© 2008 Carnegie Mellon University Combining Predicate and Numeric Abstraction for Software Model Checking Software Engineering Institute Carnegie Mellon University Pittsburgh, PA Arie Gurfinkel and Sagar Chaki

2 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Automated Software Analysis Program Automated Analysis Correct Incorrect Software Model Checking with Predicate Abstraction e.g., Microsoft’s SDV Abstract Interpretation with Numeric Abstraction e.g., ASTREE, Polyspace

3 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Predicate and Numeric Abstractions Predicate Abstraction (PA) (e.g., SDV) Typical property: no lock is acquired twice Program verification reduced to propositional reasoning with model checker Works well for control-driven programs Works poorly for data-driven programs Numeric Abstraction (NA) (e.g, ASTREE) Typical property: no arithmetic overflow Program verification reduced to arithmetic reasoning Works well for data-driven programs Works poorly for control-driven programs How to combine PA and NA to get the best of both? Precision Scalability NA PA Unrealistic

4 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Outline Predicate and Numeric Abstract for Program Analysis Strength and Weakness An “Ideal” Combination PA+NA Combination Abstract Transformers Data Structures Experimental Results Current and Future Work

5 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Predicate Abstraction: An Example Concrete Program p1:i=1 p2:i=2 p3:x1>0 p4:x2<0 Abstract Program assume (i=1 || i=2) if (i = 1) x1 := i; else if (i = 2) x2 := -4; if (i = 1) assert (x1 > 0); else if (i = 2) assert (x2 < 0); assume (p1 || p2) if (p1) p3 := ch(p1||p2,false); else if (p2) p4 := true if (p1) assert (p3); else if (p2) assert (p4); p := ch(tt,ff) if (tt) p := 1; else if (ff) p := 0; else p := *;

6 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Analysis with Predicate Abstraction p1:i=1 p2:i=2 p3:x1>0 p4:x2<0 Abstract Program assume (p1 || p2) if (p1) p3 := ch(p1||p2,false); else if (p2) p4 := true if (p1) assert (p3); else if (p2) assert (p4); p1 || p2 p1 p1&&p3 !p1&&p2&&p4 p1&&p3 || !p1&&p2&&p4 !p1&&p2 !p1&&p2&&p4 p1&&p3 Can prove safety

7 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Predicate Abstraction Strength/Weaknesses Strengths Works well for control-dependent properties Completely automated Predicates can come from any theory that has an automated (semi-)decision procedure Supports any Boolean combination of predicates Compatible with CounterExample Guided Abstraction Refinement Weaknesses Scalability (construction and analysis) Restricted to finite abstract domains

8 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Numeric Abstract Interpretation Analysis is restricted to a fixed Abstract Domain Abstract Domain ≡ “a (possibly infinite) set of predicates from a fixed theory” + efficient operations Examples of Numeric Abstract Domains Signs 0 0 Intervals c 1  x  c 2, where c 1,c 2 are a constants Octagons ± x ± y  c, where c is a constant Polyhedra a 1 x 1 + a 2 x 2 +a 3 x 3 + a 4  0, where a i are constants

9 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University AbsDom Interface interface AbsDom(V) : V – set of variables A – abstract elements, E – expressions, S – statements α : E → A γ : A → E meet : A  A → A isTop : A → bool isBot : A → bool join : A  A → A leq : A  A → bool αPost : S → (A → A) widen : A  A → A All operations are over-approximations, e.g., γ (a) || γ (b)  γ ( join (a, b) ) γ (a) && γ (b)  γ (meet (a,b) )

10 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Example: The Domain of Intervals (1, 10) meet (2, 12) = (2,10) (1, 3) join (7, 12) = (1,12) 1  x  10 (1, 10) α γ 1  x  10 (a, b) meet (c, d) = (max(a,c), min(b,d)) (a, b) join (c, d) = (min(a,c),max(b,d)) α Post (x := x + 1) ((a, b)) = (a+1, b+1)(1, 10) + 1 = (2, 11) OperationsExamples over-approximation

11 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Numeric Abstraction: An Example Concrete Program assume (i=1 || i=2) if (i = 1) x1 := i; else if (i = 2) x2 := -4; if (i = 1) assert (x1 > 0); else if (i = 2) assert (x2 < 0); assume (i=1 || i=2) if (i = 1) x1 := i; else if (i = 2) x2 := -4; if (i = 1) assert (x1 > 0); else if (i := 2) assert (x2 < 0); Abstract Program Abstraction

12 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Analysis with Intervals NA (1) assume (i=1 || i=2) if (i = 1) x1 := i; else if (i = 2) x2 := -4; if (i = 1) assert (x1 > 0); else if (i := 2) assert (x2 < 0); 1 <= i <= 2 i=1 i=1 && x1=1 i=2 i=2 && x2=-4 1 <= i <= 2 i=1 i=2 Loss of precision due to join Cannot prove safety Abstract Program

