Research Note Cyclic Abduction of Inductively Defined Safety and

14 Pages·2013·0.42 MB·English

UCLDEPARTMENTOF
COMPUTERSCIENCE
Research Note
RN/13/14
Cyclic Abduction of Inductively Defined
Safety and Termination Preconditions
July 12, 2013
James Brotherston Nikos Gorogiannis
Abstract
We describe a new method, called cyclic abduction, for automatically inferring
safetyand/orterminationpreconditionsforheap-manipulatingwhileprograms,
expressed as inductive definitions in separation logic. Cyclic abduction essen-
tially works by searching for a cyclic proof of memory safety and/or termina-
tion, abducing definitional clauses of the precondition as necessary in order to
advance the proof search process. This is achieved via a suite of heuristically
guided automatic tactics.
We have implemented our cyclic abduction procedure as an automatic tool,
Caber, that automatically infers the correct safety and termination precondi-
tions for a range of common small programs manipulating lists and trees, and
can also abduce the definitions of more exotic data structures such as cyclic or
segmented lists, or trees of linked lists. To our knowledge, cyclic abduction is
the first technique for automatically abducing such inductive definitions from
pointer programs.
Cyclic Abduction of Inductively Defined
Safety and Termination Preconditions
JamesBrotherston∗ NikosGorogiannis†
Dept.ofComputerScience,UniversityCollegeLondon,UK
Abstract describedbytheprovideddefinitions.Forexample,thewellknown
Wedescribeanewmethod,calledcyclicabduction,forautomat- SPACEINVADER [25] and SLAYER [5] analysers perform accu-
ratelyonprogramsusingcombinationsoflinkedlists,asfoundwith
ically inferring safety and/or termination preconditions for heap-
devicedrivers,butreportafalsebugiftheyencounteratree.Thus
manipulating while programs, expressed as inductive definitions
theabilitytomechanicallyinfer,orabduce,theinductivepredicates
inseparationlogic.Cyclicabductionessentiallyworksbysearch-
neededtoanalyseindividualprocedureshasthepotentialtogreatly
ingforacyclicproof ofmemorysafetyand/ortermination,abduc-
boosttheautomationofsuchverificationtools.
ingdefinitionalclausesofthepreconditionasnecessaryinorderto
In this paper, we consider the following form of the above
advance the proof search process. This is achieved via a suite of
synthesisorabductionproblem:givenaheap-manipulatingwhile
heuristicallyguidedautomatictactics.
programP,canwedefineaninductivepredicateinseparationlogic
Wehaveimplementedourcyclicabductionprocedureasanau-
suchthat,giventhispredicateasaprecondition,
tomatic tool, CABER, that automatically infers the correct safety
and termination preconditions for a range of common small pro-
• P runswithoutencounteringamemoryerror(safety);or
grams manipulating lists and trees, and can also abduce the defi-
nitionsofmoreexoticdatastructuressuchascyclicorsegmented • P eventuallyterminatessafely(termination)?
lists,ortreesoflinkedlists.Toourknowledge,cyclicabductionis
thefirsttechniqueforautomaticallyabducingsuchinductivedefi- These are highly non-trivial questions. On the one hand, the
nitionsfrompointerprograms. weakest (liberal) precondition (cf. Dijkstra [15]) can straightfor-
wardlybeextractedfromP,butisuselessforanalysis:Deciding
Categories and Subject Descriptors F.3.1 [Logics and Mean-
whichprogramstatessatisfythispreconditionisashardasdecid-
ingsofPrograms]:SpecifyingandVerifyingandReasoningabout
ingfromwhichstatesP runssafelyand/orterminates!Ontheother
Programs—Logicsofprograms,pre-andpost-conditions
hand,manypreconditions(e.g.⊥)maybecorrectbuttoostrongbe-
GeneralTerms Verification,theory causetheyruleouttheexecutionofsomeoftheprogram.Ideally,
wewouldliketofindtheweakestpreconditionthatisexpressible
Keywords abduction, shape analysis, cyclic proof, separation insidesomefairlynatural,“positive”classofinductivedefinitions
logic,inductivedefinitions,programverification andthatensuresmaximalprogramcoverage.Forcomputabilityrea-
sons,wecannothopetoobtainsuchapreconditioningeneral,so
1. Introduction wemustinsteadlookforreasonableapproximatingheuristics.
Our main contribution is a new method, cyclic abduction, for
Inthelastfewyears,anumberofprogramanalyseshaveappeared
inferring safety and/or termination preconditions for while pro-
thatemployseparationlogic[24]toestablishsafetyand/ortermi-
grams,expressedasinductivedefinitionsinseparationlogic.Our
nationpropertiesofheap-manipulatingprograms,insomecasesex-
approachisbaseduponheuristicproofsearchinaformalsystem
tendingtosubstantialcodebases(seee.g.[25,16,21,23]).Allof
ofcyclicproofs,adaptedfromthecyclicterminationproofsin[8].
these analyses rely on the use of inductive predicates to specify
Here,thecoreofacyclicproofisaderivationtreebuiltfrom
theshapeofdatastructuresstoredinmemoryandmanipulatedby
symbolic execution rules capturing the effect of program com-
theprogram,e.g.,listsortrees.However,suchpredicatesmusttyp-
mands(cf.[3]),andlogicalrulesthatmanipulatetheprecondition.
ically be hard-coded into the analysis (or in some cases must be
Cyclicproofsaresonamedbecausethisderivationtreeisallowed
providedbytheuser).Thismeansthattheanalysismusteitherfail
to contain back-links identifying leaves of the tree with arbitrary
orasktheuserforadvicewhenitencountersadatastructurenot
interior nodes, potentially creating cycles in the proof. Because
∗ResearchsupportedbyanEPSRCCareerAccelerationFellowship. such structures do not in general correspond to sound proofs, an
additional(decidable)globalsoundnessconditionmustbeimposed
†ResearchsupportedbyEPSRCgrantEP/H008373/1.
uponthesegraphstoqualifythemasgenuineproofs.Thedifference
betweencyclicproofsofmemorysafetyandthoseoftermination
isentirelyamatterofthechoiceofsoundnesscondition.
Given a program, our cyclic abduction procedure aims to si-
multaneouslyconstructtheinductivedefinitionofapreconditionin
separationlogic,underwhichthisprogramismemorysafeand/or
terminating.Broadlyspeaking,wesearchforacyclicproofthatthe
programhasthedesiredproperty,andwhentheproofsearchgets
stuck,weabduce(i.e.,guess)partofthepreconditioninorderto
[Copyrightnoticewillappearhereonce’preprint’optionisremoved.] proceed.Approximately,themainabductionprinciplesare:
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• symbolicallyexecutingbranchingcommandsinthederivation Aheapisapartialfunctionh : Loc (cid:42) (ValList)mapping
fin
leadstoconditionaldisjunctioninthedefinitions; finitelymanylocationstotuplesofvalues(i.e.records);wewrite
dom(h)forthedomainofheaph,i.e.thesetoflocationsonwhich
• symbolicallyexecutingdereferencingcommandsinthederiva-
hisdefined,andefortheemptyheapthatisundefinedeverywhere.
tionforcesustoincludepointerformulasinthedefinitions;
Ifh andh areheapswithdom(h )∩dom(h ) = ∅,wedefine
1 2 1 2
• formingback-linksinthederivationleadstotheinstantiationof h ◦h tobetheunionofh andh ;otherwise,h ◦h isundefined.
