Configuration synthesis for programmable analog devices with Arco

18 Pages·2016·0.32 MB·English

Configuration synthesis for programmable analog devices
with Arco
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Citation Achour, Sara, Rahul Sarpeshkar, and Martin C. Rinard .
"Configuration Synthesis for Programmable Analog Devices with
Arco." 13-17 June, 2016, Santa Barbara, ACM Press, 2016, pp.
177–93.
As Published http://dx.doi.org/10.1145/2908080.2908116
Publisher Association for Computing Machinery
Version Author's final manuscript
Accessed Mon Jan 07 03:05:23 EST 2019
Citable Link http://hdl.handle.net/1721.1/113652
Terms of Use Creative Commons Attribution-Noncommercial-Share Alike
Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/
Configuration Synthesis for Programmable
Analog Devices with Arco
SaraAchour RahulSarpeshkar MartinC.Rinard
MITCSAIL,USA DartmouthCollege,MITRLE,USA MITCSAIL,USA
[email protected] [email protected] [email protected]
Abstract system[6,15,40].Studyingthemodeleddynamicsondigi-
talhardwareischallengingsincethesesystemsareoftenstiff
Programmable analog devices have emerged as a powerful
andthereforepronetonumericalinstability[13,29].Inthis
computingsubstrateforperformingcomplexneuromorphic
context,astiffsystemisasystemthatispronetonumerical
and cytomorphic computations. We present Arco, a new
instability unless the time steps taken are extremely small.
solver that, given a dynamical system specification in the
form of a set of differential equations, generates physically Examples of stiff systems include biological systems with
both fast and slow dynamics. Analog devices are excellent
realizable configurations for programmable analog devices
candidatesforthesecomputations:
that are algebraically equivalent to the specified system.
On a set of benchmarks from the biological domain, Arco • Continuous Time Domain: Analog devices operate in
generatesconfigurationswith35to534connectionsand28 the continuous time domain, circumventing the time
to326componentsin1to54minutes. scaleissuesthatoftenplaguestiffdynamicalsystemcom-
putationsimplementedwithdiscretizedtimesteps.Time
Categories and Subject Descriptors D.3.4 [Processors]:
doesnot“tick”inintervalsasinstandardclockeddigital
Compilers; C.1.3[ProcessorStyles]:AnalogComputers
systems,butrunscontinuouslyandasynchronously.
Keywords Compilers,AnalogComputing,Languages
• Stochastic Behavior: The noise present in these ana-
1. Introduction log devices is directly analogous to the noise present in
biological systems as they are mapped onto the hard-
Programmable analog devices have emerged as a powerful
ware [33, 37]. This property promotes analog compu-
computingsubstrateforperformingcomplexneuromorphic
tations that accurately model the underlying biological
and cytomorphic computations [5, 7, 8, 31, 33, 34, 36, 37,
phenomena.
41]. These systems directly map state variables, transient
• DirectMapping:Sincethedynamicalsystemisdirectly
variables,andparametersintheunderlyingscientificmodel
mapped to components on the analog device, the max-
tophysicalaspectsofthecircuitsuchasvoltageandcurrent.
imum dynamical system size is directly proportional to
Programmableanalogdevicestargetlargebiologicaldy-
thescaleoftheanalogdevice.
namicalsystemssuchasgene–proteinandneuronnetworks.
Currently,thereisnocompilersupportforthesedevices,
Computations involving these networks are often used for
sotheyareprogrammedmanually.Theresultisaconfigura-
medicaldosageoptimization,diseaseprediction,andunder-
tionthatspecifieswhichanalogcomponentstoconnectand
standing biological phenomena [23, 35]. The dynamics of
which dynamical system variables and constants to assign
biologicalsystemsareoftenoversimplifiedtomakethecom-
to input and output components. The following properties
putationstractableondigitalsystems,whichreducestheac-
complicatetheautomaticgenerationoftheseconfigurations:
curacyofthemodelinrelationtothecorrespondingphysical
• Complex Analog Components: The components in
many analog devices implement complex dynamics and
nontrivialrelationssuchastranscendentalfunctions[19,
Permissiontomakedigitalorhardcopiesofpartorallofthisworkforpersonalor
classroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributed 28,41].Becausethecomponentsaretypicallyengineered
forprofitorcommercialadvantageandthatcopiesbearthisnoticeandthefullcitation for analog efficiency and generality rather than ease of
on the first page. Copyrights for components of this work owned by others than
ACMmustbehonored.Abstractingwithcreditispermitted.Tocopyotherwise,to use,theremaybeasubstantialsemanticgapbetweenthe
republish,topostonservers,ortoredistributetolists,contacttheOwner/Author. dynamicalsystemandtheanaloghardware.
[email protected].,ACM,Inc.,
fax[phone].Copyright2016heldbyOwner/Author.PublicationRights • Relation Entanglement: The complex multifunctional
LicensedtoACM. componentsinourtargetanaloghardwaredevicesimple-
PLDI’16 Monthd–d,2016,SantaBarbara,CA,USA mentmultipleinterdependentrelations.Mappingadiffer-
Copyright(cid:13)c 2016ACM978-1-nnnn-nnnn-n/yy/mm...$15.00
DOI:http://dx.doi.org/10.1145/nnnnnnn.nnnnnnn entialequationontoevenonesuchcomponentcanthere-
fore activate a cascading sequence of interactions be- • Evaluation:WeevaluateArcoonfivebiologicaldynam-
tween the component and the remaining dynamical sys- icalsystemsextractedfromanartifactrepositoryforbi-
temvariables. ological models that have been accepted into computa-
• ResourceConstraints:Analogdevicestypicallyprovide tional biology conferences [1]. Given a programmable
afinitesetofonlypartiallyconnectedcomponents.Any analog hardware description, Arco synthesizes configu-
successful mapping of the dynamical system onto the rations for the models with from 28 to 326 components
devicemustlivewithintheseconstraints. and35to534connectionsin1to54minutes.
We present Arco, a solver which, given an analog device To enable these contributions, Arco (1) works with analog
specification and dynamical system, synthesizes an alge- device descriptions written in a novel hardware specifica-
braicallyequivalentconfigurationoftheanalogdevice: tionlanguageand(2)acceptsdynamicalsystemswrittenin
anovelhighleveldynamicalsystemspecificationlanguage.
