On the inverse of reversible CA Jarkko Kari University of Iowa, USA, and University of Turku, Finland Joint work with Eugen Czeizler
Abstract Problem investigated: For a one-dimensional reversible CA with n states and size m neighborhood, how large can the neighborhood of the inverse automaton be ?
Abstract Problem investigated: For a one-dimensional reversible CA with n states and size m neighborhood, how large can the neighborhood of the inverse automaton be ? We derive a non-trivial upper bound n m − 1 − 1. The bound is tight for neighborhood size m = 2.
Preliminaries In this talk we consider rectilinear, infinite cellular spaces. In d -dimensional CA the cells are addressed by Z d . The finite state set of the CA is denoted by S , so configurations are functions c : Z d − → S that assign states to all cells. We denote by S Z d the set of all configurations.
Preliminaries In this talk we consider rectilinear, infinite cellular spaces. In d -dimensional CA the cells are addressed by Z d . The finite state set of the CA is denoted by S , so configurations are functions c : Z d − → S that assign states to all cells. We denote by S Z d the set of all configurations. The neighborhood of the CA is a vector N = ( � x n ) x 1 , � x 2 , . . . , � of n distinct elements of Z d that specify the relative positions of neighbors. x ∈ Z d , the cell in position � For every � x has n neighbors in positions x + � x i , for i = 1 , 2 , . . . , n. �
Preliminaries The local rule of the CA is a function f : S n − → S that specifies the new state of each cell as the function of the old states of its neighbors. Together N and f determine the global transition function G : S Z d − → S Z d that describes the evolution of the configurations.
Preliminaries The local rule of the CA is a function f : S n − → S that specifies the new state of each cell as the function of the old states of its neighbors. Together N and f determine the global transition function G : S Z d − → S Z d that describes the evolution of the configurations. A cellular automaton is usually identified with function G , so we talk about cellular automaton function G , or simply cellular automaton G . In algorithmic questions G is however always specified using the three finite items S , N and f .
Preliminaries Cellular automaton function G is called reversible (RCA) if there is another CA function F that is its inverse, i.e. G ◦ F = F ◦ G = identity function . RCA G and F are called the inverse automata of each other. Clearly, in order to be reversible G has to be a bijective function S Z d − → S Z d .
Preliminaries Cellular automaton function G is called reversible (RCA) if there is another CA function F that is its inverse, i.e. G ◦ F = F ◦ G = identity function . RCA G and F are called the inverse automata of each other. Clearly, in order to be reversible G has to be a bijective function S Z d − → S Z d . A CA is called • injective if G is one-to-one, • surjective if G is onto, • bijective if G is both one-to-one and onto.
Classical results It is useful to endow the set S Z d of configurations with the Cantor topology , i.e. the topology generated by the sets Cyl( c, M ) = { e ∈ S Z d | e ( � x ) = c ( � x ) for all � x ∈ M } for c ∈ S Z d and finite M ⊂ Z d . The topology is compact, and it is induced by a metric.
Classical results It is useful to endow the set S Z d of configurations with the Cantor topology , i.e. the topology generated by the sets Cyl( c, M ) = { e ∈ S Z d | e ( � x ) = c ( � x ) for all � x ∈ M } for c ∈ S Z d and finite M ⊂ Z d . The topology is compact, and it is induced by a metric. Theorem (Curtis, Hedlund, Lyndon 1969) A function G : S Z d − → S Z d is a CA function if and only if (i) G is continuous in the Cantor topology, and (ii) G commutes with shifts.
Classical results Corollary: A cellular automaton G is reversible if and only if it is bijective. Proof: If G is a reversible CA function then G is by definition bijective. Conversely, suppose that G is a bijective CA function. Then G has an inverse function that clearly commutes with the shifts. The inverse function is also continuous because the space S Z d is compact.
Classical results Corollary: A cellular automaton G is reversible if and only if it is bijective. Proof: If G is a reversible CA function then G is by definition bijective. Conversely, suppose that G is a bijective CA function. Then G has an inverse function that clearly commutes with the shifts. The inverse function is also continuous because the space S Z d is compact. The point of the corollary is that in bijective CA each cell can determine its previous state by looking at the current states in some bounded neighborhood around them.
