1.3. Objetivos de la investigación
2.1.2. Scrum
The formulae of theM-system, are calledM-formulae and are generated by the following syntax
A sequencein this system is called an M-sequence. It is denoted byΓand is a list of propositions, actions, and agents
Γ∈(LM ∪LQ∪ A)∗
withLM is the set of allM-formulae,LQthe set of allQ-formulae andA, as before, set of all agents.
Sequences that contain only propositions are denoted byΓM, sequences that contain only agents are
denoted byΓA, and same as in theQ-system, sequences of only actions are denoted byΓQ.
Sequentsof this system are calledM-sequents. They are denoted asΓ `M mwhere mis a single
propositionm ∈ LM andΓis a sequence. The empty sequence on the left hand side is the unit of
conjunction, that is>. So a sequent with an empty sequence on the left hand side `M mmeans
> `M m. Here we can have an empty sequence on the right because it corresponds to the unit of∨,
which is⊥. So we have thatΓ`M meansΓ`M ⊥.
Meaning of a Sequence. We assign meaning to the sequences of an M-system via the following operation
− M −:LM ×(LM ∪LQ∪LA)→LM
which similar to theQ-system is defined as
mM m0 = m∧m0 mM A = fAM(m)
mM q = m.q
This operation is applied to sequences inductively, for example a sequence Γ = (γ1, γ2, . . . , γn)
has the following meaning4
MΓ = (((> M γ1)Mγ2)· · · M γn)
This operation enables us to identify a sequence ofM-formulaeΓwith only oneM-formulaΓ∈LM.
For exampleΓ = (m, A, q, B, m0)has the following meaning
MΓ = (((((> M m)M A)M q)M B)M m0) =fBM((fAM(m).q))∧m
0
By adding the unit of conjunction to the left of the sequence, we identify a sequence of only one agentΓ = Awith the appearance of that agent of top of the module, which is the unit of conjunction
MΓ =fAM(>). Similarly a sequence of only actionΓ =qis identified with the update of the top of
4Here, we abuse the notation and use the same symbol
M for meaning of a sequence; in [7] we use the slightly different
symbol ofJ
module with that actionMΓ =>.q.
Interpretation.Given a distributive epistemic system(M, Q,{fA}A∈A)and its quantale interpretation
mapα:LQ → Q, we define a module interpretation mapβ from theM-formulae to the module part
of the epistemic system as follows:
β :LM →M
This maps assigns an element of the module β(m) to each M-formula m. In order to interpret a sequentΓ, we first applyM, and then theβ map. For example the interpretation of a sequence like
Γ =m, Aand will beβ((m, A)) =β(fAM(m)), and similarly the interpretation ofΓ =m, qwill be
β((m, q)) =β(m.q). This composition is well-defined since we have:
β◦ : (LM ∪LQ∪ A)∗→LM →M
Theβmap has the usual structure-preserving properties:
β(⊥) =⊥ and β(>) =>
and for the join and meet connectives we have
β(m)∨β(m0) =β(m∨m0) and β(m)∧β(m0) =β(m∧m0) and also
β(fAM(m)) =fAM(β(m)) and β(2MA m) =2MA β(m)
But moreover we have the following two for the mixed operations between quantale and module
β(m.q) =β(m).α(q) and β([q]m) = [α(q)]β(m) whereαis the interpretation map between theQ-system and the quantale.
Definition 4.1.4 For a distributive epistemic system(M, Q,{fA}A∈A), its interpretation mapsα:LQ→
Q, β:LM →M, anM-sequenceΓ, and anM-formulam0, we define a satisfaction relation as follows
Γ|=M m0 iff β(MΓ)≤β(m0)
Definition 4.1.5 A sequentΓ`M m0is valid whenever for any distributive epistemic system(M, Q,{fA}A∈A)
and its interpretation mapsα:LQ →Q, β:LM →M we haveΓ|=M m0.
Lemma 4.1.6 Every derivableM-sequent is valid.
sequents on the top line of a rule are valid (this includes validity of theQ-sequents of the mixed rules), so are the sequents on the bottom of the line.
