irreducible

Computational irreducibility as functoriality in Rust, implementing Jonathan Gorard's "A Functorial Perspective on (Multi)computational Irreducibility" (arXiv:2301.04690).

Core insight: A computation is irreducible iff a certain functor Z': T -> B (from computations to cobordisms) preserves composition. No shortcuts exist when Z' is functorial.

irreducible is the example consumer of the catgraph core + physics layer (v0.2.0): catgraph supplies the Fong-Spivak categorical infrastructure (cospans, spans, hypergraph categories, cospan-algebras), catgraph-applied the Petri-net substrate, and catgraph-physics the hypergraph DPO rewriting, multiway evolution graphs, confluence diamond detection, and branchial spectral analysis. irreducible owns the computation-facing layer -- interval algebra, adjunctions, monoidal coherence, discrete exterior calculus, trace analysis -- plus the computation models (TM, CA, SRS, NTM, Petri nets).

354 tests on default features (381 with manifold-curvature,dec), zero clippy warnings. Rust 2024 edition, MSRV 1.90.

Component Index

Module	Component	Purpose
interval.rs	DiscreteInterval, ParallelIntervals	Discrete interval algebra for the cobordism category B (re-export shim into catgraph_physics::interval as of v0.6.3)
complexity.rs	Complexity, StepCount	Sequential/parallel complexity composition
computation_state.rs	ComputationState	State lifecycle + interval-map bridge
adjunction.rs	ZPrimeOps, AdjunctionIrreducibility, AdjunctionVerification	Abstract Z' ⊣ Z adjunction traits
bifunctor.rs	TensorProduct, IntervalTransform	Bifunctor laws (associativity, unit, symmetry)
multiway_coherence.rs	AssociatorWitness, BraidingWitness, CoherenceError	Non-strict SMC coherence over multiway graphs
multiway_stokes.rs	MultiwayComplex, OneForm, TwoForm	Discrete exterior calculus on 2D multiway complexes (feature: dec)
temporal_cospan_chain.rs	TemporalComplex, ConservationResult, TemporalComplexError (StokesError deprecated alias)	Cospan chain bridge for interval sequences (re-export shim into catgraph_physics::temporal_cospan_chain as of v0.6.3)
trace.rs	StepTrace, analyze_trace, RepeatDetection, is_irreducible (IrreducibilityTrace deprecated alias)	Generic trace analysis, repeat detection (re-export shim into catgraph_physics::trace as of v0.6.3)
functor/mod.rs	IrreducibilityFunctor, MultiwayIrreducibilityResult	Functor Z': T -> B, multiway branch analysis
functor/adjunction.rs	ZPrimeAdjunction, AdjunctionVerification, CompactClosedWitness	Concrete Z' ⊣ Z adjunction + compact-closed (cup/cap zigzag, Prop 3.2) witnesses
functor/monoidal.rs	MonoidalFunctorResult, TensorCheck	Symmetric monoidal functor verification
functor/interval_algebra.rs	IntervalCospanAlgebra, multiway_step_cospans	Z' as a cospan-algebra (F&S Def 2.2): interval-bundle transport over multiway step cospans
functor/frobenius_preservation.rs	FrobeniusPreservationResult, verify_frobenius_preservation	Z' as a hypergraph functor: Eq. 12 generator preservation + per-event spider factorization (μ = merge, δ = fork, ε = death)
functor/corel.rs	Corel, step_corels, evolution_corel	Merge partitions as corelations (F&S 2018 Ex 6.64) with fingerprint gluing; hypergraph-side MergesCorelExt adds coarsest_common_refinement confluence comparison
functor/bifunctor.rs	TensorProduct, IntervalTransform	Re-exports local bifunctor laws
functor/fong_spivak.rs	FrobeniusVerificationResult, verify_cospan_chain_frobenius	Fong-Spivak Frobenius decomposition verification
functor/stokes_integration.rs	StokesIrreducibility	Stokes conservation analysis wrapper
machines/turing.rs	TuringMachine, ExecutionHistory	Deterministic Turing machines
machines/cellular_automaton.rs	ElementaryCA, Generation	1D elementary cellular automata (256 rules)
machines/trace.rs	StepTrace, TraceAnalysis (IrreducibilityTrace deprecated alias)	Generic trace analysis, repeat detection (proxy re-export of crate::trace shim)
machines/multiway/string_rewrite.rs	StringRewriteSystem, SRSState	Pattern-based multiway string rewriting
machines/multiway/ntm.rs	NondeterministicTM, NTMBuilder	Non-deterministic Turing machines
machines/multiway/manifold_bridge.rs	ManifoldCurvature, BranchialEmbedding	Regge deficit-angle curvature on branchial complexes via dc_topology (feature-gated)
machines/petri/	PetriNetMachine, PetriBuilder, PetriExecutionHistory, run_multiway_reachability	Place/transition Petri nets — linear trace, multiway reachability, cospan bridge
machines/hypergraph/catgraph_bridge.rs	MultiwayCospanExt, MultiwayCospanGraph	Hypergraph evolution cospan analysis
types.rs	ComputationDomain, ComputationContext, CausalEffect	Domain types for computation models