• AlgebraicUnification:Arcofullyexploitscomplexana-
To the best of our knowledge, Arco is the first system to
log components by using algebraic unification to find
automatically synthesize programmable analog device con-
compact and efficient isomorphisms between the rela-
figurationsfromdynamicalsystemspecifications.
tionsinthedynamicalsystemandtheprovidedhardware
primitives.
2. Example
• Materialized Constants: The Arco solver materializes
We next present an example that demonstrates (1) how to
newconstantsthatconfigureandspecializegeneralana-
logcomponentssothattheyeffectivelyimplementthere- expressadynamicalsysteminthedynamicalsystemspeci-
ficationlanguage,(2)howtodefineaprogrammableanalog
lationsinthespecifieddynamicalsystem.
• Relation Entanglement: Arco successfully manages device in the hardware specification language, and (3) the
resultingsolutiongeneratedbythesolver.
cascadedrelationentanglementsbydetectingandharmo-
nizing otherwise potentially conflicting dynamical sys-
2.1 DynamicalSystem:Enzyme-SubstrateReaction
temandhardwarevariableimplementations.
• Resource Constraints: Arco effectively works within Consider a chemical reaction where an enzyme (E) reacts
the hardware resource constraints by tracking and ex- with a substrate (S) to produce a compound (ES), where
ploiting (potentially partially) used hardware compo- the forward reaction rate is catalyzed by catalyst (Q). The
nentsandusinganSMTsolvertosolvecomponentinter- formation rate of ES (k ) is the sum of some nominal rate
f
connectionproblems. (k ) and the amount of the catalyst present multiplied by
0
its efficiency constant k , which we assume to be one. We
BecauseArcoworkswithahardwarespecificationlanguage q
thereforeexpressthereactionrateforproducingESask +Q.
thatdefinesthecapabilitiesoftheunderlyinganalogdevice, 0
Thereversereactionrateisk .
itcanworkwiththefullrangeofanalogdevices,including r
devices that provide powerful new analog primitives. The E+S →k0+Q ES ES →kr E+S
resulting analog hardware configurations are algebraically
We may express this reaction as a dynamical system com-
equivalenttothespecifieddynamicalsystems. prised of differential equations, where E and S are the
tot tot
totalamountsofEandS,respectively.Thefollowingsetof
1.1 Contributions
equationsspecifiesthebehaviorofthedynamicalsystem:
Weclaimthefollowingcontributions:
E = E −ES S =S −ES
tot tot
• Basic Approach: We present a new approach for map-
∂ES/∂t=(Q+k )·E·S −k ·ES
ping dynamical systems that model complex biological 0 r
phenomenaontoprogrammableanalogdevices.Starting Figure 1 presents the differential equations written in the
with a specification of the dynamical system, our ap- Arcodynamicalsystemspecificationlanguage(DSSL).The
proachautomaticallysynthesizesanalgebraicallyequiv- specification defines two units, seconds (s, unit of time)
alentconfigurationoftheprogrammableanalogdevice. andmolarity(M,unitofconcentration).ThecompoundQis
• AnalogCircuitSolver:Wepresentanewanalogcircuit expressed as an input quantity with unit of measure M (the
solver that uses a deductive algebraic approach to solve quantityismeasuredasaconcentration).ThecompoundsE
the analog circuit synthesis problem. Algebraic unifica- and S are expressed as intermediate (local) variables also
tionenablesArcotofullyexploitthepowerfulbutcom- measuredinmolarityM.Thereactionparametersk andk
0 r
plexanalogcomponentsthatmodernprogrammableana- areassignedappropriatevaluesandunits.Thetimevariable
log devices offer. Arco successfully manages cascading istmeasuredinseconds(s).
relationentanglements,materializesconstantsthatappro- The relation statements (rel) describe the dynamics of
priatelyconfiguregeneralanalogcomponents,andworks each local and output variable. These statements imple-
withintheresourceconstraintsoftheprogrammableana- ment the set of equations outlined above. The init clause
logdevice. inthedefinitionofESsetstheinitialvalueofESto0.423.
type s ; type M
2.2 HardwareSpecification
input Q : 1/(s*M)
Figure 2 presents the hardware specification for a simple local E : M; local S : M; output ES : M
param ETOT : M = 0.15; param STOT : M = 0.11
programmable analog device. The specification is inspired
param k0 : 1/(s*M) = 0.36; param kr : 1/s = 0.2
byexistingandenvisionedprogrammableanalogchipsthat
time t : s
Arco is designed to target [8, 37, 41]. The specification rel S = STOT - ES
definesmicroseconds(us),milliamps(mA),andmillivolts(V) rel E = ETOT - ES
rel deriv(ES,t)=((k0 + Q)*E*S)-kr*ES init 0.423
as units then maps time (t) to microseconds, the current
property (I) to milliamps, and the voltage property (V) to Figure1. EnzymeSubstrateReactionDynamicalSystem
millivolts.
The specification defines three components, a current
adder (iadd), a current multiplier (igain) and a Michaelis- type us; type mA; type V
prop I : mA; prop V : mV; time t: us
Menten component (mm) [37]. The current adder has two
comp input I
inputports(WandU)andoneoutputport(Y).Thecurrentat input X; output Z; rel I(Z) = D(X)
the output port is the sum of the currents of the two input end
ports.TheMichaelis-Mentencomponenthasfiveinputports comp output V
input X; output Z; rel D(Z) = V(X)
A0,B0,Kf,Kr,andC0andthreeoutputportsA,B,andC.The
end
component simulates a basic enzyme-substrate reaction of comp input V
theformA+B↔C: input X; output Z; rel V(Z) = D(X)
end
AV =B0V−CV BV =B0V−CV comp iadd
∂C /∂t=Kf ·A ·B −Kr ·C init C0 input W; input U; output Y; rel I(Y) = I(W)+I(U)
V I V V I V V end
inststatementsdefinehowmanyinstancesofeachcompo- comp mm
input A0; input B0; input C0;input Kf; input Kr
nentareavailable—forexample,therearetwoinstancesof
output A; output B; output C
the Michaelis-Menten (mm) component. input V and input rel V(A) = V(A0) - V(C)
I components contain a digital to analog converter (DAC) rel V(B) = V(B0) - V(C)
that converts a digital input to voltage or current, respec- rel deriv(V(C),t)=I(Kf)*V(A)*V(B)-I(Kr)*V(C)
init V(C0)
tively, for import into the analog circuit. output V compo-
end
nentscontainananalogtodigitalconverter(ADC)thatsam- schematic
ples an analog property, in this case voltage, to produce a inst mm : 2; inst iadd : 4
inst input I : 5; inst output I : 3;
sequenceofdigitalvaluesforexporttothesurroundingdig-
inst input V : 5; inst output V : 3
italcomputingenvironment.