Before starting the proof, we make the same convention as in theQ-systems, to avoid repeating the proof steps: the interpretation of eachM-formula is denoted by the formula itself. That is, we writem
instead ofβ(m). We start our proof with the axiom and rules for units, which are as follows:
m`M m id ⊥ `M m ⊥L
Γ`M
Γ`M ⊥ ⊥R Γ`M > >R
Soundness of the identity axiom follows by reflexivity of order on the modulem≤m. The left rule for
⊥is sound since⊥is less than every other element of the module⊥ ≤m. The right rule for⊥follows by the definition of empty sequence on the right hand side, which is⊥itself. The right rule for>
follows by>being the top element of the module, that is every other element is less than itMΓ≤ >.
There is no need to state a left rule for>, which would be Γ,Γ0`M m
Γ,>,Γ0`M m >L
since it is derivable from the weakening rule of the module, to be discussed below. The rules for disjunction are Γ, m1,Γ0 `M m Γ, m2,Γ0 `M m Γ, m1∨m2,Γ0 ∨L Γ`M m1 Γ`M m1∨m2 ∨R1 Γ`M m2 Γ`M m1∨m2 ∨R1
For the left rule we have M(Γ, m1,Γ0) ≤ m and M(Γ, m2,Γ0) ≤ m, from which we obtain M(Γ, m1,Γ0)∨ M(Γ, m2,Γ0) ≤ m by definition of join in the module. First consider the case
whereΓ0contains only propositions, so we have
M(Γ, m1,Γ0)∨ M(Γ, m2,Γ0) = (MΓ∧m1∧ MΓ0)∨(MΓ∧m2∧Γ0)
By distributivity of meet over join we obtain
MΓ∧(m1∨m2)∧ MΓ0 ≤m
and thusM(Γ, m1 ∨m2,Γ0) ≤ m, that is the bottom line. Now consider the case whereΓ is an
agent-only sequence and without loss of generality contains only one agentA. Then we have
M(Γ, m1, A)∨ M(Γ, m2, A) =fA(MΓ∧m1)∨fA(MΓ∧m2)
By join-preservation offAand distributivity of meet over join we obtain the following
and thus it follows thatM(Γ, m1∨m2, A)≤m. Finally, consider the case thatΓ0is an action-only
sequence. In this case we have
M(Γ, m1,Γ)∨ M(Γ, m2,Γ) = ((MΓ∧m1)· QΓ0)∨((MΓ∧m2)· QΓ0)
By join-preservation of·and distributivity of meet over join we obtain the following ((MΓ∧m1)∨(MΓ∧m2))· QΓ0 = (MΓ∧(m1∨m2))· QΓ0≤m
This it follows thatM(Γ, m1∨m2,Γ0)≤m. The soundness of cases whereΓ0is a mixture of actions,
agents, and propositions follows from the above three cases.
For the first (and similarly second) right rule assumeMΓ ≤m1 then by definition of join in the
module we havem1≤m1∨m2and thus it follows thatMΓ≤m1∨m2.
The rules for conjunction are Γ, m1,Γ0 `M m Γ, m1∧m2,Γ0`M m ∧L1 Γ, m2,Γ 0 ` M m Γ, m1∧m2,Γ0 `M m ∧L2 Γ`M m1 Γ`M m2 Γ`M m1∧m2 ∧R
For the first (and similarly the second) left rule assumeM(Γ, m1,Γ0)≤m, sincem1∧m2 ≤m1and M is order preserving we obtainM(Γ, m1 ∧m2,Γ0)≤m. For the right rule assumeMΓ ≤m1
andMΓ≤m2, from this by the definition of meet we obtainMΓ≤m1∧m2.
The rules for appearance maps are
m0, A,Γ`M m fAM(m0),Γ`M m fM A L Γ`M m Γ, A`M fAM(m) fM A R
For the left rule observe that the sequences of the top and bottom lines have the same meanings, that is
M(fAM(m0),Γ). The right rule follows by order preservation of appearance maps, ifMΓ≤mthen we can applyfAM to both sides and getfAM(MΓ)≤fAM(m)and thus the bottom sequent.
