Mikołaj Bojańczyk

## Automata, logic and games

This is a course about the interplay of automata, logic and games.

The automata are finite automata, but often on objects that are not necessarily finite words. The logic is usually monadic second-order logic. The point is that the automata and the logic define the same languages. The games are infinite duration games, which appear as technical tool in the analysis of automata and logics on infinite objects.

I will also use semigroups, as a third approach to defining languages.

Please submit any mistakes in the slides, or change requests, in this document.

I have updated the slides after the lecture. I have added examples, fixed some proofs and removed Muller from the lectures. The old slides can be found here.

### Lecture 1. Automata & logic for finite words

• the subset construction for determining automata on finite words
• monadic second-order logic (MSO) on finite words
• the equivalence of MSO and automata for finite words
• satisfiability for MSO is non-elementary

A good introduction that covers this material is Sections 2 and 3 of [Tho].

### Lecture 2. Semigroups and compositionality of logic

• monoids and semigroups
• their equivalence to automata
• Ehrenfeucht-Fraisse games
• compositionality of both first-order logic and MSO

The material of this lecture is covered in more detail in Sections 2.1 and 2.2 of [Reco].

### Lecture 3. First order logic and friends

• star-free languages and their equivalence to first-order logic
• aperiodic monoids
• linear temporal logic LTL (on finite words)
• equivalence of first order logic, aperiodic monoids, and LTL
• PSPACE completeness of LTL satisfiability, via alternating automata

A discussion of first-order logic is in Section 4 of [Tho]. The translation from aperiodic to LTL is based more on Section 2.2 of [Reco].

### Lecture 4. MSO for ω-words and Büchi automata

• MSO on infinite words
• Büchi automata and their necessary nondeterminism
• Semigroups for infinite words (lightweight approach)
• Equivalence of all models

The semigroup approach to Büchi automata is covered in more detail in Section 3.1 of [Reco].

### Lecture 5. Determinization of ω-automata

• Parity automata
• Automaton based determinization
• Semigroup based determinization

The automaton determinization is described in Chapter 1 of [Toolbox]. The semigroup determinization is based on Section 3.1 of [Reco].

### Lecture 6. Games and determinacy

• infinite games as a tool to define semantics of alternating automata
• an example of an infinite game where none of the players wins
• memoryless determinacy of parity games

A similar proof of memoryless determinacy is in Section 6.2 of [Tho].

### Lecture 7. Alternating automata for ω-words and LTL

• 3 paths from LTL to nondeterministic Büchi
• from alternating parity to nondeterministic Büchi

### Lecture 8. Algorithms for solving parity games

• a quasi-polynomial time algorithm, using universal trees

The paper with this algorithm is by Jurdziński and Lazić.

### Lecture 9. Finite trees

• finite trees and logic on them
• nondeterministic automata, including bottom-up determinisation
• equivalence of mso and automata

### Lecture 10. Infinite trees

• MSO on infinite trees, and what can be expressed in this logic

Infinite trees and MSO on them are treated in Section 6 of  [Tho].

### Lecture 11. Rabin’s Theorem

• parity automata on infinite trees: nondeterministic, and alternating
• from alternating parity to nondeterministic parity
• emptiness algorithm for nondeterministic parity, and regular trees

Automata on infinite trees, but without an emphasis on alternating automata are treated in Section 6 of  [Tho].

### Lecture 12. Some logics on Kripke structures

• LTL and PSPACE-completeness of its model checking
• modal logic
• CTL
• CTL*
• alternating automata

### Lecture 13. The μ-calculus

• fix-points and the μ-calculus
• model checking using a parity game

### Lecture 14. Graphs

• MSO on graphs, and its undecidability
• treewidth

### Lecture 15. MSO transductions

• bounded treewidth as image of MSO transductions from trees
• model checking on bounded treewidth
• decidable satisfiability iff bounded treewidth