conn iadd -> mm; conn * -> output(V)
Thespecificationalsodefinesavailableconnections.The conn input(I) -> *; conn input(V) -> *
conn iadd->mm statement, for example, states that output end
ports of iadd components can be connected to input ports Figure2. AnalogHardwareSpecification
of mm components. A * specifies either all output (* ->) or
input(-> *)portsofallcomponents.
such as voltage (V) or current (I). So A is the voltage V
V
property(inblack)oftheoutputportA:
2.3 ArcoSolver
TheArcosolverworkswithasetofgoals.Initiallythegoals C0 A0 B0
are simply the relations in the dynamical system. As the
Kf mm Kr
solver operates it maps computations within the relations
onto the hardware components, with the dynamical system C A B
variables and expressions mapped to properties (voltage, S
current) of the component ports. In our example the solver
Thenextstepistomapthedynamicalsystemexpression
startswiththefollowinggoaltable:
S −ES ontothemmcomponent.Arcoretrievesthehard-
tot
warerelationthatdefinesA andthedynamicalsystemre-
V
S =Stot−ES E = Etot−ES lationthatdefinesS:
∂ES/∂t=(k0+Q)·Stot−ES init 0.423 AV =A0V−CV S=Stot−ES
ThesolverfirstchoosestomaptheS = Stot −ES com- Arco then algebraically unifies the two relations to ob-
putation onto one of the Michaelis-Menten (mm) hardware tain the following assignments from hardware variables to
components.Theinitialstepistomapthedynamicalsystem dynamicalsystemvariablesandexpressions:
variable S onto the hardware variable A . Each hardware
V A =S A0 =S C =ES
variableisthecombinationofaport(hereA)andaproperty V V tot V
In effect, the unification maps variables and expressions At this point Arco again encounters relation entangle-
from the dynamical system onto the hardware variables of ment: it has mapped E to an output port property, specifi-
the mm component. In this case, the unification maps S callyB .Ithastwopotentiallyconflictingdefinitionsof E,
tot V
ontoA0 (thevoltagepropertyofportA0)andESontoC onefromthehardwareandonefromthedynamicalsystem.
V V
(thevoltagepropertyofportC): Itappliesthecurrentsetofassignmentstothehardwarere-
lationthatdefinesB (B =B0 −C )toobtainthefollow-
S V V V V
tot ingmixed-variablerelationthatthehardwareimplementsto
C0 A0 B0 computeE:
Kf mm Kr E=B0 −ES
V
C A B Arco retrieves the definition of E from the dynamical
ES S system (E = Etot − ES) and harmonizes the two relations
withanotheralgebraicunification.TheunificationmapsE
tot
Arconowencountersrelationentanglement.Becauseofa toB0 asfollows:
V
cyclicdependencebetweenA andC ,ithasmappedES to
V V S E
0.423 tot tot
anoutputhardwarevariableofthemmcomponent(specifi-
C0 A0 B0
callyC ).TheresultisthatArconowhastwodefinitionsof
V
ES:onefromthemmcomponent(becauseC = ES): Q + k Kf mm Kr 1
V 0
∂C /∂t=Kf ·A ·B −Kr ·C init C0 C A B
V I V V I V V
ES S E
andonefromthedynamicalsystem:
∂ES/∂t=(k +Q)·E·S−ES init 0.423 Arcohasnowsuccessfullymappedtheoriginalgoal(S =
0
S − ES) onto the mm component, including performing
tot
It is the responsibility of the Arco solver to harmonize
anycascadedmappingsrequiredtoharmonizeanygenerated
these two definitions so that the hardware dynamics of C
V potential conflicts. Because of these cascaded mappings,
correctly implement the dynamics of ES as defined in the
Arco has partially mapped all of the following dynamical
dynamicalsystem(Figure1).BecauseArcohasalreadypar-
systemrelationsontothehardware:
tially consumed the mm component, it applies the assign-
S =S −ES E = E −ES
mentsfromtheunificationofA0 −C withS −ES.There- tot tot
V V tot
sultisthefollowingmixed-variablerelation(arelationwith ∂ES/∂t=(k0+Q)·Stot−ES init 0.423
variablesfromboththedynamicalsystemspecificationand
The partial mapping generated a set of subgoals, which
theanaloghardwarespecification)thatdefineshowthehard-
consistmostlyofassignmentsofdynamicalsystemvariables
warecomputesES:
to (analog) hardware variables. Arco therefore removes the
∂ES/∂t=Kf ·S·B −Kr ·ESinitC0 partiallymappedoriginalgoalsandaddsthegeneratedsub-
I V I V
goalstoobtainthefollowinggoaltable:
Nowrelationentanglementmanifestsitselfastwopoten-
tially conflicting definitions of ∂ES/∂t: one from the above
mixed-variable relation and one from the original relation KfI =Q+k0 KrI =1 B0V =Etot
inthedynamicalsystem.Arcoharmonizesthesetwodefini- C0 =0.423 A0 =S C = ES
V V tot V
tionsbyalgebraicallyunifyingthesetworelations.Theuni-
ficationmapsdynamicalsystemvariablesandexpressionsto
themmcomponentasfollows: With the exception of the KfI goal, the new goal table
contains only assignments of dynamical system variables
S
0.423 tot to hardware variables. Arco assumes that the analog hard-
C0 A0 B0 ware device will be embedded within a larger digital com-
Q + k Kf mm Kr 1 puting framework that will supply dynamical system input
0
variablesandvaluesasdigitalnumbers.Theframeworkwill
C A B
receivedynamicalsystemoutputvariablesasasequenceof
ES S E
digitalvaluesderivedbysamplingtheanalogrepresentation
to deliver the following hardware variable assignments: of these variables at a specific frequency. The analog hard-
KfI = Q+k0,BV = E,KrI =1,andC0V =0.423.Notethat waredevicethereforecontainsinputcomponents,whichuse
KfI and KrI are current properties (not voltage properties) a digital to analog converter to import a digital input value
and are therefore colored blue (not black). Note also that intotheanalogcomputation,andoutputcomponents,which
theunificationcreatedthematerializedconstantKrI = 1,in use an analog to digital converter to produce the sampled
effectspecializingthemmcomponentforthisspecificuse. digitalsequenceofoutputvariablevalues.
