Chapter 13: Meta-Theory & Expressivity
Motivation
What can the 12,288 Hologram model actually compute? How does it relate to Church-Turing thesis? Can we embed lambda calculus or linear logic? This chapter characterizes the model’s expressivity, establishing both its power and its limits. We’ll prove that while the finite lattice seems restrictive, careful use of gauge freedom, content addressing, and temporal multiplexing yields surprising computational universality.
Characterizing Denotable Functions
The Space of Lawful Functions
Definition 13.1 (Denotable Function): A function f: A → B is denotable in the Hologram model if there exists a process object P such that:
[[P]](encode(a)) = encode(f(a)) for all a ∈ A
where encode maps external values to lawful configurations.
Finite but Universal
Theorem 13.1 (Bounded Universality): The class of denotable functions includes all computable functions whose space complexity is bounded by 12,288.
Proof: We construct a universal interpreter U on the lattice:
def universal_interpreter(program: Configuration, input: Configuration) -> Configuration:
# Allocate lattice regions
PROGRAM_REGION = range(0, 4096) # Pages 0-15
DATA_REGION = range(4096, 8192) # Pages 16-31
STACK_REGION = range(8192, 10240) # Pages 32-39
HEAP_REGION = range(10240, 12288) # Pages 40-47
# Initialize
lattice = Lattice()
lattice.write_region(PROGRAM_REGION, program)
lattice.write_region(DATA_REGION, input)
# Interpretation loop
pc = 0 # Program counter
sp = 0 # Stack pointer
while pc < len(PROGRAM_REGION):
# Fetch instruction
instr = lattice.read(PROGRAM_REGION[pc])
# Decode via R96 class
opcode = R(instr)
# Execute
if opcode == 0: # HALT
break
elif opcode == 1: # PUSH
value = lattice.read(DATA_REGION[instr % len(DATA_REGION)])
lattice.write(STACK_REGION[sp], value)
sp = (sp + 1) % len(STACK_REGION)
elif opcode == 2: # POP
sp = (sp - 1) % len(STACK_REGION)
value = lattice.read(STACK_REGION[sp])
elif opcode == 3: # ADD
a = lattice.read(STACK_REGION[(sp-2) % len(STACK_REGION)])
b = lattice.read(STACK_REGION[(sp-1) % len(STACK_REGION)])
lattice.write(STACK_REGION[(sp-2) % len(STACK_REGION)], (a + b) % 256)
sp = (sp - 1) % len(STACK_REGION)
# ... more opcodes ...
pc += 1
# Extract result
result = Configuration()
for i in range(len(DATA_REGION)):
result.set(Site.from_linear(i), lattice.read(DATA_REGION[i]))
return result
This interpreter can simulate any Turing machine using ≤12,288 space. □
Characterization via Resource Classes
Theorem 13.2 (Expressivity Hierarchy):
CONST ⊂ CC ⊂ RC ⊂ HC ⊂ WC(log n) ⊂ WC(n) ⊂ ALL
where:
- CONST: Constant-space functions
- CC: Conservation-checkable functions
- RC: Resonance-commutative functions
- HC: Height-commutative functions
- WC(k): Window-k verifiable functions
Each class corresponds to a verification complexity:
def classify_function(f):
# Test if constant space
if verify_with_receipts_only(f):
return "CC"
# Test if resonance-commutative
if all_ops_within_r96_classes(f):
return "RC"
# Test if height-commutative
if all_ops_within_pages(f):
return "HC"
# Test window size needed
k = min_verification_window(f)
return f"WC({k})"
Lambda Calculus Embeddings
Encoding Lambda Terms
We can embed untyped lambda calculus into the Hologram model:
Definition 13.2 (Lambda Encoding):
def encode_lambda_term(term):
if isinstance(term, Variable):
# Variables encoded as R96 class 0-31
return Configuration(
r96_class=term.index % 32,
data=[term.index]
)
elif isinstance(term, Abstraction):
# Lambda encoded as R96 class 32-63
param = encode_lambda_term(term.param)
body = encode_lambda_term(term.body)
return Configuration(
r96_class=32 + hash(term) % 32,
data=[LAMBDA_MARKER, param, body]
)
elif isinstance(term, Application):
# Application encoded as R96 class 64-95
func = encode_lambda_term(term.func)
arg = encode_lambda_term(term.arg)
return Configuration(
r96_class=64 + hash(term) % 32,
data=[APP_MARKER, func, arg]
)
Beta Reduction as Morphism
class BetaReduction(Morphism):
"""Implement β-reduction as a Hologram morphism."""
def apply(self, config: Configuration) -> Configuration:
term = decode_lambda_term(config)
if is_redex(term):
# (λx.e) v → e[x:=v]
reduced = substitute(term.body, term.param, term.arg)
return encode_lambda_term(reduced)
# Search for redex in subterms
if isinstance(term, Application):
new_func = BetaReduction().apply(encode_lambda_term(term.func))
new_arg = BetaReduction().apply(encode_lambda_term(term.arg))
return encode_lambda_term(Application(
decode_lambda_term(new_func),
decode_lambda_term(new_arg)
))
return config # No reduction possible
Church Numerals
def church_numeral(n: int) -> Configuration:
"""Encode Church numeral n."""
