$\require{mathtools} % %%% GENERIC MATH %%% % % Environments \newcommand{\al}[1]{\begin{align}#1\end{align}} % need this for \tag{} to work \renewcommand{\r}{\mathrm} % % Greek \newcommand{\eps}{\epsilon} \newcommand{\veps}{\varepsilon} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Th}{\Theta} \let\fi\phi % because it looks like an f \let\phi\varphi % because it looks like a p % % Miscellaneous shortcuts % .. over and under \newcommand{\ss}[1]{_{\substack{#1}}} \newcommand{\ob}{\overbrace} \newcommand{\ub}{\underbrace} \newcommand{\ol}{\overline} \newcommand{\tld}{\widetilde} \newcommand{\HAT}{\widehat} \newcommand{\f}{\frac} \newcommand{\s}[2]{#1 /\mathopen{}#2} \newcommand{\rt}[1]{ {\sqrt{#1}}} % .. relations \newcommand{\sr}{\stackrel} \newcommand{\sse}{\subseteq} \newcommand{\ce}{\coloneqq} \newcommand{\ec}{\eqqcolon} \newcommand{\ap}{\approx} \newcommand{\ls}{\lesssim} \newcommand{\gs}{\gtrsim} % .. miscer \newcommand{\q}{\quad} \newcommand{\qq}{\qquad} \newcommand{\heart}{\heartsuit} \newcommand{\es}{\emptyset} % % Delimiters % (I needed to create my own because the MathJax version of \DeclarePairedDelimiter doesn't have \mathopen{} and that messes up the spacing) % .. one-part \newcommand{\p}[1]{\mathopen{}\left( #1 \right)} \newcommand{\b}[1]{\mathopen{}\left[ #1 \right]} \newcommand{\set}[1]{\mathopen{}\left\{ #1 \right\}} \newcommand{\abs}[1]{\mathopen{}\left\lvert #1 \right\rvert} \newcommand{\floor}[1]{\mathopen{}\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\mathopen{}\left\lceil #1 \right\rceil} \newcommand{\inner}[1]{\mathopen{}\left\langle #1 \right\rangle} % .... (use phantom to force at least the standard height of double bars) \newcommand{\norm}[1]{\mathopen{}\left\lVert #1 \vphantom{f} \right\rVert} \newcommand{\frob}[1]{\norm{#1}_\mathrm{F}} %% .. two-part \newcommand{\incond}[2]{#1 \mathop{}\middle|\mathop{} #2} \newcommand{\cond}[2]{ {\left.\incond{#1}{#2}\right.}} \newcommand{\pco}[2]{\p{\incond{#1}{#2}}} \newcommand{\bco}[2]{\b{\incond{#1}{#2}}} \newcommand{\setco}[2]{\set{\incond{#1}{#2}}} \newcommand{\at}[2]{ {\left.#1\right|_{#2}}} \newcommand{\pat}[2]{\p{\at{#1}{#2}}} \newcommand{\bat}[2]{\b{\at{#1}{#2}}} % ..... (use phantom to force at least the standard height of double bar) \newcommand{\oldpara}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2} %\newcommand{\para}[2]{#1\vphantom{f} \mathop{}\middle\|\mathop{} #2} \newcommand{\para}[2]{\mathchoice{\begin{matrix}#1\\\hdashline#2\end{matrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}{\begin{smallmatrix}#1\\\hdashline#2\end{smallmatrix}}} \newcommand{\ppa}[2]{\p{\para{#1}{#2}}} \newcommand{\bpa}[2]{\b{\para{#1}{#2}}} %\newcommand{\bpaco}[4]{\bpa{\incond{#1}{#2}}{\incond{#3}{#4}}} \newcommand{\bpaco}[4]{\bpa{\cond{#1}{#2}}{\cond{#3}{#4}}} % % Levels of closeness \newcommand{\scirc}[1]{\sr{\circ}{#1}} \newcommand{\sdot}[1]{\sr{.}{#1}} \newcommand{\slog}[1]{\sr{\log}{#1}} \newcommand{\createClosenessLevels}[7]{ \newcommand{#2}{\mathrel{(#1)}} \newcommand{#3}{\mathrel{#1}} \newcommand{#4}{\mathrel{#1\!