other than terms: In particular, over elements of gs-monoidal theories, as for the deduction system presented in this section.... In PAGE 17: ... Of course, the deduction system we just presented is also valid for rewriting over monoidal theories: Since we are not interested in the eventual structure of proof terms, we just need to change the premise of the re exivity rule, re- stricting the attention to terms in ME( ). The system in Table 2 induces over terms the same rewrite relation as the one de ned in Table1 for alge- braic sequents, since algebraic theories are just gs-monoidal theories plus the naturality axioms En, that is, AE( ) = GSE[En( ). The correspondence re- sult between the two deduction systems is explicitly given by the following proposition, stated here only for rewrite theories with an empty set of axioms.... ..."

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### Table 2. Comparison of approaches for reasoning about static spatial relations.

"... In PAGE 6: ... Many of such qual- itative spatial calculi have been developed during the past decades, mainly for topological or positional reasoning; however, they are often not fully specified, and mostly no implementation is made available [14]. Table2 shows a compar- ison of popular qualitative spatial calculi which are classified according to the four relation types presented above. Many of them incorporate the spatial con-... ..."

### Table 1: Sequent proof rules and corresponding typing rules

1993

"... In PAGE 34: ...let M be P in N )l N[ ] Match(M; P) = ( P :J) of M is Q in N )l let J[ ] be Q in N [M jL N] )l M [M jR N] )l N Match(M; P1@P2) fails let M be P1@P2 in N )l let hM; Mi be hP1; P2i in N Match(M; P1@P2) fails ( P1@P2:J) of M is Q in N )l ( hP1; P2i:J) of hM; Mi is Q in N Match(M; P) fails M P ; M0 let M be P in N )l let M0 be P in N Match(M; P) fails M P ; M0 ( P :J) of M is Q in N )l ( P :J) of M0 is Q in N M P1 ; M0 inl(M) (P1j P2) ; inl(M0) M P2 ; M0 inr(M) (P1j P2) ; inr(M0) Match(M1; P1) fails M1 P1 ; M10 hM1; M2i hP1;P2i ; hM10; M2i Match(M2; P2) fails M2 P2 ; M20 hM1; M20i hP1;P2i ; hM1; M20i M )l M0 M P ; M0 Table1 0: Lazy evaluator in SOS semantics style Lemma 6.1 If M well-typed and not a lazy-canonical form then we have M )l N for some N.... In PAGE 38: ...2 and Ce ranges over eager canonical forms. Match(Ce; P) = let Ce be P in N )e N[ ] M )e M0 let M be P in N )e let M0 be P in N [M jL N] )e M [M jR N] )e N Match(Ce; P) = ( P :J) of Ce is Q in M )e let J[ ] be Q in M N )e N0 ( P :J) of N is Q in M )e ( P :J) of N0 is Q in M M )e M0 hM; Ni )e hM0; Ni M )e M0 hN; Mi )e hN; M0i M )e M0 inl(M) )e inl(M0) M )e M0 inr (M) )e inr(M0) Table1 1: Eager evaluator in SOS semantics style Well-typed closed terms can always be reduced to a eager-canonical forms with the eager evaluator appearing in gure 10. We de ne a pattern P to specify the type A if and only if P and A correspond to one of the following cases: and x satisfy any type A.... In PAGE 41: ... Of course lazy pattern-matching can o er such a rst-order algorithm8, see table 12. nat def= recX:1 + X zero def= foldX:1+X(inlnat(?)) succ(M) def= foldX:1+X(inr1(M)) zero j succ(P) def= fold((? j P)) inf def= hzero j succ(x); zero j succ(y)i: [[zero j zero] j [zero j succ(inf hx; yi)]] : nat nat ! nat Table1 2: inf in the typed pattern calculus However, Colson also shows that higher-order (Godel apos;s T) primitive recursion algorithms exist with this inten- sional behavior. It remains open then to nd better evidence for the intuition that the typed pattern calculus is intensionally more expressive.... ..."

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### Table 1: A sequent system L for PL

"... In PAGE 4: ...Table 1: A sequent system L for PL . With one exception, the rules in Table1 are the natural generalisations of the rules sug- gested by Girard for the commutative intuitionistic linear logic, cf. [19] and [20, 21], to the non-commutative case, cf [14, 15].... ..."

### Table 1 Sequent Calculus Rules for S4

"... In PAGE 5: ... More abstractly formu- lated, we will have what can be described category-theoretically as a monad, or proof-theoretically as an S4 modal operator. We should recall that S4 modalities are given by the rules in Table1 ; classically or intuitionistically, these rules give the usual modal logic [34, Section 9.1], but they can equally well be added to linear logic and they satisfy the usual proof-theoretic properties (cut elimination and so on) [18].... In PAGE 8: ...Table 2 The System LL ?; A ` @L ?; @A ` ?; @A ` B; @R1 ?; @A ` @B; ? ` A; B; @R2 ? ` A; @B; ?; A ` B; L1 ?; A ` B; ?; A; @B ` L2 ?; A; @B ` ? ` A; R ? ` A; So we are led to consider modalities given by the rules in Table 2; we will call these strong modalities (the category-theoretic counterpart of (4) is called a strength). Conversely, the usual S4 rules ( Table1 ) will be called monoidal modalities (since (2) makes a monoid in an appropriate category of endofunc- tors).This system, based on classical linear logic together with a strong modality, will be called LL ; it will be our point of departure.... In PAGE 37: ... Proof We make an induction on the length of the proof of ` ; we go by cases according to the last rule of the proof. We use a presentation of our basic system, LL 00, given in Table1 0; we can easily prove a dual form of Proposition 9 to show that they are equivalent to the rules of Table varModalRules. Axiom The sequent is of the form A ` A, so the result is clear.... In PAGE 48: ... The top diagram, for example, expresses the equality A A = A A; now the rst is given by the proof Ax A A ` A R A ` A L A ` A L A ` A A A ` A cut A ` A whereas the second is given by the proof AxA A ` A R A ` A L A ` A L A ` A R A ` A L A ` A A A ` A cut A ` A We want these two proofs to be equal. We need similar equalities for the other diagrams in Table1 1, and for the other diagrams de ning a ?-autonomous category. With this de nition of equality between proofs, then, we have de ned a category, F, in which the objects are linear logic formulae and in which the morphisms are proofs of entailments.... ..."

### Table 3: the sequent calculus BCT

1995

"... In PAGE 10: ... This should motivate the rules of the sequent calculus BCT, the \By Cases Theory quot; given in the next de nition. In Table3 we use the vector-notation described earlier: in rule ( !), the terms ~ A are of types such that (f ~ A) has product type; and in rule ByCases, the terms ~ P are of types such that (h ~ P ) has sum type. De nition 5.... ..."

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