Programming in hybrid constraint languages

1
Programming in hybrid constraint languages

• ????? M2
• ?? ??

2
The relations between the cc languages

• cc
  cc
• Default cc Timed cc Default cc
  Timed cc
• cc defaults cc time
  
• 
  Timed default cc
• Hybrid cc Timed Default cc
• cc defaults cc defaults
• cont.time discrete time
  Hybrid cc

3
Hybrid cc

• ????????????????????????????????
• ???????????????
• Example ??????????????
• ????????????????????????
• ?????????????????????????
• ?????????????

4
The cc Programming Language

• ??store?Agent(tell,ask)???????????????
• Example program x10,x0,if xgt0 then x-10

5
Hybrid cc the language and its use

• c tell the constraint c
• if d then A if d holds,reduce to A
• if d else A reduce to A unless d holds
• A,B Parallel Composition
• new V in A V is local to A
• hence A execute A at every instant
• after now

6
Hcc Example1
x10,x'0, henceif xgt0 then x''-10,
if x0 then x' -0.5x'
7
Hcc Example2

• Prey Predator(in Hcc)
• py8, //prey
• pd2, //predator
• pd0.2,
• always py0.08py 0.04pypd //prey growth
• always
• cont(pd)
• if(pdgt0.5py) then
• pd-0.1pd0.02pypd
• else
• pd-0.06pd0.02pypd
• ,
• sample(pd), sample(py)

8
PreyPredator
FigurePreyPredator
9
Computational model

• ?????????????
• 1. Point Phase(???????????)
• 2. Interval Phase(???????????)
• ? 1 , 2??????????

10
The Control Constructs and Execution of hcc

• Hcc?????

1. Point Phase????? 2. Point Phase????????????? 3.
Interval Phase??? 4. ??????????????????(4th
order Runge-Kutta?) 5. ???????????????????????????
??? ?????? 6. ??????Point Phase??? ????1
6?????
11
Execution of hcc

• Example
• X 0,hencex1,if(x2) then y1
• In the interval phase following x0,x evolves
  continuously
• according to x1,through the interval(0,2) until
  x2 is about to
• become true.
• At this point the set of constraints may
  change,so the next point
• phase is started.

12
Continuous Constraints

• ContConstr Term RelOp Term
  cont(LVariable) lcont(LVariable)
• RelOp gt lt
• Term LVariableExpr Constant
  Term BinOp Term
• UnOp(Term)
  dot(Term,Num) Term (Term)
• LVariableExpr Lvariable Uvariable.LVariable
  Expr
• BinOp - ? /
• UnOp - log exp prev
• Lvariables are strings which start with a
  lowercase character.
• Constants are floating point numbers.
• exp(x) is the exponential function ex.
• Term denotes the derivative of Term.
• dot(Term , n) denotes the nth derivative of
  Term(n is a natural number) .

13
Ask Constraints(for Continuous Constraints)

• AContConstr Term ARelOp Term
• ARelOp gt lt lt gt !

14
Non-arithmetic Constraints

• Dconstr UVariableExpr
  UVarExpr Dexpr
• UVariableExpr UVariable
  UVariableExpr.UVariableExpr
• Dexpr UVariableExpr
  prev(UVariableExpr) String
• 
  (VarList)HccAgent
• 
  (VarList)VarListHccAgent
• 
  UVariableExpr(VarList) VarListHccAgent
• VarList Uvariable Lvariable
  VarList,VarLIst
• Uvariable is a string starting with an
  uppercase character.
• HccAgent is hcc Agent.

15
Closure Class definition

• Closure definition X (A,x) Agent
• Class definition X (m,n)A,m Agent

16
Closure definition

• X (A,x)Agent
• Example
• P (n,m,Q)
• if n gt 0 then
• new m1 in Q(n-1,m1,Q), m m1n
• else
• m 1
• ,
• Fact (n,m) P(n,m,P),
• Fact(10,m)

17
Class Definition

• X (m,n)A,m Agent
• Example
• Planet (initvx,initvy,initpx,initpy,m)px,py,mas
  s
• px initpx, py initpy, px initvx,
  py initvy,
• always
• massm,
• pxsum(F.x,Force(F),F.Targe
  t Self),
• pysum(F.x,Force(F),F.Targe
  t Self)

18
Boolean Constraints

• BoolConstr Atom StrConstr ContConstr
  
• Bool BoolConstr
  (BoolConstr)
• ABoolConstr Atom StrConstr AContConstr
  
• ABool ABoolConstr
  
• ABoolConstr
  ABoolConstr
• (ABoolConstr)

19
Reduction Rules

• Tell lt(G, c),s,next, default ? ltG,s U c,next,
  default
• s- d
• Ask
• lt(G,if d then A),s,next, defaultgt ?
  lt(G,A),s,next, defaultgt
• else lt(G,if d else A),s,next,elsegt ?
  ltG,s,next,(else, if d else A)gt
  

G denotes a set of Hybrid cc program
fragments. s denotes the store.
next the set of program fragments to be run
in the next phase. default a
set of suspended else statements. s - c
denotes entailment checking.
20
Reduction rules(for Hence)

• The rule for hence A differs in point and
  interval phases
• Hence Point lt(G,hence A), s, next ,
  defaultgt ?
• ltG , s, ltnext ,
  hence Agt, defaultgt
• Hence Interval lt(G, hence A), s, next ,
  defaultgt ?
• lt(G, A), s, (next
  , A , hence A), defaultgt
• G denotes a set of Hybrid cc program
  fragments.
• s denotes the store.
• next the set of program fragments to be
  run in the next phase.
• default a set of suspended else statements.
• s - c denotes entailment checking.