Rules for the knowledge modality on the module are
m0,Γ`M m 2M Am 0, A,Γ` M m 2M AL Γ, A`M m Γ`M 2MA(m) 2M AR
For the left rule, assume the top line, which isM(m0,Γ)≤m. By composition of adjointsfAM and
2M
A we havefAM(2MAm0) ≤ m0. Since·,fAM and∧are all order preserving, we applyM withΓ
to both sides and getM(fAM(2MAm0),Γ)≤ M(m0,Γ). By the top line assumption and transitivity
we getM(fM
A (2MAm0),Γ)≤ m, which is what we want for the bottom line. The right rule follows
directly from the definition of adjunction. By the top line we have fAM(MΓ) ≤ m, which is by adjunction equivalent toMΓ ≤ 2MAm. Similar to the knowledge rule in theQ-system, this rule is
The cut rule for theM-system is as follows
Γ0`M m0 m0,Γ00 `M m
Γ0,Γ00`M m cut
By the first assumption we haveMΓ0 ≤ m0, from which we obtainM(MΓ0,Γ00) ≤ M(m,Γ00)
by order preservation of M. By the second assumption we have M(m,Γ00) ≤ m0 and thus by
transitivityM(MΓ0,Γ00)≤m0, which is what we want for the bottom line. The rule for factual propositions is
Γ`M p
Γ, q`M p f act
The validity of the top line sequent saysMΓ≤p, we update both sides withqand we getMΓ. q≤
p . q, by the definition of facts we have thatp . q≤p, and by transitivity we getMΓ. q≤p, which is
what we want for the validity of the bottom line sequent. Other structural rules of the module are
Γ,Γ0`m Γ, m0,Γ0`M m weakL Γ` ⊥ Γ`M m weakR Γ, m0, m0,Γ0 `M m Γ, m0,Γ0`M m contr Γ, m00, m0,Γ0 `M m Γ, m0, m00,Γ0 `M m exch
For the left weakening assume the top line, that isM(MΓ,Γ0) ≤ m. SinceMΓ∧m0 ≤ MΓ
andM is order preserving, we obtainM(MΓ∧m0,Γ0) ≤ m, that is the bottom line. The right
weakening follows sinceMΓ ≤ ⊥is equivalent to MΓ = ⊥and⊥ ≤ m. Contraction is sound since we haveMΓ∧m0 ∧m0 = MΓ∧m0 andM is order preserving. Exchange follows by
commutativity of meetm00∧m0 =m0∧m00and order preservation ofM. The rules for epistemic update are
m0, q,Γ`M m
m0.q,Γ`M m .L
Γ,ΓA`M m ΓQ,ΓA`Qq
Γ,ΓQ,ΓA`M m.q .R
The left rule follows from the definition of comma between a proposition and an actionm0, q=m0.q
and order preservation ofM. For the right rule, first assume that we have only one agent in our agent context, that isΓA = A. By the first assumption of the top line we havefAM(MΓ) ≤mand by the
second assumption we havefAQ(QΓQ) ≤ q. Since update is order preserving, we can update both
sides of these two assumption by each other and getfAM(MΓ).fAQ(QΓQ) ≤m.q. Now by update
inequality we havefAM(MΓ.QΓQ)≤fAM(MΓ).fAQ(QΓQ)≤m.q, which is what we want for
the bottom line and we are done. If we have more than one agent, that isΓA = A1, . . . , An, then
innermost agentA1to the outmost oneAn, that is
fAMn(fAMn−1(. . . fAM1(MΓ. QΓQ)))≤m . q
The rules for dynamic modality are
m0 `M m ΓQ`Qq
[q]m0,ΓQ`M m
DyL Γ, q`M m
Γ`M [q]m DyR
For the left rule start from the second assumptionQΓQ ≤ q, since update is order preserving, this
inequality is preserved under update of the proposition[q]m0 as follows [q]m0· QΓQ≤[q]m0·q
By adjunction between update and dynamic modality we have[q]m0. q≤m0, and thus [q]m0· QΓQ ≤m0
now by the first assumption of the top line we havem0≤mand by transitivity we get [q]m0· QΓQ ≤m
which is exactly what we want for the bottom line. The right rule follows directly by definition of adjunction. The top line assumption saysMΓ. q≤m, which by adjunction is equivalent toMΓ≤
[q]m, which is the bottom line. This rule holds in both direction.
