Propositional Proof System (PPS)

This repository provides well tested implementations of some satisfiability algorithms for propositional formulas.

• Getting Started
• Propositional Formulas
  • Formulas in textual forms
  • Formulas as CNF object
• Implemented Satisfiability Algorithms
  • Truth Table Algorithm
  • Independent Set Algorithm
  • 2-SAT
• Check satisfiability of CNF object directly
• Check if ⍺ ⊨ β
• Check if ⍺ ≈ β
• Check for minimal unsatisfiability (MU)
• Convert a formula in NNF to (Q)CNF

Getting Started

Install the package using

npm install --save pps2

Yes, there is an existing npm package pps 💔

Use it in your code as follows.

// using require
const pps2 = require("pps2");
pps2.TruthTable.isSat("x and notx"); // false
pps2.TruthTable.isSat("x or notx"); // true

// or using ES6 import
import { TruthTable } from "pps2";
TruthTable.isSat("x and notx"); // false
TruthTable.isSat("x or notx"); // true

Propositional Formulas

As said above, this repository provides implementations of some satisfiability algorithms of propositional formulas, it expects the input formulas to be in CNF. Some valid examples of CNF are x ∧ y, x ∨ y, (x ∨ y) ∧ (¬x ∨ ¬y), etc. Only the following connectives in the formula are supported:

• AND (∧)
• OR (∨)
• NOT (¬)

Implication, and equivalence in the formula are not supported, and such formulas need to be converted to NNF first.

Input formula can be a string or a CNF object.

Formulas in textual forms

If the formula is a string then it needs to be in any of these following forms:

• (x ∨ y) ∧ (¬x ∨ ¬y) - using unicode chars
• (x OR y) AND (NOTx OR NOTy) - using textual forms (connectives are case insensitive; i.e. AND, and, or AnD refers same 'AND' connective).
• (x || y) && (!x || !y)

It also supports mixed forms such as (x OR y) ∧ (NOTx ∨ ¬y) 😕 (yes, there are test cases for that 😁), however doing so only affects the readability.

Any formula in textual form can be parsed to a CNF object. Example:

import {CNF} from "pps2";
const cnf: CNF = CNF.parse("x and notx");

Formulas as CNF object

Alternatively, you can instantiate a CNF object directly. As a CNF is a conjunction of Clause(s), which are in turn disjunction of Literal(s), to instantiate a CNF object needs more or less instantiating all three classes. Below is a minimal example to instantiate x ∧ ¬x.

import {Clause, CNF, Literal} from "pps2";
const cnf: CNF = new CNF([
    new Clause([new Literal("x")]), // x
    new Clause([new Literal("x", true)]) // not x
]);

The signature of Literal's ctor is as follows:

// by default all literals as positive if not specified otherwise
constructor(variable: string, isNegative: boolean = false);

Another slightly complex example of instantiating (x ∨ ¬y) ∧ (¬x ∨ y):

import {Clause, CNF, Literal} from "pps2";
const cnf: CNF = new CNF([
    new Clause([new Literal("x"), new Literal("y", true)]), // (x OR NOTy)
    new Clause([new Literal("x", true), new Literal("y")])  // (NOTx OR y)
]);

Though these examples looks more complex than simply parsing a textual formula, these constructors can be very useful to programmatically construct any arbitrary formula (how do you think the textual formulas are parsed:wink:).

Implemented Satisfiability Algorithms

In the basic form the satisfiability algorithms decide whether a propositional formula, in CNF, is satisfiable or not. These algorithms are implemented in their separate classes, which has a common function isSat with following signature.

isSat(cnf: CNF | string): boolean;

Truth Table Algorithm

Truth table is the most basic algorithm for satisfiability checking. If there are n variables in the CNF then in worst case, this algorithm will try all 2ⁿ binary combinations (truth assignments) for the formula. However, it returns true as soon as the first satisfying truth assignment is found.

Example:

import { TruthTable } from "pps2";

// unsatisfiable; returns false after trying all 8 truth assignments:
TruthTable.isSat("x AND (NOTx OR y) AND (NOTy OR z) AND NOTz");

// tautology(always satisfiable); returns true after trying only 1 truth assignment:
TruthTable.isSat("x OR NOTx");

Additionally, TruthTable can also return the satisfiable model for a given propositional formula in CNF. Example:

TruthTable.getModel("x or notx"); // [ Map { 'x' => false }, Map { 'x' => true } ]

Independent Set Algorithm

The basic idea of independent set algorithm is to count the number of unsatisfying truth assignments. If the count is 2ⁿ for a formula with n variable then the formula is unsatisfiable.