1 2 1 2 1 2
recursioninthedefinitions;and Wewrites[x(cid:55)→v]forthestackdefinedexactlyassexceptthat
• encounteringaloopintheprogramalertsustothepossibility (s[x(cid:55)→v])(x)=v,andadoptasimilarupdatenotationforheaps.
thatwemayneedtogeneralisetheprecondition. Weemployastandardsmall-stepoperationalsemanticsofour
programs (Fig. 1). A (program) state is either a triple (C,s,h)
Wehaveimplementedourabductionprocedureasanautomatic where C is a command sequence, s a stack and h a heap, or the
tool, CABER, that builds on the generic cyclic theorem prover specialstatefault,usedtocatchmemoryerrors.Givenaprogram
CYCLIST [10]. This tool essentially comprises a number of low- fields f ,...,f ; C, we map the field names f ,...,f onto
1 k 1 k
leveltacticsimplementingtheabductionprinciplesabove,heuristi-
elementsofheaprecordsbyf = j.Thesmall-stepsemantics
j def
callyguidedbyahigh-levelproofsearchstrategy.CABERisableto ofourprogramsisthenasusualgivenbyabinaryrelation(cid:32)on
automaticallyabducesafetyand/orterminationpreconditionsfora states,presentedinFigure1.Wewrite(cid:32)nforthen-stepvariantof
fairlywidevarietyofcommonsmallprograms.Theabducedinduc- (cid:32),and(cid:32)∗ foritsreflexive-transitiveclosure.Wesaythatastate
tivepredicatesincludedefinitionsofsegmented,cyclic,nestedand (C,s,h)issafeifthereisnocomputation(C,s,h)(cid:32)∗ fault,i.e.
mutuallydefineddatastructures(overanynumberofparameters).
if running the program from (C,s,h) cannot result in a memory
Theremainderofthispaperisstructuredasfollows.Section2
error.Wesaythat(C,s,h)isterminatingifitissafeandthereis
introducestheprogramminglanguageandthefragmentofsepara- noinfinite(cid:32)-computationstartingfrom(C,s,h).
tionlogicpreconditionsweusetoexpressprogrampreconditions.
Section 3 presents the formal system of cyclic safety/termination
Proposition 2.1 (Safety / termination monotonicity). If (C,s,h)
proofsonwhichourabductiontechniqueisbased.InSection4we
issafe(resp.terminating)andh◦h(cid:48)isdefinedthen(C,s,h◦h(cid:48))
presentanoverviewofthecyclicabductionstrategy,andthende-
isalsosafe(terminating).
scribeindetailthevariousabductivetacticsfromwhichitisbuilt.
Section5describestheimplementationofourcyclicabductiontool
CABERandourexperimentalevaluationofthistool.Section6ex- Essentially,Proposition2.1holdsforthesamereasonasin[26],
aminesrelatedworkandSection7concludes. i.e.,extendingthememorycannotleadtonewmemoryfaults.Itis
possiblethatextendingthememorymightleadtonon-termination
wherepreviouslytherewasamemoryfault,butthisisnotproblem-
2. Programsandpreconditions
aticsinceterminatingstatesarealsorequiredtobesafe.
Inthissectionwepresentabasiclanguageofwhileprogramswith
heappointers(similartothelanguagein[26])andthefragmentof
separationlogicweusedtoexpressprogrampreconditions,based
2.3 Syntaxoflogicalpreconditions
uponthesymbolicheapsof[3].
Weoftenusevectornotationtoabbreviatetuplesorlists,e.g.x We express program preconditions using a simple fragment of
for(x ,...,x ).Wewritex fortheithelementofthetuplex. separationlogicwithinductivedefinitions,basedonthesymbolic
1 k i
heapsof[3].Weassumeaninfinitesetofpredicatesymbols,each
2.1 Syntaxofprograms. withanassociatedarity.
We assume infinite sets Var of variables and of field names. An
expressioniseitheravariableortheconstantnil.Branchingcondi- Definition2.2. Formulasaregivenbythefollowinggrammar:
tionsB andcommandsequencesC aredefinedasfollows,where
x,yrangeoverVar,f overfieldnamesandEoverexpressions: F ::=(cid:62)|⊥|E =E |E (cid:54)=E |emp|x(cid:55)→E|P(E)|F ∗F
B ::= (cid:63)|E =E |E (cid:54)=E
C ::= (cid:15)|x:=E; C |y:=x.f; C |x.f :=E; C | wherexrangesoverVar,Erangesoverexpressions,P overpred-
free(x); C |x := new(); C | icatesymbolsandEovertuplesofexpressions(matchingthearity
ifBthenCelseCfi; C |whileBdoCod; C ofP inP(E)).WewriteF[E/x]fortheresultofreplacingalloc-
currencesofthevariablexbytheexpressionE intheformulaF.
wherey := x.f andx.f := E(cid:48) respectivelyreadfromandwrite Substitutionisextendedpointwisetotuples;butwhenwewritean
tofieldf oftheheapcellwithaddressx,and(cid:63)representsanon- expressionoftheformF[E/x ],wemeanthatEshouldbesubsti-
i
deterministic condition. A program is a list of the field names of tutedfortheithcomponentofxonly.
heaprecordsfollowedbyacommandsequence: We define ≡ to be the least equivalence on formulas closed
underassociativityandcommutativityof∗andF ∗emp≡F.
fieldsf ,...,f ; C
1 k
2.2 Semanticsofprograms Definition2.3. Aninductiverulesetisafinitesetofinductiverules
eachoftheformF ⇒P(E),whereF andP(E)areformulas.We
We use a typical RAM model employing heaps of records. We
fix a set Val of values, of which an infinite subset Loc ⊂ Val sometimeswriteaninductiveruleasF ⇒z P(E),wherezisatuple
are locations, i.e., the addresses of heap cells. We also assume a listingthesetofallvariablesappearinginF andE.
“nullary” value nil ∈ Val\Loc which is not the address of any If Φ is an inductive rule set we define ΦP to be the set of all
heapcell.Astackisafunctions:Var→Val.Thesemantics[[E]]s inductiverulesforP inΦ,i.e.thoseoftheformF ⇒ P(E).We
ofexpressionEinstacksisdefinedasusual:[[x]]s=def s(x)forall sayP isundefinedifΦP isempty.
x∈Var,and[[nil]]s= nil.Thesemantics[[B]]s⊆{true,false}
def
ofabranchingconditionBinstacksisdefinedintheobviousway InductiverulesforapredicateP areunderstoodasdisjunctive
forequalitiesanddisequalities,and[[(cid:63)]]s= {true,false}. clausesofthedefinitionofP (cf.[8]).
def
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2.4 Semanticsoflogicalpreconditions we include an unfolding (or “case analysis”) rule for inductively
definedpredicatesthatunfoldsaformulaP(E)intheconclusion
Theforcingrelations,h|= F forsatisfactionoftheformulaF
Φ accordingtothedefinitionofP inaninductivedefinitionsetΦ:
bythestacksandheaphunderinductiverulesetΦisdefinedby
s,h|= (cid:62) ⇔ always Example3.2. Defineaunaryinductivepredicatebt(x),denoting
Φ
s,h|= ⊥ ⇔ never thoseheapsstructuredasabinarytreewithheadpointerx,by
Φ
s,h|=Φ E1 =E2 ⇔ [[E1]]s=[[E2]]sandh=e x=nil⇒bt(x)
s,h|= E (cid:54)=E ⇔ [[E ]]s(cid:54)=[[E ]]sandh=e
Φ 1 2 1 2 x(cid:54)=nil∗x(cid:55)→(y,z)∗bt(y)∗bt(z)⇒bt(x)
s,h|= emp ⇔ h=e
Φ
s,h|= E (cid:55)→E ⇔ dom(h)={[[E]]s}andh([[E]]s)=[[E]]s Theunfoldingruleforbtisthefollowing:
Φ
s,h|=Φ P(E) ⇔ (h,[[E]]s)∈[[P]]Φ E =nil∗F (cid:96)C
s,h|=Φ F1∗F2 ⇔ h=h1◦h2ands,h1 |=Φ F1 E (cid:54)=nil∗E (cid:55)→(y(cid:48),z(cid:48))∗bt(y(cid:48))∗bt(z(cid:48))∗F (cid:96)C
ands,h2 |=Φ F2 (bt)
bt(E)∗F (cid:96)C
Inordertoremovetheneedforstandardconjunction∧inourlogic
fragment, it is convenient to interpret equalities and disequalities wherey(cid:48)andz(cid:48)arefreshvariables.
as holding in the empty heap. (This does not result in a loss of
Definition3.3(Pre-proof). Apre-proof ofF (cid:96)Cisapair(D,L),
expressivity,since,e.g.,arbitrarys,hsatisfyings(x) = s(y)can
whereDisafinitederivationtreewhoserootislabelledbyF (cid:96)C,
berepresentedasx=y∗(cid:62).
