1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
|
(theory Fixed_Size_BitVectors
:written_by {Silvio Ranise, Cesare Tinelli, and Clark Barrett}
:date {May 7, 2007}
:notes
"Against the requirements of the current SMT-LIB standard this theory does
not provide a value for the formal attributes :sorts, :funs, and :preds.
The reason is that the theory has an infinite number of sort, function, and
predicate symbols, and so they cannot be specified formally in the current
SMT-LIB language. While extending SMT-LIB's type system with dependent
types would allow a finitary formal specification of all the symbols in
this theory's signature, such an extension does not seem to be worth the
trouble at the moment. As a temporary ad-hoc solution, this theory
declaration specifies the signature, in English, in the user-defined
attributes :sorts_description, :funs_description, and :preds_description.
"
:sorts_description {
All sort symbols of the form BitVec[m],
where m is a numeral greater than 0.
}
:funs_description {
Constant symbols bit0 and bit1 of sort BitVec[1]
}
:funs_description {
All function symbols with arity of the form
(concat BitVec[i] BitVec[j] BitVec[m])
where
- i,j,m are numerals
- i,j > 0
- i + j = m
}
:funs_description {
All function symbols with arity 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 arity 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
}
:preds_description {
All predicate symbols with arity of the form
(pred BitVec[m] BitVec[m])
where
- pred is from {bvult}
- m is a numeral greater than 0
}
:definition
"This is 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.
The theory is defined semantically as follows.
The sort BitVec[m] (for m > 0) is the set of finite functions
whose domain is the initial segment of the naturals [0...m), meaning
that 0 is included and m is excluded, and the co-domain is {0,1}.
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-1} -> {0,1}], the set of total
functions from {0,...,m-1} to {0,1}.
- Constant symbols bit0 and bit1 of sort BitVec[1]
[[bit0]] := \lambda x : [0,1). 0
[[bit1]] := \lambda x : [0,1). 1
- Function symbols for concatenation
[[(concat s t)]] := \lambda x : [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)]] := \lambda x : [0...i-j+1). [[s]](j+x)
where s is of sort BitVec[l], 0 <= j <= i < l.
- Bit-wise operations
[[(bvnot s)]] := \lambda x : [0...m). if [[s]](x) = 0 then 1 else 0
[[(bvand s t)]] := \lambda x : [0...m).
if [[s]](x) = 0 then 0 else [[t]](x)
[[(bvor s t)]] := \lambda x : [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
To define the semantics of the bitvector arithmetic operators, we first
introduce some additional definitions:
o (x div y) where x and y are integers with x >= 0 and y > 0 returns the
integer part of x divided by y (i.e., truncated integer division).
o (x rem y) where x and y are integers with x >= 0 and y > 0 returns the
remainder when x is divided by y. Note that we always have the following
equivalence (for y > 0): (x div y) * y + (x rem y) = x.
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
b(m-1)*2^{m-1} + ... + b(0)*2^0 = n rem 2^m
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
nat2bv[m](bv2nat([[s]]) div bv2nat([[t]]))
[[(bvurem s t)]] := if bv2nat([[t]]) != 0 then
nat2bv[m](bv2nat([[s]]) rem bv2nat([[t]]))
- 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 the binary predicate bvult:
(bvult s t) is interpreted to be true iff bv2nat([[s]]) < bv2nat([[t]])
Note that the semantic interpretation above is underspecified because it
does not specify the meaning of (bvudiv s t) or (bvurem s t) in case
bv2nat([[t]]) is 0. Since the semantics of SMT-LIB's underlying logic
associates *total* functions to function symbols, we then consider as models
of this theory *any* interpretation conforming to the specifications above
(and defining bvudiv and bvurem arbitrarily when the second argument
evaluates to 0). Benchmarks using this theory should only include a
:status sat or :status unsat attribute if the status is independent of
the particular choice of model for the theory.
"
)
|
