MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 65the complex algebras of Kripke models for PDL. But the relationship between theoperators interpreting π and π ∗ in the algebra of a Kripke model is not equationallyexpressible, and there are dynamic algebras that belong to the equational classgenerated by the algebras of Kripke models but are not themselves representablein such models.Process logic was introduced in [Pratt, 1979] by interpreting a program, notas a relation between states, but as the set of possible state-sequences that canbe generated by executing the program. In addition to “after”, he proposed thefollowing modalitiesthroughout π, α :during π, α :π preserves α :α holds at every state of any sequence generatedin executing π.every π-computation has α true at some point.in every π-computation, once α becomes trueit remains so thereafter.Parikh [1978b] developed a decidable system of second-order process logic that subsumedPratt’s, and allowed quantification over states and state-sequences. ThenNishimura [1980] combined PDL with some temporal connectives to devise a systemextending Parikh’s. All of these were subsumed by the powerful system ofprocess logic of Harel, Kozen and Parikh [1982] which was shown to be decidableby reduction to the second-order decidability results of [Rabin, 1969].The article [Harel, 1984] surveys the first decade of dynamic logic, and there isa further review in [Kozen and Tiuryn, 1990].7.2 Hennessy–Milner LogicMatthew Hennessy and Robin Milner [1980; 1985] applied modal logic to processalgebra in a manner that is reminiscent of the Kripke modelling of PDL. Theyused a modal language to express assertions about transitions between processesin such a way that two processes prove to be “observationally equivalent” justwhen they satisfy the same modal properties.A process is viewed as an agent that interacts with its environment by performingobservable actions which cause it to change its state. Processes are identifiedwith their states, so an observation changes a process into a new process. Thenotation 〈p, p ′ 〉 ∈ R i means that process p can become p ′ by performing, or participatingin, the observation i. Thus R i is a binary relation on a given set P ofprocesses, and we envisage a collection {R i : i ∈ I} of such observation relationscorresponding to a set I of “types of observation”. A particular pair 〈p, p ′ 〉 ∈ R irepresents a single observation, and is also viewed as an “experiment” performedby the observer on process p. (In subsequent literature the notation p i −→ p ′ becamestandard in place of 〈p, p ′ 〉 ∈ R i .)The Hennessy–Milner modal language has no propositional variables, but constructsformulas from the constant ⊤ by the truth-functional connectives and the