Arcosolvesallofthegoalsthatassigndynamicalsystem 2.4 ArcoSearchSpaceandTableauConfigurations
inputoroutputvariablesorvaluestoanaloghardwarevari-
Inpractice,theArcosolvertypicallyhasmanypotentialop-
ablesbyinsertingappropriateinputoroutputcomponentsto
tionsateachstepasitmapsthedynamicalsystemontothe
import or export the relevant values to or from the analog
hardware.Notalloftheseoptionsleadtoasuccessfulhard-
computation:
ware configuration. Arco therefore deploys a search algo-
0.423 Stot Etot rithm to explore a search space of partial hardware config-
urations. The algorithm uses a variety of optimizations and
X X X 1 heuristicstomakethesearchtractable(Section4.3).
in in in
Arco organizes the search with a tableau (Section 4.1).
Z Z Z X
Thisdatastructuretracksthegoals,partiallyusedhardware,
in
and generated partial hardware configurations. The search
C0 A0 B0 Z
starts with a tableau configuration containing the analog
Q + k Kf mm Kr
0 hardwareandthedynamicalsystem.Itgeneratesaspaceof
C A B tableau configurations by mapping dynamical system vari-
ES Z out X S E ables, expressions, and relations onto the analog hardware
components.Thesearchofthegeneratedtableauspaceter-
Atthispointthegoaltablecontainsonlyonegoal: minateswhenArcofindsasolvedtableauconfigurationwith
anemptygoaltable(Section4.2).
Kf =Q+k
I 0
ArcosolvesthisgoalbymappingtheQ+k computation
0
ontoaniaddhardwarecomponent,thenconnectingtheiadd 3. Languages
outputporttotheKfinputportofthemmcomponent.Itthen
Arco works with (1) a dynamical system specification that
uses appropriate input components to solve the remaining
describesthedynamicalsystemtomodeland(2)ahardware
goals,whichassigntheQandk variablestoiaddhardware
0 specificationthatdescribesthetargetprogrammableanalog
input variables. Unlike the previous collection of inputs,
device.
the iadd requires current, not voltage, analog inputs. Arco
thereforeselectsinputcomponentsthatproducecurrent,not
3.1 DynamicalSystemSpecificationLanguage
voltage,values.
AtthispointArcohassuccessfullysolvedallofthegoals The dynamical system specification language (DSSL) al-
inthegoaltable.Theresultisananaloghardwareconfigura- lows the programmer to describe the dynamical system to
tion that correctly implements the specified dynamical sys- simulate.Figure3presentsthesyntaxfortheDSSL.
tem. The configuration specifies the generated inputs, out- Unit: A unit (type x) is a named unit of measurement that
puts,andanaloghardwareconnections: maybecomposedintounitexpressions.Inputs,outputs,and
Q k0 parametersarealldeclaredwithunitexpressionsthatdefine
theunitsinwhichtheyaremeasured.
S E
X X 0.423 tot tot Time: The time (time x : x(cid:48)) is a named time variable x
in in X X X 1 withaunitexpressionx(cid:48)inwhichtimeismeasured.
Z Z in in in Variables: A variable (input|output|local x : U) is a
Z Z Z X namedquantityxwithanattachedunitexpressionU.
W U in Parameter:Aparameter(param x:U = f)issettoafixed
iadd
C0 A0 B0 Z floatingpointvalue f thatdoesnotchangeovertime.
Y
Kf mm Kr Relations:Arelation(rel x=E,rel deriv(x,x(cid:48))=E init f)
definestherelationshipbetweenalocaloroutputvariable
C A B
andtheothervariablesinthedynamicalsystem.
ES Z out X S E
TypeCheckingRules:AppendixApresentstypechecking
rulesforthedynamicalsystemspecificationlanguage.
ES (cid:55)→ outV .X mm .C (cid:22) outV .X
0 D 0 0
0.423 (cid:55)→ inV .X inV .Z (cid:22) mm .C0
0 D 0 0 3.2 AnalogHardwareSpecificationLanguage
S (cid:55)→ inV .X inV .Z (cid:22) mm .A0
tot 1 D 1 0
E (cid:55)→ inV .X inV .Z (cid:22) mm .B0 Figure 4 presents the syntax of the hardware specification
tot 2 D 2 0
1 (cid:55)→ inI .X inI .Z (cid:22) mm .Kr language. Specifications contain named component defini-
0 D 0 0
Q (cid:55)→ inI .X inI .Z (cid:22) iadd .W tions,whichspecifythecomputationalprimitivesofthepro-
1 D 1 0
k0 (cid:55)→ inI2.XD inI2.Z (cid:22) iadd0.U grammableanalogdevice.Theseprimitivesaredefinedasa
iadd0.Y (cid:22) mm0.Kf setofrelationsoverproperties(suchascurrentandvoltage)
oftheinputandoutputportsofthecomponent.
f ∈F n∈N x∈Ident f ∈F n∈N x∈Ident
U ∈ Unit ::= U*U|U/U|x|none Q ∈ Port-Prop ::= x(x(cid:48))
E ∈ Expr ::= E+E(cid:48)|E-E(cid:48)|-E|E*E(cid:48)|E/E(cid:48) U ∈ Unit ::= U*U|U/U|x|none
exp(E)|EˆE|n|x| f E ∈ Expr ::= E+E|E-E|-E|E*E|E/E
S ∈ Stmt ::= type|timex:x(cid:48)|inputx:U exp(E)|EˆE|Q| f |n|x
outputx:U|localx:U D ∈ Defn ::= inputx|outputx|paramx:U= f
paramx:U=x|relx=E relderiv(Q,x)=EinitQ(cid:48)
relderiv(x,x(cid:48))=Einit f relQ=E
P ∈ Prog ::= S∗ W ∈ Wire ::= *|x|W.x|W[n:n(cid:48)]
R ∈ Res ::= instx:n|connW->W(cid:48)
Y ∈ Comp ::= D∗
Figure3. DynamicalSystemSpecificationLanguage Z ∈ Schem ::= R∗
S ∈ Stmt ::= typex|timex:x(cid:48)|propx:x(cid:48)
compx:Yend|schematic:Zend
Units: A unit (type x) is a named unit of measurement for P ∈ Program ::= S∗
aspecificpropertyofthecircuitortime.Unitsrelatecircuit
Figure4. HardwareSpecificationLanguage
propertiestocomputationalproperties.
