A Category Theoretic Interpretation of Gandy's Principles for Mechanisms
SS. Alves and M. Pagani (Eds.): DCM 2018 and ITRS 2018EPTCS 293, 2019, pp. 85–92, doi:10.4204/EPTCS.293.7 c (cid:13)
J. Razavi & A. SchalkThis work is licensed under theCreative Commons Attribution License.
A Category Theoretic Interpretationof Gandy’s Principles for Mechanisms
Joseph Razavi Andrea Schalk
School of Computer ScienceUniversity of ManchesterManchester, UK [email protected] [email protected]
Based on Gandy’s principles for models of computation we give category-theoretic axioms describinglocally deterministic updates to finite objects. Rather than fixing a particular category of states, wedescribe what properties such a category should have. The computation is modelled by a functor thatencodes updating the computation, and we give an abstract account of such functors. We show thatevery updating functor satisfying our conditions is computable.
In a well-known paper [5], Gandy sets out principles which aim to characterize the possible behavioursof a discrete, deterministic mechanical computing device which could be realized in the physical world.Although in Gandy’s detailed axiomatization there are four principles, they can be summarized by twoconceptual insights (as emphasized by, for instance, [1, 14]): first, that states of computation should befinite objects with a bounded amount of local detail; second, that changes should only propagate with abounded velocity, and thus their effects on a given location should be determined by a neighbourhood offinite size.Gandy’s technical realization of these principles uses axioms which are set-theoretic in nature, the ideabeing that every mathematical description of a computing machine ought to correspond to a set-theoreticalone. As long as this is done sensibly, and the original model satisfies Gandy’s conceptual principles,the resulting formalization will satisfy Gandy’s axioms. Although some later studies keep within thisset-theoretic framework [14], other work inspired by Gandy’s principles replaces his arbitrary hereditarilyfinite sets with mathematical objects for which one has more direct spatial intuitions, such as graphs[1, 10, 13] or simplicial surfaces [4]. Indeed, when one has a concrete model of computation in mind, it isoften easier to conduct detailed investigations into the operation of Gandy’s principles by working withthe models directly, in an environment which respects their extra structure, rather than their set-theoreticencodings.In many of these models the dynamics are given in terms of a colimit of updated versions of certainneighbourhoods of a state. This is explicit in [9, 10], and suggested by the use of unions in [1, 4]. In[9] the colimit to be taken is strikingly similar to the one implied by Proposition 1 of this paper. It isinteresting that two quite different intuitions about the meaning of local determinism lead naturally tosome of the same categorical structures. Indeed, if one heuristically reads Gandy’s notion of restrictionof a state to a part of a state as analogous to a set of morphisms, then Gandy’s own description stronglysuggests a colimit. This is evidence that the ambient categorical structure can play an explanatory role.Rather than describing specific categories whose objects can serve as states of computation, one mightask what kinds of category could serve as a setting for Gandy-like models. We know that objects like6
A Category Theoretic Interpretation of Gandy’s Principles graphs and simplicial complexes are characterized by being colimits of a set of generating objects. Wemight wonder whether the colimit description of the dynamics follows from this, given some more directaxiomatization of local determinism.In this paper, we attempt to give just such a description of a class of categories suitable for describingstates of computation in the style of Gandy, along with a categorical description of locally deterministicdynamics. We came to this axiomatization in an attempt to better understand the phenomena of informationflow in cellular automata as described in the first author’s thesis [11]. For this purpose, they seem to beuseful for getting at the relevant aspects of the causal structure, and we hope, therefore, that they may beof use to workers engaged in similar enquiry.
