Automated Verification of Concurrent Linked Lists with Counters Tuba Yavuz-Kahveci and Tevfik Bultan Department of Computer Science University of California, Santa Barbara

General Problem Concurrent programming is difficult and error prone –Sequential programming: states of the variables –Concurrent programming: states of the variables and the processes Linked list manipulation is difficult and error prone –States of the heap: possibly infinite We would like to guarantee properties of a concurrent linked list implementation

More Specific Problem There has been work on verification of concurrent systems with integer variables (and linear constraints) –[Bultan, Gerber and Pugh, TOPLAS 99] –[Delzanno and Podelski STTT01] –Use widening based on earlier work of [Cousot and Halbwachs POPL 77] on analyzing programs with integer variables There has been work on verification of (concurrent) linked lists –[Yahav POPL’01] What can we do for concurrent systems: –where both integer and heap variables influence the control flow, –or the properties we wish to verify involve both integer and heap variables?

Our Approach Use symbolic verification techniques –Use polyhedra to represent the states of the integer variables –Use BDDs to represent the states of the boolean and enumerated variables –Use shape graphs to represent the states of the heap –Use composite representation to combine them Use forward-fixpoint computations to compute reachable states –Truncated fixpoint computations can be used to detect errors –Over-approximation techniques can be used to prove properties Polyhedra widening Summarization in shape graphs

Action Language Tool Set Action Language Specification of the Concurrency Component Action Language Parser Verifier Code Generator OmegaLibraryCUDDPackage Verified code (Java monitor classes) MONA Composite Symbolic Library

Outline Specification of concurrent linked lists –Action Language Symbolic verification –Composite representation Approximation techniques –Summarization –Widening Counting abstraction Experimental results Related Work Conclusions

Action Language [Bultan ICSE00] [Yavuz-Kahveci, Bultan ASE01] A state based language –Actions correspond to state changes States correspond to valuations of variables –Integer (possibly unbounded), heap, boolean and enumerated variables –Parameterized constants are allowed Transition relation is defined using actions –Atomic actions: Predicates on current and next state variables –Action composition: synchronous (&) or asynchronous (|) Modular –Modules can have submodules Properties to be verified –Invariant(p) : p always holds

Composite Formulas: State Formulas We use state formulas to express the properties we need to check –No primed variables in state formulas –State formulas are boolean combination ( , , , ,  ) of integer, boolean and heap formulas numItems>2 => top.next!=null integer formula heap formula

State formulas Boolean formulas –Boolean variables and constants (true, false) –Relational operators: =,  –Boolean connectives ( , , , ,  ) Integer formulas (linear arithmetic) –Integer variables and constants –Arithmetic operators: +,-, and * with a constant –Relational operators: =, , >, <, ,  –Boolean connectives ( , , , ,  ) Heap formulas –Heap variable, heap-variable.selector, heap constant null –Relational operators: =,  –Boolean connectives ( , , , ,  )

Composite Formulas: Transition Formulas We use transition formulas to express the actions –In transition formulas primed-variables denote the next-state values, unprimed-variables denote the current-sate values pc=checknull and numItems=0 and top’=add and add’.next=null and numItems’=1 and pc’=create and mutex’; current state variables next state variables

Transition Formulas Transition formulas are in the form: boolean-formula  integer-formula  heap-transition-formula Heap transition formulas are in the form: guard-formula  update-formula A guard formula is a conjunction of terms in the form: id 1 = id 2 id 1  id 2 id 1.f = id 2 id 1.f  id 2 id 1.f = id 2.f id 1.f  id 2.f id 1 = null id 1  null id 1.f = null id 1.f  null An update formula is a conjunction of terms in the form : id’ 1 = id 2 id’ 1 = id 2.f id’ 1.f = id 2 id’ 1.f = id 2.f id’ 1 = null id’ 1.f = null id’ 1 = new id’ 1.f = new

Stack Example module main() heap {next} top, add, get, newTop; boolean mutex; integer numItems; initial: top=null and mutex and numItems=0; module push() enumerated pc {create, checknull,updateTop}; initial: pc=create and add=null; push1: pc=create and mutex and !mutex’ and add’=new and pc’=checknull; push2: pc=checknull and top=null and top’=add and add’.next=null and numItems'=1 and pc’=create and mutex’; push3: pc=checknull and top!=null and add’.next=top and pc’=updateTop; push4: pc=updateTop and top’=add and numItems’=numItems+1 and mutex’ and pc’=create; push: push1 | push2 | push3 | push4; endmodule Variable declarations define the state space of the system Predicates defining Predicates defining the initial states the initial states Atomic actions: primed variables denote next sate variables Transition relation of the push module is defined as asynchronous composition of its atomic actions

