1. Daniel Kroening and Ofer Strichman Decision Procedure
Bit Vector
Daniel Kroening and Ofer Strichman
Decision Procedure

2. Decision procedures Decision procedures which we learnt..
SAT Solver
BDDs
Decision procedure for equality logic …
However, what kind of logic do we need to express bit- wise operations and bit-wise arithmetic?
Logics which we covered can not express those kind of operations.
We need bit-vector logic.

3. We need bit-vector logic
Bit-wise operators : bit-wise AND, shift …
Bit-wise arithmetic : bit addition, bit multiplication …
Since bit-vector has finite domain, so we need to consider overflow problem which can not be happened in unbounded type operations, such as integer domain.
We want to verify large formulas
Program analysis tools that generate bit-vector formulas:
CBMC
SATABS
F-Soft …

4. Contents Introduction to bit-vector logic Syntax Semantics
Decision procedures for bit-vector logic
Flattening bit-vector logic
Incremental flattening
Conclusion

5. Bit-vector logic syntax

6. Semantics Following formula obviously holds over the integer domain:
However, this equivalence no longer holds over the bit- vectors.
Subtraction operation may generate an overflow.
Example

7. Width and Encoding
The meaning of a bit-vector formula obviously depends on
the width of the expression in bits
the encoding - whether it is signed or unsigned
Typical encodings:
Binary encoding - unsigned
Two’s complement - signed

8. Examples The width of the expression in bits The encoding
unsatisfiable for one bit wide bit vectors, but satisfiable for larger widths.
The encoding
means different with respect to each encoding schemes.
Notation to clarify width and encoding
width in bits
U: unsigned binary encoding
S : signed two’s complement

9. Definition of bit-vector
Definition. A bit vector b is a vector of bits with a given length l (or dimension) :
The i-th bit of the bit vector is denoted by …
bits

10. λ - Notation for bit-vectors
A lambda expression for a bit vector with bits has the form
is an expression that denotes the value of the i-th bit.
Example
The expression above denotes the bit vector

11. Examples (cond.) The vector of length l that consists of zeros:
A function that inverts a bit vector:
A bit-wise OR:

12. Semantics for arithmetic operators (1/3)
What is the answer for the below C program ?
On 8 bits architectures, this is 44 which is not 300.
Therefore, Bit vector arithmetic uses modular arithmetic.

13. Semantics for arithmetic operators (2/3)
Semantics for addition and subtraction:
Semantics for relational operators:

14. Semantics for arithmetic operators (3/3)
Semantics for shift :
logical left shift
logical right shift
arithmetic right shift - the sign bit of a is replicated

15. Decision procedure for bit-vector
Bit-vector flattening
Most commonly used decision procedure
Transform bit-vector logic to propositional logic, which is then passed to SAT solver.
Algorithm
Input : A formula in bit-vector arithmetic
Output : An equisatisfiable Boolean formula
Convert each term into new Boolean variable
Set each bit of each term to a new Boolean variable
Add constraint for each atom
Add constraint for each term

16. Example Bit-vector formula Convert each term into new Boolean variable
Set each bit of each term to a new Boolean variable
Add constraint for each atom
Add constraint for each term

17. Example (l-bit Adder) 1-bit adder can be defined as follows:
Carry bit can be defined as follows:

18. Example (l-bit Adder) l-bit Adder can be defined as follows:
The constraints generated by algorithm for the formula is following:

19. Incremental bit flattening (1/4)
Some arithmetic operation result in very hard formulas
Multiplication
Multiplier is defined recursively for , where denotes the width of the second operand:
Therefore, we want to check satisfiability of a given formula without checking satisfiability of sub-formulas which have complicated arithmetic operations such as multiplication.

20. Incremental bit flattening (2/4)
Example
This formula is obviously unsatisfiable
Since first two conjuncts are inconsistent and last two conjuncts are also inconsistent.
SAT solver wants to make a decision of first two conjuncts because a and b are used frequently than x and y.
However, this decision isn’t good because last two conjuncts are rather easy to check satisfiability since relation bit-vector operation is less complicate than multiplication bit-vector operation.

21. Incremental bit flattening (3/4)
UNSAT
SAT
YES
Pick ‘easy’ part
convert to CNF
: Boolean part of
: set of terms that encoded to CNF formula
: set of terms that are inconsistent with the current satisfying assignment

22. Incremental bit flattening (4/4)
Idea : add ‘easy’ parts of the formula first
Only add hard parts when needed
only gets stronger - that’s why it is incremental

23. Conclusion
We can compute bit-wise operations and arithmetics using bit-vector logic.
There are decision procedures which check satisfiability of given bit-vector logic formula.