Finite Model Theory Lecture 10. Second Order Logic. Outline. Chapter 7 in the textbook: SO, MSO, 9 SO, 9 MSO Games for SO Reachability Buchi’s theorem. Second Order Logic. Add second order quantifiers: 9 X. f or 8 X. f
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Second Order Logic
Chapter 7 in the textbook:
The MSO game is the following. Spoiler may choose between point move and set move:
Theorem The duplicator has a winning strategy for k moves if A and B are indistinguishable in MSO[k]
[ What is MSO[k] ? ]
Both statement and proof are almost identical to the first order case.
Proposition EVEN is not expressible in MSO
Theorem Reachability on directed graphs is not expressible in 9 MSO
The l,k-Fagin game on two structures A, B:
Theorem If duplicator has a winning strategy for the l,k-Fagin game, then A, B are indistinguishable in MSO[l, k]
Theorem If spoiler has winning strategy, then P cannot be expressed by a formula in MSO[l, k]
Application: prove that reachability is not in 9MSO [ in class ? ]
Theorem [Buchi] On strings: MSO = regular languages.
Corollary. On strings: MSO = 9MSO = 8MSO
TheoremOn strings, MSO = TrCl1
However, TrCl2 can express an.bn [ how ? ]
Question: what is the relationship between these languages: