syntactic resources needed to create logical formulas defining them.

We've defined first-order logic, where variables represent positions in the input string, the

atomic formulas are "I_{a}(x)" (position x holds an a), x = y, and x ≤ y, we have boolean

connectives, and we have first order forall and exists quantifiers.

-- BarrinG - 2012-02-27

**Theorem**

A language is first-order definable iff it is star-free regular.

**Proof**

**Star-free → first-order**

We show that the following elements of a star-free expression can be expressed in first-order logic:

*∅ (empty set)*: ∃x: x≠x

*a (letter in Σ)*: ∃x:∀y: (y=x) ∧ I_{a}(x)

*‾ (complement)*: If Φ_{X} is the first-order formula for a star-free expression X, then ¬Φ_{X} is the formula for expression X

*∪ (union)*: If Φ_{X} and Φ_{Y} are the first-order formulas for star-free expressions X and Y, then Φ_{X}∨Φ_{Y} is the formula for expression X∪Y

*• (concatenation): *Let Φ_{X} and Φ_{Y} be the first-order formulas for star-free expressions X and Y. Informally, the formula for XY would be the following:

∃z: (w_{1}w_{2}...w_{z} *satisfies* Φ_{X}) ∧ (w_{z+1}...w_{n} *satisfies* Φ_{Y}), where w_{1}w_{2}...w_{n} is a string over Σ.

The first part (call it Φ_{≤z}) can be constructed from Φ_{X} in the following way: replace each "∃x" with "∃x: (x≤z) ∧" and each "∀x" with "∀x: (x≤z) →". Analogously we construct the second part.

-- SvorotnI - 2012-03-25

Buchi's Theorem says that a language is definable in second-order monadic logic iff it is regular. (See the page with the theorem for the proof.)

-- SvorotnI - 2012-03-12

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