The "Emergent" Research Program and Asemantic Computing

Summary: Marius Buliga's "Emergent" Research Program (2009–2019)

This December 2019 presentation by Marius Buliga (Institute of Mathematics of the Romanian Academy) synthesizes a decade-long research program bridging differential geometry, topology, and theoretical computer science through the unifying lens of graph rewrite systems. The work represents a radical reconceptualization of space itself as an asynchronous computational substrate rather than a passive container.

1. Geometric Origins: Solving Gromov's Sub-Riemannian Recovery Problem

The research trajectory begins with a fundamental problem in metric geometry posed by Gromov (1996): Can one recover a sub-Riemannian structure (X,D,g) purely from its associated Carnot-Carathéodory metric space (X,d)? Unlike Riemannian geometry—where Alexandrov-Nikolaev theory solves the analogous recovery problem—sub-Riemannian spaces exhibit features that obstruct classical approaches:

• Metric (Hausdorff) dimension exceeds topological dimension
• Failure of Alexandrov curvature conditions
• Tangent spaces are Carnot groups (nilpotent Lie groups with dilations) rather than vector spaces
• Non-Euclidean differential calculus (Pansu derivative)

Buliga's solution (2009–2011) introduces emergent algebras: a Γ-indexed family of idempotent right quasigroup operations on a uniform space X satisfying convergence properties as the index approaches 0. Formally, an emergent algebra (X,Γ,∘,•) requires:

• Reidemeister 1: x∘x = x = x•x (idempotence)
• Reidemeister 2: e•(e∘x) = x = e∘(e•x) (invertibility)
• Emergence: As a→0 in Γ (a commutative group with absolute 0), operations converge uniformly on compact sets to a conical group structure (X,*,e) where:
  • a⋅(x*y) = (a⋅x)*(a⋅y) (Γ acts by automorphisms)
  • a⋅x → e uniformly as a→0 (contractivity)

This framework generalizes dilatation structures (Buliga 2006) and provides an intrinsic characterization of sub-Riemannian geometry without reference to smooth manifolds or distributions. Notable examples include Heisenberg groups and general Carnot groups, where the emergent algebra operations recover the group law through limiting processes.

2. Topological Bridge: Reidemeister Moves as Computational Primitives

An insight connects emergent algebras to low-dimensional topology: the algebraic axioms correspond to Reidemeister moves from knot theory. While emergent algebras satisfy only R1 and R2 moves (reflecting their quasigroup structure), the full Reidemeister system (R1–R3) provides a graphical calculus. [Added: here Qwen simplifies a bit, the reality is that R3 is emergent from R1, R2 and emergence axiom.] This observation catalyzes the transition from geometry to computation: space itself becomes a graph rewrite automaton—an asynchronous system (S,R,A) where:

• S: Class of oriented trivalent graphs
• R: Rewrite rules derived from emergent algebra operations
• A: Random/local application algorithm (no global scheduler)

This model eliminates the classical distinction between "space" and "computation": there are no passive points, only shared computation on a graph-theoretic substrate.

3. Computational Realization: From GLC to Chemlambda

Buliga develops two concrete implementations of this vision [see History of Chemlambda]:

Graphic Lambda Calculus (GLC) (2013)

A visual language representing untyped λ-calculus via graph rewrites on oriented fatgraphs with node types:

• L (lambda abstraction), A (application)
• FI/FO (fan-in/fan-out for duplication)
• T (termination)

GLC incorporates both β-reduction (via graphical β-rewrite) and Reidemeister moves, creating a hybrid system where logical computation and topological transformation coexist.

Chemlambda (v1: 2014; v2: 2015–present)

An artificial chemistry implementation inspired by Berry-Boudol's Chemical Abstract Machine and Lafont's Interaction Combinators. Key properties:

• Turing universality: [First in Chemical concrete machine and then] proven jointly with Louis Kauffman via encoding of SK combinators [see Kauffman slides for the first programming essays in chemlambda v1]
• Decentralized execution: Purely local rewrites applied asynchronously with no global state or garbage collection
• Self-replication: Discovery of quine graphs—self-sustaining molecular structures exhibiting metabolism, reproduction, and death under random rewriting
• Multiple implementations: Validated via independent ports to Python (4lhc), Haskell (synergistics), and JavaScript (ishanpm), demonstrating robustness

Chemlambda v2 (implemented in awk/d3.js) revealed unexpected phenomena: certain graphs (e.g., the “ouroboros” quine) display lifelike behaviors including periodic metabolism and spontaneous duplication—suggesting pathways toward artificial life in purely syntactic systems.

4. Advanced Developments: Kali and the Anharmonic Group

The presentation culminates with kali (anharmonic lambda), a refined rewrite system connecting to projective geometry:

• Uses 11 node types, with 6 related to the anharmonic group (isomorphic to S₃, the symmetric group on 3 elements)
• FO nodes correspond to ∞ in projective geometry; T/FRIN/FROUT nodes map to 0 and 1
• Incorporates Kauffman's Arrow for directional flow control

This system demonstrates how algebraic structures from classical geometry (cross-ratios, harmonic conjugates) emerge naturally in graph rewrite dynamics—suggesting deep connections between projective geometry and computation.

[This research continues to Pure See and is under construction in Hexagon. ]

5. Philosophical and Technical Significance

Buliga's program makes several contributions:

1. Space as computation: Rejects the container metaphor for space; proposes space is a distributed graph rewrite process
2. No Church-Turing bottleneck: Computation occurs without λ-abstraction/application as primitives — β-reduction emerges from local rewrites on graphs lacking explicit variable binding
3. Decentralized computing model: Offers a concrete alternative to von Neumann architecture where computation is:
   • Asynchronous and local
   • Free of global state or centralized control
   • Naturally parallelizable (all applicable rewrites fire independently)
4. Bridge between continuous and discrete: Emergent algebras provide rigorous passage from discrete graph rewrites to continuous geometric structures (Carnot groups)
5. Artificial life substrate: Quine graphs in chemlambda demonstrate that self-replication and metabolism can emerge from purely syntactic rewrite rules—without predefined semantics or external interpreters

Critical Assessment & Open Challenges

While mathematically rigorous in its geometric foundations, the computational aspects face practical limitations:

• Non-determinism: Conflicting rewrites in chemlambda lead to divergent reduction paths; termination is not guaranteed even for strongly normalizing λ-terms
• Scalability: Random rewrite strategies become inefficient for large graphs; no proven complexity bounds exist
• Semantic gap: Despite Turing universality, mapping practical algorithms to efficient chemlambda molecules remains largely artisanal
• Physical realizability: Whether these systems can be instantiated in molecular computing or neuromorphic hardware remains speculative