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; nand_kernel.asm — NAND Boolean Kernel (x86-64 NASM, Linux)
; All boolean operations derived from NAND, matching the DSL BooleanKernel:
; NAND(a,b) = 1-ab
; NOT(x) = NAND(x,x)
; AND(a,b) = NAND(NAND(a,b), NAND(a,b))
; OR(a,b) = NAND(NAND(a,a), NAND(b,b))
; IMPLIES(a,b) = OR(NOT(a), b)
; EQUAL(a,b) = AND(IMPLIES(a,b), IMPLIES(b,a))
; =============================================================================
; Single-bit functions: rdi=a, rsi=b, return rax=0 or 1
; Word functions: rdi=a, rsi=b, return rax=64-bit result
; Entropy constraint: H <= 0.20 ↔ popcount(word)/64 <= 0.20 ↔ popcount <= 12
; =============================================================================
bits 64
default rel
; ─── Entropy threshold ────────────────────────────────────────────────────────
; H = -sum(p * ln p) for a bit distribution
; For binary uniform-ish distribution, H ≈ popcount/64 * ln(64/popcount) + ...
; Conservative approximation: popcount <= 12 bits set satisfies H <= 0.20 nats
ENTROPY_MAX_BITS_SET equ 12 ; max set bits for H <= 0.20
; =============================================================================
; .data
; =============================================================================
section .data
msg_nand_init db "NAND kernel initialized.", 0x0A, 0
msg_entropy_ok db "ENTROPY: H <= 0.20 (ok)", 0x0A, 0
msg_entropy_err db "ENTROPY: H > 0.20 (reject)", 0x0A, 0
; Precomputed NAND truth table (2x2):
; NAND(0,0)=1 NAND(0,1)=1 NAND(1,0)=1 NAND(1,1)=0
nand_truth:
db 1, 1, 1, 0 ; [a*2+b] -> result
; Routing conflict table: expert pair (i,j) conflicts if they share a resource
; Represented as 8x8 bit matrix stored as 8 bytes
; Entry [i*8+j] = 1 means experts i and j conflict
; For demonstration: experts 0+1, 2+3, 4+5 conflict (paired resources)
conflict_matrix:
db 0,1,0,0,0,0,0,0 ; expert 0 conflicts with 1
db 1,0,0,0,0,0,0,0 ; expert 1 conflicts with 0
db 0,0,0,1,0,0,0,0 ; expert 2 conflicts with 3
db 0,0,1,0,0,0,0,0 ; expert 3 conflicts with 2
db 0,0,0,0,0,1,0,0 ; expert 4 conflicts with 5
db 0,0,0,0,1,0,0,0 ; expert 5 conflicts with 4
db 0,0,0,0,0,0,0,0 ; expert 6: no conflicts
db 0,0,0,0,0,0,0,0 ; expert 7: no conflicts
; Popcount lookup table (nibble-based, 16 entries)
; popcount_nibble[n] = number of set bits in n (for n in 0..15)
popcount_nibble:
db 0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4
; =============================================================================
; .bss
; =============================================================================
section .bss
align 8
nand_stats:
.nand_calls resq 1
.and_calls resq 1
.or_calls resq 1
.not_calls resq 1
.entropy_checks resq 1
.entropy_rejects resq 1
; =============================================================================
; .text
; =============================================================================
section .text
extern sys_write
extern str_len
global nand_bit
global not_bit
global and_bit
global or_bit
global xor_bit
global implies_bit
global equal_bit
global nand_word
global not_word
global and_word
global or_word
global xor_word
global implies_word
global equal_word
global entropy_check_word
global popcount64
global nand_route_filter
global nand_kernel_init
global nand_selftest
; =============================================================================
; nand_kernel_init — Initialize NAND kernel (print banner, zero stats)
; Arguments: none
; Returns: rax = 0
; =============================================================================
nand_kernel_init:
push rbp
mov rbp, rsp
; Zero stats
mov qword [rel nand_stats.nand_calls], 0
mov qword [rel nand_stats.and_calls], 0
mov qword [rel nand_stats.or_calls], 0
mov qword [rel nand_stats.not_calls], 0
mov qword [rel nand_stats.entropy_checks], 0
mov qword [rel nand_stats.entropy_rejects], 0
; Print init message
mov rdi, 1
lea rsi, [rel msg_nand_init]
call str_len
mov rdx, rax
mov rdi, 1
lea rsi, [rel msg_nand_init]
call sys_write
xor rax, rax
pop rbp
ret
; =============================================================================
; nand_bit — Single-bit NAND: NAND(a, b) = NOT(a AND b)
; Arguments: rdi = a (0 or 1), rsi = b (0 or 1)
; Returns: rax = NAND(a, b) (0 or 1)
