Try:

$ [ "${BASH_VERSINFO:-0}" -ge 4 ] && echo "bash supports associative arrays"
bash supports associative arrays

BASH_VERSINFO is a readonly array variable whose members hold version information for this instance of bash. Since it was introduced with bash 2.0, it is likely supported by all bash versions you will encounter. But, to be cautious, we include a default value of of 0 for any earlier version bash for which this variable is unset.

Extracting version information from other programs

You asked about LibreOffice, Python, kernel, etc.

LibreOffice produces version information that looks like:

$ libreoffice --version
LibreOffice 5.2.6.2 20m0(Build:2)

To extract the version number:

$ libreoffice --version | cut -d' ' -f2
5.2.6.2

For python:

$ python -c 'import platform; print(platform.python_version())'
2.7.13

To obtain the kernel version, use uname:

$ uname -r
4.9.0-2-amd64

Instead of comparing version numbers, you could test directly for the feature itself. declare -A returns 2 (at least in Bash 3.2) if it doesn't recognize -A, so test for that (it also prints an error):

unset assoc
if ! declare -A assoc ; then
    echo "associative arrays not supported!"
    exit 1
fi

(declare -A var also fails if var is a non-associative array, so unset it first.)

While I don't really assume anyone's going to backport features in Bash, in general it's more appropriate to check for the features, not the versions. Even in Bash's case, someone might compile a version with only limited features...

The more general case of testing version numbers has two parts: 1) how to find the correct version number to test, and 2) how to compare it against another value.

The first is the more difficult one. Many programs tell their version number with a command line flag like --version or -v, but the output format varies and programmatically picking the version number may be difficult. Then there's the issue of possibly having several versions of the same program installed at the same time.

The second depends on some knowledge of the format of the version numbers. dpkg can compare Debian-style version numbers (which I think includes semver type versions as subset):

if dpkg --compare-versions 4.3.30 ge 4.0.0 ; then
    echo "it's version 4.x"
fi

Or, just to combine the above:

bashver=$( bash --version | sed -Ee 's/GNU bash, version ([0-9.]+).*/\1/;q' )
if dpkg --compare-versions "$bashver" ge 4.0.0 ; then
    echo "'bash' in your path is version 4.x"
fi

There's couple ways to approach what you want to achieve.

1. Use $BASH_VERSION

It's sufficient to just to see what's in $BASH_VERSION variable. Personally I'd use subshell like so:

$ (read -d "." version trash <<< $BASH_VERSION; echo "$version" )
4

Note that <<< syntax for here-doc is not portable, if you're going to use it with /bin/sh, which is Dash on Ubuntu and might be something else on a different system

Alternative way is via case statement or if statement. Personally, I'd do this:

bash-4.3$ case $BASH_VERSION in 4.*) echo "Can use associative arrays";; ?) echo "can't use associative arrays" ;; esac
Can use associative arrays

Probably for the sake of portability , you probably should check if such variable is even set at all in the first place with something like [ -n $BASH_VERSION ]

This totally can be rewritten as function to be used in a script. Something a long the lines of:

#!/bin/bash
version_above_4(){
    # check if $BASH_VERSION is set at all
    [ -z $BASH_VERSION ] && return 1
   
    # If it's set, check the version
    case $BASH_VERSION in 
        4.*) return 0 ;;
        ?) return 1;; 
    esac
}

if version_above_4
then
    echo "Good"
else
    echo "No good"
fi

This is not a one-liner, although this is much better. Quality over quantity.

2. Check what's installed

For that you need to filter output of apt-cache policy like so

$ apt-cache policy bash | awk -F '[:.]' '/Installed:/{printf "%s\n",substr($2,2)}'
4

dpkg-query can also come in handy with some filtering via awk.

$ dpkg-query -W bash | awk '{print substr($2,1,1)}'   
4

Note that this is not portable, since if there's no dpkg or apt installed on a system ( for instance, RHEL or FreeBSD ), it won't do you any good.

3. Use set -e to exit script if there's an error

One way to get around it is just simply go ahead an use associative arrays and quit when bash cannot use them. set -e line below #!/bin/bash will allow the script to quit if the script can't use associative array.

This will require you to explicitly tell the user: "Hey, you really need bash version 4.3 or above, otherwise the script won't work". Then the responsibility rests with the user, although some might argue that this is not really a good approach to software development.

4. Abandon all hope and write portable, POSIX-compliant scripts

bash scripts aren't portable because its syntax isn't compatible with Bourne shell. If the script that you're writing is going to be used on a range of different systems, not just Ubuntu alone, then abandon all hope, and find ways to use something other than associative arrays. That might include having two arrays or parsing a configuration file. Consider also switching to a different language, Perl or Python, where syntax is at least more portable than bash.

One-liner not possible but a bash script is possible

I developed a script that draws on answers in Stack Overflow. One of those answers led to a Dell Employee writing version number comparisons in 2004 for the DKMS application.

The code

The bash script below needs to be marked as executable using the command chmod a+x script-name. I'm using the name /usr/local/bin/testver:

#!/bin/bash

# NAME: testver
# PATH: /usr/local/bin
# DESC: Test a program's version number >= to passed version number
# DATE: May 21, 2017. Modified August 5, 2019.

# CALL: testver Program Version

# PARM: 1. Program - validated to be a command
#       2. Version - validated to be numberic

# NOTE: Extracting version number perl one-liner found here:
#       http://stackoverflow.com/questions/16817646/extract-version-number-from-a-string

#       Comparing two version numbers code found here:
#       http://stackoverflow.com/questions/4023830/how-compare-two-strings-in-dot-separated-version-format-in-bash

# Map parameters to coder-friendly names.
Program="$1"
Version="$2"

# Program name must be a valid command.
command -v $Program >/dev/null 2>&1 || { echo "Command: $Program not found. Check spelling."; exit 99; }

# Passed version number must be valid format.
if ! [[ $Version =~ ^([0-9]+\.?)+$ ]]; then
    echo "Version number: $Version has invalid format. Aborting.";
    exit 99
fi

InstalledVersion=$( "$Program" --version | perl -pe '($_)=/([0-9]+([.][0-9]+)+)/' )
# echo "Installed Version: $InstalledVersion"

if [[ $InstalledVersion =~ ^([0-9]+\.?)+$ ]]; then
    l=(${InstalledVersion//./ })
    r=(${Version//./ })
    s=${#l[@]}
    [[ ${#r[@]} -gt ${#l[@]} ]] && s=${#r[@]}

    for i in $(seq 0 $((s - 1))); do
        # echo "Installed ${l[$i]} -gt Test ${r[$i]}?"
        [[ ${l[$i]} -gt ${r[$i]} ]] && exit 0 # Installed version > test version.
        [[ ${l[$i]} -lt ${r[$i]} ]] && exit 1 # Installed version < test version.
    done

    exit 0 # Installed version = test version.
else
    echo "Invalid version number found for command: $Program"
    exit 99
fi

echo "testver - Unreachable code has been reached!"
exit 255