Bit-Vector Research Page
This page contains actual information about developments and results
concerning the theory of bit-vectors with a special focus to the
construction of solver algorithms.
Bit-vectors are - in the context of hardware verification - data
structures, which represent elements of a mutisorted logic (i.e. a
bit-vector can be of the type bvec, bvec,...). The width 0 is
illegal.The bits are counted from left to right, starting with 0. In
contrast to arrays, bit-vectors are considered finite objects, each
position containing a value out of a binary alphabet.
Typical operations are
Additionally, the following operations are found frequently:
- Extraction :: <bvec> [<nat>:<nat>]
e.g. x[0:6] means the seven leftmost bits of x (x: bvec[n], n>= 7)
- Concatenation :: <bvec> o <bvec>
t.i. the associative concatenation of bit-vectors (like x o y o z)
- Fill :: Fill[<nat>](<Bit>)
Denoting a bit-vector of the width n, containing the same bit at
each position. This bit can be a constant or a variable.
The main focus of the research in Ulm is to construct efficient
solvers for a rich theory of bit-vector
- Boolean operations :: NOT <bvec>, <bvec> AND <bvec> , ...
are of course only valid, if the arguments have the same width
- arithmetical operations :: + *
Usually, an arithmetic modulo 2^n is intended, where n is the width
of the arguments. The leftmost bit is considered to be the least
- Bitextract :: <bvec> (<nat>)
Produces - as expected- a bit
- bv2nat(<bvec>) and nat2bv[<nat>](<nat>)
Transformation of bit-vectors to natural numbers (and vice versa);
The function nat2bv gets usually the width of the resulting bit-vector as an argument
As it turnes out, it is a crucial point whether to allow variable
width (e.g. vie introducing a dependend type bvec[n], where n is a
integer variable itself) or variable extraction (terms x[i:j] where
i and j are integer variables). If one of these operations is
permitted, the theory is no longer convex in the sense that a
solution is always a conjunction of equations.
Those should be build according to the
requirements from the theory of congruence closure (Nelson-Oppen), explicitly formalised in the CADE-paper 1996 of Patt Lincoln, David Cyrluk und Natarjan Shankar).
The next step was the introduction of boolean operators (David
Cyrluk/Oliver Möller, SRI, October 96), using ordered binary
decision diagrams (for a demonstration see OBDDs ).
A solver for simple bit-vector terms (fix extraction, known length, no boolean
connectors) was given in the technical report of Harald Ruess/Oliver
Möller (both Ulm) and David Cyrluk (SRI),
called "An Efficient Decision Procedure for a Theory of Fixed-Sized Bit-vectors
with concatenation and extraction":
The algorithm (in Common Lisp) was implemented and a paper titled
"An Efficient Decision Procedure for the
theory of Fixed-Sized Bit-Vectors" is published in the CAV97:
And a recent result, adressing non-fixed size is found here:
You might also get a short overview on the topic by looking through the
In connection with the work on the diploma thesis "Solving
Bit-Vector Equations - A Decision Procedure for Hardware
Verification" a number of decision procedures were implemented in
Allegro Common Lisp and are available here:
Solver for Bit-Vector Theories [in ACL]
||WS1S with MONA
concatenation and (variable) extraction
gzipped tar archives, uncompress via gtar -xvzf
; to get all in one archive (73,4 KB), click here|
The thesis itself can be obtained at Diploma Thesis. It contains a
description of the solvers as well as some new complexity results.
E.g. it is shown that no solver can exist for the theory of
bit-vectors, if variable width, variable extraction and boolean
operations are allowed.
If you have any comments, your feedback is apprechiated: email@example.com
Last modified: Tue Aug 24 12:51:15 1999