Carry: Theories, Design & Applications

• Carry is a multifaceted concept that includes digit differences in numeration, signal propagation in adders, payload support in robotics, and state inference in sensing.
• In arithmetic and numeration, rigorous metrics like amortized carry propagation and carry value transformations isolate carry dynamics and guide algorithmic design.
• Practical applications in adder circuits, robotic payload mechanisms, and wearable exoskeletons showcase how optimizing carry management enhances performance and efficiency.

In the literature surveyed here, the term carry is used for several technically distinct phenomena. In numeration theory, it denotes the number of digit positions that differ between the representation of an integer $N$ and that of $N+1$. In computer arithmetic, it denotes the bit or block-level signal that propagates through adders and often dominates latency. In robotics and human assistance, it denotes payload carrying and carrying activities. In sensing, it becomes a recognition target: whether a moving subject is carrying an object (Berthé et al., 2019, Balasubramanian et al., 2017, Zhou et al., 2022, Lazzaroni et al., 2023, Gao et al., 2024).

1. Carry propagation in numeration systems

For an abstract numeration system ordered by radix order, the carry propagation at integer $N$ is defined by

$$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$

that is, the number of digit-positions that differ between $rep_N$ and $rep_{N+1}$. Writing

$S(N)=\sum_{i=0}^{N-1} cp(i),$

the amortized carry propagation is

$CP=\lim_{N\to\infty} \frac{S(N)}{N},$

when this limit exists (Berthé et al., 2019).

In the usual base-$p$ numeration system, the limit exists and is

$\frac{p}{p-1}.$

The paper recovers similar values for several non-standard numeration systems. For abstract numeration systems whose signature is eventually periodic, the amortized carry propagation exists and equals $N+1$0. For rational abstract systems, existence is linked to the local growth rate $N+1$1 and to the generating function being “adev”; when the relevant hypotheses hold for the language and its quotients, the value is $N+1$2. For greedy systems with exponential basis $N+1$3, one obtains $N+1$4 when the system is also pce. For $N+1$5-numeration with $N+1$6 a Parry number, the value is

$N+1$7

(Berthé et al., 2019).

A central point is that existence is not automatic. The paper gives counterexamples in which the local growth rate exists but amortized carry propagation fails, and states that the sufficient conditions established by combinatorial, algebraic, and ergodic methods are close to being necessary. This makes carry propagation not merely a local digit effect, but an asymptotic invariant sensitive to the structure of the representation system (Berthé et al., 2019).

2. Carry Value Transformation and iterative carry dynamics

Carry Value Transformation (CVT) isolates the carry structure of addition. For two nonnegative integers $N+1$8 and $N+1$9 written in binary,

$N$0

Equivalently, in integer form,

$N$1

The definitions extend to any radix $N$2 by using digit-wise carry tests and shifting the carry vector one place (Pal et al., 2011).

The fundamental identity proved in the paper is

$N$3

This decomposes ordinary addition into a carry component and an XOR component. The same framework yields an iterative map

$N$4

under which the sum $N$5 remains invariant. The paper proves that, when repeatedly adding the CVT and XOR values of two non-negative integers $N$6 and $N$7 with $N$8, the number of iterations required to get either $N$9 or $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$0 is at most the length of $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$1 when both are expressed as binary strings (Pal et al., 2011).

A modified scheme using MCVT converges even faster: repeated addition of MCVT and XOR requires a maximum of two iterations for MCVT to be zero. The same work also defines an equivalence relation on $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$2 based on convergence time, thereby dividing the CV table into disjoint equivalence classes. This suggests a view of carry not only as a by-product of addition, but as the state variable of a discrete deterministic dynamical system (Pal et al., 2011).

3. Carry as the critical path in adder design

In binary adders, carry propagation is the main source of delay. For an accurate ripple-carry adder, with

$$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$3

the standard relations are

$$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$4

Carry lookahead adders replace serial carry rippling with explicit lookahead equations over generate and propagate terms; section-carry based carry lookahead adders compute only one carry-out per block and allow intra-block ripple for sums. In a 32/28nm CMOS semi-custom ASIC flow, an optimized SCBCLA architecture reports a 9.8% improvement in figure-of-merit (FOM) compared to an optimized CCLA architecture, where the FOM is defined as the inverse of the product of power, delay, and area (Balasubramanian et al., 2016).

