FixedSizeBitVectors
(theory FixedSizeBitVectors
:smt-lib-version 2.7
:smt-lib-release "2024-07-21"
:written-by "Clark Barrett, Pascal Fontaine, Silvio Ranise, and Cesare Tinelli"
:date "2010-05-02"
:last-updated "2025-02-25"
:update-history
"Note: history only accounts for content changes, not release changes.
2025-02-25 Renamed and updated conversion operators to/from integers.
2024-07-21 Updated to Version 2.7.
2024-07-16 Added conversion operators between bitvectors and integers.
2024-07-14 Fixed minor typos.
2023-11-29 Added bvnego bvuaddo bvsaddo bvumulo bvsmulo.
2020-05-20 Fixed minor typo.
2017-06-13 Added :left-assoc attribute to bvand, bvor, bvadd, bvmul
2017-05-03 Updated to version 2.6; changed semantics of division and
remainder operators.
2016-04-20 Minor formatting of notes fields.
2015-04-25 Updated to Version 2.5.
2013-06-24 Renamed theory's name from Fixed_Size_Bit_Vectors to FixedSizeBitVectors,
for consistency.
Added :value attribute.
"
:notes
"This theory declaration defines a core theory for fixed-size bitvectors
where the operations of concatenation and extraction of bitvectors as well
as the usual logical and arithmetic operations are overloaded.
"
:sorts_description "
All sort symbols of the form (_ BitVec m)
where m is a numeral greater than 0.
"
; Bitvector literals
:funs_description "
All binaries #bX of sort (_ BitVec m) where m is the number of digits in X.
All hexadeximals #xX of sort (_ BitVec m) where m is 4 times the number of
digits in X.
"
:funs_description "
All function symbols with declaration of the form
(concat (_ BitVec i) (_ BitVec j) (_ BitVec m))
where
- i, j, m are numerals
- i > 0, j > 0
- i + j = m
"
:funs_description "
All function symbols with declaration of the form
((_ extract i j) (_ BitVec m) (_ BitVec n))
where
- i, j, m, n are numerals
- m > i ≥ j ≥ 0,
- n = i - j + 1
"
:funs_description "
All function symbols with declaration of the form
(op1 (_ BitVec m) (_ BitVec m))
or
(op2 (_ BitVec m) (_ BitVec m) (_ BitVec m))
where
- op1 is from {bvnot, bvneg}
- op2 is from {bvand, bvor, bvadd, bvmul, bvudiv, bvurem, bvshl, bvlshr}
- m is a numeral greater than 0
The operators in {bvand, bvor, bvadd, bvmul} have the :left-assoc attribute.
"
:funs_description "
All function symbols with declaration of the form
(bvult (_ BitVec m) (_ BitVec m) Bool)
where
- m is a numeral greater than 0
"
:funs_description "
If the Ints theory is also present, all function symbols with declarations of the form
(ubv_to_int (_ BitVec m) Int)
or
(sbv_to_int (_ BitVec m) Int)
or
((_ int_to_bv m) Int (_ BitVec m))
where
- m is a numeral greater than 0.
Note that only a single operator for conversions from integers to bitvectors is needed,
as the semantics are the same whether the intention is to convert to a signed or
unsigned bitvector. See the definition for more details.
"
:definition
"For every expanded signature Sigma, the instance of FixedSizeBitVectors
with that signature is the theory consisting of all Sigma-models that
satisfy the constraints detailed below.
The sort (_ BitVec m), for m > 0, is the set of finite functions
whose domain is the initial segment [0, m) of the naturals, starting at
0 (included) and ending at m (excluded), and whose co-domain is {0, 1}.
To define some of the semantics below, we need the following additional
functions :
o div and mod, binary infix functions on integers defined as the
interpretations of the corresponding operators in the Ints theory,
i.e., the operators satisfying:
(forall ((m Int) (n Int))
(=> (distinct n 0)
(let ((q (div m n)) (r (mod m n)))
(and (= m (+ (* n q) r))
(<= 0 r (- (abs n) 1))))))
Note that an important consequence of the above definition is that
for n > 0, m mod n is always in the range [0, n).
o bv2nat, which takes a bitvector b: [0, m) → {0, 1}
with 0 < m, and returns an integer in the range [0, 2^m),
and is defined as follows:
bv2nat(b) := b(m-1)*2^{m-1} + b(m-2)*2^{m-2} + ⋯ + b(0)*2^0
o nat2bv[m], with 0 < m, which takes a non-negative integer n and returns
the (unique) bitvector b: [0, m) -> {0, 1} such that:
bv2nat(b) = n mod 2^m
The semantic interpretation [[_]] of well-sorted BitVec-terms is
inductively defined as follows.
- Variables
If v is a variable of sort (_ BitVec m) with 0 < m, then
[[v]] is some element of {[0, m) → {0, 1}}, the set of total
functions from [0, m) to {0, 1}.
