# Automata, Games, and Verification

Advanced Lecture (Vertiefungsvorlesung), Summer Term 2015, 6 CP

Lecture Notes (28.07.2015)

## Lectures

**Lecture 1, Apr 21, 2015.**As motivation & overview for the course, we looked at the model checking and synthesis problems and their relationship to automata and games and discussed the connection between automata and logics.

**Lecture 2, Apr 28, 2015.**We introduced Büchi automata and omega-regular expressions. Next, we started proving closure properties of the Büchi-recognizable languages needed for Büchi’s Characterization Theorem (1962): An omega-language is Büchi-recognizable iff it is omega-regular.

**Lecture 3, May 05, 2015.**We completed the set of closure properties of the Büchi-recognizable languages and proved Büchi’s Characterization Theorem. Then, we studied deterministic Büchi automata and their characterization via the limit operator.

**Lecture 4, May 12, 2015.**We started on the proof that the Büchi-recognizable languages are closed under complement. For this purpose, we introduced the concept of ranked run DAGs and showed that their existence is a neccessary and sufficient condition for the non-acceptance of an infinite word.

**Lecture 5, May 19, 2015.**First, we completed the complementation construction for nondeterministic Büchi automata. Then, we introduced Muller automata and showed that they recognize the same languages as Büchi automata and that deterministic Muller automata are closed under boolean operations.

**Lecture 6, May 26, 2015.**Proving McNaughton’s Theorem, we studied the transformation of nondeterministic Büchi automata into semi-deterministic Büchi automata, and the transformation of semi-deterministic Büchi automata into deterministic Muller automata.

**Lecture 7, June 02, 2015.**First, we studied the alternative transformation of nondeterministic Büchi automata to deterministic Muller automata via Safra’s construction. Then, we started our discussion of logics over infinite sequences. We introduced linear-time temporal logic (LTL) and started proving that the models of an LTL formula form a non-counting language.

**Lecture 8, June 09, 2015.**After completing the proof that every LTL-definable language is non-counting, we introduced Quantified Propositional Temporal Logic (QPTL), and the second-order theory of 1 successor (S1S). Further, we showed that every Büchi-recognizable language is QPTL-definable and that QPTL-definable languages are S1S-definable.

**Lecture 9, June 16, 2015.**We completed the proof that QPTL, S1S and Büchi automata are equally expressive by showing that S1S definable langages are Büchi-recognizable. Furthermore, we introduced a weak variant of S1S (WS1S) and showed that S1S and WS1S are equally expressive. Finally, we defined alternating Büchi automata.

**Lecture 10, June 23, 2015.**We translated LTL formulas to alternating Büchi automata and proved that for each accepting run tree of an alternating Büchi automaton there is also a memoryless accepting run tree.

**Lecture 11, June 30, 2015.**We studied Miyano and Hayashi’s translation from alternating Büchi automata to nondeterministic Büchi automata and got started on infinite games. We proved that reachability games are memoryless determined.

**Lecture 12, July 07, 2015.**We proved memoryless determinancy for Büchi games and parity games.

**Lecture 13, July 14, 2015.**We introduced tree automata and proved that parity tree automata are closed under complement.

**Lecture 14, July 21, 2015.**We introduced S2S, the second-order theory of 2 successors, and showed that S2S is decidable.

**Lecture 15, July 28, 2015.**First, we looked at Computation Tree Logic (CTL) and alternating tree automata. Then we wrapped up with a summary of the course’s main points.