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310 lines
11 KiB
Text
310 lines
11 KiB
Text
import Semantics.PhysicsScalar
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import Semantics.RegimeCore
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import Semantics.BoundaryDynamics
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import Semantics.SpikingDynamics
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import Semantics.ExoticSpacetime
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import Semantics.ManifoldStructures
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namespace Semantics.CriticalityDynamics
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open Semantics.PhysicsScalar
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open Semantics.RegimeCore
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open Semantics.BoundaryDynamics
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open Semantics.SpikingDynamics
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open Semantics.ExoticSpacetime
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open Semantics.ManifoldStructures
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abbrev CriticalSiteId := UInt16
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abbrev AvalancheId := UInt16
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abbrev ToppleCount := UInt16
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inductive PotentialRegime
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| subcritical
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| nearCritical
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| critical
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| supercritical
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| dissipative
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deriving DecidableEq
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inductive ManifoldAvalancheClass
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| acNone
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| acLocal
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| acCascading
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| acSystemWide
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| acRecurrent
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deriving DecidableEq
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inductive StabilizationStatus
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| stable
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| unstable
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| stabilizing
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| dissipated
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| unresolved
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deriving DecidableEq
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structure CriticalPotential where
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load : PhysicsScalar.Q16_16
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threshold : PhysicsScalar.Q16_16
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gradient : PhysicsScalar.Q16_16
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dissipation : PhysicsScalar.Q16_16
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manifoldBias : PhysicsScalar.Q16_16
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deriving DecidableEq
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structure CriticalSite where
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siteId : CriticalSiteId
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label : String
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regionId : RegionId
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load : PhysicsScalar.Q16_16
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threshold : PhysicsScalar.Q16_16
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capacity : UInt16
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sink : Bool
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manifoldWeight : PhysicsScalar.Q16_16
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temporalDifferential? : Option TemporalDifferential
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boundaryId? : Option BoundaryId
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deriving DecidableEq
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structure RedistributionEdge where
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sourceId : CriticalSiteId
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targetId : CriticalSiteId
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weight : PhysicsScalar.Q16_16
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gated : Bool
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boundaryInfluence : PhysicsScalar.Q16_16
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deriving DecidableEq
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structure CriticalNetwork where
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sites : List CriticalSite
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edges : List RedistributionEdge
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defaultDissipation : PhysicsScalar.Q16_16
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deriving DecidableEq
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structure ToppleStep where
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siteId : CriticalSiteId
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outgoingCount : UInt16
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emittedLoad : PhysicsScalar.Q16_16
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remainingLoad : PhysicsScalar.Q16_16
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avalancheClass : ManifoldAvalancheClass
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deriving DecidableEq
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structure Avalanche where
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avalancheId : AvalancheId
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seedSiteId : CriticalSiteId
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steps : List ToppleStep
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dischargedLoad : PhysicsScalar.Q16_16
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span : UInt16
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avalancheClass : ManifoldAvalancheClass
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deriving DecidableEq
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structure StabilizationResult where
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network : CriticalNetwork
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avalanches : List Avalanche
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status : StabilizationStatus
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totalDischargedLoad : PhysicsScalar.Q16_16
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deriving DecidableEq
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def potentialOf (site : CriticalSite) : CriticalPotential :=
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{ load := site.load
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, threshold := site.threshold
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, gradient := PhysicsScalar.Q16_16.absDiff site.load site.threshold
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, dissipation := if site.sink then PhysicsScalar.Q16_16.one else PhysicsScalar.Q16_16.quarter