Approximate adder design alters carry handling explicitly. In the 32-bit approximate RCAs and CLAs considered in one study, the lower $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$5 bits are placed in approximate mode, no carry-propagation is performed in those positions, the sum is produced by a 2-input OR, and the carry is fixed to zero in the approximated segment. The paper reports that approximate RCAs achieve reductions in power-delay product ranging from 19.5% to 82% relative to the accurate RCA for approximation sizes from 4 to 20 bits, while approximate CLAs achieve reductions ranging from 16.7% to 74.2% relative to the accurate CLA; on average, approximate CLAs achieve a 46.5% reduction in PDP compared to approximate RCAs (Balasubramanian et al., 2017).

A different strategy is carry prediction. The CESA architecture partitions an $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$6-bit adder into $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$7 summation blocks, each with a Carry Estimate Unit, Propagating Error Rectification Logic, and a Selection Unit. The reported result is that the adder is 91.2% faster than the conventional ripple-carry adder, and that rectification logic makes it more accurate than state-of-the-art adders like SARA and BCSA by 74% (Bhattacharjya et al., 2020).

Carry management also appears in carry-increment structures. An improved carry increment adder replaces the ripple-carry block inside the conventional CIA with a carry lookahead block. In the 8-bit Spartan-6 implementation reported, the maximum path delay changes from 21.61 ns for CIA_RCA to 19.61 ns for CIA_CLA, while power remains essentially unchanged at 46.8 mW and 47.0 mW respectively (Devi et al., 2016).

In quaternary CNTFET carry-propagate adders, the critical design decision is to compute two intermediate carries,

$$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$8

and then multiplex them according to $$cp(N) = |rep_N| + |rep_{N+1}| - 2\cdot |lcp(rep_N,rep_{N+1})|,$$9. The paper states that, although the data path is quaternary, the carry output always has logical value 0 or 1, and that using 0 and $rep_N$0 levels instead of 0 and $rep_N$1 for input and output carries significantly reduces carry propagation (Etiemble, 2022).

4. Reversible and fault-tolerant carry logic

In reversible logic, carry generation must be realized without information loss, fan-out, or feedback. Because a reversible circuit implements only functions having one-to-one mapping between input and output vectors, every intermediate carry must itself be preserved or encoded through reversible gates. This redefines conventional carry look-ahead and carry-skip constructions in terms of gate count, garbage outputs, constant inputs, and parity preservation (Islam et al., 2010, Islam et al., 2010).

The key building blocks in the cited works are the IG and MIG gates. The MIG gate is described as universal and parity preserving, and it allows any fault that affects no more than a single signal to be readily detectable at the circuit’s primary outputs. The proposed reversible fast adders implement carry look-ahead and carry-skip logic and are reported to offer less hardware complexity while being efficient in terms of gate count, garbage outputs, and constant inputs than existing counterparts (Islam et al., 2010).

A parallel formulation uses the IG gate for a proposed 16-bit high speed reversible adder. The IG gate is likewise described as universal and parity preserving, with the same single-signal fault detectability at the primary outputs. The paper argues that the resulting carry look-ahead and carry-skip adders therefore have an inherent opportunity of detecting errors on the output side, while remaining efficient in hardware complexity, gate count, garbage outputs, and constant inputs (Islam et al., 2010).

A related paper proposes a generalized $rep_N$2 reversible gate family, discusses a $rep_N$3 gate of that family, gives a full-adder implementation using two such gates, designs a 4-bit carry skip adder, and discusses a variable block carry skip adder (Islam et al., 2010). Taken together, these works treat carry not simply as a signal to be propagated quickly, but as a quantity that must be propagated reversibly and checked through parity-preserving structure.

5. Carry as payload support and locomotor assistance

In robotics, “carry” denotes payload support during stance and locomotion. The quadruped robot Kirin is presented as a 50 kg electrically-actuated quadruped robot intended to leverage payload carrying capability. Its defining mechanism is a prismatic quasi-direct-drive (QDD) leg, which the paper states greatly augments payload carrying capability. The theoretical analysis implies that the lifting task tends to be a bottleneck for existing robots with articulated knee joints. In preliminary experiment, up to 125 kg payload lifting in static stance and 50 kg payload carrying in dynamic trotting are tested, and whole body compliance with payload carrying is also demonstrated (Zhou et al., 2022).