Thesemantics[[P]]ΦofthepredicateP underΦis,asusual,the and L is a back-link function assigning to every open leaf (cid:96) of
D a node L((cid:96)) of D such that the judgements at (cid:96) and L((cid:96)) are
leastprefixedpointofamonotoneoperatorconstructedfromΦ:
syntacticallyidentical.
Definition 2.4. Assume that Φ defines n predicates P1,...,Pn. Anypre-proofP =(D,L)canbeunderstoodasagraph,with
PartitionΦintoΦ1,...,Φn,whereΦi ⊆Φisthesetofinductive an edge from the conclusion of any rule instance to each of its
rulesoftheformF ⇒ Pix.WeleteachΦi beindexedbyj,and premises,byidentifyingeachopenleaf(cid:96)ofDwithL((cid:96)).Apath
foreachinductiveruleΦi,j oftheformF ⇒ Pix,wedefinethe inPisthenunderstoodintheobviousway.
operatorϕ by:
i,j Lemma3.4. LetΦbeaninductiveruleset.Supposetheconclu-
ϕi,j(X)=def {(s(x),h)|s,h|=X F} sion F (cid:96) C of an instance of a rule R is invalid w.r.t. Φ, so
that there exist a stack s and heap h such that s,h |= F but
where |= is the satisfaction relation defined above, except that Φ
[[P ]]X =X X ,whereX = (X ,...,X )isatupleofpairsof (C,s,h)(cid:32)n fault.
i def i 1 n Then there is a premise F(cid:48) (cid:96) C(cid:48) of this rule instance that
theappropriatetype.Wethendefine[[P]]Φby:
is invalid w.r.t. Φ, i.e. there exist stack s(cid:48) and heap h(cid:48) such that
[[P]]Φ =def µX.((cid:83)jϕ1,j(X),...,(cid:83)jϕn,j(X)) s(cid:48),h(cid:48) |=Φ F(cid:48),but(C(cid:48),s(cid:48),h(cid:48))(cid:32)m fault.Moreover,m≤n,andif
Risasymbolicexecutionrulethenm<n.
Wewrite[[P ]]Φfortheithcomponentof[[P]]Φ.
i
Proof. A straightforward verification for each proof rule in Fig-
Anextremelyusefulfactaboutthefragmentofseparationlogic
ure2.Thecaseof(Frame)reliesuponProposition2.1.
weconsideristhat,foranygiveninductiveruleset,itisdecidable
whetheraformulainthefragmentisconsistent,i.e.,whetherornot
Definition 3.5 (Cyclic proof). A pre-proof P is a cyclic (safety)
theformulaissemanticallyequivalentto⊥.Theproofofthisfact,
proof if for every infinite path in P, there are infinitely many
whichisveryhelpfulinsimplifyingabducedpreconditionsaswell
symbolicexecutionruleapplicationsalongthispath.
asassessingtheirquality,appearsin[9].
Theorem 3.6. For any inductive rule set Φ, if there is a cyclic
3. Formalcyclicsafety/terminationproofs safetyproofofF (cid:96)C,thenF (cid:96)Cisvalidw.r.tΦ.
Inthissectionwepresentaformalcyclicproofsystemforproving
Proof. Suppose for contradiction that F (cid:96) C has a cyclic safety
memory safety and/or termination of programs. The system here
proof P but is invalid. Using Lemma 3.4, we can construct an
is adapted from the system of cyclic termination proofs in [8],
infinite path (F (cid:96) C ) in P, and an an infinite sequence
withthefollowingmaindifferences:(i)wetreatwhileprograms k k k≥0
(n ) of natural numbers such that n < n whenever
rather than goto programs; and (ii) we are additionally able to k k≥0 k+1 k
F (cid:96) C istheconclusionofasymbolicexecutionruleinstance,
considermemorysafetyonlybyimposinganalternativesoundness k k
andn =n otherwise.
conditionontheproofgraph. k+1 k
SinceP isacyclicsafetyproof,thereareinfinitelymanysym-
AproofjudgementisgivenbyF (cid:96)C,whereC isacommand
bolic executions along the path (F (cid:96) C ) . This implies the
sequenceandF isaformula. k k k≥0
existenceofaninfinitedescendingchainofnaturalnumbers,which
Definition 3.1 (Validity). Let Φ be an inductive rule set. The isimpossible.HenceF (cid:96)Cmustbevalid.
judgement F (cid:96) C is valid (resp. termination-valid) w.r.t. Φ if
s,h|= F implies(C,s,h)issafe(resp.terminating). Wecanconsidercyclicproofsofterminationratherthanmem-
Φ
orysafetysimplybyreplacingthesoundnessconditionof3.5with
The proof rules for judgements are given in Fig. 2. Our nota-
the trace-based soundness condition of [8], which essentially de-
tionalconventionisthattheprimedvariablesx(cid:48),x(cid:48)(cid:48)etc.appearing
mands that some inductive predicate be unfolded infinitely often
inthepremisesofrulesarechosenfresh,andwewriteBtomean
alongeveryinfinitepathinthepre-proof.Thus,byasimpleadap-
E (cid:54)= E(cid:48) if B is (E = E(cid:48)), and vice versa. The symbolic ex-
tationofthesoundnessresultin[8],wealsohave:
ecution rules for commands are adaptations of standard rules for
separationlogic[3].Thelogicalrulesmanipulatetheprecondition Theorem 3.7. For any inductive rule set Φ, if there is a cyclic
withoutadvancingtheprogram.Inparticular,therule(Frame)can terminationproofofF (cid:96)C—i.e.,apre-proofofF (cid:96)Csatisfying
be seen as a special case of the general frame rule of separation thesoundnessconditionof[8]—thenF (cid:96)Cistermination-valid
logic(seee.g.[26]),wherethepostconditionisomitted.Crucially, w.r.tΦ.