\!#1}} \newcommand{#5}{\mathrel{#1\!\!#1\!\!#1}} \newcommand{#6}{\mathrel{(\sdot{#1})}} \newcommand{#7}{\mathrel{(\slog{#1})}} } \let\lt\undefined \let\gt\undefined % .. vanilla versions (is it within a constant?) \newcommand{\ez}{\scirc=} \newcommand{\eq}{\simeq} \newcommand{\eqq}{\mathrel{\eq\!\!\eq}} \newcommand{\eqqq}{\mathrel{\eq\!\!\eq\!\!\eq}} \newcommand{\lez}{\scirc\le} \newcommand{\lq}{\preceq} \newcommand{\lqq}{\mathrel{\lq\!\!\lq}} \newcommand{\lqqq}{\mathrel{\lq\!\!\lq\!\!\lq}} \newcommand{\gez}{\scirc\ge} \newcommand{\gq}{\succeq} \newcommand{\gqq}{\mathrel{\gq\!\!\gq}} \newcommand{\gqqq}{\mathrel{\gq\!\!\gq\!\!\gq}} \newcommand{\lz}{\scirc<} \newcommand{\lt}{\prec} \newcommand{\ltt}{\mathrel{\lt\!\!\lt}} \newcommand{\lttt}{\mathrel{\lt\!\!\lt\!\!\lt}} \newcommand{\gz}{\scirc>} \newcommand{\gt}{\succ} \newcommand{\gtt}{\mathrel{\gt\!\!\gt}} \newcommand{\gttt}{\mathrel{\gt\!\!\gt\!\!\gt}} % .. dotted versions (is it equal in the limit?) \newcommand{\ed}{\sdot=} \newcommand{\eqd}{\sdot\eq} \newcommand{\eqqd}{\sdot\eqq} \newcommand{\eqqqd}{\sdot\eqqq} \newcommand{\led}{\sdot\le} \newcommand{\lqd}{\sdot\lq} \newcommand{\lqqd}{\sdot\lqq} \newcommand{\lqqqd}{\sdot\lqqq} \newcommand{\ged}{\sdot\ge} \newcommand{\gqd}{\sdot\gq} \newcommand{\gqqd}{\sdot\gqq} \newcommand{\gqqqd}{\sdot\gqqq} \newcommand{\ld}{\sdot<} \newcommand{\ltd}{\sdot\lt} \newcommand{\lttd}{\sdot\ltt} \newcommand{\ltttd}{\sdot\lttt} \newcommand{\gd}{\sdot>} \newcommand{\gtd}{\sdot\gt} \newcommand{\gttd}{\sdot\gtt} \newcommand{\gtttd}{\sdot\gttt} % .. log versions (is it equal up to log?) \newcommand{\elog}{\slog=} \newcommand{\eqlog}{\slog\eq} \newcommand{\eqqlog}{\slog\eqq} \newcommand{\eqqqlog}{\slog\eqqq} \newcommand{\lelog}{\slog\le} \newcommand{\lqlog}{\slog\lq} \newcommand{\lqqlog}{\slog\lqq} \newcommand{\lqqqlog}{\slog\lqqq} \newcommand{\gelog}{\slog\ge} \newcommand{\gqlog}{\slog\gq} \newcommand{\gqqlog}{\slog\gqq} \newcommand{\gqqqlog}{\slog\gqqq} \newcommand{\llog}{\slog<} \newcommand{\ltlog}{\slog\lt} \newcommand{\lttlog}{\slog\ltt} \newcommand{\ltttlog}{\slog\lttt} \newcommand{\glog}{\slog>} \newcommand{\gtlog}{\slog\gt} \newcommand{\gttlog}{\slog\gtt} \newcommand{\gtttlog}{\slog\gttt} % % Miscellaneous \newcommand{\LHS}{\mathrm{LHS}} \newcommand{\RHS}{\mathrm{RHS}} % .. operators \DeclareMathOperator{\poly}{poly} \DeclareMathOperator{\polylog}{polylog} \DeclareMathOperator{\quasipoly}{quasipoly} \DeclareMathOperator{\negl}{negl} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator*{\argmax}{arg\,max} % .. functions \DeclareMathOperator{\id}{id} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\err}{err} \DeclareMathOperator{\ReLU}{ReLU} % .. analysis \let\d\undefined \newcommand{\d}{\operatorname{d}\mathopen{}} \newcommand{\dd}[1]{\operatorname{d}^{#1}\mathopen{}} \newcommand{\df}[2]{ {\f{\d #1}{\d #2}}} \newcommand{\ds}[2]{ {\s{\d #1}{\d #2}}} \newcommand{\ddf}[3]{ {\f{\dd{#1} #2}{\p{\d #3}^{#1}}}} \newcommand{\dds}[3]{ {\s{\dd{#1} #2}{\p{\d #3}^{#1}}}} \newcommand{\part}{\partial} \newcommand{\partf}[2]{\f{\part #1}{\part #2}} \newcommand{\parts}[2]{\s{\part #1}{\part #2}} \newcommand{\grad}[1]{\mathop{\nabla\!