1

Satisfiability modulothe Theory of Bit Vectors

Alessandro CimattiIRST, Trento, Italy

cimatti@irst.itc.it

Joint work with R. Bruttomesso, A. Franzen, A. Griggio, R. Sebastiani

We gratefully acknowledge support from the Academic Research Program of Intel

2

Index of the talk

• Satisfiability Modulo Theory• The theory of Bit Vectors• Satisfiability Modulo BV

– Bit blasting– Eager encoding into Linear Integer Arithmetic– A lazy approach

• Conclusions

• ( A preview of QF_UFBV32 at SMT-COMP )

3

SMT in a nutshell

• Satisfiability Modulo Theory– or: beyond boolean SAT

• Decide the satisfiability of a first order formula with respect to a background theory

• Examples of relevant theories– uninterpreted functions: x=y & f(x) != f(y)– difference logic: x – y < 7– linear arithmetic: 3x + 2y < 12– arrays: read(write(M, a0, v0) a1)– their combinations– bit vectors

4

Why SMT

• From SAT-based to SMT-based verification• Representation of interesting problems

– timed automata– hybrid automata– pipelines– software

• Efficient solving– leverage availability of structural information– hopefully retaining efficiency of boolean SAT

5

Satisfiability Modulo Theory

• Satisfiability: – is there a truth-assignment to boolean

variables – and a valuation to individual variables – such that formula evaluates to true?

• Standard semantics for FOL• Assignment to individual variables

– Induces truth values to atoms

• Truth assignment to boolean atoms• Induced value to whole formula

6

Propositionalstructure

TA TA TA TA

P P P

x y z w x x y z w x

+ - + - + - + -+ - + -

8

Eager Approach to SMT

• Main idea: compilation to SAT– STEP1: Theory part compiled to equisatisfiable

pure SAT problem– STEP2: run propositional SAT solver

9

Propositionalstructure

TA TA TA TA

P P P

x y z w x x y z w x

10

P P PTA TA TA TA

Lifted theory Propositionalstructure

11

The Lazy approach

• Ingredients– a boolean SAT solver – a theory solver

• The boolean solver is modified to enumerate boolean (partial) models

• The theory solver is used to Check for theory consistency

12

Propositionalstructure

TA TA TA TA

P P P

x y z w x x y z w x

TA TA TA TA

13

MathSAT: intuitions• Two ingredients: boolean search and theory reasoning

– find boolean model• theory atoms treated as boolean atoms• truth values to boolean and theory atoms• model propositionally satisfies the formula

– check consistency wrt theory• set of constraints induced by truth values to theory atoms• existence of values to theory variables

• Example: (P v (x = 3)) & (Q v (x – y < 1)) & (y < 2) & (P xor Q)

• Boolean model– !P, (x = 3), Q, (x – y < 1), (y < 2)– Check (x = 3), (x – y < 1), (y < 2)– Theory contradiction!

• Another boolean model– P , !(x = 3) , !Q, (x – y < 1), (y < 2)– Check !(x = 3), (x – y < 1), (y < 2)– Consistent: e.g. x := 0, y := 0

14

TSAT!

Boolean SAT: search spaceP

• The DPLL procedure• Incremental construction of satisfying assignment• Backtrack/backjump on conflict• Learn reason for conflict• Splitting heuristics

Q

R

S

S

R

T

Q

S

T

R

15

MathSAT: approach

• DPLL-based enumeration of boolean models– Retain all propositional optimizations

• Conflict-directed backjumping, learning

– No overhead if no theory reasoning• Tight integration between

– boolean reasoning and – theory reasoning

16

Bool Bool TMath

Bool Bool TMath

Bool Bool TMath TSAT!

MathSAT: search spaceP

Many boolean models are not theory consistent!

Q

R

S

S

R

T

Q

S

T

R

17

Bool Bool TMath TSAT!

EP:Math EP:Math T

EP:Math T

EP:Math T

EP:Math T

Pruned away in the EP step

Early pruning

Check theory consistency of partial assignments

P

Q

S

T

R

18

THEORY OF FIXED-WIDTH BIT VECTORS

19

Bit Vectors: Example

LTmp0 = a;LTmp1 = 2 * b;LTmp2 = LTmp0 + LTmp1;LTmp3 = 4 * c;LTmp4 = LTmp2 + LTmp3;LTmp5 = 8 * d;LOut = LTmp4 + LTmp5;

Are they equivalent?

((a + 2b) + 4c) + 8d

RTmp0 = d;RTmp1 = RTmp0 << 1;RTmp2 = c + RTmp1;RTmp3 = RTmp2 << 1;RTmp4 = b + RTmp3;RTmp5 = RTmp4 << 1;ROut = a + RTmp5;

a + ((b + ((c + (d<<1)) <<1)) <<1)

input a, b, c, d : reg[N];

I.e. LOut = ROut ?

