Equivalence and Meaning
Formalizing Semantic Filters
In the previous post I introduced the Semantic Filter as a refinement of exclusion. Exclusion filters prune Chaos by ruling out incoherent strings; semantic filters then assign lawful meaning to the survivors, interpreting them as trajectories. Here I want to formalize this idea: how semantic filters act as quotient maps that collapse strings into equivalence classes, each corresponding to a coherent world.
1. From Strings to Trajectories
Let Chaos be the set of all infinite binary strings. An exclusion filter picks out a subset F ⊆ Chaos: the strings that pass basic consistency constraints. A semantic filter is then a mapping:
S : F → T
where T is the set of lawful trajectories (quantum state evolutions, automaton runs, dynamical histories). Each string x ∈ F is interpreted as a trajectory τ = S(x).
2. Equivalence Classes of Strings
Different strings can map to the same trajectory:
S(x₁) = S(x₂) = τ
In this case, x₁ and x₂ belong to the same equivalence class under S. The preimage of a trajectory τ is the set of all strings that generate it:
S⁻¹(τ) = { x ∈ F : S(x) = τ }.
Thus, a semantic filter partitions the set F into equivalence classes, each class corresponding to one lawful world.
3. Examples
Redundancy in Encoding: Suppose the semantic filter ignores every second bit. Then strings 0101… and 0001… both map to the same trajectory. They belong to the same equivalence class.
Quantum Histories: Imagine a qubit measured repeatedly in the Z basis, but where some measurement outcomes are decohered away. Distinct bitstrings differing only in decohered positions map to the same physical trajectory.
4. Invertibility and Loss
If S is one-to-one, every coherent string corresponds to a unique trajectory. The string is the trajectory.
If S is many-to-one, strings collapse into equivalence classes. Some detail is lost; only the quotient structure remains.
If S were one-to-many (rare), then a single string could correspond to multiple possible worlds; but in most formalizations we prefer determinism at the semantic level.
5. Why This Matters
Exclusion filters define what survives.
Semantic filters define how survivors are grouped into worlds.
The equivalence class view makes clear that meaning emerges not from individual bitstrings, but from the partition structure induced by semantics.
In other words: coherence is not just survival; it is also identification — treating different raw sequences as the same world.
6. Place in the Arc
Chaos Reservoir — all random strings.
Exclusion Filters — prune incoherence.
Semantic Filters — partition survivors into equivalence classes, each a lawful world.
Constructors — stable patterns within worlds.
Life and Consciousness — self-maintaining and self-representing constructors.
Conclusion
Semantic filters are quotient maps: they collapse raw randomness into equivalence classes of coherent strings, each class corresponding to a world. This formalization makes precise the idea that semantics is not just interpretation but identification — the recognition that different chaotic traces can mean the same coherent trajectory.