Continuous Universal Analog Logic (LCUA): Smooth Functional Equivalents of Digital Gates Without Comparators

1. Introduction and Motivation

Decades of CMOS scaling made digital logic the dominant computing substrate. Yet the energy costs of ubiquitous inference and sensing have revived interest in analog and mixed-signal approaches for specific tasks (e.g., multiply–accumulate, sensing near the source, and low-latency control). Contemporary AI workloads, along with edge constraints, motivate exploring logic families that: (i) minimize conversion overhead between analog and digital domains, (ii) offer graceful degradation under noise, and (iii) can interface naturally with neuromorphic and analog ML primitives.

This paper develops Continuous Universal Analog Logic (LCUA): smooth gate realizations exact on $\{0,1\}$, stabilized by a simple booleanizer $B_n$. The contributions are:

• Smooth polynomial gates ($\mathrm{NOT}$/$\mathrm{AND}$/$\mathrm{OR}$/$\mathrm{XOR}$/$\mathrm{NAND}$/$\mathrm{XNOR}$) that coincide with Boolean truth tables on $\{0,1\}$.
• Booleanizer $B_n$: a two-attractor map on $[0,1]$ with local contraction around 0 and 1; a single scalar parameter $n>1$ trades off noise tolerance and flatness around $1/2$.
• Cascade stability: a sufficient block-contraction condition ensuring that deep interleavings of gates and booleanizers remain within a tube around the Boolean vertices and converge.
• Harmonic/phase variant: an equivalent gate family in the encoding $s=2b-1\in \{-1,+1\}$ (equivalently $s=cos\left(\pi \right(1-b\left)\right)$), relevant to oscillator/phase domains.
• Analog route-to-hardware: practical realizations using op-amp adders/scalers, analog multipliers (e.g., Gilbert-cell/OTA-based), and a log–antilog realization of $B_n$.

1.0.0.1. Why now?

Inference energy, analog sensing, and on-device AI increasingly rely on compact, low-latency primitives. LCUA eliminates hard comparators as the fundamental discretization step while still asymptotically restoring Boolean states. This can reduce analog–digital crossings and enable continuous-time pipelines (e.g., in-sensor processing) that are compatible with AI front-ends and neuromorphic back-ends.

2. Related Work

Classical Boolean algebra (Boole) and its realization via relays/switching (Shannon) underlie digital electronics. Fuzzy logic and t/s-norms (Zadeh; Hájek; Klement–Mesiar–Pap) explore continuous operations on $[0,1]$ (e.g., product t-norm $a·b$ and probabilistic s-norm $a+b-ab$). Neuromorphic computing (Mead; Indiveri & Liu) and ultra-low-power analog design (Sarpeshkar) show the efficacy of analog primitives for efficient computation. Prior analog-logic proposals (e.g., continuous-time analog logic for probabilistic inference) typically end with discretization stages or assume domain-specific comparators. Here we prove a sufficient cascade-contraction condition for a family of smooth gates that need no hard thresholds to recover Boolean behavior.

3. Formal Framework

3.1. Signal Space and Polynomial Gates

We work on the signal space $S=[0,1]$. For $a,b\in [0,1]$, define smooth gate primitives

$$\begin{array}{cc}\hfill \mathrm{NOT}\left(a\right)& =1-a,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{AND}(a,b)& =ab,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \mathrm{OR}(a,b)& =a+b-ab,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathrm{XOR}(a,b)& =a+b-2ab,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{NAND}(a,b)& =1-ab,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathrm{XNOR}(a,b)& =1-a-b+2ab.\hfill \end{array}$$

(6)

Lemma 1

(Exactness on vertices). For $a,b\in \{0,1\}$, the above formulas coincide with the corresponding Boolean truth tables.

Proof. 

Use $a^2=a$ for $a\in \{0,1\}$ and evaluate each polynomial on the vertices. □

3.2. Smooth Booleanizer

Definition 1

(Booleanizer $B_n$). For $n>1$, define $B_n:[0,1]\to [0,1]$ by

$$B_n\left(x\right)={\displaystyle \frac{x^n}{x^n+{(1-x)}^n}}.$$

(7)

Proposition 1

(Fixed points and symmetry). $B_n\left(0\right)=0$, $B_n\left(1\right)=1$, and $B_n(1-x)=1-B_n\left(x\right)$.