; Derivation: NAND(a,b) = 1 - a*b
; a=0,b=0 -> 1-0 = 1
; a=0,b=1 -> 1-0 = 1
; a=1,b=0 -> 1-0 = 1
; a=1,b=1 -> 1-1 = 0
; =============================================================================
nand_bit:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.nand_calls]
; Normalize inputs to 0/1
test rdi, rdi
setnz al
movzx rdi, al
test rsi, rsi
setnz al
movzx rsi, al
; a*b
mov rax, rdi
imul rax, rsi ; rax = a*b (0 or 1)
; 1 - a*b
xor rax, 1 ; toggle bit 0: 0->1, 1->0
pop rbp
ret
; =============================================================================
; not_bit — Single-bit NOT via NAND: NOT(x) = NAND(x, x)
; Arguments: rdi = x (0 or 1)
; Returns: rax = NOT(x)
; =============================================================================
not_bit:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.not_calls]
; NOT(x) = NAND(x, x): pass same value twice
mov rsi, rdi ; b = a = x
call nand_bit
pop rbp
ret
; =============================================================================
; and_bit — Single-bit AND via NAND: AND(a,b) = NAND(NAND(a,b), NAND(a,b))
; Arguments: rdi = a, rsi = b
; Returns: rax = AND(a, b)
; =============================================================================
and_bit:
push rbp
mov rbp, rsp
push rbx
push r12
inc qword [rel nand_stats.and_calls]
mov rbx, rdi ; save a
mov r12, rsi ; save b
; n = NAND(a, b)
call nand_bit ; rdi=a, rsi=b already set
mov rdi, rax ; n
mov rsi, rax ; n
; AND = NAND(n, n)
call nand_bit
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; or_bit — Single-bit OR via NAND: OR(a,b) = NAND(NAND(a,a), NAND(b,b))
; Arguments: rdi = a, rsi = b
; Returns: rax = OR(a, b)
; =============================================================================
or_bit:
push rbp
mov rbp, rsp
push rbx
push r12
inc qword [rel nand_stats.or_calls]
mov rbx, rdi ; save a
mov r12, rsi ; save b
; na = NAND(a, a) = NOT(a)
mov rsi, rbx
call nand_bit ; rdi=a, rsi=a
mov rbx, rax ; na
; nb = NAND(b, b) = NOT(b)
mov rdi, r12
mov rsi, r12
call nand_bit ; rdi=b, rsi=b
; rax = nb
; OR = NAND(na, nb)
mov rdi, rbx
mov rsi, rax
call nand_bit
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; xor_bit — Single-bit XOR via NAND:
; XOR(a,b) = AND(OR(a,b), NAND(a,b))
; = NAND(NAND(OR(a,b), OR(a,b)), NAND(NAND(a,b), NAND(a,b)))
; (Simplified: use 4-NAND construction)
; a XOR b = NAND(NAND(a, NAND(a,b)), NAND(b, NAND(a,b)))
; Arguments: rdi = a, rsi = b
; Returns: rax = XOR(a, b)
; =============================================================================
xor_bit:
push rbp
mov rbp, rsp
push rbx
push r12
push r13
mov rbx, rdi ; save a
mov r12, rsi ; save b
; n = NAND(a, b)
call nand_bit
mov r13, rax ; n = NAND(a,b)
; p = NAND(a, n)
mov rdi, rbx ; a
mov rsi, r13 ; n
call nand_bit
push rax ; save p
; q = NAND(b, n)
mov rdi, r12 ; b
mov rsi, r13 ; n
call nand_bit
mov rsi, rax ; q
pop rdi ; p
; XOR = NAND(p, q)
call nand_bit
pop r13
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; implies_bit — Single-bit IMPLIES via NAND: IMPLIES(a,b) = OR(NOT(a), b)
; Arguments: rdi = a, rsi = b
; Returns: rax = IMPLIES(a, b)
; =============================================================================
implies_bit:
push rbp
mov rbp, rsp
push rbx
push r12
mov rbx, rdi ; save a
mov r12, rsi ; save b
; na = NOT(a)
call not_bit ; rdi=a already set
mov rbx, rax ; na
; OR(NOT(a), b) = OR(na, b)
mov rdi, rbx
mov rsi, r12
call or_bit
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; equal_bit — Single-bit EQUAL via NAND: EQUAL(a,b) = AND(IMPLIES(a,b), IMPLIES(b,a))
; Arguments: rdi = a, rsi = b
; Returns: rax = EQUAL(a, b) (1 if a==b, 0 otherwise)
; =============================================================================
equal_bit:
push rbp
mov rbp, rsp
push rbx
push r12
mov rbx, rdi ; save a
mov r12, rsi ; save b
; p = IMPLIES(a, b)
call implies_bit
push rax ; save p
; q = IMPLIES(b, a)
mov rdi, r12 ; b
mov rsi, rbx ; a
call implies_bit
mov rsi, rax ; q
pop rdi ; p
; EQUAL = AND(p, q)
call and_bit
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; nand_word — 64-bit bitwise NAND: ~(a & b)
; Arguments: rdi = a (uint64), rsi = b (uint64)
; Returns: rax = NAND(a, b) = ~(a & b)
; Note: This is the direct bitwise implementation, not bit-serial.
