Deciding Contextual Equivalence for IMJ* Andrzej Murawski Steven - - PowerPoint PPT Presentation
Deciding Contextual Equivalence for IMJ* Andrzej Murawski Steven Ramsay Nikos Tzevelekos University of Warwick University of Oxford Queen Mary University of London For all interface tables and IMJ contexts such
Steven Ramsay Andrzej Murawski Nikos Tzevelekos University of Warwick University of Oxford Queen Mary University of London
it follows that: terminates iff terminates For all interface tables and IMJ contexts such that:
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2 + x 2 * x can be witnessed by let x = 1 in if □ == 2 then skip else (while 1 do skip) let x = 1 in if 2 + x == 2 then skip else (while 1 do skip) diverges let x = 1 in if 2 * x == 2 then skip else (while 1 do skip) terminates
{ [], [0,2], [1,3],… } { [], [0,0], [1,2],… }
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let x = new {_: IntRef;} in new {_:I; run: λ_. if x.val = 0 then x.val := 1; f.run(); if x.val = 2 then skip else div else if x.val = 1 then x.val := 2 else div } new {_:I; run: λ_. div} let z = new { _:IRef;} in let f = new {_:I; run: λ_. z.val.run() } in z.val := □ z.val.run()
fobj zobj call zobj.run() call fobj.run() call zobj.run() ret zobj.run ret fobj.run ret zobj.run
let x = new {_:IntRef;} in let c1 = new {_:ObjRef;} in let c2 = new {_:ObjRef;} in new {_:ObjCell; get: λ_.if x.val then c1.val else c2.val, getprev: λ_.if x.val then c2.val else c1.val, set: λo. if x.val then x.val := 0 else x.val := 1; if x.val then c1.val := o else c2.val := o } let last = new {_:ObjRef;} in let current = new {_:ObjRef;} in new {_:ObjCell; get: λ_.current.val, getprev: λ_.last.val, set: λo.last.val := current.val; current.val := o }
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[MRT ATVA’15] new { this: I; m1: body1, …, mk: bodyk } let x = exp in exp if x = exp then exp else exp while exp do exp exp.fld exp.fld := exp exp = exp exp + exp (I)exp exp ; exp null skip exp.m(exp1,…,expk) x n
[MRT ATVA’15]
Only finite types, ground fields Only first-order objects Only iteration Only second-order objects returning ground data
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Qu, QuItem, QuItemId
Given a queue machine
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Qu, QuItem, QuItemId
Given a queue machine
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = qI head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 b b: Qu
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = qI head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 b b: Qu b.enq(3)
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = q1 head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 prev: QuItemId = myId: QuItemId = deq () : void = ….. enqd? : int = 3 b b: Qu d: QuItem b.enq(3) ret b.enq(d) Assuming δE(qI, 3) = q1
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = q1 head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 prev: QuItemId = myId: QuItemId = deq () : void = ….. enqd? : int = 3 b b: Qu d: QuItem b.enq(3) ret b.enq(d) b.enq(6)
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = q2 head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 prev: QuItemId = myId: QuItemId = deq () : void = ….. enqd? : int = 3 enqd? : int = 6 prev: QuItemId = myId: QuItemId = deq () : void = ….. b b: Qu d: QuItem f: QuItem b.enq(3) ret b.enq(d) b.enq(6) ret b.enq(f) Assuming δE(q1 , 6) = q2
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = q2 head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 prev: QuItemId = myId: QuItemId = deq () : void = ….. enqd? : int = 3 enqd? : int = 6 prev: QuItemId = myId: QuItemId = deq () : void = ….. b b: Qu d: QuItem f: QuItem b.enq(3) ret b.enq(d) b.enq(6) ret b.enq(f) d.deq()
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Qu, QuItem, QuItemId
Given a queue machine
qState: int = q3 head: QuItemId = enq (d: int) : QuItem = ……………….. enqd? : int = 0 prev: QuItemId = myId: QuItemId = deq () : void = ….. enqd? : int = 0 enqd? : int = 6 prev: QuItemId = myId: QuItemId = deq () : void = ….. b b: Qu d: QuItem f: QuItem b.enq(3) ret b.enq(d) b.enq(6) ret b.enq(f) d.deq() ret d.deq Assuming δD(q2) = q3 and checking d.prev.enqd? = 0 and d.myId.enqd? != 0
[MRT ATVA’15]
Only finite types, ground fields Only first-order objects Only iteration Only second-order objects returning ground data
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Translate IMJ* terms into their strategies in the game model, represented as two IMJ Automata (IMJA). Reduce the equivalence problem for IMJA to the emptiness problem for Fresh Pushdown Register Automata (FPDRA). Solve the emptiness problem for FPDRA using saturation algorithm.
1 2 3
[MT POPL’14] [MRT ATVA’15] [MRT ATVA’15] [MT ICALP’12] [MRT MFCS’14]
A machine representation for strategies (sets of plays).
fobj zobj call zobj.run() call fobj.run() call zobj.run() ret zobj.run ret fobj.run ret zobj.run
Object creation Call stack discipline Finite set of possible moves modulo
Fresh-name recognition Visible pushdown stack Accepts words over a nominal alphabet
(Representation of stores not shown)
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A machine representation for strategies (sets of plays).
ν2. call 3.myMethod(2) / (k, {2,3}) q q’ ν2. ret 3.myMethod(2) / (k, {2,3}, {2}) q q’ ν2. 2 q q’
+ Bookkeeping
f:I Ⱶ new {_:I; run: λ_. div} : I ν1. 1
fobj, zobj
let x = new {_: IntRef;} in new {_:I; run: λ_. if x.val = 0 then x.val := 1; f.run(); if x.val = 2 then skip else div else if x.val = 1 then x.val := 2 else div } ν3. 3 ν1. 1
f:I Ⱶ
Simulate two sets of registers using one set equipped with a representation of one
REPRESENTATION OF SYMMETRIC DIFFERENCE (FPRDA) PLAYS OF SYSTEM 1 (IMJA) PLAYS OF SYSTEM 2 (IMJA)
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SYMMETRIC DIFFERENCE (FPRDA) ACCEPTING CONFIGURATIONS (RA) CONFIGURATIONS LEADING TO ACCEPT (RA)
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