Stack (Cont’d) module pop() enumerated pc {copyTopNext, getTop, updateTop}; initial: pc=copyTopNext and get=null and newTop=null; pop1: pc=copyTopNext and mutex and top!=null and newTop’=top.next and !mutex’ and pc’=getTop; pop2: pc=getTop and get’=top and pc’=updateTop; pop3: pc=updateTop and top’=newTop and mutex’ and numItems’=numItems-1 and pc’=copyTopNext; pop: pop1 | pop2 | pop3; endmodule main: pop() | pop() | push() | push(); spec: invariant([numItems=0 => top=null]) spec: invariant([numItems>2 => top->next!=null]) endmodule Invariants to be verified Transition relation of main defined as asynchronous composition of two pop and two push processes

Stack (with integer guards) module main() heap {next} top, add, get, newTop; boolean mutex; integer numItems; initial: top=null and mutex and numItems=0; module push() enumerated pc {create, checknull,updateTop}; initial: pc=create and add=null; push1: pc=create and mutex and !mutex’ and add’=new and pc’=checknull; push2: pc=checknull and numItems=0 and top’=add and add’.next=null and numItems’=1 and pc’=create and mutex’; push3: pc=checknull and numItems>0 and add’.next=top and pc’=updateTop; push4: pc=updateTop and top’=add and numItems’=numItems+1 and mutex’ and pc’=create; push: push1 | push2 | push3 | push4; endmodule

Outline Specification of concurrent linked lists –Action Language Symbolic verification –Composite representation Approximation techniques –Summarization –Widening Counting abstraction Experimental results Related Work Conclusions

Symbolic Verification: Forward Fixpoint Forward fixpoint for the reachable states can be computed by iteratively manipulating symbolic representations –We need forward-image (post-condition), union, and equivalence check computations ReachableStates(I: Set of initial states, T: Transition relation) { RS := I; repeat { RS old := RS; RS := RS old  forwardImage(RS old, T); } until (RS old = RS) }

Symbolic Verification: Symbolic Representations Use a symbolic representation for the sets of states –A boolean logic formula (stored as a BDD) represents the sets of states of the boolean variables: pc=create  mutex –An arithmetic constraint (stored as polyhedra) represents the sets of states of integer variables: numItems>0 –Shape graphs are used to represent the sates of the heap variables and the heapaddtop

Composite Representation Each variable type is mapped to a symbolic representation type –Boolean and enumerated types  BDD representation –Integer variables  Polyhedra –Heap variables  Shape graphs Each conjunct in a transition formula operates on a single symbolic representation Composite representation: A disjunctive representation to combine different symbolic representations Union, equivalence check and forward-image computations are performed on this disjunctive representation

Composite Representation A composite representation A is a disjunction where –n is the number of composite atoms in A –t is the number of basic symbolic representations Each composite atom is a conjunction –Each conjunct corresponds to a different symbolic representation

Composite Representation: Example pc=create  mutex numItems=2addtop    pc=checkNull   mutex  numItems=2  addtop BDD A list of polyhedra A list of shape graphs  pc=updateTop   mutex  numItems=2  addtop  pc=create  mutex  numItems=3  addtop

Composite Symbolic Library [Yavuz-Kahveci, Tuncer, Bultan TACAS01], [Yavuz-Kahveci, Bultan STTT02] Our library implements this approach using an object-oriented design –A common interface is used for each symbolic representation –Easy to extend with new symbolic representations –Enables polymorphic verification –As a BDD library we use Colorado University Decision Diagram Package (CUDD) [Somenzi et al] –As an integer constraint manipulator we use Omega Library [Pugh et al] –For encoding the states of the heap variables and the heap we use shape graphs encoded as BDDs (using CUDD)

Composite Symbolic Library: Class Diagram CUDD LibraryOMEGA Library Symbolic +union() +isSatisfiable() +isSubset() +forwardImage() CompSym –representation: list of comAtom + union() BoolSym –representation: BDD +union() compAtom –atom: *Symbolic HeapSym –representation: list of ShapeGraph +union() IntSym –representation: list of Polyhedra +union() ShapeGraph –atom: *Symbolic

Satisfiability Checking for Composite Representation boolean isSatisfiable(CompSym A) for each compAtom a in A do if a is satisfiable then return true return false boolean isSatisfiable(compAtom a) for each symbolic representation t do if a t is not satisfiable then return false return true is Satisfiable? isSatisfiable?  Satisfiable? is Satisfiable? is Satisfiable? and is Satisfiable? and or