In this section we briefly sketch some category theoretic background, loosely following [7, 15]; the readerinterested in a more comprehensive overview of category theory is directed to [8].In order to talk about things happening locally, as is required to model Gandy’s ideas, it is useful tothink of a morphism 𝑝 ∶ 𝐴 𝑋 in ℂ as a ‘shape 𝐴 located at position 𝑝 in 𝑋 ’. More generally, oneoften wants to restrict attention to the case where the source and target of the morphisms are the outputof two functors 𝐹 ∶ 𝕏 ℂ and 𝐺 ∶ 𝕐 ℂ . In this paper, we study such morphisms in the standardsetting: the comma category 𝐹 ∕ 𝐺 . The objects of 𝐹 ∕ 𝐺 are triples ( 𝑋, 𝑝, 𝑌 ) where 𝑋 is an object of 𝕏 , 𝑌 is an object of 𝕐 , and 𝑝 ∶ 𝐹 𝑋 𝐺𝑌 in ℂ . A morphism from ( 𝑋, 𝑝, 𝑌 ) to ( 𝑋 ′ , 𝑝 ′ , 𝑌 ′ ) is a pair ( 𝑓 , 𝑔 ) such that 𝐺𝑔 ∘ 𝑝 = 𝑝 ′ ∘ 𝐹 𝑓 .We may think of such an equation as telling us that the way the shape
𝐹 𝑋 is located at position 𝑝 in 𝐺𝑌 is compatible with that of 𝐹 𝑋 ′ at 𝑝 ′ in 𝐺𝑌 ′ via 𝐹 𝑓 and 𝐺𝑔 .The definition of objects of the comma category as triples allows us to define two functors, dom and cod , to 𝕏 and 𝕐 respectively, by projecting onto the first and last components. The morphisms in themiddle components of these triples assemble themselves into a natural transformation 𝐹 ∕ 𝐺 𝕏𝕐 ℂ , domcod 𝐹 mid 𝐺 and this square has a universal property which defines the comma category up to isomorphism.When giving a comma category, if we have a subcategory 𝔾 of ℂ , we often write 𝔾 to mean itsinclusion into ℂ , writing for example 𝔾 ∕ ℂ to mean the comma category from the inclusion of 𝔾 to theidentity on ℂ . Similarly, if 𝐴 is an object of ℂ , we may write 𝐴 to mean the constant functor from theone-object category to 𝐴 . As a first approximation, one can think of a comma category 𝐹 ∕ 𝐺 as a sort ofasymmetric pullback, asking not what 𝐹 and 𝐺 have in common, but what the result of probing 𝐺 with 𝐹 is. Certain properties are then unsurprising. The domain and codomain functors on ℂ ∕ ℂ both have asection, given by sending each object to its identity morphism. Moreover, if 𝐹 has a section, then so doesthe codomain functor of 𝐹 ∕ 𝐺 .In the introduction, we mentioned that in many examples we encounter functors which can becalculated by colimits. This may indicate the presence of a Kan extension, which is defined as follows.Suppose we have functors 𝐹 ∶ 𝕏 𝔻 and 𝐺 ∶ 𝕏 ℂ . A (left) Kan extension of 𝐺 along 𝐹 is afunctor 𝐿 ∶ 𝔻 ℂ equipped with a natural transformation 𝜅 ∶ 𝐺 𝐿 ∘ 𝐹 which is universal in thesense that for every 𝐾 ∶ 𝔻 ℂ with 𝛼 ∶ 𝐺 𝐾 ∘ 𝐹 there exists a unique mediator 𝜇 ∶ 𝐿 𝐾 such . Razavi & A. Schalk 𝛼 is equal to the pasting 𝕏 ℂ𝔻 ℂ𝔻 ℂ . 𝐺𝐹 𝜅𝐿 𝜇𝐾
On first encountering the definition, it is common to wonder whether when 𝐺 = 𝐿 ∘ 𝐹 , the identitytransformation exhibits L as a Kan extension. One soon realizes, however, that far away from the imageof 𝐹 , the values of 𝐿 may have little to do with 𝐺 ; this will preclude having the universal property. Anatural thing to demand to improve this situation is that 𝐹 have a section; one easily verifies that in thiscase every functor that post-composes with 𝐹 to give 𝐺 is a Kan extension of the latter along the former.We are interested in Kan extensions with additional properties.A (left) Kan extension 𝐿 of 𝐺 along 𝐹 as below is called absolute if for all 𝐻 ∶ ℂ 𝔼 the pasting
𝕏 ℂ 𝔼𝔻 ℂ 𝔼
𝐹 𝐺 𝐻𝜅𝐿 𝐻 is a Kan extension. This means that having a Kan extension for 𝐺 gives us a Kan extension for any compos-ite of 𝐺 . For example, suppose 𝐹 ∶ ℂ 𝔻 , 𝑈 ∶ 𝔻 ℂ with a natural transformation 𝜂 ∶ 1 ℂ 𝑈 ∘ 𝐹 .Then 𝜂 is the unit of an adjunction with left adjoint 𝐹 and right adjoint 𝑈 if and only if it exhibits 𝑈 asthe Kan extension of the identity along 𝐹 and this extension is absolute.A Kan extension 𝐿 of 𝐺 along 𝐹 as above is called pointwise if whenever we have a functor 𝐽 ∶ 𝕐 𝔻 and we consider the comma category 𝐹 ∕ 𝐽 , the pasting 𝐹 ∕ 𝐽 𝕏 ℂ𝕐 𝔻 ℂ cod dom