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, manifoldBias := site.manifoldWeight }
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def classifyPotentialRegime (potential : CriticalPotential) : PotentialRegime :=
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if PhysicsScalar.Q16_16.le potential.load (PhysicsScalar.Q16_16.subSaturating potential.threshold PhysicsScalar.Q16_16.quarter) then
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PotentialRegime.subcritical
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else if PhysicsScalar.Q16_16.lt potential.load potential.threshold then
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PotentialRegime.nearCritical
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else if PhysicsScalar.Q16_16.eq potential.load potential.threshold then
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PotentialRegime.critical
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else if PhysicsScalar.Q16_16.gt potential.load (PhysicsScalar.Q16_16.addSaturating potential.threshold PhysicsScalar.Q16_16.half) then
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PotentialRegime.supercritical
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else
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PotentialRegime.dissipative
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def siteUnstable (site : CriticalSite) : Bool :=
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PhysicsScalar.Q16_16.ge site.load site.threshold
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def edgeActive (edge : RedistributionEdge) : Bool :=
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edge.gated && PhysicsScalar.Q16_16.nonZero edge.weight
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def siteEdges (network : CriticalNetwork) (siteId : CriticalSiteId) : List RedistributionEdge :=
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network.edges.filter (fun edge => edge.sourceId = siteId && edgeActive edge)
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def activeNeighborCount (network : CriticalNetwork) (siteId : CriticalSiteId) : UInt16 :=
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UInt16.ofNat (siteEdges network siteId |>.length)
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def redistributedLoadPerEdge (site : CriticalSite) (network : CriticalNetwork) : PhysicsScalar.Q16_16 :=
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let count := activeNeighborCount network site.siteId
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if count = 0 then PhysicsScalar.Q16_16.zero else PhysicsScalar.Q16_16.divQ16_16 site.threshold (UInt32.ofNat count.toNat)
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def toppledLoad (site : CriticalSite) : PhysicsScalar.Q16_16 :=
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if site.sink then site.load else site.threshold
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def remainingAfterTopple (site : CriticalSite) : PhysicsScalar.Q16_16 :=
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if site.sink then PhysicsScalar.Q16_16.zero else PhysicsScalar.Q16_16.subSaturating site.load (toppledLoad site)
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def classifyAvalancheClass (toppleCount : ToppleCount) (span : UInt16) : ManifoldAvalancheClass :=
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if toppleCount = 0 then .acNone
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else if toppleCount = 1 then .acLocal
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else if toppleCount <= 4 then .acCascading
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else if span <= 2 then .acRecurrent
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else .acSystemWide
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def toppleStepOf (site : CriticalSite) (network : CriticalNetwork) : ToppleStep :=
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let count := activeNeighborCount network site.siteId
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let remaining := remainingAfterTopple site
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{ siteId := site.siteId
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, outgoingCount := count
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, emittedLoad := toppledLoad site
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, remainingLoad := remaining
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, avalancheClass := classifyAvalancheClass 1 (if count = 0 then 0 else 1) }
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def applyToppleToSite (site : CriticalSite) : CriticalSite :=
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{ site with load := remainingAfterTopple site }
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def receiveLoad (site : CriticalSite) (incoming : PhysicsScalar.Q16_16) : CriticalSite :=
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{ site with load := PhysicsScalar.Q16_16.addSaturating site.load incoming }
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def edgeContribution (source : CriticalSite) (network : CriticalNetwork) (edge : RedistributionEdge) : PhysicsScalar.Q16_16 :=
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let base := redistributedLoadPerEdge source network
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let boundaryAdjusted := PhysicsScalar.Q16_16.mulQ16_16 base (PhysicsScalar.Q16_16.subSaturating PhysicsScalar.Q16_16.one edge.boundaryInfluence)
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boundaryAdjusted
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def findSite? (network : CriticalNetwork) (siteId : CriticalSiteId) : Option CriticalSite :=
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network.sites.find? (fun site => site.siteId = siteId)
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def rewriteSite (sites : List CriticalSite) (updated : CriticalSite) : List CriticalSite :=
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sites.map (fun site => if site.siteId = updated.siteId then updated else site)
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def applyIncomingForEdge (network : CriticalNetwork) (source : CriticalSite) (edge : RedistributionEdge) : CriticalNetwork :=
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match findSite? network edge.targetId with
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| none => network
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| some targetSite =>
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let incoming := edgeContribution source network edge
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let updatedTarget := receiveLoad targetSite incoming
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{ network with sites := rewriteSite network.sites updatedTarget }
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def distributeFromSite (network : CriticalNetwork) (source : CriticalSite) : CriticalNetwork :=
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let initialNetwork := { network with sites := rewriteSite network.sites (applyToppleToSite source) }
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(siteEdges network source.siteId).foldl (fun acc edge => applyIncomingForEdge acc source edge) initialNetwork
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def firstUnstableSite? (network : CriticalNetwork) : Option CriticalSite :=
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network.sites.find? siteUnstable
