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# AGV: Lecture Notes

**Lecture 1, Oct 23, 2012.**

As motivation & overview for the course, we looked at temporal logics along the linear/branching-time spectrum and their relationship to automata and games.

Lecture Notes 1 (updated on Oct 27, 2012)

**Lecture 2, Oct 25, 2012.**

We introduced Büchi automata and omega-regular expressions and proved the 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.

Intro Slides 2, Lecture Notes 2 (updated on Oct 27, 2012)

**Lecture 3, Oct 30, 2012.**

We studied deterministic Büchi automata and got started on the proof that the Büchi-recognizable languages are closed under complement. For this purpose, we introduced the concept of ranked run DAGs.

Intro Slides 3, Lecture Notes 3

**Lecture 4, Nov 6, 2012.**

We completed the complementation construction for nondeterministic Büchi automata.

Intro Slides 4, Lecture Notes 4

**Lecture 5, Nov 13, 2012**

We introduced Muller automata and got started on the proof of McNaughton’s Theorem: Every Büchi-recognizable language is recognizable by a deterministic Muller automaton.

Intro Slides 5, Lecture Notes 5 (updated on Nov 28, 2012)

**Lecture 6, Nov 20, 2012**

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.

Intro Slides 6, Lecture Notes 6

**Lecture 7, Nov 27, 2012**

After completing the proof of McNaughton’s Theorem, we started our discussion of logics over infinite sequences. We introduced linear-time temporal logic (LTL), Quantified Propositional Temporal Logic (QPTL), and the second-order theory of 1 successor (S1S) and proved that the models of an LTL formula form a non-counting language.

Intro Slides 7, Lecture Notes 7 (updated on Feb 12, 2013)

**Lecture 8, Dec 4, 2012**

We proved that QPTL, S1S, WS1S, and Büchi automata are equally expressive (while LTL is strictly weaker). Next, we defined alternating Büchi automata.

Intro Slides 8, Lecture Notes 8

**Lecture 9, Dec 11, 2012**

We translated LTL formulas to alternating automata and proved that alternating Büchi automata recognize the same languages as nondeterministic Büchi automata (Miyano and Hayashi, 1984).

Intro Slides 9, Lecture Notes 9 (updated on Jan 7, 2013)

**Lecture 10, December 18, 2012**

We proved memoryless determinacy for reachability and Büchi games.

Intro Slides 10, Lecture Notes 10 (updated on Jan 9, 2013)

**Lecture 11, January 8, 2013**

We proved memoryless determinacy for parity games and introduced tree automata.

Intro Slides 11, Lecture Notes 11

**Lecture 12, January 15, 2013**

We got started with the proof that parity tree automata are closed under complement.

Lecture Notes 12

**Lecture 13, January 22, 2013**

We completed the proof that parity tree automata are closed under complement and introduced S2S, the second-order theory of 2 successors.

Intro Slides 13, Lecture Notes 13

**Lecture 14, January 29, 2013**

We introduced alternating tree automata and provided a translation from CTL formulas to alternating Büchi tree automata.

Intro Slides 14, Lecture Notes 14

**Lecture 15, February 5, 2013**

We looked at various game-based temporal logics and wrapped up with a summary of the course’s main points.

Lecture Notes 15