H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990. 156  Home/Search   Document Not in Database   Summary   Related Articles   Check

....exists an arrow YAf : A, such that f o Af = A category C has all fixpoints, if every A 6 C, which has at least one constant arrow a: z A which does not contain YA, has fixpoints. The condition that there is at least one normal constant arrow avoids inconsistencies such as described in [6]. If we wouldn t require this, we could construct constants like YA (idA) which can even be made for empty types Fixpoints allow us to encode recursive arrows like (f] Proposition 20 In a category which has all fixpoints, the homomorphism construction ( for a type functor t which has a ....

Hagen Huwig and Axel Poign4, A note on inconsistencies caused by fixpoints in a cartesian dosed category, Theoretical Computer Science 73 (1990), p. 101-112.

....have canonical and minimal fixed points. This has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty or binary) Also algebraic compactness (which yields zero objects) is inconsistent with cartesian closure. This, in principle, precludes a unified treatment of sums, ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....that we are dealing with a linear category. predI a is actually a linear category, and the category of free coalgebras is equivalent to coK(predI a ) which is isomorphic to predI s , the category of pre dI domains and continuous stable functions. This category has finite sums, but according to [HP90], a cartesian closed category with fixpoints and finite sums is equivalent to the category with one object and one arrow. Thus, predI s cannot have fixpoints of arbitrary maps which entails that predI a cannot have linear fixpoints of arbitrary maps. But we do only need linear fixpoints of maps ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73, 1990.

....of G: I = der ; G using subscripts to distinguish the category in which the [ operation is working. Again this is only a weak coproduct; indeed, I is a cartesian closed category with fixed points of all endomorphisms, so it is impossible for genuine coproducts to exist [15]. Recursive types can also be modelled, as outlined above. We therefore have everything we need to interpret FPC. The type constructors , and # are modelled using their counterparts in the category, and a recursive type T. is interpreted as the canonical solution of the domain equation D ....

H. Huwig and A. Poign e, A note on inconsistencies caused by fixpoints in a cartesian closed category, Theoretical Computer Science, vol. 73 (1990), pp. 101--112.

....examples, the WCPC category C (A) induced by a PCA A and the category of algebraic lattices ALat have weak finite coproducts. The category ALat does not have true finite coproducts because it is cartesian closed and it has the fixed point property (i.e. every endomorphism has a fixed point) see [11]. Thus, to get the desired Corollary (3.8) it really is important that Theorem 3.7 only requires weak coproducts. Corollary 3.8. The realizability pretriposes over the WCPC category induced by a PCA and over the WCPCcategory ALat both have split disjunction. Realizability Pretriposes and ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....examples, the WCPC category C (A) induced by a PCA A and the category of algebraic lattices ALat have weak finite coproducts. The category ALat does not have true finite coproducts because it is cartesian closed and it has the fixed point property (i.e. every endomorphism has a fixed point) see [11]. Thus, to get the desired Corollary (3.8) it really is important that Proposition 3.7 only requires weak coproducts. Corollary 3.8. The realizability pretriposes over the WCPC category induced by a PCA and over the WCPCcategory ALat both have split disjunction. Realizability Pretriposes and ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....morphisms between dcpos are embedding projection pairs. The colimits are exactly the limits of these # chains [Jung 1990] It is also known that any nontrivial cartesian closed category with the fixed point property (e.g. DCPO , the category of dcpos with least element) cannot be cocomplete [Huwig, Poigne 1990]. In the general case, where the diagram is a small category and the underlying morphisms are arbitrary continuous functions, the existence of such colimits in DCPO isn t known very well. A nonconstructive proof can be found in [Meseguer 1977] The cocompleteness of DCPO has also been proved (in a ....

Huwig H., Poigne A. [1990] A Note on Inconsistencies Caused by Fixpoints in a Cartesian Closed Category, Theoretical Computer Science 73, p.101-112.

....predI a is cartesian closed, and predI a has linear fixpoint operators if and only if the category of free coalgebras has fixpoints in the usual sense. But the category of free coalgebras is equivalent to predI a , and it is easy to see that predI a is isomorphic to predI s . Now, according to [HP90], a cartesian closed category with fixpoints and finite sums is equivalent to the category with one object and one arrow. So predI s cannot have fixpoints since it is cartesian closed and has finite sums. This entails that predI a cannot have linear fixpoints. But if we cut our model down to the ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73, 1990.