We now prove the soundness of the rules that deal with occurrences of actions in theM-sequences. Recall that actions can only occur on the left hand side ofM-sequents, so we have left rules for all the operations on actions, including the unit of sequential composition. The rule for update with unit of sequential composition in theM-system is
Γ,Γ0 `M m
Γ,1,Γ0 `M m 1M L
By unity of 1 we haveMΓ.1 =MΓ, and thusM(Γ,Γ0) =M((MΓ.1),Γ0). So by the top line assumption and transitivity we getM((MΓ.1),Γ0)≤m.
The rule for update with sequential composition of actions in the module is Γ, q1, q2,Γ0 `M m
Γ, q1•q2,Γ0 `M m
The top line assumption meansM(((MΓ. q1). q2),Γ0)≤m, by module equation we have (MΓ. q1).q2 =MΓ.(q1•q2)
and so we getM(MΓ.(q1•q2),Γ0)≤m, which is the meaning of the bottom line.
The rules for update with right and left residuals in theM-sequents are ΓQ`Q q2 Γ, q1 `M m
Γ, q1/q2,ΓQ`M m
/M L ΓQ`Qq1 Γ, q2`M m
Γ,ΓQ, q1\q2 `M m
\M L
For the right residual we have two top line assumptions:QΓQ ≤q2 and MΓ.q1 ≤m. Start from
the first assumptionQΓQ ≤q2and compose both sides withq1 on the left and we getq1• QΓQ≤
q1•q2, which is by residuation equal toq1/q2• QΓQ ≤q1. Now update the propositional sequent MΓwith this inequality and we getMΓ.(q1/q2• QΓQ)≤ MΓ. q1. By the second assumption
of the top line MΓ. q1 ≤ m and so by transitivity we have MΓ.(q1/q2 • QΓQ) ≤ m. By
the module equation this inequality is equivalent to(MΓ. q1/q2). M ΓQ ≤ m. Since we have
(MΓ. q1/q2). M ΓQ =M(Γ, q1/q2,ΓQ), we obtainM(Γ, q1/q2,ΓQ) ≤m, which is what we
need for the bottom line.
The proof of the left residual is similar, we start by the first assumption of the top lineQΓQ ≤q1
and compose it on both sides with q2 on the right and we get QΓQ •q2 ≤ q1 •q2, which is by
residuation equal toQΓQ• q1\q2≤q2, now update both sides with the propositional contextMΓ
on the left and we getMΓ.(QΓQ• q1\q2)≤ MΓ. q2. By module equation this is equivalent to (MΓ. QΓQ). q1 \q2 ≤ MΓ. q2. Now we use the second assumption of the first line that says MΓ. q2 ≤mand by transitivity of order we get(MΓ. QΓQ). q1\q2 ≤ m, which is what we
want for the bottom line, sinceM(Γ,ΓQ, q1\q2) = (MΓ. QΓQ). q1\q2.
The rule for update with choice of actions inM-sequents is Γ, q1`M m Γ, q2 `M m
Γ, q1∨q2 `M m
∨M L
Assume the top line sequents: MΓ.q1 ≤ m and MΓ.q2 ≤ m. By definition of join we obtain (MΓ. q1)∨(MΓ. q2) ≤ m. Since update is join preserving we get(MΓ. q1)∨(MΓ. q2) = MΓ.(q1∨q2), soMΓ.(q1∨q2)≤m, which is what we want for the bottom line.
We have two left rules for update with meet of actions Γ, q1,Γ0 `M m Γ, q1∧q2,Γ0 `M m ∧M L1 Γ, q2,Γ 0 ` M m Γ, q1∧q2,Γ0 `M m ∧M L2
For the left rule, assume the top sequent, that isM((MΓ.q1),Γ0)≤m. Sinceq1∧q2 ≤q1andM
is order preserving, we obtainM((MΓ.(q1∧q2)),Γ0)≤m, which is what we want for the bottom
Theorem 4.1.7 The rules of IDEAL are sound with respect to the algebraic semantics in terms of distributive epistemic systems.
Proof follows directly from lemmas 4.1.3 and 4.1.6. 2