3.3 Sequent Calculus
3.3.2 Intuitive Reading
We provide a way to read theM andQsequents in natural language. This can be seen as capturing the
intuitive meaningof a sequent.
• `M mmeans that propositionmholds in all contexts.
• `Qqmeans that actionqis less deterministic than the action that does nothing, and thus might
not change the truth-value of the proposition (on which it might act).
• Γ, A,Γ0 `M mmeans that in contextΓ, agentAknows or believes thatΓ0 `M mholds. So this captures features ofA’s own reasoning: the sequentΓ0 `M mis accepted byAin contextΓas a valid argument.
• Γ, q,Γ0 `M mmeans that, after actionqhappens on contextΓ, the sequentΓ0 `M mwill hold.
• m,Γ `M m0 means that, in context m (i.e. in any situation in whichm is true), the sequent
Γ`M m0 holds.
• Γ, A,Γ0 `Q q means agentAknows or believes that in contextΓactionq is less-deterministic
than the sequential composition of programs inΓ0.
• Finallyq,Γ`Q q0means that sequential composition ofqwith actions inΓis more deterministic than actionq.
This reading might seem backward in agent and action cases. The reason is that whenever we have an agentA, we are going to read the proposition or action to the left of it as the agent’s knowledge or belief about them, where as we have encoded the agentAas his appearance mapfAand not his knowledge in
our sequents. But the epistemic adjunction between the twofAa2Aallows us to present the intuitive
This becomes clear below where we give examples. The same holds for the actions in propositional contexts, that is whenever we have an action we are going to update its left hand side and read it as ’after’, where as the ’after’ operation is the dynamic modality and the adjoint of update.
Example.The intuitive reading of the sequentm, A, B`M m0 is
‘In contextm, agentAknows that agentB knows thatm0’. The formal meaning of this sequent will be
fB(fA(m))`M m0
which by epistemic adjunction encode in the rules, to be presented later, is equivalent to
m`M 2M A2MB m0
As another example consider the sequentm, A, q, B`M m00, which is intuitively read as:
In contextm, agentAbelieves that after actionqagentBwill believe that propositionm00must hold. and formally means
fBM(fAM(m). q)`M m00
equivalent to
m`M 2MA [q]2MB m
00
Resource Sensitivity. From this reading we can explain how our sequent calculus expresses two forms of resource sensitivity. One is the use-only-once form of Linear Logic [44]. We call these resourcesdynamic resources. They express the fact that repetition of actions matters in validity of sequents, and thus actions cannot be freely added to or deleted from the sequents. This is true in both our propositional and action sequents. For example in a propositional context, a propositionmmight not entailm0, that ism 0M m0, but if we update it with an action it will, that ism, q `M m0. An
example would be when knowledge of an agent does not entailm0, that is for example2Am0M m0,
but if we announcem0 to him via actionq, then he will know it and thus we will have2Am, q ` m0.
The same holds for repetition of actions, for example we might not havem, q `M m0, but if we doq
twice, then we will havem, q, q`M m0. A very good example is the muddy children puzzle where each repetition of the no answer yields new information in children. Another case would be if we do another action afterq, for exampleq0 then we will havem, q, q0 `M m0. Similarly in the other direction, if we
havem `M m0 and then updatem with an actionq, we might not get the same resultm, q 0M m0,
this for example the opposite ofm0is announced to an agent. The same holds for action sequents, for example we might haveq`Q q00, but after sequentially composingqwith another actionq0, we do not
more deterministic than actionq00, that isq 0Q q00, but sequentially composing it withq0 will give us
q, q0`Q q00.
The other form of our resource sensitivity deals withepistemic resources: these are resources that are available to each agent and enable him to reason in a certain way and for example to infer a conclu- sion from these resources. These resources are encoded in the way the context appears to the agent in sequents, for instanceΓin the sequentΓ, A,Γ0 `M mis the context, and hencefA(Γ)is the resource
that enables agentAto do theΓ0`M mreasoning. Note thatΓ0 `M mmight not be a valid sequent in the contextΓ, but it is valid in the context given byΓ’s appearance to agentA. That is we haveΓdoes not entailmin realityΓ 0M mbut agentAthinks it doesΓ, A`M m. Also in the other direction, a
sequent might hold in realityΓ`M m, but agentAis deceived and cannot do the same deduction, that isΓ, A0M m. Exactly the same holds for action sequents, for example in reality an actionqmight be
more deterministic than actionq0, that isq`Qq0, but not for agentA, that isq, A0Qq0. In this way we
can think ofpresence of agentsas resources that make a difference in validity of sequents and cannot be freely added to or deleted from the sequents.