Example:

import { IndependentSet } from "pps2";

IndependentSet.isSat("x AND (NOTx OR y) AND (NOTy OR z) AND NOTz"); //false

IndependentSet.isSat("x OR NOTx"); //true

2-SAT

This algorithm first transforms every clause in CNF to implication form. Form the implication forms, it generates a directed graph, and concludes the formula to unsatisfiable if the graph contains a strongly connected component (SCC), with complimentary pair of literals in it.

import { TwoSAT } from "pps2";

// use a string
TwoSAT.isSat("x AND NOTx"); //false

//or use a CNF object
TwoSAT.isSat(new CNF(...));

Check satisfiability of CNF object directly

To make things semantically pleasing, CNF also exposes a isSat method with the help duck typing of SAT checkers (SATChecker). This means that you can do following:

const cnf = new CNF([...]);

// CNF#isSat signature: isSat(satChecker: SATChecker): boolean
// so we need to pass the algorithm we want use:

cnf.isSat(TruthTable);
// OR
cnf.isSat(IndependentSet);

Neat!

Check if ⍺ ⊨ β

Check if ⍺ ⊨ β (⍺ entails β); i.e. whenever ⍺ is true, if β is true as well for those truth assignments or not.

import { IndependentSet, TruthTable, TwoSAT } from "pps2";
const alpha = CNF.parse("x");
const beta = Clause.parse("x OR y");

IndependentSet.entails(alpha, beta); // true
TruthTable.entails(alpha, beta); // true
TwoSAT.entails(alpha, beta); // true

Note: ⍺ ⊨ β iff ⍺ ∧ ¬β is not satisfiable. The CNF based algorithms such as IndependentSet, and TwoSAT leverage this fact. Therefore for these algorithms ⍺ ∧ ¬β has to be a CNF. This puts the following restrictions for the said algorithms:

• ⍺ has to be a CNF, and
• β can be one of the following:
  • a Literal,
  • a disjunctive Clause, or
  • a CNF with only unit clauses.

Check if ⍺ ≈ β

Check if ⍺ ≈ β (⍺ is equivalent to β); i.e. for all truth assignments v, whether v(⍺) == v(β).

import { TruthTable } from "pps2";

// x, and NOTx should not be equivalent
const alpha = new CNF([new Clause([new Literal("x")])]);
const beta  = new CNF([new Clause([new Literal("x", true)])]);

TruthTable.isEquivalent(alpha, beta); // false

Check for minimal unsatisfiability (MU)

A CNF formula is minimally unsatisfiable if and only if the following conditions hold.

1. The CNF is unsatisfiable, and
2. Every subsets of clauses that can be formed by excluding a clause every time, are satisfiable.

To check if a CNF is in MU or not you can use MinimalUnsatisfiabilityChecker.

Example:

import { CNF, MinimalUnsatisfiabilityChecker } from "pps2";

const cnf = CNF.parse("(x) AND (NOTx) AND (y)");
MinimalUnsatisfiabilityChecker.isMinimallyUnsatisfiable(cnf); // false

Convert a formula in NNF to (Q)CNF

Use PSGraph to convert a formula in NNF to CNF, or more specifically QCNF (Quantified CNF; read about quantified boolean formula). This uses the Parallel-Serial graph algorithm. References:

• CNF Transformation by Parallel-Serial Graphs
• Paper

Example: Converting ¬a ∧ ((b ∧ ¬c) ∨ (d ∧ e))

import { Connectives, Literal, NNF, PSGraph } from "pps2";

// Construct the NNF
const NOTa = new Literal("a", true);
const b_AND_NOTc = new NNF([new Literal("b"), new Literal("c", true)], Connectives.and);
const d_AND_e = new NNF([new Literal("d"), new Literal("e")], Connectives.and);
const nnf = new NNF([NOTa, new NNF([b_AND_NOTc, d_AND_e], Connectives.or)], Connectives.and);

// Convert to QCNF.
const qcnf = PSGraph.convertNNFToCNF(nnf, "z");
qcnf.toString(); // ∃z1: (¬a) ∧ (b ∨ z1) ∧ (¬c ∨ z1) ∧ (¬z1 ∨ d) ∧ (¬z1 ∨ e)

Note that convertNNFToCNF method optionally needs a second parameter, which is the prefix for the internal quantified variables created by the algorithm (refer the toString representation). If no prefix is passed then the string "internal" is used for this purpose, which might not be that nice. However, if a prefix is used explicitly, then it can't be a variable or prefix of a variable in the input nnf.

More features will be added soon (hopefully 😛).

Have fun ☺️