# Debug-SAT

A debuggable automatic theorem prover for boolean satisfiability problems (SAT).

Brought to you by the AdvancedResearch Community.

Designed for debugging and introspection rather than performance.

NB! This library might contain bugs. Do not use in safety critical applications!

This library can be used to:

• Learn basic logic theorem proving
• Design and verify boolean circuits
• Create machine checked proofs in propositional calculus for many variables
• Used as theorem prover assistant by using tactics step by step
• Optimize partial proofs by selecting from equivalent expressions

### How to use it

The `Graph::var` method adds a new variable. Give it a unique id.

When creating a gate, you use the variables of previously created gates.

```use debug_sat::Graph;

let ref mut graph = Graph::new();
let a = graph.var(0);
let b = graph.var(1);
let a_and_b = graph.and(a, b);```

There is one method for the following 5 logical gates (selected for readability):

• AND
• OR
• NOT
• EQ
• IMPLY

Other gates are considered less readable, but can be created by combining these 5 gates. For example, if you need XOR, use `not(eq(a, b))`.

By default, variables and expressions have no value, which are added by making assumptions. An assumption is added to a history stack and can be undoed or inverted.

There are two kinds of assumptions: Equality and inequality. Instead of saying that a variable `a` is `true`, you say that `a` is equivalent to `true` or not equivalent to `false`.

The `Graph::are_eq` method is used to check the value of an variable or expression.

```use debug_sat::Graph;

let ref mut graph = Graph::new();
let a = graph.var(0);
let tr = graph.true_();
let a_is_tr = graph.assume_eq(a, tr);
assert_eq!(graph.are_eq(a, tr), Some(true));
a_is_tr.undo(graph); // Alternative: `a_is_tr.invert(graph)````

When you add new stuff to the theorem prover, it does not automatically know the right answer. This requires executing tactics or solving (runs all tactics).

```use debug_sat::{Graph, Proof};

let ref mut graph = Graph::new();
let a = graph.var(0);
let b = graph.var(1);
let a_and_b = graph.and(a, b);
let a_is_b = graph.assume_eq(a, b);
// Does not know that `and(a, b) = a` when `a = b`.
assert_eq!(graph.are_eq(a_and_b, a), None);
// Run the tactic that checks input to AND is EQ.
// This is how the theorem prover learns, by checking stuff.
// Alternative: `graph.solve()` runs all tactics until nothing new is learned.
assert_eq!(graph.eq_and(a_and_b), Proof::True);
// Now the theorem prover knows the answer.
assert_eq!(graph.are_eq(a_and_b, a), Some(true));```

For more information about tactics do, see the `Proof` enum.

### Design

Uses a graph that links expressions together. Expressions that are grammatically identical has the same id. Commutative operators are ordered on insertion of a new expression, to make them trivial equivalent. New expressions are never deleted from the graph, even after making new assumptions. This does not affect soundness because expressions only have value by their constraints.

Tactics are based on entangled functions under equality and inequality plus normal currying. One nice mathematical property of entangled boolean functions of 2 arguments is that every equality of inequality constraint reduces the function to 0 or 1 arguments. This corresponds to natural deduction, so the tactics can be used for assisted theorem proving. You can find more information about this in papers about entangled functions in the research repository for Path Semantics.

A first-order Havox diagram is used prove equality and inequality of expressions. First order means it does not infer between edges. Law of excluded middle is added as tactic to specialize inference on boolean logic. Relations are stored on the current provable minimum ids in the moment of insertion. This solves a soundness problem when searching for representatives for equivalence classes (not sure why it works at this moment, but believe it got something to with ordering the relations). It also accelerates proof of inequality since these relations are often looked up directly. For more information about Havox diagrams, see papers about Havox diagrams in the research repostiory for Path Semantics.

## Structs

 Graph Binds expressions together with constraints and acceleration data structures.

## Enums

 Assumption Stores assumption, which is extra information added that might lead to inconsistency. Expression Stores a logical expression. Proof Stores result of a proof.