Lemma 2

(Local contraction at the attractors). There exist $\delta \in (0,\frac{1}{2})$ and $\kappa \in (0,1)$ such that for $x\in [0,\delta ]\cup [1-\delta ,1]$, one has $|B_n^{\prime }\left(x\right)|\le \kappa$. In particular, 0 and 1 are locally attractive fixed points.

Proof. 

Differentiate to obtain

$$B_n^{\prime }\left(x\right)={\displaystyle \frac{n{x}^{n-1}{(1-x)}^{n-1}}{{\left(x^n+{(1-x)}^n\right)}^2}}.$$

(8)

On $[0,\delta ]$ with $\delta <1/2$, use $(1-x)\ge (1-\delta )$ to bound $|B_n^{\prime }\left(x\right)|\le n{x}^{n-1}/{(1-\delta )}^{n+1}\to 0$ as $x\to 0$, for $n>1$. By symmetry, the same holds on $[1-\delta ,1]$. Continuity implies the existence of $\kappa <1$. □

3.3. Deep Cascades with Interleaved Projection

Let each block $G_{\ell }:{[0,1]}^m\to [0,1]$ be $L_{G_{\ell }}$-Lipschitz on its operating range $I_{\ell }\subset [0,1]$. Consider the depth-L interleaving

$${\mathrm{output}}_{\ell }=B_n\left(G_{\ell }\left({\mathrm{inputs}}_{\ell }\right)\right),\ell =1,\dots ,L.$$

(9)

Theorem 1

(Sufficient block-contraction for cascade stability). If for every subchain $[i,k]$ we have

$$\prod _{j=i}^k{L}_{G_j}·\prod _{j=i}^k\underset{x\in I_j}{sup}\left|B_n^{\prime }\left(x\right)\right|<1,$$

(10)

then the interleaved network (9) is contractive by blocks: trajectories remain in a neighborhood (“tube”) of $\{0,1\}$ and converge to Boolean vertices after sufficiently many stages.

Proof 

(Proof Sketch). A composition of Lipschitz maps is Lipschitz with constant the product of the constituent constants. The “tube” around $\{0,1\}$ is precisely $[0,\delta ]\cup [1-\delta ,1]$, where $B_n$ is contractive (Lemma 2). Even if a gate $G_{\ell }$ pushes a signal slightly away from the attractors (e.g., $\mathrm{OR}(0.9,0.9)=0.99$), the subsequent $B_n$ pulls it back with a net contraction factor. The product bound (10) ensures contraction on each subchain and thus convergence by the Banach fixed-point principle. □

4. A Harmonic (Phase) Variant

Encode $b\in \{0,1\}$ as $s=2b-1\in \{-1,+1\}$ (equivalently $s=cos\left(\pi \right(1-b\left)\right)$). This encoding is natural in oscillator/phase domains and differential signaling: it avoids DC offsets, improves common-mode noise rejection, and interfaces cleanly with coupled-oscillator and Ising-like substrates. For $s_a=2a-1$ and $s_b=2b-1$, define

$$\begin{array}{cc}\hfill {\mathrm{NOT}}_{\phi }\left(a\right)& =\frac{1-s_a}{2},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathrm{AND}}_{\phi }(a,b)& =\frac{1+s_a+s_b+s_a{s}_b}{4},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\mathrm{OR}}_{\phi }(a,b)& =\frac{3+s_a+s_b-s_a{s}_b}{4},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\mathrm{XOR}}_{\phi }(a,b)& =\frac{1-s_a{s}_b}{2}.\hfill \end{array}$$

(14)

These formulas are smooth and match the Boolean truth tables exactly on $\{0,1\}$ under the $s=2b-1$ encoding. Interleaving with $B_n$ remains recommended in noisy phase domains.

