The first attack occurred late on Saturday in the village of Sadovoye near the capital of Chechnya when unidentified assailants opened automatic fire at a policeman's car near his house, the police source said.
When a police squad arrived at the scene at midnight, an unidentified explosive device equivalent of about 400 grams of TNT went off, killing three policemen and a local resident and injuring another five policemen and also two local residents, the police source said.
In a separate attack early on Saturday, unidentified attackers opened fire at a police car near the village of Gekhi in the Urus-Martan district, southwest of Chechnya's capital, killing a policeman, the police source said.
In a separate episode, police found a large arms cache in the village of Staroshchedrinskaya in the Shelkovskoi district in northeastern Chechnya, the police source said.
The arms cache contained seven grenade launchers, four Kalashnikov automatic rifles and a large number of cartridges, the police source said.
Sporadic terrorist attacks and militant clashes remain common in Russia's North Caucasus, although the active phase of the military campaign in Chechnya officially ended several years ago.