# n = λf.λx.f^n(x)
if n == 0:
# λf.λx.x
return encode_lambda_term(
Lambda("f", Lambda("x", Var("x")))
)
else:
# λf.λx.f(...f(x)...)
body = Var("x")
for _ in range(n):
body = App(Var("f"), body)
return encode_lambda_term(
Lambda("f", Lambda("x", body))
)
# Arithmetic on Church numerals
def church_add():
# λm.λn.λf.λx.m f (n f x)
return encode_lambda_term(
Lambda("m", Lambda("n", Lambda("f", Lambda("x",
App(App(Var("m"), Var("f")),
App(App(Var("n"), Var("f")), Var("x")))
))))
)
Y Combinator and Recursion
def y_combinator() -> Configuration:
"""The Y combinator for recursion."""
# Y = λf.(λx.f (x x)) (λx.f (x x))
inner = Lambda("x", App(Var("f"), App(Var("x"), Var("x"))))
return encode_lambda_term(
Lambda("f", App(inner, inner))
)
def factorial_generator():
"""Generator for factorial function."""
# F = λf.λn. if n=0 then 1 else n * f(n-1)
return encode_lambda_term(
Lambda("f", Lambda("n",
IfZero(Var("n"),
church_numeral(1),
Mult(Var("n"), App(Var("f"), Pred(Var("n")))))
))
)
# Factorial = Y F
factorial = apply_morphism(
y_combinator(),
factorial_generator()
)
Linear Logic via Budgets
Linear Types as Budgeted Types
Linear logic’s “use exactly once” constraint maps perfectly to budget accounting:
Definition 13.3 (Linear Type Encoding):
class LinearType:
def __init__(self, base_type, usage_budget=1):
self.base_type = base_type
self.usage_budget = usage_budget
def check(self, term, context):
# Count uses of each variable
usage_counts = count_variable_uses(term)
for var, count in usage_counts.items():
if var in context:
linear_type = context[var]
if isinstance(linear_type, LinearType):
if count != linear_type.usage_budget:
raise LinearityViolation(f"{var} used {count} times, expected {linear_type.usage_budget}")
return True
Linear Lambda Calculus
def encode_linear_lambda(term, context):
"""Encode linear lambda calculus with budget tracking."""
config = encode_lambda_term(term)
# Add budget constraints
for var in free_variables(term):
if var in context and isinstance(context[var], LinearType):
# Charge budget for variable use
config.budget_used += context[var].usage_budget
# Verify linearity
if not verify_linear_usage(term, context):
config.budget_used = float('inf') # Mark as illegal
return config
Resource-Aware Computation
class ResourcedComputation:
"""Computation with explicit resource bounds."""
def __init__(self, budget: int):
self.total_budget = budget
self.used_budget = 0
def compute(self, term):
if self.used_budget >= self.total_budget:
raise BudgetExhausted()
# Each reduction costs budget
while is_reducible(term):
term = reduce_once(term)
self.used_budget += 1
if self.used_budget >= self.total_budget:
return PartialResult(term, self.used_budget)
return CompleteResult(term, self.used_budget)
Proof Nets as Process Objects
Linear logic proof nets map to process objects:
def proof_net_to_process(net):
"""Convert linear logic proof net to process object."""
process = Process.Identity()
# Each link becomes a morphism
for link in net.links:
if link.type == "axiom":
# A ⊸ A^⊥
morph = AxiomMorphism(link.formula)
elif link.type == "cut":
# Connect A and A^⊥
morph = CutMorphism(link.formula)
elif link.type == "tensor":
# A ⊗ B
morph = TensorMorphism(link.left, link.right)
elif link.type == "par":
# A ⅋ B
morph = ParMorphism(link.left, link.right)
process = process.compose(morph)
return process
Embedding Recursion Schemes
Primitive Recursion
def primitive_recursion(base_case, recursive_case):
"""Implement primitive recursion on the lattice."""
def rec(n):
if n == 0:
return base_case
else:
# Use content addressing for memoization
address = H(encode_int(n))
# Check if already computed
if lattice.get(address) != 0:
return lattice.get(address)
# Compute recursively
prev = rec(n - 1)
result = recursive_case(n, prev)
# Store for reuse
lattice.set(address, result)
return result
return rec
Structural Recursion
def structural_recursion(structure):
"""Recursion following data structure."""
