A survey of two-dimensional automata theory
Information Sciences: an International Journal
Two-dimensional on-line tessellation acceptors are not closed under complement
Information Sciences: an International Journal
Handbook of formal languages, vol. 3
Picture Languages: Formal Models for Picture Recognition
Picture Languages: Formal Models for Picture Recognition
Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures
Theoretical Computer Science
Computation: finite and infinite machines
Computation: finite and infinite machines
Automata on a 2-dimensional tape
FOCS '67 Proceedings of the 8th Annual Symposium on Switching and Automata Theory (SWAT 1967)
On Complexity of Two Dimensional Languages Generated by Transducers
CIAA '08 Proceedings of the 13th international conference on Implementation and Applications of Automata
Picture Languages Generated by Assembling Tiles
LATA '09 Proceedings of the 3rd International Conference on Language and Automata Theory and Applications
On the computation power of finite automata in two-dimensional environments
DLT'04 Proceedings of the 8th international conference on Developments in Language Theory
A survey on picture-walking automata
Algebraic Foundations in Computer Science
Simulating two-dimensional recognizability by pushdown and queue automata
CIAA'05 Proceedings of the 10th international conference on Implementation and Application of Automata
Picture Languages Generated by Assembling Tiles
Fundamenta Informaticae - Theory that Counts: To Oscar Ibarra on His 70th Birthday
Fundamenta Informaticae - Machines, Computations and Universality, Part II
Two-dimensional sgraffito automata
DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
Comparing two-dimensional one-marker automata to sgraffito automata
CIAA'13 Proceedings of the 18th international conference on Implementation and Application of Automata
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We resolve several long-standing open questions regarding the power of various types of finite-state automata to recognize "picture languages," i.e. sets of two-dimensional arrays of symbols. We show that the languages recognized by 4-way alternating finite-state automata (AFAs) are incomparable to the so-called tiling recognizable languages. Specifically, we show that the set of acyclic directed grid graphs with crossover is AFA-recognizable but not tiling recognizable, while its complement is tiling recognizable but not AFA-recognizable. Since we also show that the complement of an AFA-recognizable language is tiling recognizable, it follows that the AFA-recognizable languages are not closed under complementation. In addition, we show that the set of languages recognized by 4-way NFAs is not closed under complementation, and that NFAs are more powerful than DFAs, even for languages over one symbol.