### TABLE 2 Complexity of monadic query answering on monadic indefinite databases

### Table 11 Strong Monads

"... In PAGE 42: ...) Now a ?-autonomous category with ( nite and nullary) products and co- products (one implies the other because of the duality) is a model of classical linear logic; for convenience, we call the product N and the coproduct . We add to this the following: Definition 13 (Strong Monads) A strong monad is a monad h ( ); ; i on C, together with a natural transformation : ( ) ! ( ) such that the diagrams in Table11 commute.... 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 Table11 , 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.... In PAGE 50: ...and y commutes because of the bottom diagram in Table11 . (The small triangles commute by de nition.... ..."

### Table 11 Strong Monads

"... In PAGE 40: ...) Now a ?-autonomous category with ( nite and nullary) products and co- products (one implies the other because of the duality) is a model of classical linear logic; for convenience, we call the product N and the coproduct . We add to this the following: Definition 13 (Strong Monads) A strong monad is a monad h ( ); ; i on C, together with a natural transformation : ( ) ! ( ) such that the diagrams in Table11 commute.... In PAGE 46: ... 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 Table11 , 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.... In PAGE 47: ... A A0 f f0/ / amp; amp; L L L L L L L L L L ( B) B0 ( g) g0 / / tB;B0 ? ( C) C0 C IdC0 / / t C;C0 ( C) C0 tC;C0 / / (C C0) (B B0) (g g0) / / apos; apos; O O O O O O O O O O O ( C C0) tC;C0 y (C C0) C C0 6 6 m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m The top edge is the C-morphism corresponding to (g f) (g0 f0), whereas the composite of the bottom two edges is the C-morphism corresponding to (g g0) (f f0). However, ? commutes because t is a natural transformation, and y commutes because of the bottom diagram in Table11 . (The small triangles commute by de nition.... ..."

### Table 2: Inference Rules of Many Sorted Monadic Equational Logic

1991

"... In PAGE 7: ...omplex assertions, e.g. formulas of rst order logic, then they should be interpreted by subobjects; in particular equality = : A should be interpreted by the diagonal [[A]]. The formal consequence relation on the set of equations is generated by the inference rules for equivalences ((re ), (simm) and (trans)), congruence and substitutivity (see Table2 ). This formal consequence relation is sound and complete w.... In PAGE 12: ...7 Given a signature for the programming language, let be the signature for the metalanguage with the same base types and a function p: 1 ! T 2 for each command p: 1 * 2 in . The translation from programs over to terms over is de ned by induction on raw programs: x [x]T (let x1(e1 in e2) (letT x1(e1 in e2 ) p(e1) (letT x(e1 in p(x)) [e] [e ]T (e) (letT x(e in x) The inference rules for deriving equivalence and existence assertions of the simple programming language can be partitioned as follows: general rules (see Table 6) for terms denoting computations, but with variables ranging over values; these rules replace those of Table2 for many sorted monadic equational logic rules capturing the properties of type- and term-constructors (see Table 7) after interpretation of the programming language; these rules replace the additional rules for the metalanguage given in Table 4.... ..."

Cited by 585

### Table 1: Monad transformers

1994

"... In PAGE 10: ... Applying two (di erent) environment monad transformers to the identity monad indeed yields the double environment monad. Other monad transformers are listed in Table1 . Notice that composition of transformers is not commutative.... In PAGE 13: ... Levels may have multiple names because conceptually distinct levels may coincide. For instance, level 6 of Table1 is known to store constructs as stores and to environment constructs as env-results. Of course, distinct levels must have distinct names.... ..."

Cited by 5

### Table 1. Some classes of monads

"... In PAGE 2: ... Each class of monad has some specific operations apart from the predefined return and (greatermuch=). Table1 contains some classes of monads with their operations.... ..."

### Table 1. Some classes of monads

"... In PAGE 3: ... [39,40,5]. Table1 presents two classes of monads that will be used in the rest of the paper. Table 1.... ..."

### Table 2: The monadic typing rules

909