3 2013/7/12
(x:=E; C,s,h)(cid:32)(C,s[x(cid:55)→[[E]]s],h)
[[y]]s∈dom(h) [[x]]s∈dom(h)
(x:=y.f; C,s,h)(cid:32)(C,s[x(cid:55)→h([[y]]s).f],h) (x.f :=E; C,s,h)(cid:32)(C,s,h[[[x]]s.f (cid:55)→[[E]]s])
(cid:96)∈Loc\dom(h) v ,...,v ∈Val [[x]]s∈dom(h)
1 k
(x:=new(); C,s,h)(cid:32)(C,s[x(cid:55)→(cid:96)],h[(cid:96)(cid:55)→(v ,...,v )] (free(x); C,s,h)(cid:32)(C,s,(h(cid:22)(dom(h)\{[[x]]s}))
1 k
[[y]]s(cid:54)∈dom(h) [[x]]s∈/ dom(h) [[x]]s∈/ dom(h)
(x:=y.f; C,s,h)(cid:32)fault (x.f :=E; C,s,h)(cid:32)fault (free(x); C,s,h)(cid:32)fault
true∈[[B]]s false∈[[B]]s
(ifBthenCelseC(cid:48)fi; C(cid:48)(cid:48),s,h)(cid:32)(C; C(cid:48)(cid:48),s,h) (ifBthenCelseC(cid:48)fi; C(cid:48)(cid:48),s,h)(cid:32)(C(cid:48); C(cid:48)(cid:48),s,h)
true∈[[B]]s false∈[[B]]s
(whileBdoCod; C(cid:48),s,h)(cid:32)(C; whileBdoCod; C(cid:48),s,h) (whileBdoCod; C(cid:48),s,h)(cid:32)(C(cid:48),s,h)
Figure1. Small-stepoperationalsemanticsofprograms,givenbythebinaryrelation(cid:32)overprogramstates.
Symbolicexecutionrules:
x=E[x(cid:48)/x]∗F[x(cid:48)/x](cid:96)C
F (cid:96)(cid:15) F (cid:96)x:=E; C
x=E [x(cid:48)/x]∗(y(cid:55)→E∗F)[x(cid:48)/x](cid:96)C x(cid:55)→E[E/E ]∗F (cid:96)C
f f
|E|≥f |E|≥f
y(cid:55)→E∗F (cid:96)x:=y.f; C x(cid:55)→E∗F (cid:96)x.f :=E; C
x(cid:55)→(x(cid:48),...,x(cid:48))∗F[x(cid:48)/x](cid:96)C F (cid:96)C
1 k
F (cid:96)x:=new(); C x(cid:55)→E∗F (cid:96)free(x); C
B∗F (cid:96)C; C(cid:48)(cid:48) B∗F (cid:96)C(cid:48); C(cid:48)(cid:48) F (cid:96)C; C(cid:48)(cid:48) F (cid:96)C(cid:48); C(cid:48)(cid:48)
B∗F (cid:96)ifBthenCelseC(cid:48)fi; C(cid:48)(cid:48) B∗F (cid:96)ifBthenCelseC(cid:48)fi; C(cid:48)(cid:48) F (cid:96)if(cid:63)thenCelseC(cid:48)fi; C(cid:48)(cid:48)
B∗F (cid:96)C; whileBdoCod; C(cid:48) B∗F (cid:96)C(cid:48) F (cid:96)C; whileBdoCod; C(cid:48) F (cid:96)C(cid:48)
B∗F (cid:96)whileBdoCod; C(cid:48) B∗F (cid:96)whileBdoCod; C(cid:48) F (cid:96)while(cid:63)doCod; C(cid:48)
Logicalrules:
F (cid:96)C F (cid:96)C xnotaprogram F(cid:48) (cid:96)C G(cid:48)∗F (cid:96)C
(Frame) (Subst) F ≡F(cid:48) (Equiv) G(cid:96)G(cid:48) (Cut)
F ∗G(cid:96)C F[E/x](cid:96)C variable F (cid:96)C G∗F (cid:96)C
(t =t ∗F)[t /x,t /y](cid:96)C
1 2 2 1 (=) ((cid:54)=) ((cid:55)→)
(t =t ∗F)[t /x,t /y](cid:96)C t1 =t2∗t1 (cid:54)=t2∗F (cid:96)C x(cid:55)→E∗x(cid:55)→E(cid:48)∗F (cid:96)C
1 2 1 2
Predicateunfoldingrule:
(E=Ej[xj/zj]∗Fj[xj/zj]∗F (cid:96)C)1≤j≤k Φ ={F ⇒z1 P(E ),...,F ⇒zk P(E )}
P 1 1 k k (P)
P(E)∗F (cid:96)C ∀xj ∈{xj}.xj isfresh
Figure2. Hoarelogicrulesforproofjudgements.
4 2013/7/12
4. Cyclicabduction:basicstrategy&tactics Principle3. Beforeapplyingsymbolicexecutiontoawhileloop,
oneshouldnormallyattempttogeneralisethepreconditionF ap-
Wenowturntothemaincontributionofthispaper:ourcyclicab-
pearingatthesubgoalinquestion.Thatistosay,weshouldattempt
duction method for inferring inductive safety and/or termination
tofindsomeweakerpreconditionF(cid:48) suchthatF(cid:48) (cid:96) F isavalid
preconditions of programs. Here, we first explain the high-level
entailment,and,byapplying(Cut),proceedwiththeproofsearch
strategyforabducingsuchpreconditions,andthendevelopanum-
usingthepreconditionF(cid:48) inplaceofF.Ifnecessary,wemayab-
berofautomatic,abductivetacticsusedinimplementingthisstrat-
duceinductiverulesinordertoobtainthismoregeneralF(cid:48).
egy.Then,inSection5,wedescribehowthesetacticsareimple-
mentedandcombinedintoanabductiveproofsearchalgorithm. Generalisation is well known to be a necessary (and difficult)
stepininductivetheoremproving[11],andunsurprisinglyitshows
4.1 Overviewofabductionstrategy up in our abduction proofs also. Section 4.6 presents tactics for
Thetypicalinitialproblemwearefacedwithis:givenaprogram generalisingthepreconditionwhenweencounterawhileloop.
withcodeC andinputvariablesx,findaninductivedefinitionset
4.2 Tactics:anoverview
ΦsuchthatthejudgementP(x)(cid:96)Cis(termination-)validwrt.Φ,
whereP isafreshpredicatesymbol. A tactic in our setting is a more general version of a tactic as it
OurstrategyforfindingsuchaΦistosearchforacyclicproof usually appears in automated theorem proving. Here, a tactic is
ofthejudgementP(x) (cid:96) C (withrespecttosafetyortermination essentiallyageneralprooftransformer:itupdatesthecurrentproof
as desired). Almost invariably, this search process will become statebyapplyingsome(possiblynondeterministic)combinationof
“stuck” at some point, e.g., because the precondition does not atomic inferences. The differences between our setting and that
containenoughinformationforasymbolicexecutionruletofire. of a traditional theorem prover reside in the underlying notions
When this happens, we may abduce one or more new inductive of“proofstate”and“atomicinference”.First,sinceweemploya
rulestoenablethesearchtoproceed.Inthefollowing,wesetout cyclic notion of proof, with back-links joining leaves to arbitrary
informallythemainprinciplesgoverningthisprocess. proof nodes, it is not sufficient to restrict our attention to the
current subgoal only: our proof state must reflect the entire pre-
Principle1. Thefirstpriorityofthesearchprocedureistoclose
proof,andformingaback-linkbetweennodesmustcountasavalid
the current branch of the derivation tree, preferably by applying
atomicinferencestep.Second,sinceweareallowedtoabducenew
an axiom, or else by forming a back-link to some other node.
inductive rules in the proof search, the current inductive rule set
(The formation of back-links must respect the relevant soundness
mustformpartoftheproofstate,andaddingnewinductiverulesto
conditiononcyclicproofs.)
thissetmustalsocountasavalidinferencestep.(Asusual,allproof
If closing the branch is not possible, the second priority is to
rulesofoursystem,inFigure2,arealsovalidatomicinferences.)
applythesymbolicexecutionruleforthecommandappearingat
Takingtheaboveintoaccount,ourformalproofstatesaretriples
thecurrentsubgoal.
comprisedofthefollowingelements:
That is, if we can close the current branch somehow then we
P: A partial pre-proof, representing the portion of proof con-
doso,otherwisewetrytoadvancetheprogramalongthisbranch
structedsofar.By“partial”,wemeanthatsomeoftheleaves
via symbolic execution. (Note that when we attempt to form a
ofPmaybeopen;wecallthesetheopensubgoalsofP.
back-linkintheproof,violationsofthesoundnessconditioncanbe
detectedautomaticallybyappealingtoamodelchecker.)However, Φ: Aninductiveruleset,containingalltheinductiverulesabduced
itoftenhappensthatneitheroftheseispossible.Whenthissituation sofarintheproofsearch.
occurs, we are allowed to use the logical rules and/or to abduce (cid:96): Adistinguishedopenleafofthepre-proofP,representingthe
inductiverulesinordertoenabletheproofsearchtomakeprogress, subgoalonwhichthenextstepoftheproofsearchistooperate.
asdescribedbyoursecondprinciple.