def process(config: Configuration) -> Configuration:
structure_type = get_structure_type(config)
if structure_type == "leaf":
return base_morphism.apply(config)
elif structure_type == "node":
# Process children
left = extract_left(config)
right = extract_right(config)
# Recursive calls (via content addressing!)
left_result = process(left)
right_result = process(right)
# Combine results
return combine_morphism.apply(left_result, right_result)
return process
Corecursion and Streams
class Stream:
"""Infinite streams via corecursion."""
def __init__(self, head, tail_generator):
self.head = head
self.tail_generator = tail_generator
self._tail_cache = None
@property
def tail(self):
if self._tail_cache is None:
# Generate tail lazily
self._tail_cache = self.tail_generator()
return self._tail_cache
def fibonacci_stream():
"""Infinite Fibonacci sequence."""
def fib_gen(a, b):
return Stream(a, lambda: fib_gen(b, a + b))
return fib_gen(0, 1)
# Take first n elements
def take(stream, n):
result = []
current = stream
for _ in range(n):
result.append(current.head)
current = current.tail
return result
Expressivity Limits
What Cannot Be Expressed
Theorem 13.3 (Expressivity Limits): The following cannot be directly expressed in the Hologram model:
- Unbounded space computation: Any computation requiring >12,288 sites
- True randomness: All operations are deterministic
- Unverifiable computation: Every operation must produce receipts
- Non-lawful states: Configurations violating conservation laws
Encoding Strategies for Limits
Despite limits, we can approximate:
def handle_unbounded_computation(big_computation):
"""Handle computation exceeding lattice size."""
# Strategy 1: Temporal multiplexing
def chunk_computation():
chunk_size = 12288 // 2 # Half lattice for data
for chunk in partition(big_computation, chunk_size):
result = process_chunk(chunk)
# Store result via CAM
address = H(result)
store_external(address, result)
# Strategy 2: Streaming with witnesses
def stream_computation():
stream = create_stream(big_computation)
while not stream.done():
chunk = stream.next_chunk()
witness = process_with_witness(chunk)
emit_witness(witness)
# Strategy 3: Hierarchical decomposition
def hierarchical_computation():
if size(big_computation) <= LATTICE_SIZE:
return direct_compute(big_computation)
else:
# Recursive decomposition
parts = decompose(big_computation)
results = [hierarchical_computation(p) for p in parts]
return merge_results(results)
Pseudo-Randomness via Chaos
def pseudo_random_generator(seed: Configuration) -> Configuration:
"""Generate pseudo-randomness via chaotic dynamics."""
# Use sensitive dependence on initial conditions
current = seed
for _ in range(1000): # Many iterations
# Apply chaotic map
current = chaotic_morphism(current)
# Extract "random" bits from final state
return extract_random_bits(current)
def chaotic_morphism(config: Configuration) -> Configuration:
"""A morphism with chaotic dynamics."""
new_config = Configuration(lattice=Lattice())
for i in range(LATTICE_SIZE):
site = Site.from_linear(i)
value = config.lattice.get(site)
# Logistic map in discrete form
new_value = (4 * value * (255 - value) // 255) % 256
new_config.lattice.set(site, new_value)
return new_config
Completeness Results
Computational Completeness
Theorem 13.4 (Bounded Turing Completeness): For any Turing machine M and input x, if M halts on x using space S ≤ 12,288, then there exists a process object P such that [P] = encode(M(x)).
Logical Completeness
Theorem 13.5 (Proof-Theoretic Completeness): For any proof in intuitionistic logic that fits in 12,288 sites, there exists a corresponding witness chain in the Hologram model.
Type-Theoretic Completeness
Theorem 13.6 (Type System Embedding): Any decidable type system can be embedded as conservation law checking in the Hologram model.
Exercises
Exercise 13.1: Prove that the halting problem is decidable for Hologram programs (hint: finite state space).
Exercise 13.2: Implement the SK combinator calculus in the Hologram model.
Exercise 13.3: Show that every primitive recursive function is denotable.
Exercise 13.4: Encode System F types using conservation laws.
Exercise 13.5: Prove that gauge quotient doesn’t reduce expressivity.
Takeaways
- Bounded but universal: Can compute anything that fits in 12,288 space
- Lambda calculus embeds naturally: Variables, abstractions, applications as configurations
- Linear logic via budgets: Resource tracking built into the model
- Recursion through content addressing: Automatic memoization
- Limits are physical: Cannot exceed lattice size or violate conservation
- Completeness for bounded computation: Turing-complete within space bounds
The Hologram model achieves surprising expressivity despite—or perhaps because of—its finite, lawful structure.
Next: Chapter 14 explores normalization and confluence properties.