Time:Thetime(time x:x(cid:48))isanamedtimevariable xand
theassociatedunitsx(cid:48)inwhichitismeasured.
components oc ∈ OC, and analog computation components
Properties:Aproperty(prop x:x(cid:48)),isanamedcircuitprop-
cc ∈ CC.Togetherthesecomponentsarethehardwarecom-
erty x,suchascurrentorvoltage,andtheassociatedunit x(cid:48)
ponentsc∈C.Thehardwarespecificationlanguageenables
inwhichitismeasured.Thecombinationofapropertyxand
the designer to specify multiple instances of declared com-
aportx(cid:48)(cid:48),writtenx(x(cid:48)(cid:48)),isahardwarevariable.
ponents.Eachsuchinstanceisaseparatecomponentc∈C.
Components: Components (comp x ...end block) are the
Components have input ports i ∈ IP and output ports
basic computational primitives of the circuit. Components
o ∈ OP.Togethertheseportsaretheportsp ∈ P = IP∪OP.
have input ports (input x), output ports (output x), and
Eachportpbelongstoacomponentc,[email protected]
parameters (param x:u=f). Stateless relations rel Q=E de-
portphaspropertiesq∈Q,denotedp .Commonproperties
finethevalueofanoutputportpropertyQasanexpression q
qincludecurrentandvoltage.
E.Statefulrelationsrel deriv(Q,x)=E init Q(cid:48),definethe
The combination of a port and a property is a hardware
valueofoutputportpropertyQwithadifferentialequation.
variable. There are input variables i ∈ I = IP × Q and
Q(cid:48)providestheinitialvalue. q
outputvariableso ∈ O = OP×Q.Togetherthesevariables
Arco also supports input components, which provide a q
arethehardwarevariablesv∈V=I∪O.Eachvariablev∈V
digitalinterfaceforimportingdynamicalsystemvaluesinto
isapropertyqofaportp,denotedv:p .
the analog circuit, and digital output components, which q
Eachcomponentc∈Chasasetofvariablesvars(c)⊆V.
provide a digital interface for exporting dynamical system
Componentshavedisjointvariablessothatc (cid:44) c implies
values from the analog circuit to the surrounding digital 1 2
vars(c )∩vars(c ) = ∅. Input components ic ∈ IC have
computingenvironment. 1 2
a single digital input variable i (the input port i with the
Schematic: A schematic..end description block specifies d
digital property d). Output components oc ∈ OC have a
resourcesR,includingthenumberofinstancesofeachcom-
single digital output variable o (the output port o with the
ponent inst x : n and the potential connections between d
digitalpropertyd).
components inst x : n (at the granularity of components,
The hardware implements a set of hardware relations
instancesofcomponents,orindividualcomponentports).
r ∈ R, where each relation r is of the form r : v = E or
TypeCheckingRules:AppendixBpresentsthetypecheck-
ingrulesforthehardwarespecificationlanguage. r : ∂v/∂t = E init v(cid:48),wherev ∈ OandEisanexpression
overVsothatvars(E)⊆V.
ThehardwarealsohasasetofwiresW ⊆ OP×IP.Each
4. ArcoSolver
wireisaconfigurableconnectionbetweenanoutputportand
DynamicalSystem:TheArcosolverworkswithadynami- aninputport.
calsystemwithinputvariables(cid:98)i∈(cid:98)I,outputvariables(cid:98)o∈O(cid:98), Goals:Asthesolveroperates,itmapsthedynamicalsystem
and local variables(cid:98)l ∈ (cid:98)L. Together these variables are the relations onto the hardware (by mapping variables and ex-
dynamicalsystemvariables(cid:98)v∈(cid:98)V =(cid:98)I∪O(cid:98)∪(cid:98)L.Italsoworks pressions from the dynamical system to properties of hard-
with sets of dynamical system relations(cid:98)r ∈ (cid:98)R, where each wareports).Thesolverworkswithasetofgoals(cid:101)r∈R(cid:101),each
relation(cid:98)risoftheform(cid:98)r :(cid:98)v =(cid:98)Eor(cid:98)r : ∂(cid:98)v/∂t =(cid:98)E init x, of which is derived from an original dynamical system re-
(cid:98)v ∈ (cid:98)L ∪ O(cid:98), x ∈ R, and (cid:98)E is an expression over (cid:98)V so that lation(cid:98)r ∈ (cid:98)R and each of which captures some remaining
vars((cid:98)E)⊆(cid:98)V. unmapped part of(cid:98)r. The solver terminates when all of the
Hardware: The hardware has digital to analog (DAC) in- relations in the dynamical system have been fully mapped
put components ic ∈ IC, analog to digital (ADC) output ontothehardwareandthesetofgoalsisempty.
solve(R,W,(cid:98)R)={Z|(cid:104)R,∅,W,(cid:98)R,∅(cid:105)→∗ (cid:104)R(cid:48),R˙(cid:48),W(cid:48),∅,Z(cid:105)}
unify connect
r∈R∪R˙ (cid:101)r∈R(cid:101) unify(r,(cid:101)r,R,R˙,R(cid:101))=(cid:104)R(cid:48),R˙(cid:48),R(cid:101)(cid:48)(cid:105) (cid:101)r:iq=oq∈R(cid:101) w:(cid:104)o,i(cid:105)∈W
(cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R(cid:48),R˙(cid:48),W,R(cid:101)(cid:48),Z(cid:105) (cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R,R˙,W−{w},R(cid:101)−{(cid:101)r},Z∪{o(cid:22)i}(cid:105)
input-var-map input-val-map
(cid:101)r:id=(cid:98)i∈R(cid:101) (cid:98)i∈(cid:98)I i@c c∈IC r:id=x∈R(cid:101) i∈I x∈R i@c c∈IC
(cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R,R˙,W,R(cid:101)−{(cid:101)r},Z∪{(cid:98)i(cid:55)→id}(cid:105) (cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R,R˙,W,R(cid:101)−{(cid:101)r},Z∪{x(cid:55)→id}(cid:105)
local-var-map output-var-map
(cid:101)r:(cid:98)l(cid:55)→oq∈R(cid:101) (cid:98)l∈(cid:98)L (cid:101)r:(cid:98)o(cid:55)→o(cid:48)q∈R(cid:101) (cid:98)o∈O(cid:98) r:od=iq∈R o@c c∈OC
(cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R,R˙ ∪{oq=(cid:98)l},W,R(cid:101)−{(cid:101)r},Z∪{(cid:98)l(cid:55)→oq}(cid:105) (cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R−{r},R˙ ∪{o(cid:48)q=(cid:98)o},W,(R(cid:101)−{(cid:101)r})∪{iq=o(cid:48)q},Z∪{(cid:98)o(cid:55)→od}(cid:105)
Figure5. DefinitionoftheTableauTransitionRelation→
Each goal(cid:101)r ∈ R(cid:101) is a relation(cid:101)r of the form(cid:101)r : (cid:98)v = (cid:98)E, which contains the initial unused hardware environment R,
(cid:101)r:∂(cid:98)v/∂t=(cid:98)E init x(relationsfromthedynamicalsystem), wiresW,anddynamicalsystemrelations(cid:98)Rasgoals.