Atacticisthensimplyatransformeronproofstates.
Principle2. Wemayabduceinductiverulesand/ordeploythelog-
Example 4.1. In order to demonstrate how our abductive tactics
icalandpredicateunfoldingrulesinthefollowingcircumstances:
are applied in practice, we will frequently refer to the following
(a) inordertosymbolicallyexecuteabranchingcommand; runningexampleintheremainderofthissection.Figure3shows
(b) inordertosymbolicallyexecuteadereferencingcommand; anabductivecyclicproofofthefollowingprogram:
(c) inordertoformaback-link;
fieldsl,r;
(d) aspartofageneralisationattempt. whilex(cid:54)=nildo
if(cid:63)then x := x.l else x := x.r fi
When we abduce inductive rules for a predicate occurring in the
od
current subgoal,we alwaysimmediately apply the unfolding rule
forthatpredicatetothesubgoal. TheproofinFigure3abducesthebinarytreepredicatefromExam-
Wemayabduceinductiverulesonlyforpredicatesymbolsthat ple3.2asasafetyandterminationpreconditionforthisprogram.
arecurrentlyundefined. Foreaseofpresentation,wenumbereachindividualcommand(se-
quence)intheprogram,andwewritethejudgementsintheproof
We explain in the following subsections exactly how induc-
byreferringtotheseindicesratherthanthecommandsthemselves.
tive rules are abduced in the situations (a), (b), and (c) described
by Principle 2 (generalisation (d) being covered in Section 4.6).
4.3 Abductivetacticfordeterministicbranchingcommands
Thereasonforrestrictingabductiontoundefinedpredicatesisthat
adding inductive rules to Φ adds new premises to the unfold- Ourproofrulesfordeterministicifandwhilecommands(Fig.2)
P
ing rule (P), rendering any existing applications of (P) in the mirrortheoperationalsemanticsinthattheyrequiretheprecondi-
derivation unsound. We will indicate a combined abduction-and- tiontodeterminethestatusofthebranchingcondition.Weintro-
unfoldingstepinourderivationswithA(P). duceanabductivetactic,abduce branch,enablingustoproceed
Ourfinalprinciplesetsoutthegeneralideabehindperforming wheneverthesymbolicexecutionofsucharulefails.
generalisationinaproofsearch. Supposeabduce branchisappliedtotheproofstate(P,Φ,(cid:96))
where the command sequence C in the judgement appearing at
5 2013/7/12
P (x)(cid:96)0
0
(Frame)
x(cid:48) (cid:54)=nil∗
(cid:96)0
P0(x)(cid:96)0 x(cid:48) (cid:55)→(y,x)∗P0(y)∗P0(x)∗P5(x(cid:48),y,x)
x(cid:48) (cid:54)=nil∗ (Frame) x(cid:48) (cid:54)=nil∗ A(P4)
(cid:96)0 (cid:96)0
x(cid:48) (cid:55)→(x,z)∗P (x)∗P (x(cid:48),x,z) x(cid:48) (cid:55)→(y,x)∗P (y)∗P (x(cid:48),y,x)
0 4 0 4
x(cid:48) (cid:54)=nil∗ A(P3) x(cid:48) (cid:54)=nil∗ (P3)
(cid:96)0 (cid:96)0
x(cid:48) (cid:55)→(x,z)∗P (x(cid:48),x,z) x(cid:48) (cid:55)→(y,x)∗P (x(cid:48),y,x)
3 3
x := x.l x := x.r
x(cid:54)=nil∗ x(cid:54)=nil∗
(cid:96)2 (cid:96)3
x(cid:55)→(y,z)∗P (x,y,z) x(cid:55)→(y,z)∗P (x,y,z)
3 3
A(P2) (P2)
x(cid:54)=nil∗P (x)(cid:96)2 x(cid:54)=nil∗P (x)(cid:96)3
2 2
(cid:15) if(cid:63)...
x=nil∗P (x)(cid:96)(cid:15) x(cid:54)=nil∗P (x)(cid:96)1
1 2
whilex(cid:54)=nil whilex(cid:54)=nil
x=nil∗P (x)(cid:96)0 x(cid:54)=nil∗P (x)(cid:96)0
1 2
A(P0)
P (x)(cid:96)0
0
fieldsl,r; x=nil∗P (x)⇒P (x)
1 0
0: whilex(cid:54)=nildo
x(cid:54)=nil∗P (x)⇒P (x)
1: if (cid:63) then 2 0 x=nil∗emp⇒P (x)
0
2: x := x.l x(cid:55)→(y,z)∗P (x,y,z)⇒P (x)
3 2
x(cid:54)=nil∗x(cid:55)→(y,z)∗P (y)∗P (z)⇒P (x)
else 0 0 0
P (y)∗P (x,y,z)⇒P (x,y,z)
3: x := x.r 0 4 3
fi od P0(z)∗P5(x,y,z)⇒P4(x,y,z)
Figure3. Top:abductiveproofforabinarytreesearchprogram(shownbottomleft).Bottomcenter:inductiverulesabducedduringthe
proof.Bottomright:simplifiedinductiverules.
the current subgoal (cid:96) is of the form whileBdoCod;C(cid:48) or inductiverulesforP :
0
ifBthenCelseC(cid:48)fi;C(cid:48)(cid:48) (where B (cid:54)= (cid:63)). For simplicity we
x=nil∗P (x) ⇒ P (x)
assumeBisanequalityordisequalitybetweentwoprogramvari- 1 0
ablesx,y(thecasewhereoneofthetwotermsisnilisverysimi- x(cid:54)=nil∗P2(x) ⇒ P0(x)
lar).First,abduce branchselectsasubformulaoftheformP(E)
ThetacticthenunfoldsP (x)andappliessymbolicexecutionfor
appearingin(cid:96)suchthatthepredicatesymbolP iscurrentlyunde- 0
thewhilecommandinthetworesultingsubgoals.Thisisshownin
finedinΦ,andsuchthatxandy occurinthetupleE.Thus,we
Fig.3astheapplicationofA(P )followedbythenormalsymbolic
maywritethejudgementappearingat(cid:96)asF ∗P(E) (cid:96) C where 0
executionsforwhile).
x = E andy = E (wherek (cid:54)= j).Then,abduce branchadds
k j
thefollowinginductiverulesforP toΦ:
4.4 Abductivetacticfordereferencingassignments
B[z /x,z /y]∗P(cid:48)(z) ⇒ P(z)
k (cid:96) The symbolic execution rules for commands that dereference a
B[z /x,z /y]∗P(cid:48)(cid:48)(z) ⇒ P(z)
k (cid:96) memory address (Fig. 2) require the precondition to guarantee
where P(cid:48),P(cid:48)(cid:48) are fresh predicate symbols and z is a tuple of that this address is indeed allocated (otherwise the program will
appropriatelymanyarbitraryvariables.Next,thetacticunfoldsthe encounter a memory fault, as per the operational semantics in
indicatedoccurrenceofP(E)in(cid:96)asfollows: Fig.1).Herewepresentanabductivetactic,abduce deref,that
enablesthesymbolicexecutionofsuchcommandstoproceedby
B∗F ∗P(cid:48)(E)(cid:96)C B∗F ∗P(cid:48)(cid:48)(E)(cid:96)C abducingtheallocationoftheappropriateaddress.