Forward Image for Composite Representation CompSym forwardImage(Compsym A, transitionRelation R) CompSym C; for each compAtom a in A do for each atomic action r in R do insert forwardImage( a,r ) into C return C A: R: C:

Forward Image for Composite Atom compAtom forwardImage(compAtom a, atomic action r) for each symbolic representation type t do replace a t by forwardImage(a t, r t ) return a r: a:

Forward-Image Computation: Example pc=updateTop   mutex  numItems=2  addtop pc=updateTop and pc’=create and mutex’ pc=create  mutex numItems’=numItems+1  numItems=3 top’=add  addtop

Forward–Fixpoint Computation (Repeatedly Applies Forward-Image) pc=create  mutex numItems=0 addtop   pc=create  mutex  numItems=1  addtop  pc=checkNull   mutex  numItems=0  addtop  pc=checkNull   mutex  numItems=1  addtop

 pc=updateTop   mutex  numItems=1  addtop  pc=create  mutex  numItems=2  addtop  pc=checkNull   mutex  numItems=2  addtop  pc=updateTop   mutex  numItems=2  addtop

pc=create  mutex numItems=3addtop  ... 

Forward-Fixpoint does not Converge We have two reasons for non-termination –integer variables can increase without a bound –the number of nodes in the shape graphs can increase without a bound The state space is infinite Even if we ignore the heap variables, reachability is undecidable when we have unbounded integer variables So, we use conservative approximations

Outline Specification of concurrent linked lists –Action Language Symbolic verification –Composite representation Approximation techniques –Summarization –Widening Counting Abstraction Experimental results Related Work Conclusions

Conservative Approximations p To verify or falsify a property p RS  ) RS + ) Compute a lower ( RS  ) or an upper ( RS + ) approximation to the set of reachable states There are three possibilities: “The property is satisfied” RS  p p p p RS +

Conservative Approximations reachable sates which violate the property “The property is false” RS   p p p p RS “I don’t know” RS   p p p p RS RS +

Computing Upper and Lower Bounds for Reachable States Truncated fixpoint computation –To compute a lower bound for a least-fixpoint computation –Stops after a fixed number of iterations Widening –To compute an upper bound for the least-fixpoint computation –We use a generalization of the polyhedra widening operator by [Cousot and Halbwachs POPL’77] Summarization –Generate heap nodes in the shape graphs which represent more than one concrete node –Materialization: we need to generate concrete nodes from the summary nodes when needed

Summarization The nodes mapped to a summary node form a chain No heap variable points to any concrete node that is mapped to a summary node Each concrete node mapped to a summary node is only pointed by one pointer During summarization, we also introduce an integer variable which counts the number of concrete nodes mapped to a summary node...

Summarization Example pc=create  mutex numItems=3 add top   pc=create  mutex numItems=3  summarycount=2 add top   summary node a new integer variable representing the number of concrete nodes encoded by the summary node After summarization, it becomes:

Summarization Summarization guarantees that the number of different shape graphs that can be generated are finite However, the summary-counts can still increase without a bound We use polyhedral widening operation to force the fixpoint computation to convergence

Let’s Continue the Forward-fixpoint pc=create  mutex numItems=3  summaryCount=2 addtop pc=checkNull   mutex  addtop numItems=3  summaryCount=2  pc=updateTop   mutex   addtop numItems=3  summaryCount=2 pc=create  mutex add top   numItems=4  summaryCount=2 We need to do summarization 

Summarization pc=create  mutex add top   numItems=4  summaryCount=2 After summarization, it becomes: pc=create  mutex add top   numItems=4  summaryCount=3

Simplification After each fixpoint iteration we try to merge as many composite atoms as possible For example, following composite atoms can be merged pc=create  mutex numItems=3  summaryCount=2 addtop pc=create  mutex add top   numItems=4  summaryCount=3

Simplification pc=create  mutex numItems=3  summaryCount=2 addtop pc=create  mutex add top   numItems=4  summaryCount=3 = pc=create  mutex add top   (numItems=4   summaryCount=3   numItems=3  summarycount=2)

Simplification on the integer part pc=create  mutex add top   (numItems=4   summaryCount=3   numItems=3  summaryCount=2)= pc=create  mutex add top  numItems=summaryCount+1   3  numItems   numItems  4

