The dynamic turn in Epistemic Logic is based on the idea that notions of information should be studied together with the actions that modify them. Dynamic epistemic logics have explored how knowledge and beliefs change as consequence of, among others, acts of observation and upgrade. Nevertheless, the omniscient nature of the represented agents has kept finer actions outside the picture, the most important being the action of inference. Following proposals for representing non-omniscient agents, recent works have explored how implicit and explicit knowledge change as a consequence of acts of observation, inference, consideration and even forgetting. The present work proposes a further step towards a common framework for representing finer notions of information and their dynamics. We propose a combination of existing works in order to represent implicit and explicit beliefs. Then, after adapting definitions for the actions of upgrade and retraction, we discuss the action of inference on beliefs, analyzing its differences with respect to inference on knowledge and proposing a rich system for its representation.
Epistemic Logic [
Based on the awareness approach of [
The present work follows the previous ones, now focussing on the notion of beliefs. We combine approaches of the existing literature, proposing a setting for representing the notions of implicit and explicit belief (Section 2). Then we look into the dynamics of these notions; first, by adapting existing proposals to define the actions of explicit upgrade (explicit revision) and retraction (Section 3), and second, by discussing the action of inference on beliefs and its differences with inference on knowledge, and by proposing a rich system for its representation (Section 4). 2
Modelling implicit and explicit beliefs This section recalls a framework for implicit and explicit information and a framework for beliefs. By combining them, we will get our model for representing implicit/explicit beliefs. But before going into their details, we recall the framework on which all the others are based.
Epistemic Logic. The frameworks of this section are based on that of
Epistemic Logic (EL; [
Formulas are evaluated on pointed models (M, w) with M a Kripke model and w ∈ W a given evaluation point. Boolean connectives are interpreted as usual; the key clause is the one for ϕ, indicating that the agent is informed about ϕ at w iff ϕ is true in all the worlds the agent considers possible from w: ϕ iff for all u ∈ W, Rwu implies (M, u) ϕ 2.1
Implicit and explicit information
Non-omniscient agents. The formula (ϕ → ψ) → ( ϕ → ψ) is valid in
Kripke models: the agent’s information is closed under logical consequence.
This becomes obvious when we realize that each possible world stands for
a maximally consistent set of formulas. So if both (ϕ → ψ) and ϕ hold at
world w, both ϕ → ψ and ϕ are true in all worlds R-reachable from w. But
then ψ also holds in all such worlds, and therefore ψ holds at w. Usually
the discussion revolves around whether this is a reasonable assumption
for ‘real’ agents. Even computational agents may not have this property,
since they may lack of resources (space and/or time) to derive all the logical
consequences of their information [
One of the most influential solutions to this omniscience problem is
awareness logic [
Syntactically, awareness logic extends the EL language with formulas of the form A ϕ: “the agent is aware of ϕ”. Semantically, it extends Kripke models with a function A that assigns a set of formulas to the agent in each possible world. The new formulas are evaluated in the following way:
Implicit information about ϕ is defined as ϕ, but explicit information is defined as ϕ ∧ A ϕ. Although implicit information is still closed under logical consequence, explicit information is not. This follows from the fact that, different from the possible worlds, the A-sets do not need to have any closure property; in particular, {ϕ → ψ, ϕ} ⊆ A(w) does not imply ψ ∈ A(w). Agents with reasoning abilities. Still, though a ‘real’ agent’s information does not need to be closed under logical consequence, it does not need to be static. The more interesting approach for us is that in which the agent can extend her explicit information by the adequate actions. But, which are these actions and what does the agent needs in order to perform them?
In [
In [
The combination of the mentioned ideas have produced models for representing implicit and explicit knowledge ([5; 28; 13; 18; 10] among others). But the notion of belief is different, as we discuss in the next subsection. 2.2
Modelling beliefs
The KD45 approach. For modelling knowledge in EL, it is usually asked
for the accessibility relation R to be at least reflexive (making ϕ → ϕ valid:
if the agent knows ϕ, then ϕ is true), and often to be also transitive and
euclidean (giving the agent full positive and negative introspection). Beliefs
can be represented in a similar way, now asking for R to satisfy weaker
properties, the crucial one following the idea that, though beliefs do not
need to be true, we can expect them to be consistent. This is achieved by
asking for the relation to be serial, making the D axiom ¬ ⊥ valid. Full
introspection is usually assumed, yielding the classical KD45 approach.
Belief as what is most plausible. But beliefs are different from knowledge.