_{#1}}} % .. sets of numbers \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} % %%% SPECIALIZED MATH %%% % % Logic and bit operations \renewcommand{\and}{\wedge} \newcommand{\AND}{\bigwedge} \newcommand{\or}{\vee} \newcommand{\OR}{\bigvee} \newcommand{\xor}{\oplus} \newcommand{\XOR}{\bigoplus} \newcommand{\union}{\cup} \newcommand{\inter}{\cap} \newcommand{\UNION}{\bigcup} \newcommand{\INTER}{\bigcap} \newcommand{\comp}{\overline} \newcommand{\true}{\r{true}} \newcommand{\false}{\r{false}} \newcommand{\tf}{\set{\true,\false}} \DeclareMathOperator{\One}{\mathbb{1}} \DeclareMathOperator{\1}{\mathbb{1}} \DeclareMathOperator{\LSB}{LSB} % % Linear algebra \renewcommand{\span}{\mathrm{span}} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\Img}{Im} \newcommand{\transp}{\mathsf{T}} \renewcommand{\t}{^\transp} % ... named tensors \newcommand{\namedtensorstrut}{\vphantom{fg}} % milder than \mathstrut \newcommand{\name}[1]{\mathsf{\namedtensorstrut #1}} \newcommand{\nbin}[2]{\mathbin{\underset{\substack{#1}}{\namedtensorstrut #2}}} \newcommand{\ndot}[1]{\nbin{#1}{\odot}} \newcommand{\ncat}[1]{\nbin{#1}{\oplus}} \newcommand{\nsum}[1]{\sum\limits_{\substack{#1}}} \newcommand{\nfun}[2]{\mathop{\underset{\substack{#1}}{\namedtensorstrut\mathrm{#2}}}} \newcommand{\ndef}[2]{\newcommand{#1}{\name{#2}}} \newcommand{\nt}[1]{^{\transp(#1)}} % % Probability \newcommand{\Normal}{\mathcal{N}} \let\Pr\undefined \DeclareMathOperator*{\Pr}{Pr} \DeclareMathOperator*{\G}{\mathbb{G}} \DeclareMathOperator*{\Odds}{Od} \DeclareMathOperator*{\E}{E} \DeclareMathOperator*{\Var}{Var} \DeclareMathOperator*{\Cov}{Cov} \DeclareMathOperator*{\corr}{corr} \DeclareMathOperator*{\median}{median} \newcommand{\dTV}{d_{\mathrm{TV}}} \newcommand{\dHel}{d_{\mathrm{Hel}}} \newcommand{\dJS}{d_{\mathrm{JS}}} % ... information theory \let\H\undefined \DeclareMathOperator*{\H}{H} \DeclareMathOperator*{\I}{I} \DeclareMathOperator*{\D}{D} % %%% SPECIALIZED COMPUTER SCIENCE %%% % % Complexity classes % .. classical \newcommand{\Poly}{\mathsf{P}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\PH}{\mathsf{PH}} \newcommand{\PSPACE}{\mathsf{PSPACE}} \renewcommand{\L}{\mathsf{L}} % .. probabilistic \newcommand{\formost}{\mathsf{Я}} \newcommand{\RP}{\mathsf{RP}} \newcommand{\BPP}{\mathsf{BPP}} \newcommand{\MA}{\mathsf{MA}} \newcommand{\AM}{\mathsf{AM}} \newcommand{\IP}{\mathsf{IP}} \newcommand{\RL}{\mathsf{RL}} % .. circuits \newcommand{\NC}{\mathsf{NC}} \newcommand{\AC}{\mathsf{AC}} \newcommand{\ACC}{\mathsf{ACC}} \newcommand{\TC}{\mathsf{TC}} \newcommand{\Ppoly}{\mathsf{P}/\poly} \newcommand{\Lpoly}{\mathsf{L}/\poly} % .. resources \newcommand{\TIME}{\mathsf{TIME}} \newcommand{\SPACE}{\mathsf{SPACE}} \newcommand{\TISP}{\mathsf{TISP}} \newcommand{\SIZE}{\mathsf{SIZE}} % .. keywords \newcommand{\co}{\mathsf{co}} \newcommand{\Prom}{\mathsf{Promise}} % % Boolean analysis \newcommand{\zo}{\set{0,1}} \newcommand{\pmo}{\set{\pm 1}} \newcommand{\zpmo}{\set{0,\pm 1}} \newcommand{\harpoon}{\!