\(
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% an undirected version of \rightarrow:
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\)
What circuits produce and what they actually compute
As was shown in the book, there is a big difference between what polynomials
$P$ circuits over a given semiring $(R,\suma,\daug)$ do produce (purely syntactically,
as formal expressions), and what functions $f:R^n\to R$
they actually compute.
The point is that no cancellations are applied when producing (formal) polynomials $P$!
For example, in the circuits over the arithmetic semiring $(\RR,+,\times)$ (non-monotone arithmetic $(+,\times,-)$ circuits),
the polynomial produced by the circuit $\Phi=(x+y)(x-y)$
is $P=x^2+c_1xy+c_2y^2$ with the set $B=\{(2,0), (1,1),(0,2)\}$ of exponent vectors, and coefficients $c_1=1-1=0$ and $c_2=-1$, while the polynomial function computed by the circuit $\Phi$ is $f=x^2-y^2$ whose set of exponent vectors is $A=\{(2,0),(0,2)\}$.
In monotone Boolean $(\lor,\land)$ and tropical $(\min,+)$ or $(\max,+)$ circuits we have no cancellations like arithmetic $x-x=0$ because there are no analogs of arithmetic subtraction in the corresponding semirings.
However, in these circuits, we have cancellations via absorption
$x\lor xy=x$ or $\min\{x,x+y\}=x$ or $\max\{x,x+y\}=x+y$ when going from
the produced polynomials $P$ to the functions $f$ actually
computed by the circuits(1).
Note, however, that if $P$ is the polynomial produced by a a tropical, say by $(\min,+)$, circuit computing a tropical polynomial $f$ (as a function), then one can not necessarily obtain $f$ from $P$ by only using cancellations $\min\{x,x+y\}=x$. For example, if
$P=\min\{2x,x+y,2y\}$, then $f=\min\{2x,2y\}$. The reason is that the "exponent" vector $(1,1)$ of the polynomial $P$ is a convex combination of "exponent" vectors $(2,0)$ and $(0,2)$ of $f$. Thus, instead of cancelation
of the arithmetic term $(1-1)xy=0$, the tropical analog $0+x+y=x+y$
of this term is now eliminated via convexity. Likewise, in monotone Boolean $(\lor,\land)$ circuits,
the transition from produced polynomials
$P$ to computed polynomials $f$ uses the idempotence $x^2=x$, not only absorption $x\lor xy=x$.
Monotone arithmetic $(+,\times)$ circuits make a big exception:
there, as shown by Corollary 1.13 in the book, $f=P$ holds (as formal expressions), that is, in these circuits, we have no cancellations at all.
Footnotes:
(1)
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