𝐹 𝐺 mid 𝜅𝐽 𝐿 is a Kan extension. We can think of a pointwise Kan extension as being ‘locally a Kan extension’ in thatwhenever we ‘probe’ 𝐿 with a functor 𝐽 as above, we still get a Kan extension. One reason pointwise Kanextensions are useful is that for any object 𝐷 of 𝔻 we can compute 𝐿𝐷 as the colimit of the functor givenby precomposing 𝐺 with dom ∶ 𝐹 ∕ 𝐷 𝕏 , which means that if we take the big diagram with one copyof each object 𝑋 of 𝕏 for every morphism from 𝐹 𝑋 𝐷 , and arrows between them making commutingtriangles, then 𝐷 is the colimit.One family of examples is given by the fact that every absolute Kan extension is pointwise. Anotherexample which one frequently encounters is the idea of a dense subcategory . A subcategory 𝔾 of ℂ iscalled dense if and only if the identity on ℂ is the pointwise Kan extension of the inclusion of 𝔾 into ℂ along itself, meaning that every object of ℂ is the colimit of all ways of mapping objects of 𝔾 into it. Wethink of objects in ℂ as regions generated by the shapes in the dense subcategory 𝔾 . In the sequel, this isused to capture the notion of ‘finite detail’. Since every morphism out of a colimit is uniquely determinedby suitable morphisms out of the constituent parts this also characterizes the morphisms in ℂ , which aredetermined by the way they act on morphisms out of 𝔾 by postcomposition. Similar reasoning shows thatthe domain of every object of ℂ ∕ ℂ is the colimit of all the domains of morphisms of 𝔾 ∕ ℂ into it (to seethat one does not end up with spurious extra copies of objects of 𝔾 , the important observation is that everymorphism in ℂ ∕ ℂ factors as a morphism with an identity of ℂ in its domain, followed by one with an8 A Category Theoretic Interpretation of Gandy’s Principles identity of ℂ in its codomain). This implies that the identity natural transformation corresponding to thecommuting diagram 𝔾 ∕ ℂ ℂ ∕ ℂ ℂℂ ∕ ℂ ℂ ∕ ℂ ℂ domdom is a pointwise Kan extension. The codomains of morphisms play little role in this argument, and indeedone can replace 𝔾 ∕ ℂ and ℂ ∕ ℂ in the above with 𝔾 ∕ 𝑈 and 𝔾 ∕ 𝑈 respectively, for any functor 𝑈 into ℂ .Kan extensions enjoy the property that if one vertically composes a collection of Kan extensionsquares, the result is again a Kan extension. Moreover, if every component of the pasting is absolute orpointwise, so is the result. Suppose one has in mind a collection of spatially extended states, and incomplete parts of states, for a‘mechanical’ model of computation, along with a notion for how the various parts fit together to formlarger parts and complete states. These constitute a category ℂ . Following Gandy, we stipulate that, if themodel is to be truly mechanical, these states must be finitely big and have a finite amount of possible localdetail.Let us first consider the restriction on ℂ corresponding to the objects being ‘finitely big’. Given twofinite combinatorial objects, we expect that the number of ways in which one fits into the other to be finite.This means that all hom-sets in ℂ should be finite; one says that ℂ is ‘locally finite’.Only slightly more subtle is the issue of finite local detail. The idea is that there should be a finite setof objects such that any object 𝑋 is determined by the ways in which these objects map into 𝑋 . This isexactly the condition described above for a dense sub-category, so this is the notion we use here.One example of a locally finite category with a finite dense subcategory is the category of finitegraphs: there are finitely many graph homomorphisms between any two graphs, and every graph can beconstructed as a colimit of nodes and edges. This is, however, a somewhat unnatural example for ourpurposes, since allowing nodes of arbitrarily high degree means that the notion of ‘locally finitely detailed’can not mean ‘locally’ in the usual sense for graphs. It is more natural to take graphs with a fixed boundon the degree.Another example, which we return to below, is the category 𝖳𝖺𝗉𝖾 whose objects are strings over thealphabet { ■ , □ } , and where a morphism from 𝐴 to 𝐵 is a position in 𝐵 at which 𝐴 occurs as a substring.For example, there are two morphisms from ■□■■ to ■□■■□■■ , since it occurs at positions and . For technical reasons, it is useful to assume that the empty string is an initial object, occurring just oncein every string. The full subcategory on the objects { ■ , □ , ■■ , ■□ , □■ , □□ } is dense, because anystring is determined by knowing its consecutive pairs of letters, and which pairs share individual letters. Ina similar vein, one might think of objects made of a finite number of tiles (even of, say, a Penrose tiling),generated by gluing together neighbourhoods where they share tiles.The issue of local determinism is more interesting. First, we suppose that the way in which states areupdated by the dynamics is given by a functor 𝑈 ∶ ℂ ℂ . Functoriality is a reasonable requirement,meaning that the updated versions of the parts of a state should fit together in the updated version of thatstate. It loosely corresponds to Gandy’s idea that the update function in his model should be ‘structural’,able only to observe the way atomic parts of the state fit together, not the names we have given them. . Razavi & A. Schalk { ■ , □ } in which a white cell becomes black if any of its neighbours is black. We can model this as a functor 𝑈 ∶ 𝖳𝖺𝗉𝖾 𝖳𝖺𝗉𝖾 which removes the two outer cells in a string (returning the empty string if there are toofew cells) and updates those in the middle according to the rule. For example 𝑈 ( ■□□□■□ ) = ■□■■ Functoriality of 𝑈 corresponds roughly to the fact that the update can not make use of, for example, theposition of a cell in the sequence to determine its new colour.Now we come to local determinism itself. Suppose we update a state, and then look at a part of theupdated situation. This amounts to considering a map 𝑝 ∶ 𝐴 𝑈 𝑋 for some 𝐴 and 𝑋 in ℂ . We want toremember the state 𝑋 we are updating, so it is best to consider 𝑝 as an object of ℂ ∕ 𝑈 . Since we postulatethat the action of 𝑈 is locally deterministic, this local effect must have had some local cause 𝑓 ∶ 𝑁 𝑌 ,for some 𝑁 and 𝑌 in ℂ , which explains why 𝑝 occurs in 𝑈 𝑋 in the following sense. If we update 𝑓 toget 𝑈 𝑓 ∶ 𝑈 𝑁 𝑈 𝑌 , we should find 𝑝 inside the result via a morphism 𝜂 ∶ 𝑝 𝑈 𝑓 in ℂ ∕ 𝑈 . Becauseof the way the comma category is defined, this implies that there is a morphism cod( 𝜂 ) ∶ 𝑋 𝑌 in ℂ .We think of 𝑓 as a ‘causal neighbourhood’ of 𝑝 . Although we could, if we wished, choose any causalneighbourhood, it is best to choose one which comes with a reasoning principle. Since we are supposingthat ‘changes propagate with a bounded velocity’, there ought to be a minimal causal neighbourhood suchthat any sequence of changes which could have contributed to the updated state 𝑝 must have passed intoit. This amounts to saying that for every other causal neighbourhood, that is for every 𝑔 ∶ 𝑀 𝑍 in ℂ with a morphism 𝛾 ∶ 𝑝 𝑈 𝑔 in ℂ ∕ 𝑈 , there is a unique ̂𝛾 ∶ 𝑓 𝑔 in ℂ ∕ ℂ such that 𝛾 = 𝑈 ̂𝛾 ∘ 𝜂 .Since we want this assignment of a causal neighbourhood to vary naturally as we vary 𝑝 , it amounts toan adjunction; but an adjunction where? The simplest option is to consider the functor 𝑈 ⊲ ∶ ℂ ∕ ℂ ℂ ∕ 𝑈 which takes an object ( 𝑋, 𝑓 ∶ 𝑋 𝑌 , 𝑌 ) of ℂ ∕ ℂ to ( 𝑈 𝑋, 𝑈 𝑓 , 𝑌 ) in ℂ ∕ 𝑈 . Note that the target compon-ent 𝑌 is unchanged. It is for this functor that we demand a left adjoint, say 𝐹 ⊲ .What does this all mean in our example? Consider the occurrence of ■■ at position in ■□■■ .We must find a causal neighbourhood which explains why it occurs. We choose the occurrence of □□■□ at the end of ■□□□■□ . This updates precisely to the part we wanted, and it must be the