Combinatorial Problem Solving (CPS)
Project: Logic Synthesis.
Updated: March 19, 2024
1 Description of the Problem
In digital circuit theory, a logical circuit has a given number of inputs and
one output. By assigning 0/1 signals to the inputs, a 0/1 signal is obtained
at the output, which is a Boolean function of the input signals.
Given a specification of a logical circuit, e.g., by means of the truth
table of the output as a function of the inputs, there are several ways to
implement the circuit physically. In this project we consider the problem of
synthesizing circuits built up of NOR gates (henceforth referred to as NORcircuits). A NOR gate is a device with two inputs x1 and x2 and one output
y that behaves as described in the truth table below:
x1 x2 y
0 0 1
0 1 0
1 0 0
1 1 0
In other words, the output has signal 1 if and only if neither of the inputs
has signal 1.
It is not difficult to see that any logical circuit can be implemented as
a NOR-circuit. For example, in Figure 1 we can see the truth table of the
AND function of x1 and x2, and a NOR-circuit that implements it. Notice
that, in addition to x1 and x2, we also allow that some of the circuit inputs
are constant 0 signals.
Another example can be seen in Figure 2. Note that input signals as
well as constant 0 signals may be repeated, and not all input signals need
to be used (input signal x1 is not used in this example).
The depth of a NOR-circuit is the maximum distance (i.e., the number
of gates in the path) between any of the inputs and the output. This is
an important parameter, as the time that the circuit needs to compute the
x1 x2 y
0 0 0
0 1 0
1 0 0
1 1 1
x1 x2 00
NORNOR
NOR
Figure 1: Truth table of y = AND(x1, x2) and NOR-circuit implementing it.
x1 x2 x3 y
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
x2 00
x2
x3
x3 NORNOR
NORNOR
NOR
Figure 2: Truth table of a function and NOR-circuit implementing it.
output signal when given the input signals is proportional to this value.
For example, the depth of the circuit of Figure 1 is 2, as the paths from
the inputs to the output all cross 2 NOR gates. Similarly, the depth of the
circuit of Figure 2 is 3.
Another relevant parameter related to the performance of the circuit is
its size, i.e., the total number of NOR gates: the more gates, the larger area
the circuit needs and the more power it consumes. For example, the NOR
circuit in Figure 1 has size 3, while the one in Figure 2 has size 5.
In this project our goal is to solve the NOR Logic Synthesis Problem
(NLSP): given a specification of a Boolean function f(x1, ..., xn) in the form
of a truth table, to find a NOR-circuit satisfying the specification that minimizes depth (and, in case of a tie in depth, with minimum size).
In order to simplify the problem, several assumptions are made:
ˆ Only NOR gates with 2 inputs and 1 output can be used: more general
NOR gates with more inputs are not allowed (i.e., the fan-in of NOR
gates is always 2).
ˆ The output of a NOR gate can only be the input of a single gate:
outputs cannot be reused as inputs of more than one gate (i.e., the
fan-out of NOR gates is always 1).
ˆ In addition to the input signals of the Boolean function to be implemented, constant 0 signals can also be used as inputs of NOR gates.
Constant 0 signals as well as input signals can be used as many times
as needed as inputs of NOR gates. On the other hand, the circuit does
not need to use all input signals. Similarly, the constant 0 signal does
not have to be used if it is not needed. See Figures 1 and 2.
2 Input and Output Formats
This section describes the format in which instances of NLSP are written
(Sect. 2.1), as well as the format for the corresponding solutions (Sect. 2.2).
2.1 Input Format
An instance of NLSP consists of several lines of integer values (Booleans
are represented with 0/1 as usual). The first line consists of the number
of input signals n of the Boolean function f(x1, . . . , xn) to be implemented
(where n ≥ 2). Then 2n
lines follow, which describe the truth table of
f. For each 0 ≤ i < 2
n
, the i-th of these lines has the following form.
Let αi1 αi2
· · · αin be the binary representation of i with n bits, and let
βi = f(αi1
, αi2
, . . . , αin
). Then the i-th line contains the value βi
.
