P. J. Freyd, P. W. O'Hearn, A. J. Power, R. Street, M. Takeyama, and R. D. Tennent. Bire ectivity. Theoretical Computer Science, 228(1-2):49-76, October 1999. Preliminary version in [9].  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with no recourse to models or external structures. The upshot is that we obtain slightly stronger results, suitable for generalizing beyond the scope of the current applications of build fusion. The paranaturality condition is a variation on the theme of functorial and structural polymorphism [4, 3]. But while the dinatural transformations of [4] allow too many terms, the structor morphisms of [3] precisely correspond to the polymorphic terms, but do not stipulate which of many possible choices of structors should be used to interpret the particular polytypes. We ll this gap, presently just ....

....results, suitable for generalizing beyond the scope of the current applications of build fusion. The paranaturality condition is a variation on the theme of functorial and structural polymorphism [4, 3] But while the dinatural transformations of [4] allow too many terms, the structor morphisms of [3] precisely correspond to the polymorphic terms, but do not stipulate which of many possible choices of structors should be used to interpret the particular polytypes. We ll this gap, presently just enough to analyze the programs to the initial and from the nal datatypes as parametric paranatural ....

P. J. Freyd. Structural polymorphism. Theoretical Computer Science, 115(1):107-129, 1993.

.... 6 Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma jAj jxj jA 0 j f g f 0 g 0 jBj jyj jB 0 j S t S 0 Morphisms in the scone of a product category resemble transformations of structors, described in [Fre93]. Let us now concentrate on the subcategory g A ThetaB of d A ThetaB. Working through the definition of g A ThetaB, we may regard the objects of this subcategory as tuples hS; f; g; A; Bi, where f and g are the coordinate functions of an inclusion S , jAj Theta jBj. In other words, since S, f ....

....a to b , and similarly for S 00 . These concepts were motivated by work of Wadler [Wad89] In a modified form (see Example 7. 10) they have been applied to the so called strictness analysis [AbJ91] We also note that although relators are slightly more general than tabular structors proposed in [Fre93], the Horn clause condition involved in the definition of relator transformations is basically the same as the condition defining transformations of tabular structors given in [Fre93] Tabular structors map morphisms to relations rather than relations to relations, so the change from tabular ....

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P.J. Freyd. Structural polymorphism. Theoretical Computer Science, 115:107--129, 1993.

....for some A and B. By adjointness we obtain two maps I A B which, by well pointedness, can be distinguished by a map 1 I . For any object D we can compose that map with D 1, thus showing that I is weakly terminal. For uniqueness of the map D I we make use of the following two facts. i) [15] If C is a well pointed category then there is only one natural endomorphism on the identity functor id C : C C. ii) 14] If (C; I) is a monoidal category then there is an injective function from C[I ; I ] into C C [id C ; id C ] It follows that the identity is the only endomorphism on ....

....a more important question is whether such an extension is of any interest. Fact 21 rules out the most important models of linear type theory that have been given in the literature. An exception is the model of passivity in syntactic control of interference, based on bire ective subcategories [29, 15]. But for that model additional properties, above and beyond those associated with , are crucial. Here we have discussed models that consist of a single category, possessing all of the properties required to model both linear logic or type theory and BI or . There is another way that one ....

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P.J. Freyd, P. W. O'Hearn, A. J. Power, R. Street, M. Takeyama, and R. D. Tennent. Bire ectivity. Theoretical Computer Science, 228(1-2):49-76, October 1999. Preliminary version in [8].

....for any categories C and D, and any functors C op Theta C D. This does not take nested polymorphic functions explicitly into account. Thus, the notion is precise, but not tied in any way to particular type theories. Peter Freyd s work on structors also a general flavour of this kind [Fre93] Directions and Problems. Broadly speaking, there is room and reason to investigate both generalizations and further applications of parametricity. For example, it has been suggested that the information hiding aspect of locally encapsulated state, as found in objects or higher order ....

P. J. Freyd. Structural polymorphism. Theoretical Computer Science, 115:107--129, 1993.

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P. J. Freyd, P. W. O'Hearn, A. J. Power, R. Street, M. Takeyama, and R. D. Tennent. Bire ectivity. Theoretical Computer Science, 228(1-2):49-76, October 1999. Preliminary version in [9].

.... rules are as follows: M [i( i( Contr (where = This gives us a limited form of Contraction for , in addition to the general Contraction for ; These rules can all be interpreted using the bire ective subcategory structure found in Tennent s model [10, 4]. 7.2 Implicit State Models A very di erent semantics of SCI can be obtained using ideas of Uday Reddy. In [21] Reddy presents a novel semantics of SCI in which the state is implicit in a history of events. The model is not cartesian closed, but it was subsequently shown in [11] how a Yoneda ....

P.J. Freyd, P. W. O'Hearn, A. J. Power, R. Street, M. Takeyama, and R. D. Tennent. Bire ectivity. Theoretical Computer Science, 228(1-2):49-76, October 1999. Preliminary version in [3].

....free expressions) use the state in a strictly read only or passive way. Reynolds also showed [50, 52] how to extend Hoare s logic [20] to full Algol like languages by allowing formal reasoning about non interference properties. Several researchers have been kept occupied for many years [70, 54, 71, 35, 30, 31, 14] addressing the syntactic, semantic, and even categorical challenges raised by this work. 7 8. Irreversibility In 1973, Strachey [69] pointed out a fundamental property of imperative programming that, it seems, had not previously been noted: The state transformation produced by obeying a ....

P. Freyd, P. W. O'Hearn, A. J. Power, M. Takeyama, R. Street, and R. D. Tennent. Bireflectivity. Theoretical Computer Science, 228:5--47, 1999.

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