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Satisfiability modulo the Theory of Bit Vectors

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Title: Satisfiability modulo the Theory of Bit Vectors


1
Satisfiability modulothe Theory of Bit Vectors
  • Alessandro Cimatti
  • IRST, Trento, Italy
  • cimatti_at_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 xy f(x) ! f(y)
  • difference logic x y lt 7
  • linear arithmetic 3x 2y lt 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
7
Two Main Approaches to SMT
  • the eager approach
  • the lazy approach
  • theory independent view
  • theory specific view

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
(No Transcript)
10
Lifted theory
Propositionalstructure
P P P
TA TA TA TA
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
TA TA TA TA
x y z w x
x y z w x
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 lt 1)) (y lt
    2) (P xor Q)
  • Boolean model
  • !P, (x 3), Q, (x y lt 1), (y lt 2)
  • Check (x 3), (x y lt 1), (y lt 2)
  • Theory contradiction!
  • Another boolean model
  • P , !(x 3) , !Q, (x y lt 1), (y lt 2)
  • Check !(x 3), (x y lt 1), (y lt 2)
  • Consistent e.g. x 0, y 0

14
Boolean SAT search space
P
Q
Q
R
S
S
T
S
T
R
R
?
?
?
T SAT!
?
?
  • The DPLL procedure
  • Incremental construction of satisfying assignment
  • Backtrack/backjump on conflict
  • Learn reason for conflict
  • Splitting heuristics

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
MathSAT search space
P
Q
Q
R
S
S
T
S
T
R
R
Bool ?
Bool T Math ?
Bool ?
Bool T Math T SAT!
Bool T Math ?
Bool ?
  • Many boolean models are not theory consistent!

17
Early pruning
  • Check theory consistency of partial assignments

P
EPMath ?
EPMath T
Q
EPMath T
S
Pruned away in the EP step
EPMath T
T
EPMath T
R
Bool ?
Bool T Math T SAT!
18
THEORY OF FIXED-WIDTH BIT VECTORS
19
Bit Vectors Example
input a, b, c, d regN
  • 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 ltlt 1
  • RTmp2 c RTmp1
  • RTmp3 RTmp2 ltlt 1
  • RTmp4 b RTmp3
  • RTmp5 RTmp4 ltlt 1
  • ROut a RTmp5
  • a ((b ((c (dltlt1)) ltlt1)) ltlt1)

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 x150
  • concatenation y z
  • bitwise operators x y, z w,
  • arithmetic operators x y, z w,
  • shifting x ltlt 2, y gtgt 3
  • Predicate symbols
  • comparators , ? , gt , lt , ,

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 lt-gt y0) (x1 lt-gt y1) (x63 lt-gt 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
input a, b, c, d regN
  • 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 ltlt 1
  • RTmp2 c RTmp1
  • RTmp3 RTmp2 ltlt 1
  • RTmp4 b RTmp3
  • RTmp5 RTmp4 ltlt 1
  • ROut a RTmp5
  • a ((b ((c (dltlt1)) ltlt1)) ltlt1)

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 lt 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 310 as x in range 0, 232)
  • Assignments treated as equality, e.g. x y
  • Arithmetic, e.g. z x y
  • Linear arithmetic? not quite! BV Arithmetic is
    modulo 2N
  • z x y - 2N s, with z in 0, 2N)
  • Concatenation x y as 2n x y
  • Selection relational encoding (based on
    integrity)
  • x2316 as xm, where
  • x 224 xh 216 xm xl, xl in 0, 216), xm
    in 0, 28), xl in 0, 28)
  • 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 2N
  • 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 x70 y30 else x52
    y103
  • 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
The architecture
Boolean enumeration
BV solver
EUF reasoning
LIAencoding
BV rewriter
39
Rewriting rules
  • evaluation of constant terms
  • 0b8_0101010142 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 zh_n y zl_n

40
Rewriting rules
  • pushing selections
  • (x y)70 becomes (x70 y70)
  • (x y)238 becomes (x70 y158)
  • pigeon-hole rules
  • from (x ! 0 x ! 1 x ! 2 x lt 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
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48
(No Transcript)
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_UFBV32 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_UFBV32 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
  • Ackermans elimination of function symbols
  • rewriting simplification
  • bit blasting
  • Core call SAT solver underlying MathSAT
  • every SAT problem in lt 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
  • Ackermans 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
SAT instances
56
UNSAT instances
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