\upharpoonright\!} \newcommand{\rr}[2]{#1\harpoon_{#2}} \newcommand{\Fou}[1]{\widehat{#1}} \DeclareMathOperator{\Ind}{\mathrm{Ind}} \DeclareMathOperator{\Inf}{\mathrm{Inf}} \newcommand{\Der}[1]{\operatorname{D}_{#1}\mathopen{}} \newcommand{\Exp}[1]{\operatorname{E}_{#1}\mathopen{}} \DeclareMathOperator{\Stab}{\mathrm{Stab}} \DeclareMathOperator{\T}{T} \DeclareMathOperator{\sens}{\mathrm{s}} \DeclareMathOperator{\bsens}{\mathrm{bs}} \DeclareMathOperator{\fbsens}{\mathrm{fbs}} \DeclareMathOperator{\Cert}{\mathrm{C}} \DeclareMathOperator{\DT}{\mathrm{DT}} \DeclareMathOperator{\CDT}{\mathrm{CDT}} % canonical \DeclareMathOperator{\ECDT}{\mathrm{ECDT}} \DeclareMathOperator{\CDTv}{\mathrm{CDT_{vars}}} \DeclareMathOperator{\ECDTv}{\mathrm{ECDT_{vars}}} \DeclareMathOperator{\CDTt}{\mathrm{CDT_{terms}}} \DeclareMathOperator{\ECDTt}{\mathrm{ECDT_{terms}}} \DeclareMathOperator{\CDTw}{\mathrm{CDT_{weighted}}} \DeclareMathOperator{\ECDTw}{\mathrm{ECDT_{weighted}}} \DeclareMathOperator{\AvgDT}{\mathrm{AvgDT}} \DeclareMathOperator{\PDT}{\mathrm{PDT}} % partial decision tree \DeclareMathOperator{\DTsize}{\mathrm{DT_{size}}} \DeclareMathOperator{\W}{\mathbf{W}} % .. functions (small caps sadly doesn't work) \DeclareMathOperator{\Par}{\mathrm{Par}} \DeclareMathOperator{\Maj}{\mathrm{Maj}} \DeclareMathOperator{\HW}{\mathrm{HW}} \DeclareMathOperator{\Thr}{\mathrm{Thr}} \DeclareMathOperator{\Tribes}{\mathrm{Tribes}} \DeclareMathOperator{\RotTribes}{\mathrm{RotTribes}} \DeclareMathOperator{\CycleRun}{\mathrm{CycleRun}} \DeclareMathOperator{\SAT}{\mathrm{SAT}} \DeclareMathOperator{\UniqueSAT}{\mathrm{UniqueSAT}} % % Dynamic optimality \newcommand{\OPT}{\mathsf{OPT}} \newcommand{\Alt}{\mathsf{Alt}} \newcommand{\Funnel}{\mathsf{Funnel}} % % Alignment \DeclareMathOperator{\Amp}{\mathrm{Amp}} % %%% TYPESETTING %%% % % In text \renewcommand{\th}{^{\mathrm{th}}} \newcommand{\degree}{^\circ} % % Fonts % .. bold \newcommand{\BA}{\boldsymbol{A}} \newcommand{\BB}{\boldsymbol{B}} \newcommand{\BC}{\boldsymbol{C}} \newcommand{\BD}{\boldsymbol{D}} \newcommand{\BE}{\boldsymbol{E}} \newcommand{\BF}{\boldsymbol{F}} \newcommand{\BG}{\boldsymbol{G}} \newcommand{\BH}{\boldsymbol{H}} \newcommand{\BI}{\boldsymbol{I}} \newcommand{\BJ}{\boldsymbol{J}} \newcommand{\BK}{\boldsymbol{K}} \newcommand{\BL}{\boldsymbol{L}} \newcommand{\BM}{\boldsymbol{M}} \newcommand{\BN}{\boldsymbol{N}} \newcommand{\BO}{\boldsymbol{O}} \newcommand{\BP}{\boldsymbol{P}} \newcommand{\BQ}{\boldsymbol{Q}} \newcommand{\BR}{\boldsymbol{R}} \newcommand{\BS}{\boldsymbol{S}} \newcommand{\BT}{\boldsymbol{T}} \newcommand{\BU}{\boldsymbol{U}} \newcommand{\BV}{\boldsymbol{V}} \newcommand{\BW}{\boldsymbol{W}} \newcommand{\BX}{\boldsymbol{X}} \newcommand{\BY}{\boldsymbol{Y}} \newcommand{\BZ}{\boldsymbol{Z}} \newcommand{\Ba}{\boldsymbol{a}} \newcommand{\Bb}{\boldsymbol{b}} \newcommand{\Bc}{\boldsymbol{c}} \newcommand{\Bd}{\boldsymbol{d}} \newcommand{\Be}{\boldsymbol{e}} \newcommand{\Bf}{\boldsymbol{f}} \newcommand{\Bg}{\boldsymbol{g}} \newcommand{\Bh}{\boldsymbol{h}} \newcommand{\Bi}{\boldsymbol{i}} \newcommand{\Bj}{\boldsymbol{j}} \newcommand{\Bk}{\boldsymbol{k}} \newcommand{\Bp}{\boldsymbol{p}} \newcommand{\Bq}{\boldsymbol{q}} \newcommand{\Br}{\boldsymbol{r}} \newcommand{\Bs}{\boldsymbol{s}} \newcommand{\Bt}{\boldsymbol{t}} \newcommand{\Bu}{\boldsymbol{u}} \newcommand{\Bv}{\boldsymbol{v}} \newcommand{\Bw}{\boldsymbol{w}} \newcommand{\Bx}{\boldsymbol{x}} \newcommand{\By}{\boldsymbol{y}} \newcommand{\Bz}{\boldsymbol{z}} \newcommand{\Balpha}{\boldsymbol{\alpha}} \newcommand{\Bbeta}{\boldsymbol{\beta}} \newcommand{\Bgamma}{\boldsymbol{\gamma}} \newcommand{\Bdelta}{\boldsymbol{\delta}} \newcommand{\Beps}{\boldsymbol{\eps}} \newcommand{\Bveps}{\boldsymbol{\veps}} \newcommand{\Bzeta}{\boldsymbol{\zeta}} \newcommand{\Beta}{\boldsymbol{\eta}} \newcommand{\Btheta}{\boldsymbol{\theta}} \newcommand{\Biota}{\boldsymbol{\iota}} \newcommand{\Bkappa}{\boldsymbol{\kappa}} \newcommand{\Blambda}{\boldsymbol{\lambda}} \newcommand{\Bmu}{\boldsymbol{\mu}} \newcommand{\Bnu}{\boldsymbol{\nu}} \newcommand{\Bxi}{\boldsymbol{\xi}} \newcommand{\Bomicron}{\boldsymbol{\omicron}} \newcommand{\Bpi}{\boldsymbol{\pi}} \newcommand{\Brho}{\boldsymbol{\rho}} \newcommand{\Bsigma}{\boldsymbol{\sigma}} \newcommand{\Btau}{\boldsymbol{\tau}} \newcommand{\Bupsilon}{\boldsymbol{\upsilon}} \newcommand{\Bphi}{\boldsymbol{\phi}} \newcommand{\Bfi}{\boldsymbol{\fi}} \newcommand{\Bchi}{\boldsymbol{\chi}} \newcommand{\Bpsi}{\boldsymbol{\psi}} \newcommand{\Bomega}{\boldsymbol{\omega}} % .. calligraphic \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CD}{\mathcal{D}} \newcommand{\CE}{\mathcal{E}} \newcommand{\CF}{\mathcal{F}} \newcommand{\CG}{\mathcal{G}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CI}{\mathcal{I}} \newcommand{\CJ}{\mathcal{J}} \newcommand{\CK}{\mathcal{K}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CN}{\mathcal{N}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CQ}{\mathcal{Q}} \newcommand{\CR}{\mathcal{R}} \newcommand{\CS}{\mathcal{S}} \newcommand{\CT}{\mathcal{T}} \newcommand{\CU}{\mathcal{U}} \newcommand{\CV}{\mathcal{V}} \newcommand{\CW}{\mathcal{W}} \newcommand{\CX}{\mathcal{X}} \newcommand{\CY}{\mathcal{Y}} \newcommand{\CZ}{\mathcal{Z}} % .. typewriter \newcommand{\TA}{\mathtt{A}} \newcommand{\TB}{\mathtt{B}} \newcommand{\TC}{\mathtt{C}} \newcommand{\TD}{\mathtt{D}} \newcommand{\TE}{\mathtt{E}} \newcommand{\TF}{\mathtt{F}} \newcommand{\TG}{\mathtt{G}} \newcommand{\TH}{\mathtt{H}} \newcommand{\TI}{\mathtt{I}} \newcommand{\TJ}{\mathtt{J}} \newcommand{\TK}{\mathtt{K}} \newcommand{\TL}{\mathtt{L}} \newcommand{\TM}{\mathtt{M}} \newcommand{\TN}{\mathtt{N}} \newcommand{\TO}{\mathtt{O}} \newcommand{\TP}{\mathtt{P}} \newcommand{\TQ}{\mathtt{Q}} \newcommand{\TR}{\mathtt{R}} \newcommand{\TS}{\mathtt{S}} \newcommand{\TT}{\mathtt{T}} \newcommand{\TU}{\mathtt{U}} \newcommand{\TV}{\mathtt{V}} \newcommand{\TW}{\mathtt{W}} \newcommand{\TX}{\mathtt{X}} \newcommand{\TY}{\mathtt{Y}} \newcommand{\TZ}{\mathtt{Z}}$