5. Circuit Synthesis Examples

5.1. Continuous Half-Adder

For $a,b\in [0,1]$ let

$$\begin{array}{cc}\hfill \mathrm{SUM}& =B_n\left(a+b-2ab\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \mathrm{CARRY}& =B_n\left(ab\right).\hfill \end{array}$$

(16)

5.2. Continuous SR Latch (Smooth ODEs)

For states $x,y\in [0,1]$ and inputs $S,R\in [0,1]$, with time constant $\tau >0$,

$$\begin{array}{cc}\hfill \tau \dot{x}& =-x+B_n\left((1-R)(1-y)\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \tau \dot{y}& =-y+B_n\left((1-S)(1-x)\right).\hfill \end{array}$$

(18)

With $S=R=0$, the system has two stable equilibria near $(x,y)\approx (1,0)$ and $(0,1)$; smooth pulses on S or R toggle the state, as in a standard SR latch built from cross-coupled NOR gates.

6. Toward Analog Hardware

6.0.0.2. Blocks.

(i) Op-amp adders/subtractors and scalers; (ii) analog multipliers (e.g., AD633-class, OTA/Gilbert cells); (iii) a log–antilog or exponential cell to realize $z\mapsto z^n$ and ${(1-z)}^n$; (iv) low-pass RC for glitch suppression.

6.0.0.3. Scaling and range.

The polynomial gates map ${[0,1]}^2\to [0,1]$ for inputs in range. In the presence of noise or offsets, apply $B_n$ periodically to restore margins.

6.0.0.4. Realizing $B_n$.

Options include: (a) explicit log–antilog stages to compute $x^n$ and ${(1-x)}^n$, followed by a soft division; (b) rational/Padé approximations near $x\in [0,1]$; (c) piecewise-smooth approximations tuned for hardware efficiency.

7. Discussion: Sensitivity, Cost, Speed, Robustness

Sensitivity to n. Small n may not contract enough; large n flattens near $x=1/2$ and can amplify offsets. Choose n to balance RMS error in cascades vs. settling time and bandwidth.

Cost of the booleanizer.$B_n$ requires exponentiation or an accurate surrogate. Compare area/energy to a CMOS inverter baseline; explore Padé approximants and log–antilog implementations. The implementation cost of $B_n$ is likely higher than a simple digital comparator/inverter in standard CMOS, but the trade-off is the elimination of ADC/DAC boundaries in fully analog pipelines.

Speed and bandwidth. Settling is set by the booleanizer pole and multiplier bandwidth. Small-signal/Bode analysis can bound maximum rate and accumulated jitter in deep chains.

Robustness to variation. Monte Carlo (SPICE) with $\pm 5\%$–$\pm 10\%$ tolerances on gains/resistors and input noise: measure truth-table error, $0/1$ margins, and switching-time distributions.

Comparison with CMOS. Define a minimal NAND/half-adder benchmark to estimate static/dynamic energy, area, and speed relative to a CMOS baseline.

8. Protocol for Validation

• Numerical: compare deep cascades with and without $B_n$; log-relative error and settling time vs. n.
• SPICE: implement AND/OR/XOR and $B_n$; AC and transient analysis; full Monte Carlo.
• Prototype: PCB with op-amps and OTA/Gilbert multipliers, plus a log–antilog stage for $B_n$; validate truth tables, robustness, and throughput.

9. Limitations and Future Work

Global contraction without input-range assumptions remains open; the cost of $B_n$ versus CMOS in scaled technologies must be characterized; the phase variant needs a complete phase-noise analysis. Future work: alternative booleanizers with the same fixed points but lower cost; more sequential elements (JK/T) and continuous memory; co-design with neuromorphic arrays and oscillator networks; and system-level studies for in-sensor and edge-AI pipelines.

10. Conclusions

Boolean logic can be realized with smooth functions without hard comparators, exactly preserving truth tables on $\{0,1\}$ while achieving stability through interleaved projection. This opens a path for analog-universal architectures that trade some generality for significant efficiency and noise tolerance in AI-adjacent workloads.