The Moore-Myhill Garden of Eden theorem asserts that a cellular automaton with finite alphabet over a free abelian group of rank 2 is surjective if and only if it is pre-injective. Here, pre-injectivity means that two configurations which coincide outside of a finite subset of the group must coincide everywhere if they have the same image under the cellular automaton. The Garden of Eden theorem...

We consider the notion of rationality in algebras with a designated binary associative operation called the series operation, or the sequential product. We define automata operating in these algebras and rational expressions matching their expressive power, and we show that this expressive power equals that of algebraic recognizability. The framework which we consider encompasses both the free ...

The states of a finite-state automaton \(\mathcal {N} \) can be identified with collections words in the prefix closure regular language accepted by . But ordered, and among many possible orders very natural one is co-lexicographic order. Such naturalness stems from fact that it suggests transfer order to automaton’s states. This suggestion is, fact, concrete number papers automata admitting to...

Finite automata have been recently used as alternative, discrete models in theoretical physics, especially in problems related to the dichotomy between endophysical/intrinsic and exophysical/extrinsic perception (see, for instance [15, 18, 16, 7, 17, 4]). These studies deal with Moore experiments; the main result states that it is impossible to determine the initial state of an automaton, and, ...

Moore and Myhill showed that Garden-of-Eden theorem [2], [3]. A binary relation over the configurations is said to be "parallel" if it is induced by a cellular (tessellation) automaton. Richardson showed the equivalence between a parallel relation (a nondeterministic parallel map) with the quiescent state to be injective and its inverse to be parallel by the Garden-of-Eden theorem plus compactn...

Motivation Minimizing deterministic automata −→ Merging states with the same future a b a a b b (Nerode equivalence) −→ Hopcroft algorithm O(n log n) Motivation Minimizing deterministic automata −→ Merging states with the same future a b a a b b (Nerode equivalence) −→ Hopcroft algorithm O(n log n)

In this paper we define a model theory and give a semantic proof of cut-elimination for ICTT, an intuitionistic formulation of Church’s theory of types defined by Miller et. al. and the basis for the λProlog programming language. Our approach, extending techniques of Takahashi and Andrews and tableaux machinery of Fitting, Smullyan, Nerode and Shore, is to prove a completeness theorem for the c...

The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and others and it is known that the theorem does not a...