Quasi-birth–death process

A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science.

A modal logic with n primitive unary modal operators ${◻}_i,i\in \{1,\dots ,n\}$ is called an n-modal logic. Given these operators and negation, one can always add ${◊}_i$ modal operators defined as ${◊}_{i}P$ if and only if $\mathrm{\neg }{◻}_i\mathrm{\neg }P$.

The first substantive example of a 2-modal logic is perhaps Arthur Prior's tense logic, with two modalities, F and P, corresponding to “sometime in the future” and “sometime in the past”. A logic with infinitely many modalities is (propositional) dynamic logic, introduced in 1976 and having a separate modal operator for every regular expression. A version of temporal logic introduced in 1977 and intended for program verification has two modalities, corresponding to dynamic logic's [A] and [A*] modalities for a single program A, understood as the whole universe taking one step forwards in time. The term "multimodal logic" itself was not introduced until 1980. Another example of a multimodal logic is the Hennessy-Milner logic, itself a fragment of the more expressive modal μ-calculus, which additionally is also a fixed-point logic.

Multimodal logic can be used also to formalize a kind of knowledge representation: the motivation of epistemic logic is allowing several agents (they are regarded as subjects capable of forming beliefs, knowledge); and managing the belief or knowledge of each agent, so that epistemic assertions can be formed about them. The modal operator $◻$ must be capable of bookkeeping the cognition of each agent, thus ${◻}_i$ must be indexed on the set of the agents. The motivation is that ${◻}_i\alpha$ should assert "The subject i has knowledge about $\alpha$ being true". But it can be used also for formalizing "the subject i believes $\alpha$". For formalization of meaning based on the possible world semantics approach, a multimodal generalization of Kripke semantics can be used: instead of a single "common" accessibility relation, there is a series of them indexed on the set of agents.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Stanford Encyclopedia of Philosophy: "Modal logic" – by James Garson.