Keyboard Mnemonics: SAT problem with Z3 (TypeScript)

Assign keyboard mnemonics to a list of options. Given a set of options in a menu, we want to assign
a mnemonic to every option such that there are not two options with the same mnemonic. The solution
has to meet the following constraints:

1. Each option must have a mnemonic.
2. An option cannot have more than one mnemonic.
3. A given character cannot be a mnemonic of two different options.

The problem can be reduced to a SAT. This repository includes the formalization and the implementation. It uses Z3's bindings in TypeScript.

Formalization

Formalizing the problem

Each option i in the menu has a set of characters associated with it, a set called $Chars(i)$. For
instance, if option 1 is "Undo", $Chars(1)$ might be the set of characters ${U, n, d, o}$.

The task is to determine whether a particular character c is the mnemonic for a specific i.

The menu with $n$ options such that $i \in {1,...,n}$. We denote the set of characters $Chars(i)$
that belong to the $i$-th option. Therefore, our problem can be defined as:

$$ {U_{c,i} | 1 \leq i \leq n, c \in Chars(i)} $$

Here, $U_{c,i}$ is true if the character c is the mnemonic of the i -th option.

Formalization with examples

Let's consider options "undo", "copy", "mod" which can be written as:

Uu1, Un1, Ud1, Uo1
Uc2, Uo2, Up2, Uy2
Um3, Uo3, Ud3

graph TD
    A[Menu Options] --> B["Option 1: Chars(1) = {u, n, d, o}"]
    A --> C["Option 2: Chars(2) = {c, o, p, y}"]
    A --> D["Option 3: Chars(3) = {m, o, d}"]

    B --> |Mnemonic: U| E["U(u,1) = True"]
    C --> |Mnemonic: Y| F["U(y,2) = True"]
    D --> |Mnemonic: D| G["U(d,3) = True"]

Each option must have a mnemonic

This constraint ensures that every option has at least one character assigned as its mnemonic. Formally, for each option i, at least one of the Boolean variables $U_{c,i}$ must be true:

$$ \land_{i = 1}^n \lor_{c \in Chars(i)} U_{c,i} $$

Formalization according to our example:

$$ U_{u,1} \lor U_{n,1} \lor U_{d,1} \lor U_{o,1} $$

$$ U_{c,2} \lor U_{o,2} \lor U_{p,2} \lor U_{y,2} $$

$$ U_{m,3} \lor U_{o,3} \lor U_{d,3} $$

An option cannot have more than one mnemonic

If a character is chosen, no other character for the same option can be a mnemonic. Formally, for each option i and each character c in Chars(i), if $U_{c,i}$ is true, then all
other characters t in Chars(i) must have $U_{t,i}$ set to false:

$$ \land_{i = 1}^n \land_{c \in Chars(i)} (U_{c,i} \implies \land_{t \in {Chars(i) - c}} \lnot U_{t,i}) $$

Formalization according to our example:

$$ U_{u,1} \land U_{n,1} \land U_{d,1} \land U_{o,1} $$

$$ U_{c,2} \land U_{o,2} \land U_{p,2} \land U_{y,2} $$

$$ U_{m,3} \land U_{o,3} \land U_{d,3} $$

A given character cannot be a mnemonic of two different options

This constraint ensures that no single character is used as the mnemonic for more than one option. Formally, for each option i and each character c in Chars(i), if $U_{c,i}$ is true, then c
cannot be the mnemonic for any other option j where j ≠ i:

$$ \land_{i = 1}^n \land_{c \in Chars(i)} (U_{c,i} \implies \land_{1\leq j \leq n \land i \neq j c \in Chars(j)} \lnot U_{c,j}) $$

Formalization according to our example:

$$ U_{u,1} \implies \lnot U_{n,1} \land \lnot U_{d,1} \land \lnot U_{o,1} $$

$$ U_{n,1} \implies \lnot U_{u,1} \land \lnot U_{d,1} \land \lnot U_{o,1} $$

$$ U_{d,1} \implies \lnot U_{u,1} \land \lnot U_{n,1} \land \lnot U_{o,1} $$

$$ U_{o,1} \implies \lnot U_{u,1} \land \lnot U_{n,1} \land \lnot U_{d,1} $$

$$ U_{c,2} \implies \lnot U_{o,2} \land \lnot U_{p,2} \land \lnot U_{y,2} $$

$$ U_{o,2} \implies \lnot U_{c,2} \land \lnot U_{o,2} \land \lnot U_{y,2} $$

$$ U_{p,2} \implies \lnot U_{c,2} \land \lnot U_{o,2} \land \lnot U_{y,2} $$

$$ U_{y,2} \implies \lnot U_{c,2} \land \lnot U_{p,2} \land \lnot U_{p,2} $$

$$ U_{m,3} \implies \lnot U_{o,3} \land \lnot U_{d,3} $$

$$ U_{o,3} \implies \lnot U_{m,3} \land \lnot U_{d,3} $$

$$ U_{d,3} \implies \lnot U_{m,3} \land \lnot U_{o,3} $$

Installation

OS X & Linux:

Install Node with asdf (version manager):

asdf install

Install dependencies:

npm install

Usage example

Define options in index.ts:

// Menu options
const OPTIONS = ["cut", "copy", "cost"];

Run program:

npm start

The program will print the answers:

Starting...
Problem was determined to be sat in 252 ms
---- Result: option [mnemonic] ----
cut u
copy y
mod d

Implementation in TypeScript

For more details on the problem and the implementation, visit the
blogpost.

Acknowledgment

Assignment from "Static Program Analysis and Constraint Solving" at Universidad Complutense de Madrid. Prof. Manuel Montenegro.