  Online trading invariably involves dealings between strangers, so it is
important for one party to be able to judge objectively the trustworthiness of
the other. In such a setting, the decision to trust a user may sensibly be
based on that user's past behaviour. We introduce a specification language
based on linear temporal logic for expressing a policy for categorising the
behaviour patterns of a user depending on its transaction history. We also
present an algorithm for checking whether the transaction history obeys the
stated policy. To be useful in a real setting, such a language should allow one
to express realistic policies which may involve parameter quantification and
quantitative or statistical patterns. We introduce several extensions of linear
temporal logic to cater for such needs: a restricted form of universal and
existential quantification; arbitrary computable functions and relations in the
term language; and a "counting" quantifier for counting how many times a
formula holds in the past. We then show that model checking a transaction
history against a policy, which we call the history-based transaction
monitoring problem, is PSPACE-complete in the size of the policy formula and
the length of the history. The problem becomes decidable in polynomial time
when the policies are fixed. We also consider the problem of transaction
monitoring in the case where not all the parameters of actions are observable.
We formulate two such "partial observability" monitoring problems, and show
their decidability under certain restrictions.
