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Journal Articles (Review Article) Theoretical Computer Science Year : 2024

Inductive definitions in logic versus programs of real-time cellular automata

Abstract

Descriptive complexity provides intrinsic, that is,machine-independent, characterizations of the major complexity classes. On the other hand, logic can be useful for designing programs in a natural declarative way. This is particularly important for parallel computation models such as cellular automata, because designing parallel programs is considered a difficult task.This paper establishes three logical characterizations of the three classical complexity classes modeling minimal time, called real-time, of one-dimensional cellular automata according to their canonical variants: unidirectional or bidirectional communication, input word given in a parallel or sequential way.Our three logics are natural restrictions of existential second-order Horn logic with built-in successor and predecessor functions. These logics correspond exactly to the three ways of deciding a language on a square grid circuit of side n according to one of the three canonical locations of an input word of length n: along a side of the grid, on the diagonal that contains the output cell, or on the diagonal opposite to the output cell.The key ingredient of our results is a normalization method that transforms a formula from one of our three logics into an equivalent normalized formula that faithfully mimics a grid circuit.Then, we extend our logics by allowing a limited use of negation on hypotheses like in Stratified Datalog. By revisiting in detail a number of representative classical problems - recognition of the set of primes by Fisher’s algorithm, Dyck language recognition, Firing Squad Synchronization problem,etc. - we show that this extension makes easier programming and we prove that it does not change the complexity of our logics in real-time.Finally, starting from our experience in expressing those representative problems in logic, we argue that our logics are high-level programming languages: they allow to express in a natural,precise and synthetic way the algorithms of literature, based on signals, and to translate them automatically into cellular automata of the same complexity.
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Dates and versions

hal-02474520 , version 1 (11-02-2020)

Identifiers

  • HAL Id : hal-02474520 , version 1

Cite

Etienne Grandjean, Théo Grente, Véronique Terrier. Inductive definitions in logic versus programs of real-time cellular automata. Theoretical Computer Science, 2024, 987, pp.1-69. ⟨hal-02474520⟩
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