Widening Forward-fixpoint computation still will not converge since numItems and summaryCount keep increasing without a bound We use the widening operation: –Given two composite atoms c 1 and c 2 in consecutive fixpoint iterates, assume that c 1 = b 1  i 1  h 1 c 2 = b 2  i 2  h 2 where b 1 = b 2 and h 1 = h 2 and i 1  i 2 –Also assume that i 1 is a single polyhedron (i.e. a conjunction of arithmetic csontraints) and i 2 is also a single polyhedron

Widening Then –i 1  i 2 is defined as: all the constraints in i 1 which are also satisfied by i 2 Replace i 2 with i 1  i 2 in c 2 This gives a majorizing sequence to the forward-fixpoint computation

Widening Example pc=create  mutex add top  numItems=summaryCount+1   3  numItems   numItems  4 pc=create  mutex add top  numItems=summaryCount+1   3  numItems   numItems  5  pc=create  mutex add top  numItems=summaryCount+1   3  numItems= Now, the forward-fixpoint converges

Dealing with Arbitrary Number of Processes Use counting abstraction [Delzanno CAV’00] –Create an integer variable for each local state of a process –Each variable will count the number of processes in a particular state Local states of the processes have to be finite –Shared variables of the monitor can be unbounded Counting abstraction can be automated

Stack After Counting Abstraction module main() heap top, add, get, newTop; boolean mutex; integer numItems; integer CreateC, ChecknullC,UpdateTopC; parameterized integer numProcesses; initial: top=null and mutex and numItems=0 and CreateC=numProcesses and ChecknullC=0 and UpdateTopC=0; restrict: numProcesses>0; module push() //enumerated pc {create, checknull,updateTop}; initial: add=null; push1: CreateC>0 and mutex and !mutex' and add'=new and CreateC'=CreateC-1 and ChecknullC'=ChecknullC+1; push2: ChecknullC>0 and top=null and top'=add and add'->next=null and numItems'=1 and ChecknullC'=ChecknullC-1 and CreateC'=CreateC+1 and mutex'; push3: ChecknullC>0 and top!=null and add'->next=top and ChecknullC'=ChecknullC-1 and UpdateTopC'=UpdateTopC+1; push4: UpdateTopC>0 and top'=add and numItems'=numItems+1 and mutex' and UpdateTopC'=UpdateTopC-1 and CreateC'=CreateC+1; push: push1 | push2 | push3 | push4; endmodule Parameterized constant representing the number of processes Variables for counting the number of processes in each state When local state changes, decrement current local state counter and increment next local state counter Initialize initial state counter to the number of processes. Initialize other states to 0.

Verified Properties SPECIFICATIONVERIFIED INVARIANTS Stack top=null  numItems=0 top  null  numItems  0 numItems=2  top.next  null Single Lock Queue head=null  numItems=0 head  null  numItems  0 (head=tail  head  null)  numItems=1 head  tail  numItems  0 Two Lock Queue numItems>1  head  tail numItems>2  head.next  tail

Experimental Results - Verification Times Number of Processes Queue HC Queue IC Stack HC Stack IC 2Lock Queue HC 2Lock Queue IC 1P-1C P-2C P-4C  1P-PC  PP-1C 

Related Work There is a lot of work on Shape analysis, I will just mention the ones which directly influenced us: –[Sagiv,Reps, Wilhelm TOPLAS’98], [Dor, Rodeh, Sagiv SAS’00] Verification of concurrent linked lists with arbitrary number of processes in [Yahav POPL’01] [Lev-Ami, Reps, Sagiv, Wilhelm ISSTA 00] use 3-valued logic and instrumentation predicates to verify properties that cannot be expressed in our framework, however, our approach does not require instrumentation predicates Deutch used integer constraint lattices to compute aliasing information using symbolic access paths [Deutch PLDI’94] Use of BDDs goes back to symbolic model checking [McMillan’93] and verification with arithmetic constraints goes back to [Cousot and Halbwachs’77]

Conclusions and Future Work One of the weakness of the summarization algorithm we used is the fact that it only works on singly linked lists –We need to find a more general summarizaton algorithm which counts the number of summary nodes Implementation is not efficient, we are working on improving the performance Liveness properties? –We would like to do full CTL model checking –Need to implement the backward image computation

APPENDIX

Action Language Verifier An infinite state symbolic model checker Composite representation –uses a disjunctive representation to combine different symbolic representations Computes fixpoints by manipulating formulas in composite representation –Heuristics to ensure convergence Widening & collapsing Loop closure Approximate reachable states

