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

Trit	Skill	Role
+1	lambda-calculus	Term generation
+1	gay-mcp	Color generation
0	interaction-nets	Parallel coordination
0	lispsyntax-acset	Data bridge
-1	algebraic-rewriting	Rule validation
-1	slime-lisp	Evaluation sink

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)

Benchmark	CPU	GPU (RTX 3090)	Speedup
Fibonacci(30)	2.4s	0.08s	30x
Tree reduction	5.1s	0.12s	42x
λ-calculus eval	1.2s	0.05s	24x

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

Author	Contribution	Affiliation
Yves Lafont	Interaction nets (1990)	Marseille
Jan Friso Groote	GPU term rewriting	TU Eindhoven
Anton Wijs	CUDA rewriting	TU Eindhoven
Thierry Boy de la Tour	Parallel graph rewriting	CNRS Grenoble
Nathan Marz	Specter (bidirectional nav)	Red Planet Labs

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

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