### Table 2: Sequent Symmetry Timings in Seconds

1990

"... In PAGE 6: ... Sequent Symmetry Results For the Sequent Symmetry, the mesh size chosen was h = 1 20, and the stopping test was = 6 10?5. The conclusions to be drawn from Table2 are similar to those made for the Alliant runs.... ..."

Cited by 8

### 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 2: Sequent rules for negating indicatives

### 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.... ..."

Cited by 9

### Table 2: Sequent calculus for an intuitionistic fragment of INLL. Theorem 1 The sequent calculus given in Table 2 enjoys cut-elimination. This is a direct consequence of the cut-elimination theorem given in [27] for the sequent calculus 6

### 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.... ..."

Cited by 14

### Table 2: Sequent Calculus of Linear Logic De nition 5.5. A typed expression of l-calculus can be derived using the rules shown in Table 3. If t ` X is derivable we say that the term t has type X. Remark 5.1. Assume that the statement has been derived (x1; : : : ; xn)fj t1; : : : ; tk; a1 b1; : : : ; ar br jg ` A1; : : : ; Ak Its derivation induce a 1 ? 1 correspondence between the proper terms (that are not interactions) and the formulae of the sequent. The correspondence does not depend on the particular derivation of the term. Lemma 5.6. Every term constructed in l-calculus is closed. The demonstration consists only in a routine veri cation on the derivation of the term.

### Table 7: Single Conclusion Sequent Rules for linear logic

2000

"... In PAGE 37: ...5 it is shown that theorems in the ND system are also theorems in LL. A single conclusion presentation of the sequent calculus for LL, presented in Table7 , is the most appropriate version to use in this work since it mirrors the natural deduction rules most closely. See [Gol90] for a study of the relationship between various sequent presentations.... ..."

Cited by 1

### Table 1: Sequent Calculus formalisation of Intuitionistic Linear Logic

"... In PAGE 7: ... a b x x a b x x 2.2 Linear Logic Table1 shows the basic axiom and rules for in- tuitionistic linear logic in Gentzen style. Due to limited space, we cannot go into the detail of the system.... ..."