Intuition The ONE core ideaA computer normally keeps its "brain" (the processor) and its "notebook" (the memory) in separate places, and wastes most of its time and energy just carrying numbers between them . Processing-in-Memory says: stop carrying — do a little of the thinking right where the numbers already live.
Before you can enjoy Processing-in-memory (PIM) , you must own every word and symbol it throws at you. This page builds each one from nothing — plain words first, then a picture, then why the topic needs it . Read top to bottom; each block leans only on the ones above it.
Definition Processor (CPU) and Memory (DRAM)The processor is the part that does arithmetic — add, multiply, compare. The memory (usually DRAM , dynamic random-access memory) is a huge grid of tiny cells, each holding one bit (a 0 or a 1). Think of the processor as a fast chef and memory as a giant warehouse of labelled ingredient boxes.
Picture two boxes on a table joined by a long wire.
Figure s01 — the processor (left, yellow) and memory (right, blue) drawn as two separate boxes joined by one thin "bus" wire; the pink double-arrow marks the slow traffic path every number must cross.
Intuition Why we even name these two thingsEvery symbol in the topic is about the distance and traffic between these two boxes. If you don't see them as separate with a wire between them , nothing else will land. That wire is the villain of the whole story.
The bus is the wire (really a bundle of wires) that carries data between memory and the processor. Every number the processor wants must ride this bus.
B Bandwidth is how many bytes per second the bus can carry. We write it B byte is 8 bits — enough to store one small number or one character.
Picture a water pipe: bandwidth is the pipe's thickness , not the water's speed. A fat pipe (B
B B around this thin pipe.
N N total number of bytes a job must move over the bus. If you read three numbers and write one for every element, N
C C total number of arithmetic operations the job performs (adds, multiplies...). One a + b is one op.
R R how many operations per second the processor can do. Like B moving , R thinking .
Figure s02 — two stacked time-bars. Top (data-hungry job): the blue move-term N / B C / R C / R which term wins is what matters.
Intuition Why we split time into two piecesWe want to know which piece dominates . If N / B C / R R Memory Wall .
Definition Arithmetic intensityI I = C / N operations per byte moved . It answers: "for every byte I drag over the bus, how much useful math do I get out of it?"
⚠️ Symbol note: in this section the letter I arithmetic intensity (a pure count, ops per byte). Later, in the electricity section, we will retire this meaning and let I electrical current . Same letter, two totally different worlds — we will flag the switch when it happens.
Intuition The picture: work-per-tripImagine each byte is a heavy trip to the warehouse. High I I great PIM candidate ).
Low I bandwidth-bound → the wire limits you → PIM helps a lot.
High I compute-bound → the chef limits you → PIM helps little.
This single ratio is the "80/20 test" the parent note runs first. It also feeds the Roofline Model , which plots performance against I
A DRAM chip is split into many independent sub-grids called banks . Each bank can be read at the same time as the others — like many warehouse aisles being searched in parallel. (More in DRAM Organization (banks, rows) .)
k B bank B in t
k number of banks working in parallel.B bank inside one bank — how fast that aisle moves data internally.B in t total internal bandwidth when all banks work together:
B in t = k ⋅ B bank
Figure s03 — k B B in t B
Intuition Why internal bandwidth is secretly enormousInside a bank, thousands of bits are pulled from a row at once . That internal flow B in t B before it ever has to squeeze onto the bus. The speedup is roughly B in t / B = k ⋅ ( B bank / B )
E o p E m o v e
E o p one arithmetic operation , measured in joules per operation (J/op) .E m o v e fetch the operands from DRAM across the bus. When quoted per bit it is joules per bit (J/bit) ; per operand-fetch it is joules per access (J/access) . Concretely, a single arithmetic op costs on the order of a few picojoules (1 pJ = 1 0 − 12 J
Empirically E m o v e ≈ 100 – 1000 × E o p far more expensive than computing on it — like the tiring part of cooking being the running to the warehouse, not the chopping.
Intuition Why the topic obsesses over movementBoth time (T = N / B + C / R energy (E ≈ N ⋅ E m o v e + C ⋅ E o p move term for data-hungry jobs. So the payoff isn't a faster chef — it's fewer trips. PIM attacks exactly the term that dominates.
The parent's crossbar example borrows two laws of electricity. Both must be earned here.
I = G V Symbol switch: from here on, I electrical current (flow of charge). We are now inside electricity, not workload accounting.
Voltage V Conductance G G Current I I = G V
Why this tool? In a crossbar we store a weight as a conductance G input as a voltage V G × V is the physics.
Definition The summation symbol∑ ∑ j x j add up all the x j j ." For example ∑ j = 1 3 x j = x 1 + x 2 + x 3
Definition Kirchhoff's current law (KCL)Currents entering a shared wire add up . If three cells dump I 1 , I 2 , I 3 I 1 + I 2 + I 3
Why this tool? Multiply gave us each G ij V j sum , which — now that we know the ∑ ∑ j G ij V j = G i 1 V 1 + G i 2 V 2 + … is a dot product. Multiplication (Ohm) + addition (Kirchhoff) = a whole matrix–vector multiply in one analog instant. See Memristor / ReRAM Crossbars .
Figure s04 — a crossbar grid. Blue vertical wires carry input voltages V 1 , V 2 , V 3 G ij I i
Definition Dot product & matrix–vector multiplyA dot product of two lists multiplies matching entries and sums: a 1 b 1 + a 2 b 2 + … matrix–vector multiply does one dot product per row of a grid of numbers. This is the core operation of Neural Network Accelerators — which is why PIM loves neural nets.
Definition Amdahl's Law (the sanity brake)If only a fraction p S whole job's speedup is capped:
S t o t a l = ( 1 − p ) + S p 1 p S ( 1 − p ) S → ∞ S t o t a l = 1/ ( 1 − p )
So even a huge B in t C / R Amdahl's Law ; they never reach infinity.
Processor and Memory as two boxes
Time = N over B plus C over R
Arithmetic intensity I = C over N
Banks k and internal bandwidth B_int
Dot product and matrix vector multiply
Amdahl Law S total capped
Test yourself — cover the right side and answer before revealing.
What are the two "boxes" PIM cares about and what joins them? The processor (does math) and the memory (stores bits), joined by the bus (the slow wire).
What does bandwidth B Bytes per second the bus can carry — the thickness of a pipe, not the water's speed.
Write the classic time equation and say what each term means. T v N = N / B + C / R
Define arithmetic intensity and what low vs high I I = C / N I I
What is a bank, and why does B in t = k ⋅ B bank B A bank is an independent DRAM sub-grid; many run in parallel, so total internal bandwidth vastly exceeds the single external bus.
What units carry E o p E m o v e E o p E m o v e E m o v e ≈ 100 – 1000 × E o p
State Ohm's law and Kirchhoff's current law in one line each. Ohm: I = G V
What does the symbol ∑ j x j Add up all the x j j
How do those two laws combine into a matrix–vector multiply? Ohm gives each product G ij V j I = G V
Write Amdahl's Law and its ceiling. S t o t a l = 1/ (( 1 − p ) + p / S ) S → ∞ 1/ ( 1 − p )