(cid:101)r : iq = E (a mixed-variable derived relation that may Figure 5 presents the rules that define the solver steps.
include both dynamical system and hardware variables), or Rule unify maps a (potentially already partially mapped)
(cid:101)r : (cid:98)v (cid:55)→ oq (mapping a dynamical system local or output relation(cid:101)r ∈ R(cid:101) onto a hardware relation r ∈ R ∪ R˙. Rules
variable to a hardware output variable), where(cid:98)v ∈ (cid:98)L∪O(cid:98), connect through output-var-map solve goals that directly
x ∈R,(cid:98)Eisanexpressionover(cid:98)V sothatvars((cid:98)E)⊆(cid:98)V,andE configurethehardware:
isanexpressionoverV=V∪(cid:98)Vsothatvars(E)⊆V=V∪(cid:98)V, • connect: The connect rule solves a goal of the form
iq ∈I,andoq ∈O. iq = oq by adding a wire o (cid:22) i to the configuration
Notethatthesolvermaygeneratepartiallysolvedmixed-
Z.Thewireconnectstheoutputportototheinputporti.
variablegoalsthatincludebothdynamicalsystemandhard- • input-var-map or input-val-map: The input port mapping
warevariables. rulessolvegoalsoftheformi = xori =(cid:98)i.Therules
q q
Partially Used Hardware: As the solver operates, it con-
addthemapping x (cid:55)→ i or(cid:98)i (cid:55)→ i totheconfiguration
q q
sumes the starting hardware R to produce partially used
Z. The mapping binds the value x or dynamical system
hardware˙r∈R˙.Eachrelation˙r∈R˙ isoftheform˙r:o =E
q input(cid:98)itothehardwarevariablei .
or ˙r : ∂oq/∂t = E init iq(cid:48), where vars(E) ⊆ V = • local-var-map: The local variablqe mapping rule solves
V∪(cid:98)V.Each˙r∈R˙ isthereforeapartiallyinstantiatedmixed- goals of the form(cid:98)l (cid:55)→ o . It adds the mapping to the
q
variablerelationderivedfromanoriginalhardwarerelation. configurationZ andaddso =(cid:98)ltoR˙.Arcosolvesresult-
q
HardwareConfiguration:Asitoperates,thesolvergener-
ingfuturesubgoalsoftheformi =(cid:98)lthroughunification,
atesahardwareconfigurationZ.Eachz∈Zisoftheform: q
providedi =o holds.
q q
• o (cid:22) i:Connectawirefromhardwareoutputportoto • output-var-map:Theoutputvariablemappingrulesolves
hardwareinputporti. goalsoftheform(cid:98)o (cid:55)→ o(cid:48)q.Theruleusesadigitaloutput
• x(cid:55)→i :Setadigitalinputi ∈Itoavaluex∈R. componentc : od = iq.Italsoaddsthemapping(cid:98)o (cid:55)→ od
d d
•(cid:98)i(cid:55)→id:Mapadynamicalsysteminputvariable(cid:98)i∈(cid:98)Itoa toX,therelationo(cid:48)q =(cid:98)otoR˙,andthegoaliq =oqtoR(cid:101).
digitalinputid ∈I Theunifyruleusestheunify(r,(cid:101)r,R,R˙,R(cid:101))function,which
•(cid:98)v(cid:55)→oq:Mapadynamicalsystemoutputorlocalvariable unifies r with (cid:101)r and resolves any entangled and affected
(cid:98)v∈(cid:98)L∪O(cid:98)toahardwareoutputportpropertyoq ∈O. relations. Figure 6 presents the transition rules that define
theunifyfunction.Eachrulefirstunifiesrand(cid:101)rtoobtaina
setofassignmentsA.ItthenappliestheassignmentsAtoR˙
4.1 TheTableauTransitionRelation→
(ifr∈R˙)ortoRif(r∈R).
Weformalizetheoperationofthesolverasatransitionrela- The solver steps unify relations r. But the basic Arco
tion(cid:104)R,R˙,W,R(cid:101),Z(cid:105)→(cid:104)R(cid:48),R˙(cid:48),W(cid:48),R(cid:101)(cid:48),Z(cid:48)(cid:105)ontableauconfig- unification algorithm A ∈ unifyExpr(E,E(cid:48),V) operates on
urations (cid:104)R,R˙,W,R(cid:101),Z(cid:105), where each transition corresponds expressions E and E(cid:48). It produces sets of assignments A
to a solver step on the tableau. The solver implements the that unify E and E(cid:48) so that sub(E,A) = sub(E(cid:48),A), where
solve(R,W,(cid:98)R) function, which returns a set of solutions sub(E,A) applies the assignments A to the expression E.
Z, where each solution is reachable by applying → transi- The rules in Figure 7 lift the unification algorithm from
tionrulestothestartingtableauconfiguration(cid:104)R,∅,W,(cid:98)R,∅(cid:105), expressionsEtorelationsr.
