Tottenham would have moved ahead of Chelsea into third had they won on Saturday, but they ended the afternoon in fifth after their north London neighbours and Everton both recorded victories against Wigan and West Ham respectively.
"No one enjoys being level on points with Arsenal at this club," Villas-Boas said. "I have experienced this rivalry for the first time this year and I have spoken very highly about it before. It is an extremely healthy competition between both teams, but we have to continue now."
He went on: "Today was a difficult game. Stoke are an extremely competitive and physical team who are difficult to break down.
"I want to be extremely positive as the players really tried hard."
The game threatened to boil over at times, particularly in the second half when Emmanuel Adebayor appeared to square up to Ryan Shawcross as they grappled in the box.
The issue of grappling in the box is a hot topic in football, and Villas-Boas thinks referees need to be more consistent.
He said: "You have to take out the law (that punishes grappling) as if you pretend to be blind you have to take out the situation and move on. We had some given for us when Stoke were attacking but none in their box that could have given us a penalty. Those decisions are very hard to take, and maybe those ones are the stepping stone for people to start learning."
The officials were at the centre of another controversy in the second half when Bale, who has been booked four times this year for diving, went down easily looking for a foul just outside the box.
Pulis launched an angry tirade at Bale from the sidelines when he went down, but the Stoke boss refused to be drawn on the matter afterwards, saying: "I will leave that. It's Christmas."
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