# Shape Grammar Production Rules # Rules transform primitives within and across layers within_layer: backend: - id: R1 name: Tile Contraction rule: "T(M,N) × T(N,K) → T(M,K)" description: Matrix multiplication as tile contraction tilegym_op: matmul - id: R2 name: Kernel Fusion rule: "F(linear ∘ activation ∘ linear) → T_fused" description: Fuse multiple ops into single kernel pass tilegym_op: get_fused_swiglu_module - id: R3 name: Attention Decomposition rule: "A(Q,K,V) → softmax(QKᵀ/√d) · V" description: Decompose attention into score computation and value aggregation tilegym_op: fmha - id: R4 name: Backend Dispatch rule: "D(op, *) → D(op, cutile) | D(op, pytorch)" description: Select implementation based on available backend tilegym_op: dispatch - id: R5 name: Split-K Reduction rule: "R(T₁‖T₂‖…‖Tₖ) → T_final" description: Parallel split, independent compute, final reduction tilegym_op: splitk_reduce - id: R5b name: Student-Teacher Alignment rule: "align(student, teacher_ema) ∘ Centering(center) ∘ Sharpening(τ) → representation" description: Distill representations using exponential moving average and collapse prevention tilegym_op: ema_distill interface: - id: R6 name: Hopf Criticality rule: "O(μ→0⁺, ω₀) → bifurcation → oscillation" description: Cochlear oscillator crosses bifurcation point reference: "Eguíluz et al. (2000)" - id: R7 name: Auditory Scene Analysis rule: "Σ(mixed) → S₁ + S₂ + … + Sₙ" description: Decompose acoustic mixture into perceptual streams reference: "Bregman (1990)" - id: R8 name: Spectral Inference rule: "K(ω₁,ω₂,…) × data → P(posterior)" description: GP inference with spectral mixture kernels reference: "Wilkinson et al. (2019)" - id: R9 name: Generative Activation rule: "P(prior) + silence → I(imagery)" description: Internal predictive model generates percept without input connection: "Musical imagery, earworms" - id: R10 name: Predictive Excess rule: "I(imagery) + loss_of_constraint → hallucination" description: Unconstrained generative model produces false percepts connection: "Computational psychiatry" - id: R10b name: Learnable Filterbank rule: "dense(waveform) → filterbank(sinusoids, windowing, onsets)" description: Adaptable time-frequency representation learned directly from raw signals reference: "Verma & Berger (2021)" frontend: - id: R11 name: Texture Equivalence rule: "X(stats₁) ≈ X(stats₂) → perceptual_equivalence" description: Sounds with matched statistics are perceptually interchangeable demo: "Model-Matched Sounds" - id: R12 name: Cue Integration rule: "G(harmonicity) + G(onset) → S(grouped)" description: Multiple grouping cues combine to form a stream demo: "Auditory Grouping Cues" - id: R13 name: Illusory Continuation rule: "signal + noise_mask → C(illusory_continuation)" description: Masked signal is perceived as continuing behind the mask demo: "Spectral Completion, Illusory Texture" - id: R14 name: Generative Listening rule: "Ω(BASS_model) → Σ(inferred_scene)" description: Bayesian model infers scene structure from mixture demo: "Listening with Generative Models" reference: "Cusimano, Hewitt, & McDermott (2024)" - id: R15 name: Illusion as Probe rule: "Ψ(inharmonic_speech) → Δ(segregation_failure)" description: Manipulating harmonicity reveals segregation mechanisms demo: "Inharmonic Speech Segregation" - id: R15b name: Illusory Restoration rule: "predict(snippets) + mask → restoration" description: Reconstruct global acoustic object from partial masked snippets demo: "Illusory Continuation" cross_layer: - id: X1 name: Dispatch ↔ Tuning ↔ Framing mapping: backend: "D(op, backend) — route to correct implementation" interface: "O(μ, ω₀) — tune oscillator to characteristic frequency" frontend: "Ψ(stimulus, percept) — select perceptual interpretation" principle: "Selective routing of information to specialized processors" - id: X2 name: Fusion ↔ Spectral Composition ↔ Texture mapping: backend: "F(op₁ ∘ op₂) — fuse operations into single pass" interface: "K(spectral_mixture) — compose spectral components" frontend: "X(texture_statistics) — perceive fused texture as whole" principle: "Combining multiple components into a unified representation" - id: X3 name: Reduction ↔ Scene Integration ↔ Segregation mapping: backend: "R(split → merge) — split-K parallel reduction" interface: "Σ(S₁+S₂+…) → scene — integrate streams" frontend: "Δ(mixture → sources) — segregate sources" principle: "Parallel decomposition followed by coherent recomposition" - id: X4 name: Attention ↔ Prediction ↔ Grouping mapping: backend: "A(Q,K,V) — selective attention over sequences" interface: "P(prior, likelihood) — Bayesian selective updating" frontend: "G(cue₁, cue₂) — perceptual