Intuitively, we do not believe something because it is true in all possible
situations; we believe it because it is true in those we consider most likely
to be the case [19; 26]. This idea has led the development of variants of
Kripke models [12; 4; 3]. Here we recall the plausibility models of [
A plausibility model is a Kripke model in which the accessibility relation, denoted now by ≤, is interpreted as a plausibility relation ordering possible worlds. This relation is assumed to be a preorder (a reflexive and transitive relation). Moreover, since the idea is to define the agent’s beliefs as what is true in the most plausible worlds from the evaluation point, ≤ should satisfy an important extra property: for any possible world w, the set of worlds that are better than w among those comparable to it should have maximal worlds. In order to state this property formally, denote by Vw the set of worlds comparable to w (its comparability class: Vw := {u | w ≤ u or u ≤ w }) and by Max≤(U) the set of ≤-maximal worlds of U (Max≤(U) := {v ∈ U | for all u ∈ U, u ≤ v }). Then, in a plausibility model, the accessibility relation ≤ is asked to be a locally well-preorder: a reflexive and transitive relation such that, for each comparability class Vw and for every non-empty U ⊆ Vw, Max≤(U) ∅. Note how the existence of maximal elements in every U ⊆ Vw implies the already required reflexivity, but also connectedness inside Vw. In particular, if two worlds w2 and w3 are more plausible than a given w1 (w1 ≤ w2 and w1 ≤ w3), then these two worlds should be ≤-related (w2 ≤ w3 or w3 ≤ w2 or both).
Interestingly, the agent’s indistinguishability relation can be derived from the plausibility one. If two worlds are ≤-related, then though the agent considers one of them more plausible than the other, she cannot discard one of them when the other one is given. In other words, worlds that are ≤ related are in fact epistemically indistinguishable.
For the language we have two options.1 We can extend the propositional language with formulas of the form Bϕ, semantically interpreted as for all u ∈ W, u ∈ Max≤(R≤(w)) implies (M, u) where R≤(w) := {u ∈ W | w ≤ u}. ϕ, 1In fact, the mentioned works, [12; 4; 3], use the notion of conditional belief as the primitive one, rather than plain belief. We have chosen to stick with the notion of plain belief through the present notes, leaving an analysis of the notions of implicit/explicit conditional beliefs for further work. The second option is to use a standard modal language with [≤] standing for the relation ≤, and then define beliefs in terms of it. Given the properties of ≤ (in particular, reflexivity, transitivity and connectedness), it is not hard to see that ϕ is true in the most plausible worlds from w iff w can see a better world from which all successors are ϕ worlds. This yields the following definition for “the agent believes ϕ”:
Bϕ := Our framework for representing implicit and explicit beliefs combines the mentioned ideas. The language has two components: formulas and rules. Formulas are given by a propositional language extended, first, with modalities ≤ and , and second, with fourlmas of the form A ϕ and R ρ, where ϕ is a formula and ρ a rule. Rules, on the other hand, are pairs consisting of a set of formulas, the rule’s premises, and a single formula, the rule’s conclusion. The formal definition of our language is as follows. Definition 2.1 (Language L). Given a set of atomic propositions P, formulas ϕ and rules ρ of the plausibility-access language L are given, respectively, by ϕ ::= p | A ϕ | R ρ | ¬ϕ | ϕ ∨ ψ | ρ ::= ({ϕ1, . . . , ϕnρ }, ψ) ϕ | ≤ ϕ where p ∈ P. Formulas of the form A ϕ are read as “the agent has acknowledged that formula ϕ is true”, and formulas of the form R ρ as “the agent has acknowledged that rule ρ is truth-preserving”. For the modalities, ≤ ϕ is read as “there is a more plausible world where ϕ holds”, and ϕ as “there is an epistemically indistinguishable world where ϕ holds”. Other boolean connectives as well as the box modalities [ ] and [≤] are defined as usual. We denote by L f the set of formulas of L, and by Lr its set of rules.
Though rules are usually presented as schemas, our rules are defined as particular instantiations (e.g., the rule ({p ∧ q}, p) is different from the rule ({q ∧ r}, q)). Since they will be applied in a generalized modus ponens form (if the agent has all the premises, she can derive the conclusion), using concrete formulas avoids details of instantiation, therefore facilitating the definition. When dealing with them, the following definitions will be useful. Definition 2.2. Given a rule ρ, we will denote its set of premises by pm(ρ), its conclusion by cn(ρ), and its translation (an implication whose antecedent is the finite conjunction of ρ’s premises and whose consequent is ρ’s conclusion) by tr(ρ).
For the semantic model, we will extend the described plausibility models with two functions. Definition 2.3 (Plausibility-access model). With P the set of atomic propositions, a plausibility-access (PA) model is a tuple M = W, ≤, V, A, R where W, ≤, V is a plausibility modeolver P and • A : W → ℘(L f ) is the access set function, assigning to the agent a set of formulas of L in each possible world, • R : W → ℘(Lr) is the rule set function, assigning to the agent a set of rules of L in each possible world.
Functions A and R can be seen as valuations with a particular range, assigning to the agent a set of formulas and a set of rules at each possible world, respectively. Moreover, recall that two worlds that are ≤ related are epistemically indistinguishable, so we define as the union of ≤ and its converse ( := ≤ ∪ ≥): the agent cannot distinguish between two worlds if she considers one of them more plausible than the other.
A pointed plausibility-access model (M, w) is a plausibility-access model with a distinguished world w ∈ W.
Here it is important to emphasize our interpretation of the A-sets.