In wearable robotics, the focus shifts from transporting payloads to assisting humans while carrying them. The active back-support exoskeleton XoTrunk uses two control strategies for carrying assistance: a foot-sensors mode, based on force-sensing resistors under the user’s shoe, and an actuator-velocity mode, based on the angular velocity of each hip actuator. The exoskeleton produces a total assistive torque $rep_N$4 about the wearer’s lumbosacral joint and divides it asymmetrically between the legs according to gait phase: $rep_N$5 The total torque magnitude is modulated by forearm EMG: $rep_N$6 This makes the support stance-aware and effort-aware rather than symmetric (Lazzaroni et al., 2023).

The reported experiment involved ten healthy male volunteers performing treadmill walking at self-selected speed while carrying a 10 kg box in three randomized conditions: no exoskeleton, exoskeleton with foot-sensors control, and exoskeleton with actuator-velocity control. Stride length, hip range of motion, and speed showed no significant changes. Knee range of motion showed a significant reduction, with both exoskeleton modes reducing it by 8% versus no-exo. For erector spinae activation, RMS showed a downward trend that did not reach significance, whereas the 90th-percentile value showed a significant main effect: foot-sensors mode reduced peak by 11% versus no-exo, and actuator-velocity mode reduced peak by 10% versus no-exo. For subjective perception, 100% of users reported benefit and 0% hindrance for back/trunk in both exoskeleton modes (Lazzaroni et al., 2023).

A common simplification is to treat carrying assistance as equivalent to lifting assistance. The control strategies reported here differ from symmetric lifting-designed assistance by delivering torque only to the stance limb. The paper states that this preserves stride length and hip motion, limits knee range-of-motion reduction to 8%, and yields significant lumbar-load reduction. It also notes that the actuator-velocity framework is convenient for real-world deployment because it uses only exoskeleton encoders, but must still be validated in unstructured, non-treadmill motion (Lazzaroni et al., 2023).

6. Carry as an observable state in multimodal sensing

In sensing systems, “carry” can denote the probability that a person is carrying an object. MMW-Carry is designed to predict this probability using millimeter-wave radar signals, complemented by camera input, and is framed as a rapid and cost-effective preliminary screening solution, specifically for non-super-sensitive scenarios. The system addresses localization challenges in complex indoor environments caused by multi-path reflections by integrating camera-based human detection and tracking with a radar-camera plane transformation, followed by a zoom-in operation on radar images (Gao et al., 2024).

The camera stream is processed with a Deformable Part Model (DPM) detector and a Kalman filter under a constant-velocity state model. The radar-camera relation is represented by a $rep_N$7 homography $rep_N$8 mapping image pixels to ground-plane coordinates, estimated from manually labeled correspondences; in practice, 19 points are collected and leave-one-out cross-validation yields an average projection error of 3.4 inches. The resulting occupancy region is cropped from the radar range–azimuth–elevation image and normalized to a $rep_N$9 tensor (Gao et al., 2024).

For recognition, the system uses a lightweight 3D-ResNet plus a location embedder and three separate 5-layer MLPs that output class existence scores for laptop, phone, and knife. Temporal aggregation is handled by the late-fusion knwlTrf module, which maintains instantaneous and global states per class and produces a fused output

$rep_{N+1}$0

The paper emphasizes long-term observation: as tracked subjects move through the field of view, multiple radar viewpoints are fused rather than classified independently (Gao et al., 2024).

The evaluation uses 3.4 hours of synchronized radar and stereo-RGB data, about 367 K frames, collected at 12 indoor sites. Subjects walk randomly, alone or in pairs, carrying 0–3 objects chosen from $rep_{N+1}$1, in open or concealed form, with 48% concealed, over a 1–15 m range. The reported performance for the full system is 0.2522 FPR, 0.2171 MR, and 0.7619 accuracy; equivalently, the abstract reports an average error rate of 25.22\% false positives and a 21.71\% missing rate. Increasing the observation window from 1 to 150 frames raises accuracy from about 60% to 80%, while FPR and MR both fall from about 40% to 20%, after which gains saturate (Gao et al., 2024).

This use of “carry” is semantically different from digit carry or mechanical carrying, but it preserves the same structural idea of state inference under constraints. The paper also makes its scope explicit: it concerns a small set of common-in-everyday-life metallic objects and non-super-sensitive screening. A plausible implication is that “carry” in this context is an operational label, not a generic threat descriptor (Gao et al., 2024).