- Constant symbols
The constant symbols #b0 and #b1 of sort (_ BitVec 1) are defined as follows
[[#b0]] := λx:[0, 1). 0
[[#b1]] := λx:[0, 1). 1
More generally, given a string #b followed by a sequence of 0's and 1's,
if n is the numeral represented in base 2 by the sequence of 0's and 1's
and m is the length of the sequence, then the term represents
nat2bv[m](n).
The string #x followed by a sequence of digits and/or letters from A to
F is interpreted similarly: if n is the numeral represented in hexadecimal
(base 16) by the sequence of digits and letters from A to F and m is four
times the length of the sequence, then the term represents nat2bv[m](n).
For example, #xFF is equivalent to #b11111111.
- Function symbols for concatenation
[[(concat s t)]] := λx:[0, n+m). if (x < m) then [[t]](x) else [[s]](x - m)
where
s and t are terms of sort (_ BitVec n) and (_ BitVec m), respectively,
0 < n, 0 < m.
- Function symbols for extraction
[[((_ extract i j) s))]] := λx:[0, i-j+1). [[s]](j + x)
where s is of sort (_ BitVec l), 0 ≤ j ≤ i < l.
- Bit-wise operations
[[(bvnot s)]] := λx:[0, m). if [[s]](x) = 0 then 1 else 0
[[(bvand s t)]] := λx:[0, m). if [[s]](x) = 0 then 0 else [[t]](x)
[[(bvor s t)]] := λx:[0, m). if [[s]](x) = 1 then 1 else [[t]](x)
where s and t are both of sort (_ BitVec m) and 0 < m.
- Arithmetic operations
Now, we can define the following operations. Suppose s and t are both terms
of sort (_ BitVec m), m > 0.
[[(bvneg s)]] := nat2bv[m](2^m - bv2nat([[s]]))
[[(bvadd s t)]] := nat2bv[m](bv2nat([[s]]) + bv2nat([[t]]))
[[(bvmul s t)]] := nat2bv[m](bv2nat([[s]]) * bv2nat([[t]]))
[[(bvudiv s t)]] := if bv2nat([[t]]) = 0
then λx:[0, m). 1
else nat2bv[m](bv2nat([[s]]) div bv2nat([[t]]))
[[(bvurem s t)]] := if bv2nat([[t]]) = 0
then [[s]]
else nat2bv[m](bv2nat([[s]]) mod bv2nat([[t]]))
We also define the following predicates
[[(bvnego s)]] := bv2int([[s]]) == -2^(m - 1)
[[(bvuaddo s t)]] := (bv2nat([[s]]) + bv2nat([[t]])) >= 2^m
[[(bvsaddo s t)]] := (bv2int([[s]]) + bv2int([[t]])) >= 2^(m - 1) or
(bv2int([[s]]) + bv2int([[t]])) < -2^(m - 1)
[[(bvumulo s t)]] := (bv2nat([[s]]) * bv2nat([[t]])) >= 2^m
[[(bvsmulo s t)]] := (bv2int([[s]]) * bv2int([[t]])) >= 2^(m - 1) or
(bv2int([[s]]) * bv2int([[t]])) < -2^(m - 1)
- Shift operations
Suppose s and t are both terms of sort (_ BitVec m), m > 0. We make use of
the definitions given for the arithmetic operations, above.
[[(bvshl s t)]] := nat2bv[m](bv2nat([[s]]) * 2^(bv2nat([[t]])))
[[(bvlshr s t)]] := nat2bv[m](bv2nat([[s]]) div 2^(bv2nat([[t]])))
Finally, we can define bvult:
[[bvult s t]] := true iff bv2nat([[s]]) < bv2nat([[t]])
If the Ints theory is also present, let [[_]] be extended to additionally
interpret terms of sort Int. For operations in that theory, [[_]]
interprets them according to the semantics defined there. We further
define:
[[(ubv_to_int s)]] := bv2nat([[s]])
[[(sbv_to_int s)]] := if [[s]](m-1) = 0 then bv2nat([[s]])
else bv2nat([[s]]) - 2^m
where s is of sort (_ BitVec m) and 0 < m. We also define:
[[((_ int_to_bv N) x)]] := nat2bv[[[N]]]([[x]] mod 2^[[N]])
where N is a numeral greater than 0 and x is a term of sort Int.
Notice that the semantics are correct regardless of whether the intended
target is a signed or unsigned bitvector. For example:
[[((_ int_to_bv 4) -7)]] = [[((_ int_to_bv 4) 9)]] = 1001.
In both cases, the mod operation (as defined above to always return a
nonnegative value) leads to the correct result.
"
:values
"For all m > 0, the values of sort (_ BitVec m) are all binaries #bX with m digits.
"
:notes
"After extensive discussion, it was decided to fix the value of
(bvudiv s t) and (bvurem s t) in the case when bv2nat([[t]]) is 0.
While this solution is not preferred by all users, it has the
advantage that it simplifies solver implementations. Furthermore,
it is straightforward for users to use alternative semantics by
defining their own version of these operators (using define-fun) and
using ite to insert their own semantics when the second operand is
0.
"
)
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