A(P)
Formally,supposeabduce derefisappliedtotheproofstate
F ∗P(E)(cid:96)C
(P,Φ,(cid:96)), where the command C in the judgement labelling the
Finally,thetacticappliestheappropriatesymbolicexecutionrule current subgoal (cid:96) is of the form free(x); C, x.f := E; C or
for C to each of the new subgoals (note that this step is now y :=x.f; C.First,abduce derefnon-deterministicallyselectsa
guaranteedtosucceed). subformulaoftheformP(E)appearingat(cid:96),whereP iscurrently
The choice of subformula P(E) and indices k,j is nondeter- undefined in Φ, and such that x occurs in the tuple E. Thus, we
ministic;abduce branchreturnstheresultsofallsuchchoices. maywritethejudgementappearingat(cid:96)asF ∗P(E) (cid:96) C where
x = E (for some k). Then abduce branch adds the following
Example4.2. WhensearchingforaproofoftheprograminEx- k
inductiveruleforP toΦ:
ample4.1,ourinitialgoalisoftheform
P0(x)(cid:96)whilex(cid:54)=nildo... P(cid:48)(x(cid:116)y)∗xk (cid:55)→y⇒P(x)
where P is undefined. As the symbolic execution of the while where (cid:116) is tuple concatenation, P(cid:48) is a fresh predicate symbol,
0
commandfails,wecancallabduce branchwhichselectsP (x) and x and y are tuples of distinct, arbitrary variables such that
0
astheonlyviablesubformulaofthegoalandabducesthefollowing |x| = |E|, and |y| is the number of fields in the program. The
6 2013/7/12
tacticthenunfoldstheselectedoccurrenceofP(E)in(cid:96)asfollows: ofthefollowingform:
F ∗x(cid:55)→y(cid:48)∗P(cid:48)(E(cid:116)y(cid:48))(cid:96)C F(cid:48)∗P(cid:48)(z)⇒P(z)
A(P)
F ∗P(E)(cid:96)C whereP(cid:48)isafreshpredicate,andF(cid:48)ischosensoastosatisfy
wherey(cid:48) isatupleof|y|freshvariables.Finally,abduce deref F2[θ]⊆multiset F1∗F(cid:48)[E/z]
updates P again by applying the symbolic execution rule for C forsomesubstitutionθ ofexpressionsfornon-programvariables
(whichisnowguaranteedtosucceed). only,whereweviewspatialformulasas∗-separatedmultisets.Pro-
Similarly to abduce branch in the previous subsection, the vidingwecanfindsuitableF(cid:48)—whichisessentiallyaunification
selection of subformula P(E) and index k is essentially non- problem—abduce backlinktransformsP byapplyingrulesto
deterministic, so abduce deref returns the list of proof states (cid:96)andinsertinganewback-linkasfollows:
correspondingtotheresultsofallsuchselections.
. F2 (cid:96)C
Example 4.3. When searching for an abductive proof of the bi- .. (Subst)
nary tree traversal program in Example 4.1, after applying the F2[θ](cid:96)C
abduce branch tactic we arrive at the following subgoal on the F2 (cid:96)C F ∗F(cid:48)[E/z]∗P(cid:48)(E)(cid:96)C (Frame)
middlebranch(seeFigure3): 1
. A(P)
.
x(cid:54)=nil∗P2(x)(cid:96)x := x.l;... . F1∗P(E)(cid:96)C
where P is undefined (at the current proof state). We cannot TheroleofthefreshpredicateP(cid:48)istoallowapartoftheheaptobe
2
applythesymbolicexecutionruleforthedereferencingassignment abducedatalaterstage,possiblyonacompletelydifferentbranch
x := x.l,asitwouldrequireaformulaoftheformx (cid:55)→ (E,E(cid:48)) oftheproof(seethesecondpartofExample4.4).
toappearintheprecondition.Thuswecallabduce deref,which As with our other abductive tactics, abduce backlink acts
selects the only suitable formula P (x) in the precondition and nondeterministically: there may be several viable choices of un-
abducesthefollowinginductiverule2forP : definedpredicateP,“matchingformula”F(cid:48)andsubstitutionθ.
2
Two further caveats apply. First, according to Principle 1, we
x(cid:55)→(y,z)∗P3(x,y,z)⇒P2(x). must ensure that the proposed back-link does not violate the rel-
evant soundness condition on cyclic proofs. We verify this is the
ThetacticthenunfoldsP (x)andappliesthesymbolicexecution
2 casebycallingamodelcheckerwhenabduce backlinkattempts
ruleforthecommandx := x.lasshowninFigure3.
toformaback-link.Second,abduce backlinkisalsoallowedto
Asimilarsituationoccurswhenwereachthesimilarsubgoalon
tryunfoldingadefinedpredicateinthesubgoal(cid:96)ifnoundefined
therighthandbranch:
predicatesareavailable(similarlytoabduce deref).
x(cid:54)=nil∗P (x)(cid:96)x := x.r
2 Example4.4. InExample4.1,uponreachingthesubgoal
However, there is one crucial difference: having already abduced
x(cid:48) (cid:54)=nil∗x(cid:48) (cid:55)→(x,z)∗P (x(cid:48),x,z)(cid:96)0
a definition for P when tackling the left-hand subgoal, it is no 3
2
longer undefined. In such situations, when no suitable undefined on the middle branch of the proof in Fig. 3, we notice that the
predicateisavailable,abduce derefattemptstoapplysymbolic whilecommandinthesubgoalmatchesthatinearliernodes,sowe
execution for x := x.r by first unfolding a previously defined decidetotrytoformaback-link.Thuswecallabduce backlink,
predicateinstance,inthiscaseP2(x).(Calcagnoetal.[12]employ whichselectstheonlysuitableformulaP3(x(cid:48),x,z)inthesubgoal
asimilarabductiontacticfortheirfixedinductivepredicates.) andabducesthefollowinginductiveruleforP :
3
4.5 Abductivetacticforformingback-links P0(y)∗P4(x,y,z)⇒P3(x,y,z)
In principle, we may attempt to form a back-link from an open ThetacticthenunfoldsP3(x(cid:48),x,z),applies(Frame)andformsa
subgoallabelledbyF (cid:96) C toanyothernodeinthecurrentpre- back-linkfromtheresultingsubgoaltotherootnodeoftheproof
prooflabelledbyajudgementoftheformF(cid:48) (cid:96)C,providedthat: asshowninFig.3.Notethat,inthiscase,theuseofsubstitutionis
notrequiredandthechosen“matchingformula”F(cid:48)isP (y).
1. the judgement F (cid:96) C is derivable from F(cid:48) (cid:96) C, so that the 0
Whenwearriveatasimilarsubgoalontherightmostbranch,
source and target of the back-link can be made syntactically
identicalasrequiredbyDefinition3.3;and x(cid:48) (cid:54)=nil∗x(cid:48) (cid:55)→(y,x)∗P3(x(cid:48),y,x)(cid:96)0
2. the addition of this back-link does not create an infinite path thesituationisslightlydifferentbecauseP isnolongerundefined.
3
violating the soundness condition on cyclic proofs (for safety In this situation, abduce backlink first unfolds P according
3
ortermination,asappropriate). to its existing definition, and then proceeds in the usual way by
abducingasuitableinductiverulefortheundefinedP .(Itisforthis
Herewepresentatactic,abduce backlink,thatattemptstoform 4
reasonthatitwasessentialtoabducethefreshpredicatecomponent
suchback-linksautomaticallyduringtheproofsearch.