....forward. Filinski uses this duality to define the typed symmetric lambda calculus (SLC) in which functions can either be seen as value abstractions or continuation abstractions. He models this using a bicartesian closed category. We know that no nontrivial biCCC can support fixpoints (see [17]) but Filinski assures us that as the SLC is typed, we need a special operator to write recursive definitions. By doing so we will lose the strong normalization property. However, this presents surprisingly few new problems, as nontermination can be viewed as a special case of escaping. Thus, all ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....of the recursive type constructor . The categorical structure for interpreting the sum type constructor is standard; that is, given by binary coproducts. It then follows that the interpretation of the product type constructor Theta cannot be standard, viz. given by products (cf. [20, 16]) However, the category of games G admits a pretensor constructor Omega (see Section 2, x Type structure) which is a product in the subcategory, G t , of total innocent strategies [15] This situation, which is typical in models of FPC [9, 10] is the one that we axiomatise. The structure for ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....has a fixpoint, Theorem 2.1.19. One therefore looks for substitutes for disjoint union which retain pointedness, but, of course, one cannot expect a clean categorical characterization such as for cartesian product or function space. See also Exercise 3.3. 12(12) In fact, it has been shown in [Huwig and Poign e, 1990] that we cannot have cartesian closure, the fixpoint property and coproducts in a non degenerate category. So let us now restrict attention to pointed dcpo s. One way of putting a family of them together is to identify their bottom elements. This is called the coalesced sum and denoted D Phi E. ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....has a fixpoint, Theorem 2.1.19. One therefore looks for substitutes for disjoint union which retain pointedness, but, of course, one cannot expect a clean categorical characterization such as for cartesian product or function space. See also Exercise 3.3. 12(12) In fact, it has been shown in [Huwig and Poign e, 1990] that we cannot have cartesian closure, the fixpoint property and coproducts in a non degenerate category. So let us now restrict attention to pointed dcpo s. One way of putting a Domain Theory 43 family of them together is to identify their bottom elements. This is called the coalesced sum and ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....have canonical and minimal fixed points. This has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty or binary) Also algebraic compactness (which yields zero objects) is inconsistent with cartesian closure. This, in principle, precludes a unified treatment of ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

....domain equations involving the coproduct can be problematic, however. There are categorical impediments to the solution of some equations. For example, the equation D = 1 (D D) where 1 is the terminal object) has no solution in a any non trivial bicartesian closed category (see [12] and [6]) Moreover, there are equations which have a non trivial solution in a bicartesian closed category but have no non trivial solution over the profinites. We provide a condition which, in effect, reduces the problem of solving an equation over the profinite domains to one of getting a finite poset ....

Huwig, H. and Poign' e, A., A note on inconsistencies caused by fixpoints in a cartesian closed category. Manuscript, 1986, 16 pp.

....the Curry Howard isomorphisms be understood as having logical meaning. The role of the extensions are to extend the translation from terms in the calculus to terms in the linear calculus to a translation from terms in the rec calculus to terms in the linear rec calculus. It is shown in [HP90] that a cartesian closed category with finite sums and a fixpoint operator is inconsistent, that is, it is equivalent to the category consisting of one object and one arrow. But the category of CPOs and strict continuous functions is a consistent linear category with finite sums and a linear ....

....sums, which is easily seen not to have an internal fixpoint operator. The reader is invited to figure out how a fixpoint of the twist map [in 2 ; in 1 ] 1 1 1 1 might look in this category. The result that a cartesian closed category with finite sums and a fixpoint operator is inconsistent, [HP90], also implies that the category of bottomless dI domains and non empty join preserving stable functions can not have a fixpoint operator. It should be noted that the category of dI domains and non empty join preserving stable functions as well as the category of bottomless dI domains and ....

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H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73, 1990.

....the standard axiomatization of its equational theory is sufficient for the coherence theorem. But when we add variants, the standard axiomatization of these features, while sufficient for coherence, clashes with the standard axiomatization of recursive types, yielding an inconsistent theory (see [Law69, HP89a] for variants, that is, coproducts) The solution lies in two observations: 1) the (too) strong axioms are only needed for coercion terms , and (2) in the various models we examined these coercion terms have special interpretations (such as strict, or linear maps) so special in fact, that they ....

....case M of l 1 ) inj l 1 ; Q) l n ) inj l n ; Q) where M : l 1 : t 1 ; l n : t n ] Q: l 1 : t 1 ; l n : t n ] t . More precisely, it is possible to check that the system fVART BETAg fVART COPg is equivalent to fVART BETAg fVART CRNg fVART ETAg. However, it is known [Law69, HP89a] that fVART BETAg fVART COPg is inconsistent with the existence of fixed points. In fact, this may be refined: Proposition 5 The system fVART BETAg fVART CRNg is (equationally) inconsistent with the existence of fixed points. Proof: The categorical equation f VART COP g may be thought of as ....

H. Huwig and A. Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, ??:??--??, 1989. To appear.

....B # u C (A B) # # 1 ( #(u) #(v) C which can be shown to be natural in # . This would be equivalent to using the natural isomorphism (A B) # # = A # ) B # ) given by the observation that the functor ( # has a right adjoint and thus preserves sums. But in [HP90] a cartesian closed category with finite sums together with a fixpoint operator is shown to be inconsistent , that is, it is equivalent to the category consisting of one object and one map. So we have to be content with a less demanding assumption than finite sums: Definition16. A categorical ....

H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73, 1990.

....CBN category is in fact cartesian closed, with 1 as terminal object, A Theta B as product and [A B] as exponential. However, the behavior of A B is not what we would expect it is too lazy. In fact, there is no way to add proper coproducts to a language where every function has a fixpoint [HP90], but just as for lazy products in CBV, we can define the more conventional eager coproducts A qB : 0 A] 0 B] with the interpretation [ A q B] A] B] A] Phi [ B] which make it possible to evaluate a coproduct typed datum until its injection tag is known but ....

Hagen Huwig and Axel Poign'e. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990.

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H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73:101--112, 1990. 156

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H. Huwig and A. Poigne. A note on inconsistencies caused by fixpoints in a cartesian closed category. Theoretical Computer Science, 73(1):101--112, 1990.

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H. Huwig and A. Poigne, A Note on the Inconsistencies Caused by Fixpoints in a Cartesian Closed Category, Theoretical Computer Science 73, 1990.

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