Mikołaj Bojańczyk

In this page we prove the implication from 2 to 3 in the Schützenberger theorem. This implication says that if a language is definable in first-order logic, then its syntactic monoid is aperiodic. The key lemma is given below.

Lemma 0. If L \subseteq A^* is definable in first-order logic, then it is recognised by a morphism into a finite and aperiodic monoid.

From the above lemma it follows that if L is definable in first-order logic, then its syntactic monoid is aperiodic. This is because, by Theorem 2 on syntactic monoids, the syntactic monoid is a homomorphic image of a submonoid of any monoid recognising L. Since taking submonoids and homomorphic images preserves aperiodicity, Lemma 0 implies that the syntactic monoid of L is aperiodic.


Types. Define the quantifier rank of a formula of first-order logic to be the nesting depth of quantifiers. Formulas of quantifier rank at most k are closed under Boolean combinations, while applying a quantifier causes the rank to increase. For a natural number k, define the k-type of a word w \in A^* to be the set of sentences (i.e. formulas without free variables) of quantifier rank at most k that are true in w. By convention, we assume that all words have the same 0-type.

Theorem. For every alphabet A and natural number k,  the function that maps a word to its k-type is compositional.

Proof hint.  One way to prove this would be using the Ehrenfeucht-Fraisse method, which can be found in Phokion Kolatis’ slides here, beginning with page 16.  \Box

Corollary.  For every k, there are finitely many different k-types, and w^{2^k} and w^{2^k+1} have the same k-type for every word w.

Proof.  Define the k-profile of a word w \in A^* to be  the set of triples (\tau,a,\sigma) such that there exists a decomposition w=vau where the (k-1)-types of v and u are \tau and \sigma, respectively. From compositionality of k-types it follows that for every k \ge 1, the k-type is determined by the k-profile. The two statements in the corollary can then be proved by induction on k. \Box.

We can now prove Lemma 0. Suppose that L \subseteq A^* is definable in first-order logic, by a formula of quantifier rank k. Let M be the set of k-types. By the Theorem on compositionality above, and by the lemma on compositional mappings here, it follows that there is a monoid operation on M such that the  function which maps a word to its k-type is a monoid morphism. Furthermore, by the corollary above, the monoid M is finite and aperiodic.

 

 

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