universalexplanation since anything smaller would only be updated to a single cell. Now the reader may be troubledby a subtle point: one might have expected that the explanation for the above should be the occurrence of □■ at position in the input, since we know intuitively that its left-hand square will turn black. If wehad defined 𝑈 to make use of this knowledge, however, then we could not have defined 𝐹 ⊲ in a functorialmanner, since we would have had 𝑈 ( ■□■ ) = ■■■ . But then the middle ■ could be explained just aswell by ■□ on the left as by □■ on the right. This is why we think in terms of neighbourhoods throughwhich all causal influences on an updated part must have passed. This issue is related to the problem of‘overdetermination’ familiar to philosophers (see e.g. [12]).To enforce the bounded speed of propagation, we need to stipulate that 𝐹 ⊲ have a certain finitenessproperty. For any given object 𝐴 , there are many possible shapes of causal neighbourhood for parts ofshape 𝐴 , depending on the context in which we find them. However, since there is a bound on the size ofthese, and there can only be finitely many objects of this size, we know that any part 𝑝 ∶ 𝐴 𝑈 𝑋 hasonly finitely many different shapes of possible causal neighbourhoods. Therefore we demand wheneverwe have a subcategory 𝕊 of ℂ ∕ 𝑈 whose image under the domain functor, dom[ 𝕊 ] , is finite, the domain ofits image under 𝐹 ⊲ , which is (dom ∘ 𝐹 ⊲ )[ 𝕊 ] , is also finite.In our example, consider all morphisms out of the object ■ into outputs of 𝑈 . Although this is aninfinite set of morphisms (because there are an infinite number of strings containing ■ ), each of theseoccurrences is explained by a substring in the input of one of the forms ■■■ , ■■□ , ■□■ , ■□□ , □■■ , □■□ , or □□■ . If we did not stipulate that this collection be finite, then different occurrencesof ■ in different outputs of 𝑈 could have been explained by substrings of arbitrary length in the input.0 A Category Theoretic Interpretation of Gandy’s Principles
This would allow such behaviour as outputting a single cell which is black if and only if the input codes ahalting Turing machine!The example we have given is somewhat restrictive, since the partial states we mention always shrink,and otherwise do not change shape very much. This is not a necessary limitation. For instance, we couldintroduce special ‘blank’ cells on the ends of the strings which, rather than being removed by 𝑈 , areduplicated to allow the working area to grow. More radical changes to the intuitive shape of parts arepossible, but harder to describe. Putting all our conditions together we get the following. Definition 1. A categorical Gandy machine is an endofunctor 𝑈 ∶ ℂ ℂ where ℂ is a locally finitecategory with a finite dense subcategory, such that: • the induced functor 𝑈 ⊲ ∶ ℂ ∕ ℂ ℂ ∕ 𝑈 has a left adjoint 𝐹 ⊲ , and • for all subcategories 𝕊 of ℂ ∕ 𝑈 such that dom[ 𝕊 ] , is finite, we have that (dom ∘ 𝐹 ⊲ )[ 𝕊 ] is also finite.We are now in a position to prove that every 𝑈 satisfying these conditions is computable. We do thisin two steps. Proposition 1.
Let 𝑈 ∶ ℂ ℂ be a categorical Gandy machine. Let 𝔾 be a dense subcategory of ℂ which gives an induced inclusion 𝔾 ∕ 𝑈 ℂ ∕ 𝑈 . Then 𝑈 is the pointwise (left) Kan extensionof dom along (dom ∘ 𝐹 ⊲ ) when both are precomposed with this inclusion.Proof. Let 𝜂 be the unit of the adjunction between 𝑈 ⊲ and 𝐹 ⊲ , and consider the diagram 𝔾 ∕ 𝑈 ℂ ∕ 𝑈 ℂℂ ∕ 𝑈 ℂ ∕ 𝑈 ℂℂ ∕ ℂ ℂ ∕ 𝑈 ℂℂ ℂ . dom 𝐹 ⊲ 𝜂 domdom 𝑈 ⊲ dom 𝑈 The top row is a pointwise Kan extension by density of 𝔾 . The middle row is a pointwise Kanextension since its left-hand square comes from an adjunction. It is thus an absolute Kan extension,preserved by dom . The bottom row is a pointwise Kan extension since it commutes and dom has a section.Then the result follows by pasting of pointwise Kan extensions. Corollary 2.
