### 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 12. ws-calculus syntax

2007

"... In PAGE 33: ... 5.3 Encoding ws-calculus The syntax of (a simplified version of) ws-calculus, given in Table12 , is parameterized with respect to the following syntactic sets: properties (sorts of late bound constants storing some relevant values within service instances, ranged over by p), values (basic values and addresses, ranged over by u), partner links (variables storing addresses used to identify service partners within an interaction), operation parameters (basic vari- ables, partner links and properties, ranged over by w), and service identifiers (ranged over by A). Notationally, we will use a to range over addresses and r to range over addresses and partner links.... ..."

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### Table 4: A Reduction Calculus for DILL

"... In PAGE 9: ...?; x: A; ?0j ` x: A ?jx: A ` x: A ?j ` : I ?j 1 ` t: I ?j 2 ` u: A ?j ` let t be in u: A ?j 1 ` t: A ?j 2 ` u: B ?j ` t u: A B ?j 1 ` u: A B ?j 2; x: A; y: B ` t: C ?j ` let u be x y in t: C ?j ; x: A ` t: B ?j ` x: A:t: A? B ?j 1 ` t: A? B ?j 2 ` u: A ?j ` tu: B ?j ` t: A ?j `!t: !A ?j 1 ` t: !A ?; x: Aj 2 ` u: B ?j ` let t be !x in u: B In those rules with multiple premises, 1, 2 are disjoint and is a permutation of 1; 2. Formally, the reduction relation ) is the least congruence on terms con- taining the basic redexes given in Table4 . In the absence of proofs in the literature, we proved the basic results concerning reduction in DILL.... ..."

### 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 1.6. The concurrent object calculus with method guards The encoding of objects. To implement method guards in our basic cal- culus we use the full expressive power of name comparison. We also augment the calculus with integers and summations, with the expected semantics. The following translations [[ ]] and [[ ]]o take a process p and a record [ ], respectively, and give a process or a record in the calculus in Sections 1.2.4 and 1.3. We focus on the unique cases where [[ ]] and [[ ]]o are not homo- morphic:

### Table 2: The CPS translation of the call-by-name -calculus

"... In PAGE 3: ... For each type A of the -calculus, we define a pair of types KA and CA of the target calculus, called, respectively, the type of continuations and of computations of type A: KA = ~ A; if A is a basic type; K1 = 0; KA^B = KA + KB; KA!B = CA KB; K? = 1; KA_B = KA KB; CA = KA ! R: For each variable x and each name of the -calculus, we assume a distinct variable ~ x, respectively ~ , of the target calculus. The call-by-name CPS translation M of a typed term M is given in Table2 . It respects the typing in the following sense: x1:B1; : : : ; xn:Bn ` M : A j 1:A1; : : : ; m:Am ~ x1:CB1; : : : ; ~ xn:CBn; ~ 1:KA1; : : : ; ~ m:KAm ` M : CA : (1)... In PAGE 6: ... Continuation and computation types are defined as before: KA = VA ! R; if A is a basic type; K1 = 0; KA^B = KA + KB; KA!B = CA KB; K? = 1; KA_B = KA KB; CA = KA ! R: Notice that the value types VA are only defined at basic types. To the interpretation of terms from Table2 , we add the following interpretation of basic constants c : A and basic functions f : B1 ! : : : ! Bn ! A: c = k:k~ c; f = hx1; : : : ; xn; ki:x1( v1:x2( v2: : : : xn( vn:k( ~ fv1 : : : vn)))): Here k : KA, xi : CBi, and vi : VBi. Notice that the interpretation of c is actually a special case of that of f for n = 0.... ..."

### Table 4. Transition rules for unicast communication in !1-calculus.

"... In PAGE 8: ... Hence, the execution of a unicast send action of value x on channel z by a basic node with process interface G is represented by action label z:Gx; the corresponding receive action is labeled z:G(x). The semantic rules for unicast send (UNI-SEND), receive (UNI-RECV), and synchro- nization (UNI-COM) are given in Table4 . Scope extrusion via unicast communication is accomplished by naturally extending their -calculus counterparts (OPEN/CLOSE) rules as follows.... ..."

### 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.... ..."

### Table 1: Relations de nable in terms of C 2 The RCC Theory The RCC formalism (Randell et al. 1992) (when I refer to RCC I always refer to the theory presented in that paper) is an axiomatisation of certain spatial concepts and relations in classical 1st-order predicate calculus4. The basic theory assumes just one primitive dyadic relation: C(x; y) read as `x connects with y apos;. Individuals can be interpreted as denoting spatial regions. The C relation is re exive and symmetric, which is ensured by the following two axioms:

1995

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### Table 1: Composition table for 8 basic relations de nable in the RCC calculus. If R1(a,b) and R2(b,c), it follows that R3(a,c) where R3 is looked up in the table. \no info. quot; means that no base relation is excluded. Multiple entries in a cell are interpreted as disjunctions. Note that DR stands for DC and EC, PP for TPP and NTPP, PP?1 for TPP?1 and NTPP?1, TP?1 for TPP?1 and =, and O for PO, TPP, NTPP, TPP?1, NTPP?1, and =.

1994

"... In PAGE 2: ... A composition table gives compositions of all pairs of relations in some basis; and is a central component of any composition-based reasoning system. Table1 shows the composition table for eight basic topological relations de ned in the calculus of Randell, Cui and Cohn (1992). Relational composition (as de ned above) can be regarded as a special case of the more general notion of the locus of an unspeci ed relation.... In PAGE 4: ...i.e. a and b are related by any one of the base relations making up the (generally) disjunctive relation Comp(R1; R2)(a; c) then there mush exists some regions, say b, such that R1(a; b) and R2(b; c). But consider, for example, the composition table entry given in Table1 for the relations EC and TPP. This tells us that for any two regions, a and b, if there is some third region c such that EC(a; c) and TPP(c; b), then the relation between a and c must be either EC,PO or PP (PP means TPP or NTPP).... ..."

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