# HO (complexity)

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High-order logic is an extension of first-order and second-order with high order quantifiers. In descriptive complexity we can see that it is equal to the ELEMENTARY functions. There is a relation between the i th order and non determinist algorithm the time of which is with i 1 level of exponentials.

## Definitions and notations

We define high-order variable, a variable of order i > 1 has got an arity k and represent any set of k -tuples of elements of order i 1 . They are usually written in upper-case and with a natural number as exponent to indicate the order. High order logic is the set of FO formulae where we add quantification over higher-order variables, hence we will use the terms defined in the FO article without defining them again.

HO i is the set of formulae where variable's order are at most i . HO j i is the subset of the formulae of the form ϕ = X 1 i ¯ X 2 i ¯ Q X j i ¯ ψ where Q is a quantifier, Q X i ¯ means that X i ¯ is a tuple of variable of order i with the same quantification. So it is the set of formulae with j alternations of quantifiers of i th order, beginning by and , followed by a formula of order i 1 .

Using the tetration's standard notation, exp 2 0 ( x ) = x and exp 2 i + 1 ( x ) = 2 exp 2 i ( x ) . exp 2 i + 1 ( x ) = 2 2 2 2 2 x with i times 2

## Normal form

Every formula of i th order is equivalent to a formula in prenex normal form, where we first write quantification over variable of i th order and then a formula of order i 1 in normal form.

## Relation to complexity classes

HO is equal to ELEMENTARY functions. To be more precise, H O 0 i = N T I M E ( exp 2 i 2 ( n O ( 1 ) ) ) , it means a tower of ( i 2 ) 2s, ending with n c where c is a constant. A special case of it is that S O = H O 0 2 = N T I M E ( n O ( 1 ) ) = N P , which is exactly the Fagin's theorem. Using oracle machines in the polynomial hierarchy, H O j i = N T I M E ( exp 2 i 2 ( n O ( 1 ) ) Σ j P )

## References

HO (complexity) Wikipedia

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