Let 𝑈 ∶ ℂ ℂ be a categorical Gandy machine. Suppose that objects of ℂ are representedby diagrams in the finite dense subcategory for which they are the colimit. Then 𝑈 is computable.Proof. Suppose for some object 𝑋 of ℂ , we want to compute 𝑈 𝑋 . The above Proposition implies that
𝑈 𝑋 is the colimit of all objects which can occur as the domain of an object 𝑔 of 𝔾 ∕ 𝑈 such that the domainof 𝐹 ⊲ 𝑔 admits a morphism into 𝑋 . Since 𝔾 is finite, dom[ 𝔾 ∕ 𝑈 ] = 𝔾 is finite. Hence, dom[ 𝐹 ⊲ [ 𝔾 ∕ 𝑈 ]] isfinite. This is the category in which the diagram whose colimit is 𝑈 𝑋 must take its values. Note that, inaddition to being finite, it is independent of 𝑋 . Therefore, it can be ‘precomputed’ for use in evaluating 𝑈 .In order to evaluate 𝑈 𝑋 , all commuting triangles with one side in this precomputed category and apex 𝑋 must be computed. Since ℂ is locally finite, this is a finite diagram (and it can actually be computedSince a morphism into 𝑋 is determined by its action on morphisms in 𝔾 , and one can ‘try all possibilities’for such actions; if desired, a similar idea can be used to ensure we have the ‘standard’ diagram whosecolimit is 𝑋 ). . Razavi & A. Schalk The present work is the first step in a programme to study spatial models of computation from a categoricalperspective. Our original motivation was to study the flow of information in such models along the linesof [11]. We are particularly interested in the question of where in the spatially extended state is storedwhich piece of information about the computation. In previous work we lacked a good axiomatization ofsuitable structures which we now provide. We believe this framework to be quite robust, and in this paperwe show how its definition may be thought to arise from Gandy’s principles.A natural next step is to strengthen the connections with these principles by investigating whetherevery example satisfying Gandy’s original axioms can be interpreted in this framework. One obvioushurdle to overcome is that one has to choose a sensible category of states. This can not always haveGandy’s set of states as objects. For example, Gandy points out that the set of finite structures for afirst-order logical signature satisfies his axioms. However, if the signature contains function symbols, thenthe category of structures, while locally finite, does not have a finite dense subcategory. For example,consider a signature with a single function symbol, and interpret this by successor taken modulo 𝑛 onthe set {0 , ..., 𝑛 − 1} . These structures each admit no incoming morphisms from any other structure, andform an infinite family. The solution is to consider some notion of partial structure. This corresponds toadding objects not only for Gandy’s states, but also enough parts to cover at least the ones important forthe structure of the machine.It may also happen that the definition we give corresponds to a well-behaved subclass of Gandy’smachines (and perhaps to well-behaved classes of examples of the other models inspired by Gandy). Anexample of the limitations of the present model is given by the functor on the category of finite directedgraphs which, wherever it finds a path of length two 𝑥 → 𝑦 → 𝑧 in its input, adds a direct edge 𝑥 → 𝑧 ,taking a single step carried out by the obvious algorithm for transitive closure. This is not an instance ofthe present framework, since it will add the diagonal to a rectangle like ∙ ∙∙ ∙ but there is no unique smallest causal neighbourhood. This is related to the problem of overdeterminationmentioned above. This operation is, intuitively, locally deterministic for the usual notion of ‘locally’ in agraph. However, as we discussed earlier, if nodes of arbitrarily high degree are allowed, then our definitiontakes a strange view of the meaning of ‘local’.Comparisons with later models, especially [9], may be more direct. In [1], a definition of localdeterminism is given in which a phenomenon in an updated graph is explained by a sub-graph of boundedradius around an ‘antecedent’ in the previous state. This is shown to be equivalent to taking a union oflocal updates. The analogy between this result and Proposition 1 above raises the question whether theother results of [1] have counterparts in the present abstract setting. A different abstract perspective onthis idea, based on topology, is presented in [2, 3]; this may aid the comparison with [1]. To facilitate thisstudy, a full version of the present work containing more detailed proofs and examples will be presentedin a forthcoming paper.Other important questions are suggested by the key role played by finite categories in Definition 1. Itwould be interesting if the present extrinsic notion of finite category could be replaced by a more intrinsiccondition. In [6], the impact on models of computation of varying the notion of finiteness is discussed.Beyond this, in future work we plan to return to our original aims, and put this axiomatization to useto give a cleaner account of [11], and the flow of information in spatial models.2 A Category Theoretic Interpretation of Gandy’s Principles
Acknolwedgements.
We would like to thank Pouya Adrom for a timely pointer to Kan extensionsand Francisco Lobo for a number of very fruitful discussions about the structures we use.
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