21
The algorithm for interpreter

• 1. Run the reduction rules on the current
  ltG,s,next defaultgt,
• till no further reductions can take place.
• 2. If s is inconsistent , return 0 (false).
• 3. If default is empty , return 1 (true).
• 4. Remove one statement from default if c
  else A.
• If s - c , go to step 3.
• 5. Add A to G. Run the interpreter on the
  current state.
• If the result is 1 and s -/ c , return 1.
• 6. Undo the effects of the previous step by
  backtracking.
• Run the interpreter on the current state.
• If the result is 1 and s -/ c , return 1.
  Otherwise return 0.

22

• A Hybrid cc program A is run as follows.
• 1. Run interpreter with G A , and empty s,
  next and default
• in the point phase. If the result is 0,
  abort.
• 2. Run the interval phase with G next , as
  returned by the point
• phase.
• s , next and default are again empty. If the
  result is 0, abort.
• Record all the tells , and also the ask
  constraints that were checked
• during the phase.
• 3. Integrate the arithmetic constraints that were
  told in the previous
• step,until one of the ask constraints changes
  status(i.e.goes from
• false to true or unknown, etc.). Go to step 1
  with G next.

23
Nonlinear equations

• Indexicals
• Interval splitting
• The Newton-Raphson method
• The Simplex method
• ???4???????????????????????

24
Indexicals

• ??f(x,y)0?xg(y)????????
• Example
• x y 0, x ? 0,3, and y ? -1, -2.
• Then the indexical x -y is used to set to 1,
  2.

25
Interval splitting

• ??a,b???????????
• a1smallest number 0?f(a1,b) and a?a1.
• Hence,if 0?f(a,(ab)/2),then
  a1?a,(ab)/2,
• otherwise a1?(ab)/2,b
• b1??????
• Example x21 and x ?-8 , 8.
• It follows that 0 ?-8 , 0.
• 0 ?/ -8 , -100. a1 is determined to be in
  -100 , 0.
• Eventually, a1 is determined to be 1.

26
The Newton-Raphson method

• ???f(x)0??????????
• Interval?????????????
• ?????f(x)?????f(x)0???x?????
• ???????????x0????f(x)????????
• ???????y-f(x0)f(x0)(x-x0)??????????
• ?????X?????x1?y0?????x1x0-f(x0)/f(x0)
• ???????
• ??x1??f(x)?????x??????x2?????????
• ??????????f(x)0???????i?????????
• xi1xi-f(xi)/f(xi)?????????e???????
• xi1-xilte??????xi1???????

27
Splitting Newton-Raphson method

• Splitting?Newton-Raphson method???????(clp
  Newton)?
• Example
• x21 and x?I-8, 8
• 1. Split I recursively until I is split down to
  -2,-1.
• 2. Applying the N-R method-1,-1 is produced.
• 3. Similarly,1,1 is produced.
• 4. Only 1 and 1 are solutions to the equation.

28
The Control Constructs and Execution of hcc

• Hcc?????

1. Point Phase????? 2. Point Phase????????????? 3.
Interval Phase??? 4. ??????????????????(4th
order Runge-Kutta?) 5. ???????????????????????????
??? ?????? 6. ??????Point Phase??? ????1
6?????
29
Runge-Kutta algorithm
4?????????
30
Runge-Kutta algorithm
1. A????????? k1??? 2. B?(xih/2,yik1/2)???
?????k2??? 3. C?(xih/2,yik2/2)???
?????k3??? 4. D?(xih,yik3)???? ????k4??? 5.
k1 k4???????? ?????????yi????
?????yi1???? k1h(xi,yi),k2hf(xih/2,yik1/2),
k3hf(xih/2,yik2/2), k4hf(tih,xik3)
yi1yi(k12k22k3k4)/6
31
Program Example

• ball.hcc
• / A simple example of a ball bouncing on the
  floor.
• /
• y 10, y' 0, // initial
  conditions
• hence
• if y gt 0 then y'' -10, // free fall
• if y 0 then // bounce
• if (prev(y') gt -0.000001) then always y'
  0 // end of bouncing
• else y' -0.5 prev(y')
• ,
• sample(y)

32
??????(5??)
33
????

• B.Carlson and V.Gupta.Hybrid cc with interval
  constraints.
• V.Gupta,R.Jagadeesan,V.A.Saraswat,and D.Bobrow.
• Programming in hybrid concurrent constraint
  languages.In Panos
• Antsakis,Wolf Kohn,Anil Nerode,and Sankar
  Sastry,editors,Hybrid
• Systems ?,volume999 of Lecture Notes in
  Computer Science,1995
• V.Gupta,R.Jagadeesan,V.Saraswat,and D.Bobrow.
• Programming in hybrid constraint languages