As an example, the instance that corresponds to the function AND(x1, x2)
of Figure 1 is:
2
0
0
0
1
And the instance that corresponds to the function of Figure 2 is:
3
0
1
1
0
0
1
1
0
2.2 Output Format
The output starts with a copy of the input data. Then there is a line with
the integer values d and s, respectively the depth and the number of NOR
gates of an optimal circuit, separated with a blank space.
Then several lines describing each of the nodes of the circuit follow. Here
by node we mean a NOR gate, an input signal or a constant 0 signal.
Now, each line that describes a node should have the following format:
<id> <code> <left> <right>
where represents a blank space and:
ˆ id is an integer identifier of the node, which should range from 1 to
2
d+1 − 1 (note that a circuit with depth d can have at most 2d+1 − 1
nodes). The node whose output is the result of the Boolean function
should have identifier 1.
ˆ code is an integer code meaning:
• −1: NOR gate
• 0: constant 0 signal
• i (with 1 ≤ i ≤ n): input signal xi
ˆ left is the identifier of the node whose output is the left input of the
node under consideration. If the node is not a NOR gate, and hence
it has no inputs, then left is equal to 0 (which is an invalid node
identifier).
ˆ right is the same but for the right input.
For instance, the NOR-circuit in Figure 1 is optimal, and the output
could be:
2
0
0
0
1
2 3
1 -1 2 3
2 -1 4 5
4 1 0 0
5 0 0 0
3 -1 6 7
6 2 0 0
7 0 0 0
For the function in Figure 2 the circuit in Figure 2 is optimal, and the output
could be:
3
0
1
1
0
0
1
1
0
3 5
1 -1 2 3
2 -1 4 5
4 2 0 0
5 3 0 0
3 -1 6 7
6 -1 12 13
12 2 0 0
13 0 0 0
7 -1 14 15
14 3 0 0
15 0 0 0
3 Project
The purpose of this project is to model and solve NLSP with the three problem solving technologies considered in the course: constraint programming
(CP), linear programming (LP) and propositional satisfiability (SAT).
For the development of the project, in addition to this document, students will be provided with the following materials:
ˆ a suite of problem instances (in the format specified in Sect. 2.1). Files
are named nlsp n β.inp, where n is the number of input signals of
the circuit and β is the decimal representation of the truth table.
ˆ a checker that reads the output of solving a problem instance (following the format given in Sect. 2.2) from stdin, verifies that the circuit
fulfills its specification1 and plots it in PNG format. Use option -h or
--help to see all available options.
Note: GraphViz tools are required for the visualization. Although this
software is already installed in the lab machines, it can also be easily
installed in personal laptops. E.g., in Ubuntu, one just needs to type:
sudo apt-get install graphviz
ˆ a result table with the depth and size of the optimal solution of some
of the problem instances, to make debugging easier.
As a reference, solving processes that exceed a limit of 60 seconds of wall
clock time should be aborted (Linux command timeout may be useful). Take
into account that, depending on the solving technology, on the machine, etc.,
some of the instances may be too difficult to solve within this time limit.
For this reason, it is not strictly necessary to succeed in solving all instances
to pass the project. However, you are encouraged to solve as many instances
as possible.
There are three deadlines, one for each problem solving technology:
ˆ CP: 18 Apr.
ˆ LP: 16 May.
ˆ SAT: 17 Jun.
1The checker does not check that the circuit is optimal.
For each technology, a tgz or zip compressed archive should be delivered
via Rac´o (https://raco.fib.upc.edu) with the following contents:
ˆ a directory out with the output files of those instances that could be
solved. Please only provide the outputs of those instances for which
an optimal solution could be found. The output file corresponding to
an instance nlsp n β.inp should be named nlsp n β.out.
ˆ a directory src with all the source code (C++ programs, scripts,
Makefile, etc.) used to solve the problem, together with a README
file with basic instructions for compiling and executing so that results
can be reproduced.
Programs should read the instance from stdin and write the
solution to stdout. Other output (debugging information, etc.)
should be written to stderr. In particular, suboptimal solutions
should not be written to stdout, only the optimal solution (if found).
If say your executable is named p, then in the evaluation of the project
your results will be recomputed by calling p < nlsp n β.inp .
ˆ a document in PDF describing the variables and constraints that were
used in the model, as well as any remarks or comments you consider
appropriate.
Please follow these indications when submitting your deliveries.

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