Different from [
Now the semantic evaluation. The modalities ≤ and are interpreted via their correspondent relation in the usual way, and formulas of the form A ϕ and R ρ are interpreted with our two new functions. Definition 2.4 (Semantic interpretation). Let (M, w) be a pointed PA model with M = W, ≤, V, A, R . Atomic propositionsand boolean operators are interpreted as usual. For the remaining cases, (M, w) (M, w) (M, w) (M, w)
A ϕ R ρ ≤ iff ϕ ∈ A(w) iff ρ ∈ R(w) ϕ iff there is a u ∈ W such that w ≤ u and (M, u) ϕ iff there is a u ∈ W such that w u and (M, u) ϕ ϕ
For characterizing valid formulas, an important observation is that a
locally well-preorder is a locally connected and conversely well-founded
preorder [
If ϕ → ψ and ϕ, then ψ K≤ Dual≤ ≤ ϕ ↔ ¬[≤] ¬ϕ Nec≤ If ϕ, then [≤] ϕ
[≤] (ϕ → ψ) → ([≤] ϕ → [≤] ψ) T≤ 4≤ LC Inc with respect to plausibility-access models follows from the fact that every strict preorder is conversely well-founded.
Note how the axiom system does not have axioms for formulas of the form A ϕ and R ρ. This is because, as mentioned before, such formulas are simply special atoms for the dedicated valuation functions A and R. Moreover, we have not asked for the A- and R-sets to have any special closure property and there is no restriction in the way they interact with each other.2 Just like axiom systems for Epistemic Logic do not require special axioms describing the behaviour of atomic propositions (unless, of course, they have special properties, like q being true every time p is, characterized by p → q), our system does not require special axioms for these special atoms. More precisely, in the canonical model construction, we only need to define access and rule sets in the proper way:
A(w) := {ϕ ∈ L f | A ϕ ∈ w}
R(w) := {ρ ∈ Lr | R ρ ∈ w}
Then, formulas of the form A ϕ and R ρ also satisfy the crucial Truth Lemma,
and completeness follows. Again, see Chapter 4 of [
It is time to define the notions of implicit and explicit beliefs. Our
definitions, shown in Table 2, combine ideas from [
Explicit beliefs are implicit beliefs, witness the following validities: BExϕ → BImϕ
BExρ → BImρ
2In [
BImϕ := BExϕ := BImρ := BExρ := ≤ [≤]ϕ ∧ A ϕ ≤ [≤] tr(ρ) ≤ [≤]tr(ρ) ∧ R ρ
A possibly more interesting point is the following. An agent in [17; 10]
is non-omniscient due to lack of attention; she does not need to be aware
of every formula. On the other hand, our agent is aware of all formulas,
but still she is non-omniscient because she does not need to be aware that a
formula is true. This may seem a small difference, but the interpretation of
the A sets determines the reasonable operations over them. An agent can
become aware of any formula at any time, so any formula can be added
to the A-sets without further requirement [
We finish this section by mentioning some properties of implicit and explicit beliefs about formulas. (Rules behave in a similar way.) Implicit beliefs are closed under logical consequence: if the most plausible worlds satisfy both ϕ → ψ and ϕ, then they also satisfy ψ. But explicit beliefs do not need to have this property because the A-sets do not need to have any closure property: having ϕ and ϕ → ψ does not guarantee to have ψ.
Though ≤ is reflexive, neither implicit nor explicit beliefs have to be true because the real world does not need to be among the most plausible ones. Nevertheless, reflexivity makes implicit (and therefore explicit) beliefs consistent. Every world has at least one ≤-successor, so ¬BIm⊥ is valid.
Implicit beliefs are positively and negatively introspective. This is the case because the notion of ‘most plausible worlds’ is global inside the same comparability class. For positive introspection, if the set of maximal worlds contains only ϕ-worlds (BImϕ), so does the maximal from the maximal ones (BImBImϕ). And for negative introspection, if there is a ¬ϕ-world u in the maximal worlds, (¬BImϕ), then u is also in those maximal from the maximal ones (BIm¬BImϕ). But this does not extend to explicit beliefs, again because the A-sets do not need to have any closure property. Having ϕ does not guarantee to have BExϕ (so BExϕ → BExBExϕ is not valid), and not having ϕ does not guarantee to have ¬BExϕ (so ¬BExϕ → BEx¬BExϕ is not valid).
Dynamics part one: upgrade and retraction We have a framework for representing implicit and explicit beliefs. We now look at their dynamics by introducing two actions that modify them. The χ-upgrade operation [4; 3] modifies the plausibility relation ≤ to put the χ-worlds at the top, therefore revising the agent’s beliefs. Here we have two possibilities, depending on whether it also adds χ to the A-sets (explicit upgrade) or not (implicit upgrade). Here is the definition of the first case. Definition 3.1 (Explicit upgrade). Let M = W, ≤, V, A, R χ a formula in L. The PA model Mχ⇑+ = W, ≤, V, A , R the plausibility relation and in the access set function: be a PA model and differs from M in ≤ := (≤; χ?) ∪ (¬χ?; ≤) ∪ (¬χ?; ; χ?) A (w) := A(w) ