Combining a SAT solver with HardCaml to prove circuit equivalence.

## SAT solving

Given a boolean function $f(x,y,..)$ a SAT solver will attempt to find assignment to the variables $x,y,..$ that make the function $f(x,y,..)=1$. The SAT solver will provide one of three possible answers.

**SAT**the solver found a solution and will return values for the input variables**UNSAT**no solution exists**FAILURE**the solver failed to find an answer

### Equivalence checking

An interesting application of a SAT solver is to prove that two functions are equivalent, even if their implementations are different. For example, consider the function $f$

$$f(x,y) = (x ∧ y)′$$

by Demorgans law we know the funtion $g$ is equivalent

$$g(x,y) = x′ ∨ y′$$

We somehow need to *connect* these functions and the obvious way would be to compare the
functions for equality.

$$f=g$$

This is wrong, however. If the solver returns SAT then all it has shown is that there exists at least one assignment to $x$ and $y$ for which $f$ and $g$ produce the same result. UNSAT would tell us there are no possible assignments to $x$ and $y$. In fact we want to compare the functions for inequality

$$f≠g$$

Now if we receive SAT we will get an assignment to $x$ and $y$ for which the functions produce different results, while UNSAT will tell us that there is no possible assignment to $x$ and $y$ that makes the functions differ i.e. they are functionally the same.

## HardCamlBloop

The HardCamlBloop library provides an interface between HardCaml designs and the MiniSat solver. Any design expressed with the HardCaml combinatorial API can be converted to a SAT problem.

The library has not been released to OPAM at this time as the minisat library it depends on is not in OPAM. If anyone is interested in using the library please start an issue on Github and I will provide instructions for getting it to work.

## Example; instruction trap decoding

My first proper use of a SAT solver was to show that the trap signal in a RISC-V instruction decoder was generated correctly.

Each RISC-V instruction can be specified with a 32 bit mask and match pair. Some parts of the instruction pattern, such as a register address or immediate, can take any value while others must match a certain bit pattern.

If we repeat this for each instruction, or them all together then not the result we have a function indicating that no instruction was correctly decoded and a trap should be generated.

This provides a simple and understandable implementation for deriving the trap signal.

The actual RTL implementation of the instruction decoder is a bit more complex; it decodes various signals to control the internal pipeline and attempts to optimise the critical timing path and area. None-the-less it needs to generate the trap signal correctly. To prove this we can use the SAT solver from HardCamlBloop.

It is very useful that if the SAT solver finds a case where the implementations do not match it will report the instruction bit pattern that causes the problem.

## Example; Comparing HardCaml and Yosys techlibs

One of my current projects is to use the open source synthesis tool Yosys to convert a verilog design into a JSON netlist then load that into HardCaml.

The Yosys tool can represent the design at various abstraction levels from a very low level version made up of primitive gates to a higher level roughly consistent with HardCamls structural representation.

In the structural representation the design is made up of modules from the Yosys tech library called simlib. Basically, the behavioural constructs of verilog are compiled into registers and muxes and verilog operators are represented by simlib modules.

In order to load this into HardCaml we need a HardCaml version of simlib. Here is an example module from simlib.

One thing Yosys has not done for us is abstract away the fiendish details of the verilog resizing rules; both operands and the result may be different widths and there are implicit rules regarding how sign extension should happen. One small mistake in the HardCaml implementation of these rules will lead to VERY hard to find bugs later on.

### Enter the yosys sat solver

I decided to use SAT to prove that my simlib implementation was consistent with the yosys version. To do this I generated a verilog file which takes the same arguments as the simlib module and produces one signal called check which compares the hardcaml and yosys implementations. We can then run yosys as follows

```
yosys> read_verilog simlib.v my_sat_checking_design.v
yosys> hierarchy -top my_sat_checking_design; proc; flatten
yosys> sat -prove check 0 my_sat_checking_design
SAT proof finished - no model found: SUCCESS!
```

One complication is that this only proves the circuits are equivalent for some given combination of parameters (A_WIDTH, B_SIGNED etc). Thus I set up a bulk test which ran about 200 different parameter combinations (a mixture of handcrafted and random tests) for each module. In total I ran about 6000 tests over 35 modules in about 5 minutes.