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def temporalPotentialBias (site : CriticalSite) : PhysicsScalar.Q16_16 :=
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match site.temporalDifferential? with
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| none => PhysicsScalar.Q16_16.zero
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| some differential => temporalGradient differential
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def manifoldPotentialOf (site : CriticalSite) : PhysicsScalar.Q16_16 :=
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PhysicsScalar.Q16_16.addSaturating site.manifoldWeight (temporalPotentialBias site)
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def stabilizeStep (network : CriticalNetwork) : StabilizationResult :=
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match firstUnstableSite? network with
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| none =>
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{ network := network
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, avalanches := []
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, status := StabilizationStatus.stable
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, totalDischargedLoad := PhysicsScalar.Q16_16.zero }
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| some unstableSite =>
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let step := toppleStepOf unstableSite network
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let nextNetwork := distributeFromSite network unstableSite
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let avalanche : Avalanche :=
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{ avalancheId := unstableSite.siteId
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, seedSiteId := unstableSite.siteId
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, steps := [step]
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, dischargedLoad := step.emittedLoad
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, span := step.outgoingCount
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, avalancheClass := step.avalancheClass }
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{ network := nextNetwork
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, avalanches := [avalanche]
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, status := StabilizationStatus.stabilizing
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, totalDischargedLoad := step.emittedLoad }
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def stabilizationStatusOf (network : CriticalNetwork) : StabilizationStatus :=
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match firstUnstableSite? network with
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| none => StabilizationStatus.stable
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| some site => if site.sink then StabilizationStatus.dissipated else StabilizationStatus.unstable
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def stableByRepeatedTopple (fuel : Nat) (network : CriticalNetwork) : StabilizationResult :=
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match fuel with
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| 0 =>
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{ network := network
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, avalanches := []
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, status := StabilizationStatus.unresolved
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, totalDischargedLoad := PhysicsScalar.Q16_16.zero }
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| fuel + 1 =>
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let stepResult := stabilizeStep network
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match stepResult.status with
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| StabilizationStatus.stable => stepResult
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| _ =>
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let recursive := stableByRepeatedTopple fuel stepResult.network
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{ network := recursive.network
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, avalanches := stepResult.avalanches ++ recursive.avalanches
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, status := recursive.status
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, totalDischargedLoad := PhysicsScalar.Q16_16.addSaturating stepResult.totalDischargedLoad recursive.totalDischargedLoad }
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def abelianInvariantHoldsByLoadSum (before after : CriticalNetwork) : Bool :=
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let sumLoads := fun (sites : List CriticalSite) => sites.foldl (fun acc site => PhysicsScalar.Q16_16.addSaturating acc site.load) PhysicsScalar.Q16_16.zero
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PhysicsScalar.Q16_16.le (sumLoads after.sites) (sumLoads before.sites)
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def sinkSites (network : CriticalNetwork) : List CriticalSite :=
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network.sites.filter (fun site => site.sink)
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def criticalFrontier (network : CriticalNetwork) : List CriticalSite :=
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network.sites.filter (fun site => classifyPotentialRegime (potentialOf site) = PotentialRegime.nearCritical)
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def manifoldCriticalityScore (site : CriticalSite) : PhysicsScalar.Q16_16 :=
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PhysicsScalar.Q16_16.addSaturating (manifoldPotentialOf site) (PhysicsScalar.Q16_16.absDiff site.load site.threshold)
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def criticalityCompatibleWithSpike (site : CriticalSite) (state : MembraneState) : Bool :=
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PhysicsScalar.Q16_16.ge site.load state.threshold || PhysicsScalar.Q16_16.ge (manifoldCriticalityScore site) state.potential
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def defaultCriticalSite (siteId : CriticalSiteId) (regionId : RegionId) : CriticalSite :=
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{ siteId := siteId
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, label := "critical-site"
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, regionId := regionId
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, load := PhysicsScalar.Q16_16.zero
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, threshold := PhysicsScalar.Q16_16.one
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, capacity := 4
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, sink := false
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, manifoldWeight := PhysicsScalar.Q16_16.quarter
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, temporalDifferential? := none
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, boundaryId? := none }
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def defaultCriticalNetwork : CriticalNetwork :=
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{ sites := []
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, edges := []
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, defaultDissipation := PhysicsScalar.Q16_16.quarter }
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end Semantics.CriticalityDynamics
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