unify(r,(cid:101)r,R,R˙,(cid:101)R)={(cid:104)R(cid:48),R˙(cid:48).(cid:101)R(cid:48)(cid:105)|(cid:104)r,(cid:101)r,R,R˙,(cid:101)R(cid:105)→u (cid:104)R(cid:48),R˙(cid:48),(cid:101)R(cid:48)(cid:105)}
unify-partial unify-unused
r∈R˙ (cid:104)A,a,r(cid:48),(cid:101)r(cid:48)(cid:105)∈unifyRelation(r,(cid:101)r,V) r∈R (cid:104)A,a,r(cid:48),(cid:101)r(cid:48)(cid:105)∈unifyRelation(r,(cid:101)r,V)
(cid:104)R˙(cid:48),R(cid:101)(cid:48)(cid:48),D(cid:105)∈apply(R˙ −{r},(R(cid:101)∪{(cid:101)r(cid:48)})−{(cid:101)r},A,a) (cid:104)R(cid:48),R(cid:101)(cid:48)(cid:48),D(cid:105)∈apply(R˙ −{r},(R(cid:101)∪{(cid:101)r(cid:48)})−{(cid:101)r},A,a)
(cid:104)r,(cid:101)r,R,R˙,R(cid:101)(cid:105)→u(cid:104)R,R˙(cid:48)∪D∪{r(cid:48)},R(cid:101)(cid:48)(cid:48)(cid:105) (cid:104)r,(cid:101)r,R,R˙,R(cid:101)(cid:105)→u(cid:104)R(cid:48),R˙ ∪D∪{r(cid:48)},R(cid:101)(cid:48)(cid:48)(cid:105)
Figure6. DefinitionoftheUnifyRelation→
u
unifyRelation(r,r(cid:48),(cid:101)r,V)={(cid:104)A,a,r(cid:48),(cid:101)r(cid:48)(cid:105)|(cid:104)r,(cid:101)r,V(cid:105)→f (cid:104)A,a,r(cid:48),(cid:101)r(cid:48)(cid:105)}
unify-hw-rel unify-ds-rel
oq∈O iq(cid:48) ∈I o@c V=vars(c) A∈unifyExpr(E,E(cid:48),V) oq∈O (cid:98)v∈(cid:98)V o@c V=vars(c) A∈unifyExpr(E,(cid:98)E(cid:48),V)
(cid:104)oq=E,iq(cid:48) =E(cid:48),V(cid:105)→f (cid:104)A,oq=iq(cid:48),oq=E(cid:48),iq(cid:48) =oq(cid:105) (cid:104)oq=E,(cid:98)v=(cid:98)E(cid:48),V(cid:105)→f (cid:104)A,oq=(cid:98)v,oq=(cid:98)E(cid:48),(cid:98)v(cid:55)→oq(cid:105)
unify-ds-deriv
oq∈O iq(cid:48) ∈I (cid:98)v∈(cid:98)V x∈R o@c V=vars(c) A∈unifyExpr(E,(cid:98)E(cid:48),V)
(cid:104)∂oq/∂t=Einitiq(cid:48),∂(cid:98)v/∂t=(cid:98)E(cid:48)initx,V(cid:105)→f (cid:104)A∪{iq(cid:48) =x},oq=(cid:98)v,oq=(cid:98)v,(cid:98)v(cid:55)→oq(cid:105)
Figure7. DefinitionoftheRelationUnificationRelation→
f
apply(R,(cid:101)R,A,a)={(cid:104)R(cid:48),(cid:101)R(cid:48),D(cid:48)(cid:105)|(cid:104)R,(cid:101)R,A∪{a},A,∅(cid:105)→∗d→c→∗x (cid:104)R(cid:48),(cid:101)R(cid:48),∅,D(cid:48)(cid:105)}
apply-inp-assign apply-out-assign
a:iq=E∈A(cid:101) iq∈I a:oq=E∈A(cid:101) oq∈O r:oq=E(cid:48)∈R A(cid:48)∈unifyExpr(sub(E(cid:48),A),E,V)
(cid:104)R,R(cid:101),A,A(cid:101),D(cid:105)→d(cid:104)R,R(cid:101)∪{a},A,A(cid:101)−{a},D(cid:105) (cid:104)R,R(cid:101),A,A(cid:101),D(cid:105)→d(cid:104)R−{r},R(cid:101),A∪A(cid:48),(A(cid:101)∪A(cid:48))−{a},D∪{a}(cid:105)
apply-out-assign2
a:oq=(cid:98)v∈A(cid:101) oq∈O (cid:98)v∈(cid:98)V r:oq=E(cid:48)∈R (cid:101)r:(cid:98)v=(cid:98)E∈R(cid:101) oq@c V=vars(c) A(cid:48)∈unifyExpr(sub(E(cid:48),A),(cid:98)E,V)
(cid:104)R,R(cid:101),A,A(cid:101),D(cid:105)→d(cid:104)R−{r},R(cid:101)−{(cid:101)r},A∪A(cid:48),(A(cid:101)∪A(cid:48))−{a},D∪{a,oq=(cid:98)E}(cid:105)
conv xform-unused-rel
A(cid:101)=∅ r:oq=E∈R vars(E)∩dom(A)(cid:44)∅
(cid:104)R,R(cid:101),A,A(cid:101),D(cid:105)→c(cid:104)R,R(cid:101),A,D(cid:105) (cid:104)R,R(cid:101),A,D(cid:105)→r(cid:104)R−{r},R(cid:101),A,D∪{oq=sub(E,A)}(cid:105)
apply-unused-deriv apply-unused-done
r:∂oq/∂t=E init iq∈R vars(E)∩dom(A)(cid:44)∅ iq(cid:60)dom(A) {r|r∈R∧vars(r)∩dom(A)(cid:44)∅}=∅
(cid:104)R,R(cid:101),A,D(cid:105)→r(cid:104)R−{r},R(cid:101),A,D∪{∂oq/∂t=sub(E,A)initiq}(cid:105) (cid:104)R,R(cid:101),A,D(cid:105)→r(cid:104)R,R(cid:101),∅,D(cid:105)
Figure8. DefinitionoftheDetanglingRelation→ andtheResolvingRelation→
d r
Relation Entanglement: Unifying a partially used (r ∈ R˙) theassignmentsAtoE(cid:48) tocreatethespecializedexpression
or unused (r ∈ R) hardware relation with some goal(cid:101)r ∈ R(cid:101) E(cid:48)(cid:48) = sub(E(cid:48),A). It unifies expressions E and E(cid:48)(cid:48) (assign-
(wherethevariablesv ∈ V arethehardwarevariablesinc : ing only variables V), then incorporates the resulting set of
r @ c) may produce assignments which are entangled with assignments A(cid:48) (chosen from the space of possible assign-
other relations that belong to c in R. A hardware relation ments)intoA.
is entangled with an assignment a : o = E if the relation The→ relationdetanglesr∈RfromAbyincrementally
q d
defines o (for example, o = E(cid:48)). A hardware relation is applying assignments a ∈ A(cid:101) to the tableau and detangling
q q
affected by a if o is used (but not defined) by the relation anyentangledrelationsasneeded,whereA(cid:101) aretheremain-
q
(forexample,o = Ewhereo ∈vars(E)). ingassignments.TherulesmovedetangledrelationsintoD:
q q
DetanglingRelations:Figure8presentsthetransitionrela-
• apply-inp-assign:Assignmentsoftheformi = E create
tions that apply a set of assignments A to the tableau. The q
noentanglements.Therulethereforemovessuchassign-
rules (1) detangle any relations entangled with A (→ ) and
d mentsdirectlyintothegoalsetR(cid:101).