13 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Analysis with Intervals NA (2) if (3 <= y1 <= 4) { x1 := y1-2; x2 := y1+2; } else if (3 <= y2 <= 4) { x1 := y2-2; x2 := y2+2; } else return; assert (5 <= x1 + x2 <= 10); 3 <= y1 <= 4 1 <= x1 <= 2 5 <= x2 <= 6 3 <= y2 <= 4 1 <= x1 <= 2 5 <= x2 <= 6 1<=x1<=2 5<=x2<=6 Using PA expensive! Abstract Program

14 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Strength/Weakness of Numeric Abstraction Strength Fully Automated Scalable Supports infinite abstract domains (Supports) Automated Refinement Weakness Limited to a few theories (intervals, octagons, polyhedra) Restricted to conjunctions of terms Looses precision very quickly (join, widen, etc.)

15 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Predicates: p: A[y1+y2]=3 q: A[x1+x2]=3 assume (x1 = x2); if (p) { x1 := y1 – 2 && q := *; x2 := y2 + 2 &&& q := ch ((x1=y1-2)&&p,f) } else q := false; if (q) { x1 := x1 + x2; x2 := x2+y1-2; } assert (x1 = x2) “Ideal” combination of PA + NA assume (x1 = x2); if (A[y1+y2] = 3) { x1 := y1 – 2; x2 := y2 + 2; } else A[x1+x2] := 5; if (A[x1+x2] = 3) { x1 := x1 + x2; x2 := x2+y1-2; } assert (x1 = x2) Predicates: p: A[y1+y2]=3, q: A[x1+x2]=3 Concrete ProgramAbstract Program Abstraction

16 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Abstract with Predicates p: A[y1+y2]==3 q: A[x1+x2]==3 “Ideal” combination of PA + NA assume (x1 = x2); if (A[y1+y2] = 3) { x1 := y1 – 2; x2 := y2 + 2; } else A[x1+x2] := 5; if (A[x1+x2] = 3) { x1 := x1 + x2; x2 := x2+y1-2; } assert (x1 = x2) assume (x1 = x2); if (p) { x1 := y1 – 2 && q := *; x2 := y2 + 2 &&& q := ch ((x1=y1-2)&&p,f) } else q := false; if (q) { x1 := x1 + x2; x2 := x2+y1-2; } assert (x1 = x2) Predicates: p: A[y1+y2]=3, q: A[x1+x2]=3 Concrete ProgramAbstract Program Abstraction Concrete transformer a.k.a. program statement Abstract transformer Parallel composition

17 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Analyzing with PA + NA assume (x1 = x2); if (p) { x1 := y1 – 2 && q := *; x2 := y2 + 2 &&& q := ch ((x1+2 = y1)&&p,f) } else q := false; if (q) { x1 := x1 + x2; x2 := x2+y1-2; } assert (x1 = x2) x1=x2 p && x1=x2 p && x1=y1-2 p && q && x1=y1-2 && x2=y2+2 !p && !q && x1=x2 p && q && x1=y1-2 && x2=y2+2 || !p && !q && x1=x2 p && q && x1=y1-2 && x2=y2+2 p && q && x1=y1+y2 && x2=y2+2 p && q && x1=y1+y2 && x2=y2+y1 && x1=x2 Predicates: p: A[y1+y2]=3, q: A[x1+x2]=3 p && q && x1=y1+y2 && x2=y2+y1 && x1=x2 || !p && !q && x1=x2 Abstract Program

18 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University NumPredDom Interface interface NumPredDom(VN,VP) extends AbsDom VN – numeric variables, VP – predicates, e – conjunctive expression proj P (VP,e) = conjunction of predicates from VP s.t. e  proj P (VP,e) proj N (VN,e) = conjunction of constraints over VN s.t. e  proj N (VP,e) α P : E → A α N : E → Aunprime : A → A reduce : A → A exists : 2 VP  A → AαPost N : S → (A → A) Information Sharing between NA and PA: Factored out for efficiency Lift numeric-only abstract transformer Lift propositional-only abstract transformer Implementation-dependent. Influences precision and scalability.

