# Two-automaton emptiness check in Spot

### From LRDE

- Authors
- Clément Gillard
- Type
- techreport
- Year
- 2017
- Number
- 1706

## Abstract

Spot is library that handles **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega}**
-automata whose acceptance conditions are expressed with at most 32 acceptance sets. Since the acceptance condition of the product of two automata has to use the sum of their setswe cannot produce a product whose operands use more than 32 sets in total. A typical operation on automata is to compute **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A\times B)\ne\emptyset}**
to know if **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(A)}**
intersects **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(B)}**
. When it is implemented as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle empty(product(A, B))}**
, the computation of the product throttles the amout of acceptance sets A and B can use. We propose a new function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle empty(A, B)}**
which does the emptiness check of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\times B}**
without actually building an automaton and hence without any limit on the acceptance conditions. The ltlcross tool can now compare automata using a total of more than 32 acceptance sets.