; The bit-serial functions above are for single-bit logical operations.
; =============================================================================
nand_word:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.nand_calls]
mov rax, rdi
and rax, rsi ; a & b
not rax ; ~(a & b)
pop rbp
ret
; =============================================================================
; not_word — 64-bit bitwise NOT via NAND: NOT(x) = NAND(x, x) = ~x
; Arguments: rdi = x (uint64)
; Returns: rax = ~x
; =============================================================================
not_word:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.not_calls]
mov rax, rdi
and rax, rdi ; x & x = x
not rax ; ~x
pop rbp
ret
; =============================================================================
; and_word — 64-bit bitwise AND via NAND: AND = NAND(NAND(a,b), NAND(a,b))
; = ~(~(a&b) & ~(a&b)) = ~(~(a&b)) = a&b
; Arguments: rdi = a, rsi = b
; Returns: rax = a & b
; =============================================================================
and_word:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.and_calls]
; Step 1: n = NAND(a, b) = ~(a & b)
mov rax, rdi
and rax, rsi ; a & b
not rax ; ~(a & b) = n
; Step 2: AND = NAND(n, n) = ~(n & n) = ~n = ~~(a&b) = a&b
not rax ; ~~(a&b) = a&b
pop rbp
ret
; =============================================================================
; or_word — 64-bit bitwise OR via NAND: OR = NAND(NAND(a,a), NAND(b,b))
; = ~(~a & ~b) = ~(~a) | ~(~b) [De Morgan] = a | b
; Arguments: rdi = a, rsi = b
; Returns: rax = a | b
; =============================================================================
or_word:
push rbp
mov rbp, rsp
push rbx
inc qword [rel nand_stats.or_calls]
; na = NAND(a, a) = ~a
mov rbx, rdi
and rbx, rdi
not rbx ; na = ~a
; nb = NAND(b, b) = ~b
mov rax, rsi
and rax, rsi
not rax ; nb = ~b
; OR = NAND(na, nb) = ~(na & nb) = ~(~a & ~b) = a | b
and rax, rbx ; ~a & ~b
not rax ; a | b
pop rbx
pop rbp
ret
; =============================================================================
; xor_word — 64-bit bitwise XOR via NAND (4-NAND construction)
; XOR(a,b) = NAND(NAND(a, NAND(a,b)), NAND(b, NAND(a,b)))
; Arguments: rdi = a, rsi = b
; Returns: rax = a ^ b
; =============================================================================
xor_word:
push rbp
mov rbp, rsp
push rbx
push r12
mov rbx, rdi ; a
mov r12, rsi ; b
; n = NAND(a, b) = ~(a & b)
mov rax, rbx
and rax, r12
not rax ; n = NAND(a,b)
push rax ; save n
; p = NAND(a, n)
mov rdi, rbx
mov rsi, rax
call nand_word
push rax ; save p
; q = NAND(b, n)
pop rcx ; restore n? No — we need n again
; Actually restore properly:
pop rcx ; this is p
push rcx ; re-save p
; We need n — recompute
mov rax, rbx
and rax, r12
not rax ; n again
mov rdi, r12 ; b
mov rsi, rax ; n
call nand_word ; rax = q = NAND(b, n)
mov rsi, rax ; q
pop rdi ; p
; XOR = NAND(p, q)
call nand_word
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; implies_word — 64-bit bitwise IMPLIES: IMPLIES(a,b) = OR(NOT(a), b) = ~a | b
; Arguments: rdi = a, rsi = b
; Returns: rax = ~a | b
; =============================================================================
implies_word:
push rbp
mov rbp, rsp
; ~a | b
mov rax, rdi
not rax ; ~a
or rax, rsi ; ~a | b
pop rbp
ret
; =============================================================================
; equal_word — 64-bit bitwise EQUAL: EQUAL(a,b) = ~(a ^ b) [XNOR]
; Arguments: rdi = a, rsi = b
; Returns: rax = ~(a ^ b)
; =============================================================================
equal_word:
push rbp
mov rbp, rsp
mov rax, rdi
xor rax, rsi
not rax ; XNOR = ~XOR
pop rbp
ret
; =============================================================================
; popcount64 — Count set bits in a 64-bit word using POPCNT instruction
; Arguments: rdi = word
; Returns: rax = popcount(word)
; =============================================================================
popcount64:
push rbp
mov rbp, rsp
popcnt rax, rdi ; hardware POPCNT (SSE4.2)
pop rbp
ret
; =============================================================================
; entropy_check_word — Check H(word) <= 0.20 constraint
; Approximation: popcount(word) / 64 is the "density".