(2) move any relations affected by A out of R (→ ). Har-
r • apply-out-assign:Theruleharmonizesentangledassign-
monization detangles entangled relations. Given an assign-
ments of the form a : o = E where r : o = E(cid:48) ∈ R.
ment o = E and a relation o = E(cid:48), Arco first applies q q
q q Itthenaddstheresultingsetofassignments A(cid:48) toA(cid:101) and
addsthefullyspecializedrelationoq =sub(E(cid:48),A∪A(cid:48))= 1 search((cid:98)R0,R0,W0)
E to D. Because the harmonization satisfies assignment 2 F = {(cid:104)R ,∅,W ,R(cid:101) ,∅(cid:105)}
0 0 0
a,theruledoesnotaddatothegoalsetR(cid:101). 3 while choose t:(cid:104)R,R˙,W,R(cid:101),Z(cid:105) ∈ F
• apply-out2-assign: The rule harmonizes entangled as- 4 if R(cid:101) = ∅ return Z
signments of the form a : oq = (cid:98)v where there is an 5 select T ⊆ {t(cid:48) | t → t(cid:48) }
entangled r : oq = E(cid:48) ∈ R, and a definition of(cid:98)v of the 76 endF = (F − {t}) ∪ T
form(cid:101)r :(cid:98)v =(cid:98)E ∈ R(cid:101).Theruleharmonizes(cid:98)EandE(cid:48) (with 8 return nosolution
V). The rule makes the same changes to A(cid:101), A, and D
as apply-out-assign, but also removes the now–mapped Figure9. ArcoSearchAlgorithm
relation(cid:101)rfromthegoalsetR(cid:101).
Resolving Affected Relations: The →r relation resolves 4.3 SearchOptimizations
any affected relations r ∈ R after all potentially entangled
Wenextdescribeseveraloptimizationsthatsignificantlyim-
relations have been resolved (when A(cid:101) = ∅). Each → rule
r provetheperformanceoftheArcosearchalgorithm.
findsarelationr ∈ Raffectedbytheassignments A.Itthen
ComponentAggregation:Ourtargetprogrammableanalog
appliestheassignmentsAtortocreateaspecializedrelation
hardware platforms typically contain multiple instances of
whichitaddstoD.ItthenremovesrfromR.Itappliesthese
a given component. Arco reduces the size of the search
transitionsuntilnoaffectedrelationsareleft.
space by materializing component instances on demand to
4.2 TheSearchAlgorithm
unifythechosengoalagainstatmostoneunusedcomponent
Figure9presentstheArcosearchalgorithm.Thealgorithm instance(asopposedtounifyingagainstallavailableunused
explores the search space defined by the tableau configu- componentinstances).
ration transition relation →. It maintains a set of tableau Abstract and Concrete Hardware Configurations: Be-
configurations F at the frontier of the explored space. At cause of component aggregation, the Arco tableau search
each step, the algorithm chooses a tableau configuration algorithm does not distinguish between different instances
t : (cid:104)R(cid:48),R˙,W(cid:48),R(cid:101)(cid:48),Z(cid:105) ∈ F to explore. If all of the goals in of the same component. But the hardware may enforce
t have been solved (i.e., R(cid:101) = ∅), the algorithm returns the instance-specificwiringconstraintssothatdifferentcompo-
generated configuration Z. Otherwise, it selects a subset T nentinstancesarenotinterchangable.Arcothereforeimple-
of the set of tableau configurations directly reachable from mentsastagedsynthesisalgorithm.Thetableausolverfirst
t under → and replaces t in F with T. The decisions that synthesizesanabstracthardwareconfigurationthattreatsall
Arco makes when it chooses a tableau configuration t (line componentinstancesuniformly.Aconcretizationalgorithm
3)andselectsasubsetT toexplore(line5)haveasignificant then maps each abstract component instance to a concrete
impactontheeffectivenessofthesearchalgorithm. componentinstance.Theresultisaconcretehardwarecon-
Choosing t: At line 3, Arco applies a heuristic designed to figuration. Arco formulates the concretization problem as
findthetableauconfigurationtthatisclosesttobeingsolved. an SMT problem whose solution maps each abstract com-
Thisheuristicscoreseachtableauconfigurationbasedonthe ponent instance to a corresponding concrete component in-
number and complexity of the remaining nontrivial goals stance.TheSMTproblemhastwokindsofclauses:
(goals solved only by rule unify in Figure 5). It chooses
• Instance Assignment Clauses: These clauses ensure
the configuration t with the smallest score (i.e., fewest and
thatexactlyoneabstractcomponentmapstoexactlyone
simplestremainingnontrivialgoals).
physicalcomponent.
Selecting T: Arco first selects a goal(cid:101)r ∈ R(cid:101), prioritizing
• ConnectionClauses:Theseclausesensurethatallofthe
trivial goals (solved by rules connect through output-var-
connections in the abstract hardware configuration map
mapinFigure5)overnontrivialgoals(solvedbyruleunifyin
torealizablehardwareconnections.
Figure5).Itthencomputesthesetoftableauconfigurations
generatedbyapplyingtheappropriaterulefromFigure5to IfthegeneratedSMTproblemisunsatisfiable,thenthereis
(cid:101)rtoobtainT.Ifanyofthetrivialgoalsareunsolvable,Arco nophysicalconfigurationoftheanaloghardwarethatimple-
sets T = ∅, effectively pruning the entire search subspace ments the abstract hardware configuration. In this case the
rooted at t. The rationale is that the unsolvable trivial goal Arco search algorithm removes the corresponding tableau
ensuresthattheprunedsubspacecontainsnosolvedtableau configuration from the tableau frontier. The tableau explo-
configurations. rationalgorithmcontinuesontofindnewabstracthardware
TheunifyrulefromFigure5maygeneratealargenumber configurations.
oftableauconfigurations.Tomaintainthetractabilityofthe Partial Configuration Caching: Arco caches generated
searchalgorithm,thecurrentArcoimplementationdiscards partialconfigurationsforthedifferentdynamicalsystemre-
generated tableau configurations so that T contains at most lations(cid:98)r ∈ (cid:98)R. It reuses the cached partial configurations
fivetableauconfigurationsfromeachcombinationofrand(cid:101)r whenever it subsequently encounters the relation again in
intheunifyrule. anotherregionofthesearchspace.