Adapted from Section 1.6 of “Boolean function complexity” by Stasys Jukna.

We’re still far from proving super-polynomial lower bounds on circuit size for an explicit function. In fact, we haven’t even managed to show super-linear lower bounds yet (even within $\NC^1$)! But we can prove very crummy, linear lower bounds.

For now, we have:

  • $3.1n-o(n)$ lower bounds for circuits over the basis with all fanin-2 gates;1
  • $5n - o(n)$ lower bounds when parity and its negations are removed from the basis (and thus in particular over $\and,\or,\neg$).2

Idea

The basic idea of gate elimination is very simple: find a variable that you can fix to some value in such a way that the circuit loses $q$ gates in the process (assuming the circuit was large enough). For this to be possible, the circuit must have had $qn-O(1)$ gates to start with.

It’s interesting to think about whether this technique could give super-linear lower bounds: what would it take for a variable to eliminate more than a constant number of gates? This paper3 argues this is unlikely to work (but I’ve yet to read it).

Let’s go over two simple examples! As far as I know, both the $3.1n-o(n)$ result and the $5n-o(n)$ result are instantiations of the same idea, just with more elaborate case analysis.

Eliminating two general gates

We’ll first prove a lower bound over the basis of all fanin-2 gates. Note that in this basis, fanin-1 gates are useless (unless they’re the output gate): they can just be merged into the gates that read it. If you fix one of the inputs of a fanin-2 gate, that makes it functionally fanin-1, so you can eliminate it. So to eliminates many gates, it’s enough to find a variable that’s queried by many gates, and then fix it.4

Let $\Thr_k^n: \zo^n \to \zo$ be the threshold function: $\Thr_k^n(x) = \One[x_1 + \cdots + x_n \geq k]$. We’ll show a $2n-O(1)$ lower bound for $\Thr_2^n$ by eliminating two gates with just one variable. The main property we’ll use is that if you hardcode two variables $x_i$ and $x_j$, this can produce three different subfunctions depending on the value you give to $x_i+x_j$:

  • if $x_i+x_j = 0$, then we get $\Thr_2^{n-2}$;
  • if $x_i + x_j = 1$, then we get $\Thr_1^{n-2}$ (aka OR);
  • if $x_i + x_j = 2$, then we get $\Thr_0^{n-2}$ (aka the all-ones function).