20

Fixed Width Bit Vectors• Constants

– 0b00001111, 0xFFFF, …

• Variables– valued over BitVectors of corresponding width– implicit restriction to finite domain

• Function symbols– selection: x[15:0]– concatenation: y :: z– bitwise operators: x && y, z || w, …– arithmetic operators: x + y, z * w, … – shifting: x << 2, y >> 3

• Predicate symbols– comparators: =, ≠ , > , < , ≥ , ≤

21

Fragments of BV theory• Core

– selection– concatenation

• Bitwise operators– x && y, x || y, x ^ y

• Arithmetic operators– x +y, x – y, c * x

• Core + Bitwise + Arithmetic

• Complexity of equality between BV terms– Core is in P– Core + B + A in NP

• Variable width bit vectors: not covered here– core is in NP– small additions yield undecidability

22

Decision procedures for BV

• Many approaches– Cyrluk, Moeller, Ruess– Moeller, Ruess– Bjørner, Pichora– Barrett, Dill, Levitt

• Focus on deciding conjunctions of literals• Emphasis on proof obligations in ITP

– some emphasis on variable width, generic wrt N

• Shostak-style integration– canonization – solving

23

SATISFIABILITY MODULO THEORY OF BIT VECTORS

24

Satisfiability modulo Bit Vectors

• Applications of interest– RTL hardware descriptions essentially bit vectors– assembly-level programs– software with finite precision arithmetic

• Key feature– combination of control flow and data flow

• In principle, boolean logic can be encoded into BV– control (boolean logic) encoded into width 1 BVs.– Likely inefficient in comparison to SAT

• More natural to keep them separate at modeling– structural info can be exploited for verification

25

Approaches to SMT(BV)

• Bit blasting• Eager Encoding into LA• Lazy approach

26

SMT(BV) via Bit Blasting

27

SMT(BV) via Bit Blasting

• Boolean variables: untouched• Bit vector variables as collections of (unrelated)

boolean variables– [x0, x1, …, x63]

• Selection/concatenations are trivial– static detection

• Equalities / Assignments: x = y– (x0 <-> y0) & (x1 <-> y1) & … & (x63 <-> y63)

• Bitwise operators: x && y– [x0 & y0, x1 & y1, …, x63 & y63]

• Arithmetic operators: x + y– BVADD([x0, …, x63], [y0, …, y63])

28

Comparison of Data Paths

LTmp0 = a;LTmp1 = 2 * b;LTmp2 = LTmp0 + LTmp1;LTmp3 = 4 * c;LTmp4 = LTmp2 + LTmp3;LTmp5 = 8 * d;LOut = LTmp4 + LTmp5;

Are they equivalent?

((a + 2b) + 4c) + 8d

RTmp0 = d;RTmp1 = RTmp0 << 1;RTmp2 = c + RTmp1;RTmp3 = RTmp2 << 1;RTmp4 = b + RTmp3;RTmp5 = RTmp4 << 1;ROut = a + RTmp5;

a + ((b + ((c + (d<<1)) <<1)) <<1)

input a, b, c, d : reg[N];

I.e. LOut = ROut ?

29

Bit Blasting Words• a,b,c,d,…

– blasted to [a1,…aN], [b1,…bN], [c1,…cN], [d1,…dN], …• LTmp6 != RTmp6

– (LOut.1 != ROut.1) or … or (LOut.N != ROut.N)• LTmp1 = 2 * b

– formula in 2N vars, conjunction of N iffs• LTmp2 = LTmp0 + LTmp1

– formula relating 3N vars– possibly additional vars required (e.g. carries)

• N = 16 bits?– 13 secs

• N = 32 bits?– 170 secs

• “But obviously N = 64 bits!”– stopped after 2h CPU time

Scalabilitywith respect

to N???

30

Bit-Blasting: Pros and Conses

• Bottlenecks– dependency on word width– “wrong” level of abstraction

• boolean synthesis of arithmetic circuits• assignments are pervasive• conflicts are very fine grained

– e.g. discover x < y

• Advantages– let the SAT solver do all the work

• and nowadays SAT solvers are tough nuts to crack– amalgamation of the decision process

• no distinction between control and data• conflicts can be as fine grained as possible

– built-in capability to generate “new atoms”

31

Enhancements to BitBlasting

• Tuning SAT solver on structural information– e.g. splitting heuristic for adders

• Preprocessing + SAT [GBD05]– rewrite and normalize bit vector terms – bit blasting to SAT

32

SMT(BV) via reduction to SMT(LA)

33

From BV to LIA• RTL-Datapath Verification using Integer Linear Programming

[BD01]• BV constants as integers

– 0b32_1111 as 15• BV variables as integer valued variables, with range constraints

– reg x [31:0] as x in range [0, 2^32)• Assignments treated as equality, e.g. x = y• Arithmetic, e.g. z = x + y

– Linear arithmetic? not quite! BV Arithmetic is modulo 2^N– z = x + y - 2^N s, with z in [0, 2^N)

• Concatenation: x :: y as 2^n x + y• Selection: relational encoding (based on integrity)

– x[23:16] as xm, where– x = 2^24 xh + 2^16 xm + xl,

xl in [0, 2^16), xm in [0, 2^8), xl in [0, 2^8) • Bitwise operators

– based on selection of individual bits

• SOLVER– the omega test

34

From SMT(BV) into SMT(LIA)

• Generalizes [BD01] to deal with boolean structure

• Eager encoding into SMT(LIA)

• Unfortunately, not very efficient• More precisely, a failure

35

Retrospective Analysis• Crazy approach?