P (x,y,z)ofthepreviousinductiveruleforP ,eventhoughon
Formally, suppose that abduce backlink is applied to the 4 3
themiddlebranchthiscomponentwasdiscardedby(Frame).)
proofstate(P,Φ,(cid:96)).First,thetacticselectsanodenofP,distinct
from(cid:96),suchthatthecommandsinthejudgementslabellingnand Weobservethatabduce backlinkis“forgetful”inthatituses
(cid:96)areidentical.Thenittriestomanipulate(cid:96)usinglogicalrulesso theframeruletodiscardpartsoftheprecondition.Anotheralter-
astoobtainapreconditionidenticaltotheoneatn.Moreprecisely, native would instead be to use the (Cut) rule to establish logical
foranypredicateP in(cid:96)thatisundefinedinΦ,abduce backlink entailmentsenablingthecurrentsubgoaltobeidentifiedwiththe
attemptstoabduceinductiverulesforP suchthatafterunfolding desired target node. In some cases, such reasoning is necessary
P, the logical rules (Frame) and (Subst) can be used to obtain a in order to abduce the “ideal” predicate as precondition. We im-
copyofn. plementedanalternativeback-linkingtacticthatattemptstoapply
Wewrite(cid:96)asF ∗P(E)(cid:96)C,whereP isundefinedinΦ,andn (Cut) in order to unify the current subgoal with a chosen target
1
asF (cid:96)C.Thenabduce backlinkwillabduceaninductiverule node.Theresultinglogicalentailmentsarepassedtotheseparation
2
7 2013/7/12
logicinstantiationofCYCLIST(describedin[10])whichattempts Ourtacticseg genidentifiesyasapotentialcursorvariable,since
todischargethem.Unfortunately,becausesuchentailmentsmight itismodifiedbytheloopbodyandappearsintheloopguard,and
beinvalid,andoftenrequireinductivereasoningtoestablisheven abducesthefollowinginductiverulefortheundefinedP :
0
inthecasethattheyarevalid,suchcallstoCYCLISTareextremely P (x,x)∗P (x,nil)⇒P (x)
expensive(manysuchcallsmightbemadeduringaproofattempt 1 1 0
andanyofthemmighttime-out).Thus,atthetimeofwriting,we ThetacticthenunfoldsP0(x)andrewritesusingtheequalityx=y
havefoundthecomputationalcostofthistactictobeprohibitive. asshownbytheapplicationofSEG-GENinFig.5.Thisenablesus
tolaterformaback-linkwiththemoregeneralP (x,y)∗P (y,nil)
1 1
4.6 Existentialgeneralisation as the precondition which describes the general invariant of the
while loop (i.e., a list segment from x to y together with a list
Symbolically executing while loops creates a potentially infinite
segmentfromy tonil).However,thisrequiresustoestablishthe
branchoftheproofsearch,unlessitcanbeclosedeitherbyanax-
entailment P (x,y(cid:48)) ∗ y(cid:48) (cid:55)→ y (cid:96) P (x,y), as shown by the
iomor,morecommonly,byformingaback-link.However,naive 1 1
applicationof(Cut)onbranch,whichrequiresinductivereasoning.
attemptstoback-linktoaspecifiedtargetjudgementoftenfailbe-
InordertofindtheproofinFig.5automaticallyinreasonabletime,
cause the judgement specifies a too-precise relationship between
werequireabetterlemmaspeculationmechanism,asdiscussedin
programvariableswhichisnotpreservedbythebodyoftheloop.
theprevioussubsection.
Onepossiblesolution,whichistypicalofinductivetheoremprov-
ingingeneral,istogeneralisethepreconditionuponencountering 4.7 Simplificationofinductiverulesets
thewhileloopsoasto“forget”suchvariablerelationships.Here
Whenanabductiveproofsearchusingthetacticsabovesucceeds,
wepresentatactic,ex gen,thatimplementsthisprinciple.
the returned set of abduced inductive rules will typically be too
Formally,supposeex genisappliedtotheproofstate(P,Φ,(cid:96)),
large and unwieldy for human consumption. We can apply some
wherethejudgementlabellingcurrentsubgoal(cid:96)isoftheform
fairlystraightforwardsimplificationstoourabducedrulesetsinor-
F (cid:96)whileB doC od; C(cid:48) dertomakethemmoreeasilyreadable.Oursimplificationmethod,
outlined below, is used to produce the simplified inductive rules
ThenforeveryprogramvariablexmodifiedbytheloopbodyC,
showninFigures3–5.
ex gen replaces every occurrence of x in a subformula of F of
the form E = E(cid:48) or y (cid:55)→ E by a fresh (implicitly existentially
1. Elimination of undefined predicates. When our search tactics
quantified)variablew.(Thisstepimplicitlyusesanapplicationof
abduce inductive rules, these rules always contain an undefined
(Cut),andiseasilyseentobesound.)Thistacticalsoexhibitsnon-
componentinvolvingafreshpredicatesymbol.Thus,thefinalset
determinisminthatitmaygeneraliseoveranysubsetofvariables
ofabducedinductiveruleswilltypicallycontainseveralundefined
modifiedbytheloopbody,andpresentintheprecondition.
predicates.Clearly,suchpredicatesmaybeinterpretedhoweverwe
Example 4.5. Figure 4 shows an abductive cyclic proof of the wish;thusweinterpretthemallastheemptyheapemp.
followingprogram,whichtraversesalistoflists: For example, during the abductive proof in Figure 3 we ab-
ducetheinductiveruleP (z)∗P (x,y,z) ⇒ P (x,y,z),where
fieldsnext,down; 0 5 4
P is still undefined at the end of the proof. Thus we instantiate
whilex(cid:54)=nildo 5
P (x,y,z)toempwhereby,takingintoaccountthatF∗emp≡F,
z := x.down; 5
weobtainthesimplifiedruleP (z)⇒P (x,y,z).
whilez(cid:54)=nildo z := z.down od; 0 4
x := x.next
2. In-lining. In most cases, our abductive tactics abduce a single
od
inductive rule for an undefined predicate. Consequently, the final
Whentheproofsearchforthisprogramreachesthesubgoal inductiverulesettypicallycontainsmanymutuallyrecursivepred-
x(cid:54)=nil∗x(cid:55)→(y,z)∗P (x,y,z)(cid:96) icates,manyofwhichcanbestraightforwardlyeliminatingbyin-
3
whilez(cid:54)=nildo z := z.down od lining.First,weorderthepredicatesymbolsbytheorderinwhich
their definitions were abduced during the proof search. Then, for
we call the tactic ex gen. Since z is modified by the body of anypredicateQwithonlyasingleinductiverule,wheneverQ>P
the while loop, and the precondition contains the subformula accordingtoourordering,wereplaceallsubformulasoftheform
x(cid:55)→(y,z), the tactic replaces the z in this subformula by the Q(E)intheinductiverulesforP bythedefinitionofQ(E).
freshvariablew,asshownintheapplicationofEX-GENinFig.4. Forexample,intheabductiveproofinFigure3,thepredicate
Observethatthisgeneralisationstepisneededinordertoformthe P isdefinedbyasingleinductiverule,
2
back-linkontheright-handbranch(asx (cid:55)→ (y,z)doesnothold
afterexecutionoftheloopbody,butx(cid:55)→(y,w)does). x(cid:55)→(y,z)∗P3(x,y,z)⇒P2(x)
andP > P accordingtotheabductionorder.Thus,wereplace
Other, more complex types of generalisation are also possible 2 0
P (x)byitsdefinitioninthesecondinductiveruleforP toobtain
(and indeed might be needed for some proofs). For example, we 2 0
thein-linedinductiverule
implemented a “segmenting” generalisation tactic, seg gen, that
identifiesa“cursor”variableintheloopbodyandabducesinduc- x(cid:54)=nil∗x(cid:55)→(y,z)∗P (x,y,z)⇒P (x)
3 0
tiverulesforanundefinedpredicateintheprecondition,basedon
RepeatingthisprocessforthepredicatesP andP producesthe
forming segments from this predicate with this cursor as a mid- 3 4
inductiveruleshowninFigure3,
point.Figure5showsanapplicationofthistacticinthecontextof
anabductivecyclicproofofthefollowingprogram,whichtraverses x(cid:54)=nil∗x(cid:55)→(y,z)∗P (y)∗P (z)⇒P (x).