Fong-Spivak Feature Map

Re-exports from catgraph v0.2.0 implementing Fong & Spivak, Hypergraph Categories SS2-3:

Paper Reference	Re-exported Type	Purpose
Def 2.12	HypergraphCategory	Symmetric monoidal + Frobenius structure (eta, epsilon, mu, delta)
Def 2.2	CospanAlgebra, PartitionAlgebra, NameAlgebra	Lax monoidal functors Cospan -> Set
Def 2.12, Eq 12	HypergraphFunctor, RelabelingFunctor	Structure-preserving maps between hypergraph categories
Prop 3.8	CospanToFrobeniusFunctor	Decomposes cospans into Frobenius generators
SS3.1	cup, cap, name, unname	Self-dual compact closed structure
Thm 3.14	Cospan<Lambda>: HypergraphCategory	Free hypergraph category

Frobenius verification: verify_cospan_chain_frobenius() decomposes each cospan in a chain via CospanToFrobeniusFunctor and checks that composition is preserved -- a stronger categorical check than monoidal coherence.

Alignment with Gorard's Paper

Paper Concept	Implementation	Location
Cobordism category B	DiscreteInterval, ParallelIntervals	irreducible::interval
Functor Z': T -> B	IrreducibilityFunctor	functor/mod.rs
Adjunction Z' ⊣ Z	ZPrimeAdjunction, triangle identities	functor/adjunction.rs
Coherence (alpha, lambda, rho, sigma)	verify_associator, verify_braiding, verify_all_coherence	multiway_coherence.rs
Stokes integration	TemporalComplex, ConservationResult	temporal_cospan_chain.rs
Discrete exterior calculus	MultiwayComplex, OneForm, TwoForm	multiway_stokes.rs (feature: dec)
Frobenius structure	FrobeniusVerificationResult, verify_cospan_chain_frobenius	functor/fong_spivak.rs
DPO rewriting as spans	RewriteRule::to_span()	catgraph::hypergraph
Evolution as cospan chain	HypergraphEvolution::to_cospan_chain()	catgraph::hypergraph
Causal invariance	Wilson loops, holonomy analysis	catgraph::hypergraph
Branchial curvature	OllivierRicciCurvature	catgraph::multiway
Complexity algebra	Complexity, StepCount	irreducible::complexity

Quick Start

[dependencies]
irreducible = { git = "https://github.com/tsondru/irreducible" }

use irreducible::machines::{TuringMachine, Direction};
use irreducible::machines::trace::analyze_trace;

let bb = TuringMachine::busy_beaver_2_2();
let history = bb.run("", 20);
let analysis = analyze_trace(&history);
assert!(analysis.is_irreducible);
assert_eq!(analysis.step_count, 6);

Three Perspectives on Irreducibility

use irreducible::{
    IrreducibilityFunctor, StokesIrreducibility, TuringMachine,
};

let bb = TuringMachine::busy_beaver_2_2();
let history = bb.run("", 20);
let intervals = history.to_intervals();

// 1. Functorial: contiguous intervals under Z'
let functorial = IrreducibilityFunctor::is_sequence_irreducible(&intervals);

// 2. Stokes: conservation laws hold
let stokes = StokesIrreducibility::analyze(&intervals).unwrap();
let stokes_ok = stokes.is_irreducible();

// 3. Frobenius: valid decomposition into generators (Fong-Spivak)
let frobenius = stokes.verify_frobenius();
let frobenius_ok = frobenius.all_valid && frobenius.composition_preserved;

assert!(functorial && stokes_ok && frobenius_ok); // all agree

Feature Flags

Feature	Gates	Dependencies
(none)	Core library (TM, CA, SRS, NTM, functor, cobordism)	catgraph, catgraph-physics, serde
dc-geometry	dc_topology Regge + DEC substrate	deep_causality_topology, deep_causality_tensor, deep_causality_sparse
dec	Discrete exterior calculus on multiway complexes	dc-geometry, nalgebra
manifold-curvature	Regge deficit-angle curvature on branchial complexes	dc-geometry, nalgebra
lapack	LAPACK-accelerated eigendecomposition for MDS	nalgebra-lapack (implies manifold-curvature; requires libopenblas-dev)

The former persist feature (SurrealDB evolution-trace storage) was removed pending the catgraph-surreal reboot; see issue #15 for the restore path.