Readers Writers Monitor in Action Language module main() integer nr; boolean busy; restrict: nr>=0; initial: nr=0 and !busy; module Reader() boolean reading; initial: !reading; rEnter: !reading and !busy and nr’=nr+1 and reading’; rExit: reading and !reading’ and nr’=nr-1; Reader: rEnter | rExit; endmodule module Writer() boolean writing; initial: !writing; wEnter: !writing and nr=0 and !busy and busy’ and writing’; wExit: writing and !writing’ and !busy’; Writer: wEnter | wExit; endmodule main: Reader() | Reader() | Writer() | Writer(); spec: invariant([busy => nr=0]) endmodule

Action Language Verifier An infinite state symbolic model checker Uses composite symbolic representation to encode a system defined by (S,I,R) –S: set of states, I: set if initial states, R: transition relation Maps each variable type to a symbolic representation type –Maps boolean and enumerated types to BDD representation –Maps integer type to arithmetic constraint representation Uses a disjunctive representation to combine symbolic representations –Each disjunct is a conjunction of formulas represented by different symbolic representations

Conjunctive Decomposition Each composite atom is a conjunction Each conjunct corresponds to a different symbolic representation –x: integer; y: boolean; h heap – x>0 and x’=x+1 and y´  y Conjunct x>0 and x´  x+1 will be represented by arithmetic constraints Conjunct y´  y will be represented by a BDD –Advantage: Image computations can be distributed over the conjunction (i.e., over different symbolic representations).

BDDs Efficient representation for boolean functions Disjunction, conjunction complexity: at most quadratic Negation complexity: constant Equivalence checking complexity: constant or linear Image computation complexity: can be exponential

Arithmetic Constraint-Based Verification Can we use linear arithmetic constraints as a symbolic representation? –Required functionality Disjunction, conjunction, negation, equivalence checking, existential variable elimination Advantages: –Arithmetic constraints can represent infinite sets –Heuristics based on arithmetic constraints can be used to accelerate fixpoint computations Widening, loop-closures

Linear Arithmetic Constraints Disjunction complexity: linear Conjunction complexity: quadratic Negation complexity: can be exponential –Because of the disjunctive representation Equivalence checking complexity: can be exponential –Uses existential variable elimination Image computation complexity: can be exponential –Uses existential variable elimination

Linear Arithmetic Constraints Can be used to represent sets of valuations of unbounded integers Linear integer arithmetic formulas can be stored as a set of polyhedra c kl is a linear equality or inequality constraint and each where each c kl is a linear equality or inequality constraint and each is a polyhedron is a polyhedron

A Linear Arithmetic Constraint Manipulator Omega Library [Pugh et al.] –Manipulates Presburger arithmetic formulas: First order theory of integers without multiplication –Equality and inequality constraints are not enough: Divisibility constraints are also needed (a variable is divisible by a constant) Existential variable elimination in Omega Library: Extension of Fourier-Motzkin variable elimination to integers Eliminating one variable from a conjunction of constraints may double the number of constraints Integer variables complicate the problem even further

Fourier-Motzkin Variable Elimination Given two constraints   bz and az   we have a   abz  b  We can eliminate z as:  z. a   abz  b  if and only if a   b  Every upper and lower bound pair can generate a separate constraint, the number of constraints can double for each eliminated variable real shadow

Integers are More Complicated If z is integer  z. a   abz  b  if a  + (a - 1)(b - 1)  b  Remaining solutions can be characterized using periodicity constraints in the following form:  z.  + i = bz dark shadow

 y. 0  3y – x  7  1  x – 2y  5 Consider the constraints: 2x  6y2x  6y We get the following bounds for y: 6y  2x y  3x - 33x - 15  6y When we combine 2 lower bounds with 2 upper bounds we get four constraints: 0  14, 3  x, x  29, 0  12 Result is: 3  x  29

2y  x – 1 x – 5  2y 3y  x + 7 x  3y dark shadow real shadow 293 y x

Temporal Properties  Fixpoints Invariant(p) pppp Initialstates initial states that violate Invariant(p) Backwardfixpoint Forwardfixpoint Initialstates states that can reach  p i.e., states that violate Invariant(p) reachable states of the system pppp backwardImage of  p of  p reachable states that violate p forward image of initial states

Simplification Example (y  z´ = z + 1)(x  z´ = z + 1)((x  y)  z´ > z)    ((y  x)  z´ = z + 1)   ((x  y)  (z´ = z + 1  z´ > z))  ((x  y)  z´  z)

Polymorphic Verifier Symbolic TranSys::check(Node *f) { Symbolic s = check(f.left) case EX: s.backwardImage(transRelation) case EF: do snew = s sold = s snew.backwardImage(transRelation) s.union(snew) while not sold.isEqual(s) }  Action Language Verifier is polymorphic  When there are no integer variable it becomes a BDD based model checker