grouping by cues" principle: "Selective weighting of information based on relevance" - id: X5 name: Normalization ↔ Compression ↔ Continuity mapping: backend: "N(x, ε) — normalize/stabilize activations" interface: "O(μ→0, compressive) — cochlear compression and gain" frontend: "C(continuity) — perceptual filling-in stabilizes percept" principle: "Stabilizing representations against variation and noise" - id: X6 name: Representation Learning ↔ Learned Cochlea ↔ Illusory Restoration mapping: backend: "EMA(student, teacher) — distill robust representations" interface: "dense(waveform) — learn sinusoids, windows, and onsets" frontend: "restore(snippets) — predict global object from partial views" principle: "Adapting sensory interfaces to predict global coherence from local parts" meta_rule: name: Selective Decomposition and Recomposition pattern: | INPUT (complex, entangled) → DECOMPOSE into tiles / oscillators / streams → DISPATCH to specialized processors / channels / frames → TRANSFORM via fused kernels / spectral inference / prediction → REDUCE via split-K / scene integration / perceptual binding → OUTPUT (structured, interpreted) causal_uncertainty: description: > Manipulating causal uncertainty in sound objects — the degree to which the auditory system's causal inference about sound sources is underdetermined. rules: - id: CU1 name: Causal Ambiguity rule: "Ω(model₁) ≈ Ω(model₂) → causal_ambiguity" description: Different physical models produce perceptually similar sounds - id: CU2 name: Bayesian Causal Inference rule: "P(cause|sound) = P(sound|cause)·P(cause) / P(sound)" description: Prior beliefs about causes shape perception - id: CU3 name: Perceptual Multistability rule: "Ψ(ambiguous_cause) → bistable_percept" description: Causal uncertainty produces alternating interpretations computational_psychiatry: description: > The grammar extends into computational psychiatry via predictive processing. Perceptual disorders are production rule failures. conditions: - name: Normal Perception rule: "P(prior) + data → P(posterior)" description: Balanced Bayesian update backend_analogue: Correct dispatch, balanced fusion - name: Musical Imagery rule: "P(prior) + ∅ → I(vivid)" description: Prior-driven generation without input backend_analogue: Generative mode (no input tensor) - name: Auditory Hallucination rule: "P(prior) ≫ data → Ψ(false_percept)" description: Prior dominates over sensory evidence backend_analogue: Dispatch override — prior overrides data - name: Tinnitus rule: "O(locked) → persistent_activation" description: Oscillator locked in limit cycle without stimulus backend_analogue: Stuck dispatch — persistent activation loop improvisation: description: > Real-time sequential generation under constraints — the structural pattern shared by autoregressive LLM decoding, jazz improvisation, and generative listening. Inspired by Stephon Alexander's "The Jazz of Physics" (2016). rules: - id: IMP1 name: Autoregressive Generation rule: "context(t-1) + prior → sample(t) → context(t)" description: Each step conditioned on accumulated context and shaped by learned prior layers: backend: "KV cache + model weights → sample from logits → extend cache" interface: "Chord changes + melodic knowledge → play next note → update context" frontend: "Acoustic mixture + source priors → infer scene → update belief" - id: IMP2 name: Coltrane Substitution rule: "I → I · T₄ · T₄ · T₄" description: > Transpose resolution target by major thirds (Z₃ acting on Z₁₂). Coltrane Changes as group-theoretic production rule on pitch space. reference: "Alexander (2016), Coltrane — Giant Steps (1960)" - id: IMP3 name: Temperature / Inside-Outside rule: "P(next | context, τ) where τ controls exploration-exploitation" description: > Low temperature = predictable, inside the changes (bebop). High temperature = surprising, outside the changes (free jazz). Maps to sampling temperature in LLM decoding and posterior sharpness in Bayesian scene inference. - id: IMP4 name: Compressed Context rule: "full_history → compressed_representation → attend" description: > MLA compresses KV cache into latent representations. A musician internalizes the changes rather than reading the chart. A listener builds a statistical summary rather than storing the waveform. - id: IMP5 name: Cosmic Inharmonicity rule: "CMB_peaks ≈ [1, 2.45, 3.68] · f₀ → not_quite_harmonic" description: > The CMB acoustic oscillation peaks are shifted from integer harmonics by the baryon-to-photon ratio — the universe's first sound is slightly inharmonic, a cosmic instance of the inharmonicity manipulation in auditory grouping experiments. reference: "Alexander (2016)"