Examples

cargo run --example gorard_demo           # 9-part presentation demo
cargo run --example builders              # TuringMachineBuilder + NTMBuilder
cargo run --example bifunctor_tensor      # Tensor products, monoidal laws
cargo run --example fong_spivak           # Fong-Spivak three-perspective agreement
cargo run --example lattice_gauge         # Wilson loops, plaquette action
cargo run --example multiway_coherence    # Non-confluent fragment failing coherence
cargo run --example multiway_stokes --features dec  # Closed vs non-closed 1-forms

Testing

cargo test --workspace                               # 354 tests (default features)
cargo test --workspace --features manifold-curvature,dec  # 381 tests (CI feature leg)
cargo test --features dc-geometry                    # dc_topology smoke + bridge tests
cargo clippy --workspace --all-targets -- -D warnings  # CI gate
cargo clippy --workspace -- -W clippy::pedantic      # advisory, zero warnings

Integration suites cover: adjunction laws (Z' ⊣ Z triangle identities), the catgraph cospan bridge, computation-model classification (TM, CA, SRS, NTM, Petri), Fong-Spivak Frobenius verification, functor laws, hypergraph DPO rewriting + Wilson loops, multiway coherence (associator/braiding), branchial evolution + curvature foliation, DEC closed forms (dec), and temporal cospan-chain conservation.

Key Concepts

The Categories

Category	Objects	Morphisms	Composition
T (Computation)	Data structures / states	Computations / transitions	Sequential execution
B (Cobordism)	Step numbers (N)	Discrete intervals [n,m]	Union of contiguous intervals

The Functor Z'

Z' maps states to step numbers and computations to intervals. Theorem: Z' is a functor iff all computations in T are irreducible.

For non-deterministic (multiway) systems, both categories gain symmetric monoidal structure. Theorem: Z' is a symmetric monoidal functor iff the system is multicomputationally irreducible.

The Adjunction Z' ⊣ Z

From the paper (Section 4.2): computational irreducibility is dual/adjoint to locality of time evolution in quantum mechanics. For multiway systems, Z' is adjoint to the functor defining functorial quantum field theory (Atiyah-Segal axioms).

Multiway Coherence (v0.4.3)

Coherence verification now operates on real multiway evolution graphs from catgraph-physics -- a genuine non-strict symmetric monoidal category where confluence up to causal equivalence is a falsifiable property. verify_associator checks that parallel independent events at a fork point are pairwise confluent; verify_braiding checks that two events commute (share a common descendant). Non-confluent fragments produce CoherenceError::NonConfluent.

Discrete Exterior Calculus (v0.4.3, feature: dec)

Confluence diamonds in a multiway graph are 2-simplices. A 1-form assigns costs to edges; the exterior derivative d: Omega^1 -> Omega^2 evaluates on each diamond. A closed 1-form (d_omega = 0) means cost is path-independent across every diamond -- the categorical manifestation of causal invariance.

Stokes Integration and Cospan Composability

For 1D simplicial complexes, Stokes conservation reduces to contiguity + monotonicity -- exactly the conditions for cospan composability in B. This bridges differential geometry (Stokes theorem) to category theory (cospan composition).

Frobenius Decomposition (Fong-Spivak)

Cospan<Lambda> implements HypergraphCategory (Thm 3.14). CospanToFrobeniusFunctor decomposes each cospan into Frobenius generators. If composition is preserved through decomposition, the cospan chain has valid Frobenius structure -- a categorical invariant complementing the Stokes and functorial perspectives.

Dependencies

• catgraph v0.2.0 -- category theory infrastructure (cospans, spans, Fong-Spivak hypergraph categories); workspace tag shared with catgraph-applied (Petri nets) and catgraph-physics (hypergraph DPO rewriting, multiway evolution, confluence diamonds, discrete curvature, branchial spectral analysis)
• serde + serde_json -- serialization
• Optional: deep_causality_topology + deep_causality_tensor + deep_causality_sparse (Regge curvature + DEC substrate; currently git-rev-pinned pre-release), nalgebra (matrix ops), nalgebra-lapack (LAPACK)

References

• Gorard (2022): "A Functorial Perspective on (Multi)computational Irreducibility"
• Fong & Spivak (2019): "Hypergraph Categories"
• Mac Lane (1998): Categories for the Working Mathematician
• Wolfram (2002): A New Kind of Science
• Abramsky & Coecke (2004): "A categorical semantics of quantum protocols"
• Atiyah (1988): "Topological quantum field theories"

Contributors

• tsondru
• Claude (Anthropic)

License

MIT