## A CATEGORICAL OUTLOOK ON CELLULAR AUTOMATA

Citations: | 2 - 0 self |

### BibTeX

@MISC{Capobianco_acategorical,

author = {Silvio Capobianco and Tarmo Uustalu},

title = {A CATEGORICAL OUTLOOK ON CELLULAR AUTOMATA},

year = {}

}

### OpenURL

### Abstract

Abstract. In programming language semantics, it has proved to be fruitful to analyze context-dependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors of cellular automata with coKleisli maps of the exponent comonad on the category of uniform spaces and uniformly continuous functions and exploit this equivalence to conclude some standard results about cellular automata as instances of basic category-theoretic generalities. In particular, we recover Ceccherini-Silberstein and Coornaert’s version of the Curtis-Hedlund theorem. 1.

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Citation Context ...ndeterminism, probabilistic choice, state, continuations using monads. Specifically, effectful functions are identified with Kleisli maps of a suitable monad on the category of pure functions. Wadler =-=[18]-=- put this view to further use in programming methodology by extracting from it a very useful programming idiom for purely functional languages like Haskell. The dual view of context-dependent function... |

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Citation Context ...ances of basic category-theoretic generalities. In particular, we recover Ceccherini-Silberstein and Coornaert’s version of the Curtis-Hedlund theorem. 1. Introduction Since the seminal work of Moggi =-=[13]-=-, it has become standard in programming language semantics to analyze functions producing effects such as exceptions, input, output, interactive input-output, nondeterminism, probabilistic choice, sta... |

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Citation Context ...w can be useful. We see that it is: we can conclude some standard results about cellular automata as instances of category-theoretic generalities. In particular, we recover the Curtis-Hedlund theorem =-=[10]-=-—a characterization of global behaviors of cellular automata—in the version of Ceccherini-Silberstein and Coornaert [4] (this applies to general discrete alphabets rather than finite alphabets only). ... |

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Citation Context ...e exists U ∈ U such that {y ∈ A | (x, y) ∈ U} ⊆ Ω. Such topology is Hausdorff if and only if ⋂ U∈U U = ∆. The discrete uniformity induces the discrete topology, but is not the only one that does (cf. =-=[11, I-5]-=-), i.e., uniform spaces may be discrete without being uniformly discrete. A map f : A → B between uniform spaces is uniformly continuous (briefly, u.c.) if, for every entourage V on B, there is an ent... |

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Citation Context ...ul programming idiom for purely functional languages like Haskell. The dual view of context-dependent functions as coKleisli maps of a comonad is equally useful, but less well known. Brookes and Geva =-=[2]-=- explained the “intensional” aspect of denotational semantics in terms of the ω-chain comonad on the category of ω-cpos. More recently, Uustalu and Vene [15, 16, 17] employed comonads to analyze dataf... |

5 |
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Citation Context ...B. Then 1D views compose exactly as their 2D originals. We see that it makes sense to consider maps of categories other than Unif! 1D views of 2D cellular automata were of interest to Dennuzio et al. =-=[6]-=-�� �� �� �� � �� A CATEGORICAL OUTLOOK ON CELLULAR AUTOMATA 11 8. Comonad maps and point-dependent cellular automata To make a category out of comonads over a fixed category C one needs a suitable no... |

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Citation Context ...: in this case, the exponential topology on the space of continuous functions from C to A is the compact-open topology generated by the sets {f : C → A | f(K) ⊆ U} with K compact in C and U open in A =-=[7, 8]-=-. In particular, discrete spaces are exponentiable (which also follows from the discrete topology making every function from it continuous) and their compactopen topology is in fact the product topolo... |

3 |
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Citation Context ...eful, but less well known. Brookes and Geva [2] explained the “intensional” aspect of denotational semantics in terms of the ω-chain comonad on the category of ω-cpos. More recently, Uustalu and Vene =-=[15, 16, 17]-=- employed comonads to analyze dataflow computation and attribute grammars and Hasuo et al. [9] treated tree transducers. Characteristic of many context-dependent notions of computation is shapepreserv... |

2 |
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Citation Context ...ntics in terms of the ω-chain comonad on the category of ω-cpos. More recently, Uustalu and Vene [15, 16, 17] employed comonads to analyze dataflow computation and attribute grammars and Hasuo et al. =-=[9]-=- treated tree transducers. Characteristic of many context-dependent notions of computation is shapepreserving transformation of some datastructure based on a value update rule which is local in nature... |

1 |
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Citation Context ...tor from coKl(D ′ ) to coKl(D) and a functor from coEM(D) to coEM(D ′ ). We now introduce point-dependent cellular automata (studied under the name of non-uniform cellular automata by Cattaneo et al. =-=[3]-=-). For a set G, the local behavior of a point-dependent cellular automaton between uniform spaces A, B is a uniformly continuous function k : A G × G → B. Note the added second argument compared to th... |

1 |
A generalization of the Curtis-Hedlund theorem
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(Show Context)
Citation Context ...ry-theoretic generalities. In particular, we recover the Curtis-Hedlund theorem [10]—a characterization of global behaviors of cellular automata—in the version of Ceccherini-Silberstein and Coornaert =-=[4]-=- (this applies to general discrete alphabets rather than finite alphabets only). This theorem turns out to be an instance of the basic category-theoretic fact that the coKleisli category of a comonad ... |

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1 |
sigfpe” (2006) Evaluation of cellular automata is comonadic. Entry on the author’s blog, A Neighborhood of Infinity. http://blog.sigfpe.com/2006/12/ evaluating-cellular-automata-is.html
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Citation Context ...IANCO AND T. UUSTALU It should therefore be worthwhile to test the slogan that context-dependent computation is comonadic also on cellular automata. To a degree, this has already been done, as Piponi =-=[14]-=- programmed cellular automata in Haskell using the comonadic interface. However, he did not use his modeling of cellular automata to prove properties about them and also dropped the classical requirem... |

1 |
The essence of dataflow programming. (2006
- Uustalu, Vene
(Show Context)
Citation Context ...eful, but less well known. Brookes and Geva [2] explained the “intensional” aspect of denotational semantics in terms of the ω-chain comonad on the category of ω-cpos. More recently, Uustalu and Vene =-=[15, 16, 17]-=- employed comonads to analyze dataflow computation and attribute grammars and Hasuo et al. [9] treated tree transducers. Characteristic of many context-dependent notions of computation is shapepreserv... |

1 |
Comonadic evaluation of attribute grammars
- Uustalu, Vene
(Show Context)
Citation Context ...eful, but less well known. Brookes and Geva [2] explained the “intensional” aspect of denotational semantics in terms of the ω-chain comonad on the category of ω-cpos. More recently, Uustalu and Vene =-=[15, 16, 17]-=- employed comonads to analyze dataflow computation and attribute grammars and Hasuo et al. [9] treated tree transducers. Characteristic of many context-dependent notions of computation is shapepreserv... |