nLab
temporal logic

Contents

Idea

Temporal logics, as their name suggests, are logics that involve time. They form a very large and important class of modal logics, but here we will, to start with, only look at some simple cases. Temporal logics are very closely related to tense logic?s.

Basic temporal language, (BTL)

First the basic temporal language is built with two binary model operators, which instead of being denoted ‘box’ and ‘diamond’, are written FF and PP. The intended interpretation of these are:

  • FϕF\phi is ϕ\phi will be true at some future time;

  • PϕP\phi is ϕ\phi was true at some past time.

The reason for the notation FF and PP should be clear.

  • The dual of FF is GG, so Gϕ=¬F¬ϕG\phi = \neg F\neg \phi. This means that we read GϕG\phi as ‘at no future time is ϕ\phi not true’, i.e., ϕ\phi is always going to be true. (GG for ‘going’.)

  • The dual of PP is written HH, whence Hϕ=¬P¬ϕH\phi = \neg P\neg \phi and HϕH\phi interprets as ’ϕ\phi has always been true’. (HH is for ‘has’ - which is a bit weak, but that is life!)

Examples
  • If for all ϕ\phi we have PϕGPϕP\phi \to G P\phi, then whatever has happened will always have happened, which seems reasonable so this might be a suitable ‘general truth’ we might want a temporal logic to contain.

  • If we had in our logic that FϕFFϕF\phi \to F F\phi, then this would means that if ϕ\phi is true at some future time, then at some future time it will be true that ϕ\phi will be true at some future time, so between any two instants there must be a third, and the general truth of this would say something about the ‘internal structure’ of time as modelled by such a logic.

This basic temporal language does not yet really constitute a sensible logic that could model important features of ‘time’. but we can still look at models so as to see what properties of the models can be identified as corresponding to seemingly feasible formulae in the language.

Frames and Models for BTL

A frame for BTL will be a set, TT, with two binary relations R FR_F and R PR_P. We have R FxyR_F x y reads that ‘at xx, yy is in the future’ - but, in the ordinary common sense meaning of words, this should mean R PyxR_P y x and conversely. In other words R PR_P should be the converse or opposite relation of R FR_F.

(In general, if RR is a binary relation on a set XX, then its opposite relation is defined by

(y,x)R op(x,y)R.)(y,x) \in R^{op} \Leftrightarrow (x,y)\in R.)
Definition

A frame of the form (T,R,R op)(T,R,R^{op}) is called a bidirectional frame.

If we have any model 𝔐=(T,R,V)\mathfrak{M} = (T,R,V) with a valuation, VV, then we can interpret BTL in terms of it by specifying the interpretation of PP in terms that of RR. explicitly we define

  • 𝔐,tFϕ\mathfrak{M}, t\models F\phi if and only if s(Rts𝔐,sϕ);\exists s (R t s \wedge \mathfrak{M}, s\models \phi);

  • 𝔐,tPϕ\mathfrak{M}, t\models P\phi if and only if s(Rst𝔐,sϕ)\exists s (R s t \wedge \mathfrak{M}, s\models \phi),

but, of course, this is still a long way from having a logic that looks as if it captures ‘time-like’ behaviour. We could have models with ‘circular time’, and ‘branching time’. Both of these correspond to various situations in computational applications so should be kept in mind, both to be able to identify their occurrence and to build them in or to avoid them in the logics.

(4)

One condition it would be natural to impose is transitivity of R FR_F, since if FϕF\phi is true at some future time, then clearly, ϕ\phi itself must also be true at some future time, i.e., later on still! This leads one to ask that

(4)FFϕFϕ(4) \quad\quad F F\phi \to F\phi

should be in our logic, (and similarly for PP, but that will hold in the bidirectional models since if RR is transitive then so will be R opR^{op}).

References

Generally this entry is based on

  • P. Blackburn, M. de Rijke and Y. Vedema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

but see also:

Revised on December 14, 2012 13:01:15 by David Corfield (129.12.18.29)