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Kari, Jarkko - Department of Mathematics, University of Turku
4 Recursive and recursively enumerable languages In this last part of the course we want to show that certain decision problems can not be
Cellular Automata Jarkko Kari
4 Quantization Next we start to apply the theoretical concepts of previous section into lossy image com-
The element k, j of the n n discrete since transform (DST) matrix is Skj = C sin
4.10 Undecidability and incompleteness in arithmetics In this section we show that it is undecidable whether a given first-order mathematical statement
The initialization tiles allow the horizontal row with t to contain any initial tape content f with f(0) = a. The machine tiles then force the rows above to simulate machine MR. If MR halts then the tiling becomes
1. The original Image is partitioned into 8 8 blocks. The blocks are transformed with 2. The transform coefficients are quantized using uniform quantizers. Different transform
2.6 De Bruijn -graphs Let us continue with one-dimensional cellular automata. In this section we introduce a new
2 Injectivity and surjectivity properties 2.1 Basic facts
7 A brief revisit to tilings by polygons In the beginning of Section 4 we showed how any Wang protoset can be converted into an equivalent
Deriving the 17 wallpaper groups Jarkko Kari, University of Turku
5 Transform coding 5.1 Linear image transformations
1 Information theory and coding 1.1 Introduction
2 Lossless image compression 2.1 Binary images
3 Lossy compression 3.1 "Lossy" information theory
3 Context-free languages Superficially thinking, one might view modern computers as deterministic finite automata: Com-
4.6 Universal Turing machines Recall the language
4.11 Computable functions and reducibility So far we have considered decision problems and associated languages. In many setups it would be
Tilings and Patterns by Jarkko Kari, University of Turku
Let us first show that there exists a shortest translation in G. Lemma 2.28 Wallpaper group G has a shortest non-trivial translation. More generally, any non-empty
"dents" that exactly fit the bumps. The bump/dent pairs are different in the horizontal and the vertical directions, and they are asymmetric so that flipped and non-flipped tiles do not match
5.4 The periodic tiling problem Next we consider the problem of deciding if a given protoset admits a periodic tiling. There is an obvious
Cellular Automata: Tutorial Jarkko Kari
3 Algorithmic aspects Next we turn to algorithmic aspects of cellular automata. These are two-fold: On one side,
semi-decidable problem U r.e.-complete if for every semi-decidable problem P there exists an algorithm that converts instances of P into equivalent instances of U, that is, positive and
Figure 42: The value of x in the two-dimensional case. Lemma 55 Function d : SZd
Automata and formal languages Jarkko Kari
Automata and formal languages Jarkko Kari
Automata and Formal Languages. An answer to problem 7. (a) It is clear from the productions of the grammar that in every sentential form every A is imme-