– Arithmetic• Linear arithmetic? not quite! BV Arithmetic is modulo 2^N

– Selection and Concatenation• an easy problem becomes expensive!

– Bitwise operators• HARD!!!

• Available solvers not adequate– integers with infinite precision– reasoning with integers may be hard (e.g. BnB within real relaxation)

• Functional dependencies are lost!

• A clear culprit: static encoding– depending on control flow, same signal is split in different parts– z = if P then x[7:0] :: y[3:0] else x[5:2] :: y[10:3]

• x, y and also z are split more than needed• the notion of “maximal chunk” depends on P !!!

36

SMT(BV) via online BV reasoning

37

A lazy approach

• Based on standard MathSAT schema– DPLL-based model enumeation– Dedicated Solver for Bit vectors

• The encoding leverages information resulting from decisions– Given values to control variables, the data path

is easier to deal with (e.g. maximal chunks are bigger)

• Layering in the theory solver– equality reasoning– limited simplification rules– full blown bit vector solver only at the end

38

BV solver

The architecture

Boolean enumeration

EUF reasoning

BV rewriter LIAencoding

39

Rewriting rules

• evaluation of constant terms– 0b8_01010101[4:2] becomes 0b3_101

• rules for equality– x = y and Phi(x) becomes Phi(y)– based on congruence closure

• splitting concatenations– (x :: y) = z becomes x = z[h_n] && y == z[l_n]

40

Rewriting rules

• pushing selections– (x && y)[7:0] becomes (x[7:0] && y[7:0])– (x :: y)[23:8] becomes (x[7:0] :: y[15:8])

• “pigeon-hole” rules– from (x != 0 & x != 1 & x != 2 & x < 3) derive

false

41

BV rewriter

• Rules are applied until– fix point reached– contradiction found

• Implementation based on EUF reasoner– rules as merges between eq classes

• Open issues– incrementality/backtrackability– selective rule activation– conflic set reconstruction

• When it fails …

42

LIA encoding (the last hope)

• LIA encoding– idenfication of maximal slices– “purification”: separating out arithmetic and

BW by introduction of additional variables

• NB: on resulting problems– LIA encoding always superior to bit blasting!!!– cfr [DB01]

43

Status of Implementation

• Implementation still in prototypical state• “Does a lot of stupid things”

– conflict minimization by deletion filtering– checking that conflict are in fact minimal– unnecessary calls to LA for SAT clusters– calling LA solver implemented as dump on file,

and run external MathSAT– huge conflict sets

44

A very very preliminary evaluation

45

Competitors

• Run against MiniSAT 1.14– ~ winner of SAT competition in 2005

• KEY REMARK: – boolean methods are very mature

• A good reason for giving up?

46

Test benches

• 74 benchmarks from industrial partner– would have been ideal for SMT-COMP

• QF_UFBV32

• Unfortunately– can not be disclosed– “will have to be destroyed after the

collaboration”– hopefully our lives will be spared

47

48

49

Conclusions

• A “market need” for SMT(BV) solvers• Bit Blasting: tough competitors• After a failure, …• Preliminary results are encouraging

• Future challenges– optimize BV solver– better conflict sets– tackle some RTL verification cases– extension to memories

50

A small digression on QF_UFBV32 at SMT-COMP

51

QF_UFBV[32] at SMT-COMP

• the MathSAT you will see there IS NOT the one I described

• We currently have no results for QF_UFBV

• Easy benchmarks:– QF_UFBV[32] not particularly “SMT”– the boolean component is nearly missing– the BV part is “easily” solvable by bit blasting

• We entered SMT-COMP QF_UFBV32– MathSAT based on BIT BLASTING to SAT– NuSMV based on bit blasting to BDDs

52

QF_UFBV: Bit Blasting to SAT

• Preprocessing based on– Ackerman’s elimination of function symbols– rewriting simplification– bit blasting

• Core: call SAT solver underlying MathSAT– every SAT problem in < 0.3 secs– most UNSAT within seconds– a handful of hard ones between 300 and 500

secs

53

BDDs (???) on SMT-COMP tests

• Even NuSMV entered SMT-COMP• Ackerman’s elimination of functional symbols• Rewriting preprocessor• Core solver

– based on BDDs– conjunctively partitioned problem– structural BDD-based ordering (bit interleaving)– (almost) no dynamic reordering– affinity-based clustering, threshold 100 nodes– early quantification

• Seems to work well both on SAT and UNSAT instances

54

RESULTS

• first STP• then YICES• then NuSMV• then CVC3 (but no results on two samples)• then MathSAT BITBLASTING

– 3rd on SAT– last on UNSAT

55

0.01

0.1

1

10

100

1000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

YICES

NUSMV

CVC3

MATHSAT

SAT instances

56

UNSAT instances

0.01

0.10

1.00

10.00

100.00

1000.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

YICES

NUSMV

CVC3

MATHSAT