Roll-over tenancy agreements

Once you have given a tenant a six month (AST), do you have to renewal the tenancy every six or twelve months, or can you just provide a verbal roll-on contract, and would the original 4 weeks notice still apply?

None of what you say makes sense (to me).

There should be no AST with a "4 week notice period"... in either direction. An AST has a fixed term - often 6 months, as you cite, but can be other agreed lengths (up to 3 years).

After the fixed term expires, if nothing is done then a SPT automatically arises. You do not have to renew anything, but you can choose to if you want to. A SPT is the same agreement, periodic or monthly-rolling (whatever you want to call it), but the notice periods are defined by statute as being "at least 2 months" from Landlord to Tenant and "at least 1 month" from Tenant to Landlord.

There shouldn't be any verbal contracts when letting property.

What most Landlords do is the following... give the minimum length fixed term at the start (this helps everyone find their feet and neither side ends up locked-in to a longer fixed term) of 6 months, then let the tenancy go periodic, if everything is going well the Tenants can stay as long as they want, but there is flexibility, from their side, to leave with around a month of notice, if they decide to buy or rent elsewhere. What annoys some Tenants is amateur Landlords who've dipped their feet into this area and end up wanting the property back at relatively short notice - to either live in it, or sell it. That's the downside of flexibility - lack of security.

An AST has a fixed term - often 6 months

There is no such requirement at law, though most ASTs are initially granted for that period. They can be granted for any term length or even as a periodic tenancy from day1.

After the fixed term expires, if nothing is done then a SPT automatically arises. You do not have to renew anything, but you can choose to if you want to. A SPT is the same agreement, periodic or monthly-rolling (whatever you want to call it)

Sort-of. The period of the SPT is defined as equal to the rent-payment frequency and that is usually (but not always) monthly.

There shouldn't be any verbal contracts when letting property.

But an AST can be granted orally- I agree that this isn't a good idea, mind!