0 0 0
alistfromyafterinitiallyassigningxtoy:
3.Eliminationofredundantparameters.Aparameterofapredicate
fieldsnext; P issaidtoberedundant(intheinductiverulesetΦ)ifitdoesnot
y := x; whiley(cid:54)=nildo y := y.next od occur on the left hand side of any inductive rule in Φ , except
P
possiblyasthesameparameterofP itself.Ourfinalsimplification
Aftersymbolicallyexecutingy := xwearriveatthesubgoal:
stepistoremoveanysuchredundantparametersfromtheinductive
x=y∗P (x)(cid:96)whiley(cid:54)=nildoy := y.nextod ruleset.
0
8 2013/7/12
x(cid:54)=nil∗x(cid:55)→(y,w)∗P3(x,y,z)(cid:96)2
(Frame)
x(cid:54)=nil∗z(cid:48)(cid:54)=nil∗
P0(x)(cid:96)0 x(cid:55)→(y,w)∗z(cid:48)(cid:55)→(v,z)∗P3(x,y,z)∗P8(x,y,z(cid:48),v,z) (cid:96)2
x(cid:48)(cid:54)=nil∗z=nil∗ (Frame) x(cid:54)=nil∗z(cid:48)(cid:54)=nil∗ A(P7)
(cid:96)0 (cid:96)2
x(cid:48)(cid:55)→(x,w)∗P0(x)∗P6(x(cid:48),x,z) x(cid:55)→(y,w)∗z(cid:48)(cid:55)→(v,z)∗P7(x,y,z(cid:48),v,z)
x(cid:48)(cid:54)=nil∗z=nil∗ A(P4) x(cid:54)=nil∗z(cid:54)=nil∗ z := z.down
(cid:96)0 (cid:96)3
x(cid:48)(cid:55)→(x,w)∗P4(x(cid:48),x,z) x(cid:55)→(y,w)∗z(cid:55)→(v,v(cid:48))∗P7(x,y,z,v,v(cid:48))
x := x.next A(P5)
x(cid:54)=nil∗z=nil∗ x(cid:54)=nil∗z(cid:54)=nil∗
(cid:96)4 (cid:96)3
x(cid:55)→(y,w)∗P4(x,y,z) x(cid:55)→(y,w)∗P5(x,y,z)
whilez(cid:54)=nil whilez(cid:54)=nil
x(cid:54)=nil∗z=nil∗ x(cid:54)=nil∗z(cid:54)=nil∗
(cid:96)2 (cid:96)2
x(cid:55)→(y,w)∗P4(x,y,z) x(cid:55)→(y,w)∗P5(x,y,z)
A(P3)
x(cid:54)=nil∗x(cid:55)→(y,w)∗P3(x,y,z)(cid:96)2
EX-GEN
x(cid:54)=nil∗x(cid:55)→(y,z)∗P3(x,y,z)(cid:96)2
z := x.down
x(cid:54)=nil∗x(cid:55)→(y,y(cid:48))∗P3(x,y,y(cid:48))(cid:96)1
(cid:15) A(P2)
x=nil∗P1(x)(cid:96)(cid:15) x(cid:54)=nil∗P2(x)(cid:96)1
whilex(cid:54)=nil whilex(cid:54)=nil
x=nil∗P1(x)(cid:96)0 x(cid:54)=nil∗P2(x)(cid:96)0
A(P0)
P0(x)(cid:96)0
x=nil∗P1(x)⇒P0(x)
x(cid:54)=nil∗P2(x)⇒P0(x)
fieldsnext,down
0: whilex(cid:54)=nildo x(cid:55)→(y,y(cid:48))∗P3(x,y,y(cid:48))⇒P2(x) x=nil⇒P0(x)
1: z := x.down; z=nil∗P4(x,y,z)⇒P3(x,y,z) x(cid:54)=nil∗x(cid:55)→(y,y(cid:48))∗P3(y,y(cid:48))⇒P0(x)
2: whilez(cid:54)=nildo
3: z := z.down od; z(cid:54)=nil∗P5(x,y,z)⇒P3(x,y,z) z=nil∗P0(y)⇒P3(y,z)
4: x := x.next P0(y)∗P6(x,y,z)⇒P4(x,y,z) z(cid:54)=nil∗z(cid:55)→(v,v(cid:48))∗P3(y,v(cid:48))⇒P3(y,z)
od
z(cid:55)→(v,v(cid:48))∗P7(x,y,z,v,v(cid:48))⇒P5(x,y,z)
P3(x,y,w)∗P8(x,y,z,v,w)⇒P7(x,y,z,v,w)
Figure4. Top:abductiveproofforlist-of-liststraversal.Bottom,lefttoright:program;predicatesfound;simplifiedpredicates.
P (x,y)∗P (y,nil)(cid:96)1
1 1
(Cut)
y(cid:48) (cid:54)=nil∗P (x,y(cid:48))∗y(cid:48) (cid:55)→y∗P (y,nil)∗P (x,y,z)(cid:96)1
1 1 5
y(cid:48) (cid:54)=nil∗P (x,y(cid:48))∗y(cid:48) (cid:55)→y∗P (y(cid:48),nil,y)(cid:96)1 A(P4)
1 4
y := y.next
y(cid:54)=nil∗P (x,y)∗y(cid:55)→z∗P (y,nil,z)(cid:96)2
1 4
(cid:15) A(P3)
y=nil∗P (x,y)∗P (y,nil)(cid:96)(cid:15) y(cid:54)=nil∗P (x,y)∗P (y,nil)(cid:96)2
1 2 1 3
whiley(cid:54)=nil whiley(cid:54)=nil
y=nil∗P (x,y)∗P (y,nil)(cid:96)1 y(cid:54)=nil∗P (x,y)∗P (y,nil)(cid:96)1
1 2 1 3
A(P1)
P (x,y)∗P (y,nil)(cid:96)1
1 1
SEG-GEN
x=y ∗P (x)(cid:96)1
0
y := x
P (x)(cid:96)0
0
P (x,x)∗P (x,nil)⇒P (x)
1 1 0
fieldsnext;
x=y∗P (x,y)⇒P (x,y) P (x,x)∗P (x,nil)⇒P (x)
0: y := x; 2 1 1 1 0
1: whiley(cid:54)=nildo x(cid:54)=y∗P (x,y)⇒P (x,y) x=y ∗emp⇒P (x,y)
3 1 1
2: y := y.next
x(cid:55)→z∗P (x,y,z)⇒P (x,y) x(cid:54)=y ∗x(cid:55)→z∗P (z,y)⇒P (x,y)
od 4 3 1 1
P (z,y)∗P (x,y,z)⇒P (x,y,z)
1 5 4
Figure5. Top:abductiveproofforlisttraversalprogram(bottomleft)employingsegmentgeneralisation.Bottomcentre:inductiverules
abducedduringtheproof.Bottomright:thesimplifiedpredicates.
9 2013/7/12