Zoo Operators

co: Complements

Definition: A language L is in $\exists \cdot {\mathcal {C}}$ if $L^{c}=\{x:x\notin L\}$ is in ${\mathcal {C}}$.

Properties:$co\cdot co\cdot {\mathcal {C}}={\mathcal {C}}$

Prominent examples: $co\cdot NP=coNP$.

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$\exists$: Existential (polynomial)

Definition: A language L is in $\exists \cdot {\mathcal {C}}$ if there exists a polynomial p and a language $V\in {\mathcal {C}}$ such that, for all strings $x$, $x$ is in L if and only if there exists a string y, of length $|y|\leq p(|x|)$ such that $(x,y)\in V$.

Properties:

• $co\cdot \exists =\forall \cdot co$ and vice versa.
• $\exists \cdot \exists =\exists$

Prominent examples: $\exists \cdot$ P = NP.

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$\forall$: Universal (polynomial)

Definition: A language L is in $\forall \cdot {\mathcal {C}}$ if there exists a polynomial p and a language $V\in {\mathcal {C}}$ such that, for all strings $x$, $x$ is in L if and only if for all strings y of length $|y|\leq p(|x|)$ such that $(x,y)\in V$.

Properties:

• $co\cdot \forall =\exists \cdot co$ and vice versa.
• $\forall \cdot \forall =\forall$

Prominent examples: $\forall \cdot$ P = coNP

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BP: Bounded-error probability (two-sided)

Definition: A language L is in $BP\cdot {\mathcal {C}}$ if there exists a polynomial p and a language $V\in {\mathcal {C}}$ such that, for all strings $x$, $Pr_{y:|y|\leq p(|x|)}[L(x)=V(x,y)]\geq 3/4$.

Properties:

• $co\cdot BP=BP\cdot co$
• If ${\mathcal {C}}$ is closed under majority reductions, then $BP\cdot {\mathcal {C}}$ admits probability amplification, so we get $BP\cdot BP\cdot {\mathcal {C}}=BP\cdot {\mathcal {C}}$, and we can replace the probability of 3/4 with $1/2+\varepsilon$ for any constant (i.e., independent of the input size |x|) $\varepsilon >0$, as well as with $1-2^{poly(n)}$.
• If ${\mathcal {C}}$ is closed under majority reductions, then $BP\cdot {\mathcal {C}}\subseteq \exists \cdot \forall \cdot {\mathcal {C}}\cap \forall \cdot \exists \cdot {\mathcal {C}}$.
• If ${\mathcal {C}}$ is closed under majority reductions, then $\exists \cdot BP\cdot {\mathcal {C}}\subseteq BP\cdot \exists \cdot {\mathcal {C}}$.
• Note that, because of the semantic nature of the defining condition for the BP operator, it is possible that some languages $V\in {\mathcal {C}}$ do not define a language $L\in BP\cdot {\mathcal {C}}$ using the defining formula above. Only those V which satisfy the required condition can be used.

Prominent examples:

• $BP\cdot$ P = BPP.
• $BP\cdot NP=AM$
• $\exists \cdot BPP=MA$ (not to be confused with $\exists$BPP!)

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