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One source of complexity in the -calculus is its ability to specify an unbounded number of switches between universal (AX) and existential (EX) branching modes. We therefore study the problems of satisfiability, validity, model checking, and implication for the universal and existential fragments of the -calculus, in which only one branching mode is allowed. The universal fragment is rich enough to express most specifications of interest, and therefore improved algorithms are of practicaldoi:10.1016/j.tcs.2005.11.015 fatcat:zulrihp2krb6bgjxfworh4upwm