| A Simple Pair of Rules |
| Result from Deterministic Rules |
| Result from Probabilistic Rules |
| Example Rules for a More Complex Architecture |
| The following stimulus configurations cause the agent to deposit a type-1 brick: |
| Second Group of Rules |
| For these configurations, deposit a type-2 brick |
| Result |
| 20«20«20 lattice | |
| 10 wasps | |
| After 20 000 simulation steps | |
| Axis and plateaus | |
| Resembles nest of Parachartergus |
| Architectures Generated from Other Rule Sets |
| More Examples |
| An Interesting Example |
| Includes | ||
| central axis | ||
| external envelope | ||
| long-range helical ramp | ||
| Similar to Apicotermes termite nest | ||
| Similar Results with Hexagonal Lattice |
| 20«20«20 lattice | |
| 10 wasps | |
| All resemble nests of wasp species | |
| (d) is (c) with envelope cut away | |
| (e) has envelope cut away |
| Effects of Randomness (Coordinated Algorithm) |
| Specifically different (i.e., different in details) | |
| Generically the same (qualitatively identical) | |
| Sometimes results are fully constrained |
| Effects of Randomness (Non-coordinated Algorithm) |
| Non-coordinated Algorithms |
| Stimulating configurations are not ordered in time and space | |
| Many of them overlap | |
| Architecture grows without any coherence | |
| May be convergent, but are still unstructured |
| Coordinated Algorithm |
| Non-conflicting rules | ||
| canÕt prescribe two different actions for the same configuration | ||
| Stimulating configurations for different building stages cannot overlap | ||
| At each stage, ÒhandshakesÓ and ÒinterlocksÓ are required to prevent conflicts in parallel assembly | ||
| More FormallyÉ |
| Let C = {c1, c2, É, cn} be the set of local stimulating configurations | |
| Let (S1, S2, É, Sm) be a sequence of assembly stages | |
| These stages partition C into mutually disjoint subsets C(Sp) | |
| Completion of Sp signaled by appearance of a configuration in C(Sp+1) |
| Example |
| Modular Structure |
| Recurrent states induce cycles in group behavior | |
| These cycles induce modular structure | |
| Each module is built during a cycle | |
| Modules are qualitatively similar |
| Possible Termination Mechanisms |
| Qualitative | ||
| the assembly process leads to a configuration that is not stimulating | ||
| Quantitative | ||
| a separate rule inhibiting building when nest a certain size relative to population | ||
| Òempty cells ruleÓ: make new cells only when no empties available | ||
| growing nest may inhibit positive feedback mechanisms | ||
| Observations |
| Random algorithms tend to lead to uninteresting structures | ||
| random or space-filling shapes | ||
| Similar structured architectures tend to be generated by similar coordinated algorithms | ||
| Algorithms that generate structured architectures seem to be confined to a small region of rule-space | ||
| Analysis |
| Define matrix M: | ||
| 12 columns for 12 sample structured architectures | ||
| 211 rows for stimulating configurations | ||
| Mij = 1 if architecture j requires configuration i | ||
| Factorial Correspondence Analysis |
| Conclusions |
| Simple rules that exploit discrete (qualitative) stigmergy can be used by autonomous agents to assemble complex, 3D structures | |
| The rules must be non-conflicting and coordinated according to stage of assembly | |
| The rules corresponding to interesting structures occupy a comparatively small region in rule-space |
| LangtonÕs Vants (Virtual Ants) |
| Vants |
| Square grid | |||
| Squares can be black or white | |||
| Vants can face N, S, E, W | |||
| Behavioral rule: | |||
| take a step forward, | |||
| if on a white square then | |||
| paint it black & turn 90¡ right | |||
| if on a black square then | |||
| paint it white & turn 90¡ left | |||
| Example |
| Time Reversibility |
| Vants are time-reversible | |
| But time reversibility does not imply global simplicity | |
| Even a single vant interacts with its own prior history | |
| But complexity does not always imply random-appearing behavior |
| Digression: Time-Reversibility and the Physical Limits of Computation |
| Irreversible logic gate loses one bit of information | ||
| This equals entropy decrease of kT ln 2 | ||
| Therefore a conventional gate must dissipate at least kT ln 2 joules | ||
| typical transistors dissipate about 108kT | ||
| Reversible gates can dissipate arbitrarily little energy | ||
| Charles H. Bennett (1973). See also Feynman Lectures on Computation, ch. 5 | ||
| Demonstration of Vants |
| Run vants from CBN website |
| Conclusions |
| Even simple, reversible local behavior can lead to complex global behavior | |
| Nevertheless, such complex behavior may create structures as well as apparently random behavior | |
| Perhaps another example of Òedge of chaosÓ phenomena |