19 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University NumPredDom: Definition of α α(e1  e2) = meet ( α(e1), α(e2) ) α(e1  e2) = join ( α(e1), α(e2) ) α(e) = meet ( α P (proj P (VP  VP’, e)), α N (proj N (VN, e)) ) project abstract combine conjunctions disjunctions

20 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University NumPredDom: Abstract Transformers v ::= numeric expression e ::= Boolean expression over predicate and numeric terms τ ::= τ N | τ a | τ c | τ P | τ NP | τ || τ | τ ; τ τ NP ::= (e ? τ N ) &&& τ P (numeric + predicate) τ P ::= p := ch (e, e) | τ P &&& τ P (predicate) τ c ::= e ? τ N (conditional) τ a ::= assume (e)(assume) τ N ::= x := v | τ N &&& τ N (numeric) sequence non-det base-case

21 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Transformer Examples Predicates: p 1 :z=&x, p 2 :z=&y, p 3 :y=1 Concrete Transformer Abstract Transformer assume (*z > 0)(p 1 &&x>0 || p 2 &&y>0 || !p 1 &&!p 2 )? skip *z = u + 1 (p 1 ? x := u + 1) || (p 2 ? y := u+1) || (!p 1 && !p 2 ? skip) y = x &&& x = (y-1? v : w) (p 3 ? x := v || !p 3 ? x := w) &&& p 3 := ch (x=1,x!=1)

22 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University NumPredDom: Four Implementations NameValueExample Num. Terms NEXPoint 2 2^P  N (p||q) && (0 <= x <= 5) Explicit NEX2 P → N(p&& 0<=x<=3) || (!p && (1<=x<=5)) MTBDD2 P → N(p&& 0<=x<=3) || (!p && (1<=x<=5)) Symbolic NDD2 P → 2 N (p && (x=0 || x=3)) || (!p && (x=1 || x=5)) Precision

23 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University NEXPoint (P, N) NEXPoint elements are of the form: BDD over predicates Element of numeric abstract domain All operations are pairwise 2 2^P  N

24 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Numeric EXplicit (NEX) NEX elements are lists of NEXPoint [(P 1, N 1 ),…, (P k,N k )] Satisfying the partitioning condition P i ∩ P j = { } Operations are done using NEXPoint, but respect the partitioning condition 2 P → N

25 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University The Partitioning Condition p !p q !q x>0 y>0

26 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Multi-Terminal Numeric Decision Diagrams b1b1 b2b2 x>0 && x=y 1-edges are black, 0-edges are red edges to 0 node are not shown p 1 && !p 2 && (x>0) && (x=y) p 1 : x>0, p 2 : z<y b 1 : p 1, b 2 : p 2 MTNBDD MTNDD elements are Decision Diagrams with Numeric values at the terminals 2 P → N

27 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Numeric Decision Diagrams (p 1 &&p 2 ) || (x<0 && y=z) (x>=0 && z>0) || (!(x>=0) && y=z) p 1 : x>=0, p 2 : z>0 b1:x>=0, b2:z>0, b3:y=z b1b1 b2b2 b3b3 1 1-edges are black, 0-edges are red edges to 0 node are not shown normalize NDD elements are BDDs over Predicate and Numeric Terms 2 P → 2 N

28 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Summary of the Data Structures PrecisionScalabilityPA aloneNA aloneProp OpNum Op NEXPoint NEX MTNDD NDD

29 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Tool Implementation Java Implementation Numeric domains implemented on top of Apron library Frontend Source Annotated CFG + Initial Node + Target Node Reachability Analysis Report Fixed point computation with widening “Simple” automated abstraction-refinement 1.Check is CE is spurious using WP + TP 2.Add new numeric variables in UNSAT core first 3.Add new predicates only if all numeric variables in UNSAT core already present

30 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Experimental Results Synthetic examples used to validate specific conjectures NEX & MTNDD better than NDD when numeric joins are exact — Since NDD uses exact unions while others use numeric join NDD better than others when invariants are propositionally complex — Since NDD has the most sharing capability Realistic examples used to gauge overall performance Medley: 22 examples: Zitser buffer overflow (3), OpenSSL (2), engine controller (9), Micro-C OS (2), Device drivers (6)

31 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Negative running time means failure

32 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Experimental Results: Medley (2) Domain#Exp.TotalGammaJoinalphaPostApply Numeric Predicate NEXPoint NEX MTNDD NDD (Total # of examples = 22; all times are in seconds)

33 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Negative running time means failure

34 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Related Work Abstract Interpretation [CC’92] Our domain ≈ reduced direct product of Predicate and Numeric domains Jain et al. [CAV’06] Applies numeric invariants to simplify predicate abstraction Weaker than NEXPoint Fischer et al. [FSE’05], Beyer et al. [CAV’07,CAV’06] Predicate abstraction + Abstract Domain Similar to NEXPoint, but with simpler transfer functions Bultan et al. [TOSEM’00] MC of programs with Boolean and numeric variables using Omega library Similar to NEX, but with simpler transfer functions

35 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University Current and Future Work Augmenting our benchmark suite Better automated abstraction-refinement for PA + NA In this work, the abstract domain is treated as a black box. Exploring tighter integration between predicate and numeric domains smarter numeric transfer functions, smarter DD variable ordering, etc. We support intervals, octagons and polyhedra. Looking for combinations with other (numeric) domains

36 Combining PA and NA for Soft MC Gurfinkel and Chaki © 2008 Carnegie Mellon University