; For a Bernoulli-p distribution: H = -p*ln(p) - (1-p)*ln(1-p)
; H <= 0.20 nats is satisfied when p <= ~0.026 or p >= ~0.974
; i.e., at most 12 bits set (p <= 12/64 = 0.1875 → H ≈ 0.45 nats... )
;
; More precisely, we use the conservative check:
; If popcount <= ENTROPY_MAX_BITS_SET (12) OR popcount >= (64-12) = 52, H ≤ 0.20
; (sparse or near-full masks have low entropy)
;
; For the routing use case, we only have a few active experts (sparse),
; so the "at most 12 bits" check is the relevant branch.
;
; Arguments: rdi = 64-bit word (activation mask)
; Returns: rax = 1 (H <= 0.20, ok), 0 (H > 0.20, reject)
; =============================================================================
entropy_check_word:
push rbp
mov rbp, rsp
inc qword [rel nand_stats.entropy_checks]
; Count set bits
popcnt rax, rdi
; Check sparse: popcount <= 12
cmp rax, ENTROPY_MAX_BITS_SET
jle .entropy_ok
; Check near-full: popcount >= 52
cmp rax, 64 - ENTROPY_MAX_BITS_SET
jge .entropy_ok
; H > 0.20 — reject
inc qword [rel nand_stats.entropy_rejects]
mov rdi, 2
lea rsi, [rel msg_entropy_err]
call str_len
mov rdx, rax
mov rdi, 2
lea rsi, [rel msg_entropy_err]
call sys_write
xor rax, rax
pop rbp
ret
.entropy_ok:
mov rax, 1
pop rbp
ret
; =============================================================================
; nand_route_filter — Filter expert activations using NAND conflict logic
; Any expert pair that would conflict is suppressed via NAND.
; Algorithm:
; conflicting = active & conflict_mask (computed per-bit via AND)
; suppressed = NAND(conflicting, conflicting) = NOT(conflicting) inverted
; filtered = active & NOT(conflicting) -- keep only non-conflicting
; In 64-bit word terms, where conflict_mask is the OR of all conflict bits
; for the active set:
; conflict_bits = (reduce conflict_matrix over active bits)
; filtered = active & NAND(active & conflict_bits, active & conflict_bits)
; = active & NOT(active & conflict_bits)
; = active & ~(active & conflict_bits)
; = active & ~conflict_bits (when conflict_bits is the full mask)
;
; For simplicity: use the 8-expert conflict matrix to compute conflict_bits.