Classical results Let q ∈ S be an arbitrary fixed state. Let us call a configuration c finite if all but a finite number of cells are in state q , and let us denote by G F the restriction of G on the finite configurations. (Often it is required that f ( q, q, . . . , q ) = q , but the next results do not require this.) Garden of Eden -theorem (Moore 1962, Myhill 1963): G F is injective if and only if G is surjective.
Classical results Let q ∈ S be an arbitrary fixed state. Let us call a configuration c finite if all but a finite number of cells are in state q , and let us denote by G F the restriction of G on the finite configurations. (Often it is required that f ( q, q, . . . , q ) = q , but the next results do not require this.) Garden of Eden -theorem (Moore 1962, Myhill 1963): G F is injective if and only if G is surjective. Trivially the injectivity of G implies the injectivity of its restriction G F so Corollary: Injective CA are also surjective. Hence injectivity, bijectivity and reversibility are equivalent.
Algorithmic questions How does one determine if a given CA is injective? surjective?
Algorithmic questions How does one determine if a given CA is injective? surjective? Theorem (Amoroso, Patt 1972): There are polynomial time algorithms to determine if a given one-dimensional CA is injective/surjective. Theorem (Kari 1990): It is undecidable if a given two-dimensional CA is injective/surjective.
Algorithmic questions How does one determine if a given CA is injective? surjective? Theorem (Amoroso, Patt 1972): There are polynomial time algorithms to determine if a given one-dimensional CA is injective/surjective. Theorem (Kari 1990): It is undecidable if a given two-dimensional CA is injective/surjective. Note: We know that the in any bijective CA each cell can determine its previous state based on the states in some bounded neighborhood. Previous theorem implies that the size of this inverse neighborhood can be very large in two- and higher dimensional cases: there is no computable upper bound on the extend of the inverse neighborhood.
Inverse neighborhood More precisely, we define the inverse neighborhood of a RCA as follows: The minimum neighborhood N min of a CA function G is the smallest neighborhood that a CA with global function G can have. In other words, (1) there is CA with neighborhood N min whose global function is G , and (2) if there is a CA with neighborhood N that specifies G then we must have N min ⊆ N . It is clear that such minimum neighborhood exists for each CA function G .
Inverse neighborhood More precisely, we define the inverse neighborhood of a RCA as follows: The minimum neighborhood N min of a CA function G is the smallest neighborhood that a CA with global function G can have. In other words, (1) there is CA with neighborhood N min whose global function is G , and (2) if there is a CA with neighborhood N that specifies G then we must have N min ⊆ N . It is clear that such minimum neighborhood exists for each CA function G . If G is reversible then the minimum neighborhood of its inverse G − 1 is called the inverse neighborhood of G .
Inverse neighborhood In the case of one-dimensional RCA polynomial upper bounds on the size of the inverse neighborhood can be easily established. In this work we derive the following precise formula for the maximum size of the inverse neighborhood in the one-dimensional case: Theorem (Czeizler, Kari 2005): If an RCA has n states and it uses the two element neighborhood (0 , 1) then the inverse neighborhood consists of at most n − 1 consecutive positions. This bound is tight.
Inverse neighborhood In the case of one-dimensional RCA polynomial upper bounds on the size of the inverse neighborhood can be easily established. In this work we derive the following precise formula for the maximum size of the inverse neighborhood in the one-dimensional case: Theorem (Czeizler, Kari 2005): If an RCA has n states and it uses the two element neighborhood (0 , 1) then the inverse neighborhood consists of at most n − 1 consecutive positions. This bound is tight. Note: A quadratic upper bound O ( n 2 ) is very easy to establish, for example, using the Amoroso-Patt algorithm for deciding reversibility. However, for the precise bound n − 1 we were not able to find a simple combinatorial argument. Our proof is based on dimensionality arguments in linear algebra.
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