Take the deepest5 gate $g$, and say it queries variables $x_i$ and $x_j$. These must be two distinct variables, because otherwise $g$ is effectively a fanin-1 gate, which can be eliminated for free. We’ll show that either $x_i$ or $x_j$ is queried somewhere else, and therefore we can fix that variable to $0$ to eliminate two gates and induce on the function $\Thr_2^{n-1}$.

Assume for the sake of contradiction that $x_i$ and $x_j$ are queried only by $g$. Then $g$ is the only way for the circuit to know anything about $x_i$ and $x_j$. But fixing $x_i$ and $x_j$ can only make $g$ take two different values ($0$ or $1$), while (assuming $n\geq 3$) there are three possible subfunctions with $x_i$ and $x_j$ fixed: $\Thr_2^{n-2}$, $\Thr_1^{n-2}$, and $\Thr_0^{n-2}$. So we get a contradiction.

Eliminating three $\and,\or$ gates

If we restrict the gates to just $\and,\or,\neg$ gates,6 then we get one more trick up our sleeve: we can fix any gate to a constant by setting one of its inputs to the right value. For example, if we set one of the inputs of an AND to $0$, the output will definitely be $0$. This then allows us to eliminate more gates downstream.

Here we’ll show that any $\and,\or,\neg$ circuit for the parity function on $n$ variables must have at least $3n-O(1)$ AND and OR gates. It turns out this is tight, because you can represent a fanin-2 parity gate $x \xor y$ as $(x \and \neg y) \or (\neg x \and y)$, which uses only three AND and OR gates.

Again, let $g$ be the deepest AND or OR gate, and say it queries variables $x_i$ and $x_j$.7 These must be distinct variables, otherwise the gate is useless. We claim that either $x_i$ or $x_j$ must also be queried by some other gate $g' \neq g$. Suppose they’re not: $x_i$ and $x_j$ are only queried by $g$. Then you can set $x_i$ in a way that fixes $g$ to a constant and thus make the output of the circuit independent of $x_j$. This is impossible if the circuit computes parity.

Without loss of generality, say $x_i$ is queried by $g'$. Note that gate $g$ must be read8 by at least one other gate $h$. If as above we set $x_i$ in a way sets $g$ to a constant, then this eliminates all three gates $g$, $g'$, and $h$ (since one of $h$’s inputs would get fixed). But $g'$ and $h$ might be the same gate! If that’s the case, then you should instead set $x_i$ to whatever value fixes $h$ (not $g$): this would eliminate $g$, $h$, and whichever gate reads $h$.8

  1. Jiatu Li and Tianqi Wang, “$3.1−o(n)$ circuit lower bounds for explicit functions”. 

  2. Kazuo Iwama and Hiroki Morizumi, “An explicit lower bound of $5n − o(n)$ for boolean circuits”. 

  3. Alexander Golovnev, Edward Hirsch, Alexander Knop, and Alexander Kulikov, “On the limits of gate elimination”. 

  4. Actually, in this case you don’t even really need to make sure gates get eliminated. If you can keep finding variables queried by many gates, that also implies a lower bound simply because a (connected) circuit of size $s$ can only make $\le s +1$ queries to variables. 

  5. I.e. the first one that gets computed. This is just to make sure that it’s inputs are both variables and not previous gates. 

  6. Or indeed, as long as we eliminate parity and its negation from the list of allowable gates 

  7. It might instead query a $\neg$ gate that queries a variable, but the argument goes the same way. We’ll just pretend that when a gate queries such a $\neg$ gate, that counts as querying the variable itself. 

  8. If it’s not read by any other gate, it’s either useless (so we get to eliminate a gate for free!), or it’s the output (so fixing it to a constant gives us an obvious contradiction!).  2