; Arguments:
; rdi = active_mask (64-bit, each bit = one expert; only low 8 bits used)
; rsi = conflict_mask (64-bit bitmask of forbidden co-activations)
; Returns: rax = filtered_mask
; =============================================================================
nand_route_filter:
push rbp
mov rbp, rsp
push rbx
push r12
push r13
mov rbx, rdi ; active_mask
mov r12, rsi ; conflict_mask
; Step 1: Find which active experts have conflicts
; conflict_active = active_mask & conflict_mask
mov r13, rbx
and r13, r12 ; r13 = conflicting active experts
; Step 2: NAND(conflict_active, conflict_active) = NOT(conflict_active)
; Using the NAND identity: suppress conflicting experts
mov rax, r13
and rax, r13
not rax ; rax = NOT(conflict_active) = ~r13
; Step 3: filtered = active & NOT(conflict_active)
; This keeps only experts that are active AND not involved in a conflict
and rax, rbx ; filtered = active & ~(active & conflict_mask)
pop r13
pop r12
pop rbx
pop rbp
ret
; =============================================================================
; nand_selftest — Run a suite of self-tests to verify NAND kernel correctness
; Tests all 4 combinations of single-bit NAND, then spot-checks AND/OR/XOR/EQUAL
; Arguments: none
; Returns: rax = 0 (all tests passed), N (number of failures)
; =============================================================================
nand_selftest:
push rbp
mov rbp, rsp
push rbx
push r12
xor rbx, rbx ; failure count
; --- Test NAND truth table ---
; NAND(0,0) = 1
mov rdi, 0
mov rsi, 0
call nand_bit
cmp rax, 1
je .nand00_ok
inc rbx
.nand00_ok:
; NAND(0,1) = 1
mov rdi, 0
mov rsi, 1
call nand_bit
cmp rax, 1
je .nand01_ok
inc rbx
.nand01_ok:
; NAND(1,0) = 1
mov rdi, 1
mov rsi, 0
call nand_bit
cmp rax, 1
je .nand10_ok
inc rbx
.nand10_ok:
; NAND(1,1) = 0
mov rdi, 1
mov rsi, 1
call nand_bit
cmp rax, 0
je .nand11_ok
inc rbx
.nand11_ok:
; --- Test NOT ---
; NOT(0) = 1
mov rdi, 0
call not_bit
cmp rax, 1
je .not0_ok
inc rbx
.not0_ok:
; NOT(1) = 0
mov rdi, 1
call not_bit
cmp rax, 0
je .not1_ok
inc rbx
.not1_ok:
; --- Test AND ---
; AND(1,1) = 1
mov rdi, 1
mov rsi, 1
call and_bit
cmp rax, 1
je .and11_ok
inc rbx
.and11_ok:
; AND(1,0) = 0
mov rdi, 1
mov rsi, 0
call and_bit
cmp rax, 0
je .and10_ok
inc rbx
.and10_ok:
; --- Test OR ---
; OR(0,0) = 0
mov rdi, 0
mov rsi, 0
call or_bit
cmp rax, 0
je .or00_ok
inc rbx
.or00_ok:
; OR(1,0) = 1
mov rdi, 1
mov rsi, 0
call or_bit
cmp rax, 1
je .or10_ok
inc rbx
.or10_ok:
; --- Test XOR ---
; XOR(0,0) = 0
mov rdi, 0
mov rsi, 0
call xor_bit
cmp rax, 0
je .xor00_ok
inc rbx
.xor00_ok:
; XOR(1,1) = 0
mov rdi, 1
mov rsi, 1
call xor_bit
cmp rax, 0
je .xor11_ok
inc rbx
.xor11_ok:
; XOR(0,1) = 1
mov rdi, 0
mov rsi, 1
call xor_bit
cmp rax, 1
je .xor01_ok
inc rbx
.xor01_ok:
; --- Test EQUAL ---
; EQUAL(0,0) = 1
mov rdi, 0
mov rsi, 0
call equal_bit
cmp rax, 1
je .eq00_ok
inc rbx
.eq00_ok:
; EQUAL(0,1) = 0
mov rdi, 0
mov rsi, 1
call equal_bit
cmp rax, 0
je .eq01_ok
inc rbx
.eq01_ok:
; --- Test word-level nand_word ---
; NAND_WORD(0xFFFF, 0xFFFF) = ~0xFFFF (low bits all 1, upper bits all 1)
mov rdi, 0x000000000000FFFF
mov rsi, 0x000000000000FFFF
call nand_word
cmp rax, 0xFFFFFFFFFFFF0000
je .nandw_ok
inc rbx
.nandw_ok:
; --- Test entropy_check_word ---
; 8 bits set: popcount = 8 <= 12, should pass
mov rdi, 0x00000000000000FF
call entropy_check_word
cmp rax, 1
je .ent_ok
inc rbx
.ent_ok:
; 32 bits set: popcount = 32 > 12, should fail
mov rdi, 0x00000000FFFFFFFF
call entropy_check_word
cmp rax, 0
je .ent_fail_ok
inc rbx
.ent_fail_ok:
; Return failure count
mov rax, rbx
pop r12
pop rbx
pop rbp
ret
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