Homoiconic Rewriting

Code = Data = Graph = Parallel Reduction

Trit: 0 (ERGODIC - coordinates the stack)

Core Synthesis

┌─────────────────────────────────────────────────────────────────┐
│              HOMOICONIC REWRITING PIPELINE                      │
├─────────────────────────────────────────────────────────────────┤
│                                                                  │
│   λ-term ──quote──→ S-exp ──parse──→ INet ──CUDA──→ Result     │
│     │                  │               │              │         │
│   typed              data            graph         parallel     │
│   code            (homoiconic)     rewriting      reduction     │
│                                                                  │
└─────────────────────────────────────────────────────────────────┘

GF(3) Balanced Dependencies

lambda-calculus

gay-mcp

interaction-nets

lispsyntax-acset

algebraic-rewriting

slime-lisp

Sum: (+1+1) + (0+0) + (-1-1) = 0 ✓

The Homoiconic Property

Level 1: S-expressions (Lisp)

;; Code
(+ 1 2)

;; Data (same representation!)
'(+ 1 2)

;; Transform code as data
(map inc '(+ 1 2))  ; → (1 2 3)

Level 2: Interaction Nets (Graphs)

Code (λ-term):     Data (graph):         Rewrite (reduction):
  λx. x x          ┌───┐                 ┌───┐     ┌───┐
                   │ λ │──┬──┐           │ @ │─────│ @ │
                   └─┬─┘  │  │           └─┬─┘     └─┬─┘
                     │    │  │             │         │
                   ┌─┴─┐  │  │     →       └────┬────┘
                   │ @ │──┘  │                  │
                   └─┬─┘     │               result
                     └───────┘

Level 3: ACSets (Algebraic Databases)

# Code: rewrite rule
rule = Rule(L, K, R)

# Data: same representation!
rule_data = @acset RuleSchema begin
    L = [...]; K = [...]; R = [...]
end

# Transform rules as data
composed_rule = compose_rules(rule1, rule2)

Key Algorithms

1. λ → Interaction Net Compilation

function compile_lambda_to_inet(term::LambdaTerm)::InteractionNet
    match term
        Var(x)      => wire_to(x)
        Lam(x, body) => agent(:λ, [x], compile(body))
        App(f, arg)  => agent(:@, [compile(f), compile(arg)])
    end
end

2. Parallel Reduction (CUDA-ready)

function parallel_reduce!(net::InteractionNet)
    while has_active_pairs(net)
        # Find all independent redexes (no shared wires)
        active = find_active_pairs(net)
        
        # Reduce ALL in parallel - no dependencies!
        @cuda threads=length(active) reduce_kernel!(net, active)
    end
end

3. ACSet Rewriting (DPO)

using AlgebraicRewriting

# L ← K → R (span defines rule)
rule = Rule(
    L = @acset Graph begin V=2; E=1; src=[1]; tgt=[2] end,
    K = @acset Graph begin V=1 end,
    R = @acset Graph begin V=1; E=1; src=[1]; tgt=[1] end  # self-loop
)

# Apply via double pushout
result = rewrite(rule, graph, match)

CUDA Integration (Groote et al.)

GPU Kernel for Active Pair Reduction

__global__ void reduce_active_pairs(
    Agent* agents,
    Wire* wires, 
    int* active_pairs,
    int n_active
) {
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx >= n_active) return;
    
    int pair_id = active_pairs[idx];
    Agent* a1 = &agents[wires[pair_id].from];
    Agent* a2 = &agents[wires[pair_id].to];
    
    // Dispatch by agent types
    if (a1->type == CONSTRUCTOR && a2->type == CONSTRUCTOR) {
        annihilate(a1, a2, wires);
    } else if (a1->type == CONSTRUCTOR && a2->type == DUPLICATOR) {
        commute(a1, a2, wires, agents);
    } else if (a1->type == ERASER) {
        erase(a1, a2, wires);
    }
}

Performance (from Eindhoven team)

Typed Decomposition

Simply Typed λ-Calculus → Linear Logic → Interaction Nets

Type System          Linear Logic         Interaction Net
────────────         ────────────         ───────────────
A → B                !A ⊸ B               λ-agent + !-box
A × B                A ⊗ B                Constructor pair
A + B                A ⊕ B                Case agent

GF(3) Type Assignment

# Types carry trits
struct TypedTerm
    term::LambdaTerm
    type::SimpleType
    trit::Int  # -1, 0, +1
end

# Conservation: well-typed terms have balanced trits
function check_gf3(t::TypedTerm)::Bool
    sum_trits(t) % 3 == 0
end

Practical Commands

# Parse λ-term to S-expression
echo "(lambda (x) (x x))" | bb -e '(read)'

# Compile to interaction net (Bend/HVM)
bend compile program.bend -o program.hvm

# Run with GPU parallelism
hvm run program.hvm --cuda

# Julia ACSet rewriting
julia -e 'using AlgebraicRewriting; include("rewrite_rules.jl")'

Triadic Pipelines

Pipeline 1: λ-Calculus Optimization

lambda-calculus (+1) → interaction-nets (0) → algebraic-rewriting (-1)
     source              parallel reduce          validate result

Pipeline 2: Colored Term Rewriting

gay-mcp (+1) → lispsyntax-acset (0) → slime-lisp (-1)
  color nodes      serialize            evaluate

Pipeline 3: Full Stack

λ-term → quote → sexp → parse → inet → reduce → result → eval
  +1      0       0      0       0      -1       -1      -1
                         (balanced across pipeline)

Key Authors

Literature

1. Lafont (1990) - "Interaction Nets"
2. Groote et al. (2022) - "Innermost many-sorted term rewriting on GPUs"
3. Boy de la Tour & Echahed (2020) - "Parallel rewriting of attributed graphs"
4. Mazza (2007) - "Symmetric Interaction Combinators"

Related Skills

• lambda-calculus

  (+1) - Term generation

• interaction-nets

  (0) - Parallel semantics

• algebraic-rewriting

  (-1) - DPO/SPO rules

• lispsyntax-acset

  (0) - Sexp ↔ data bridge

• sexp-neighborhood

  (0) - Sexp skill index

• gay-mcp

  (+1) - Deterministic coloring

Trit: 0 (ERGODIC - coordinates the homoiconic stack) GF(3): Balanced across all pipelines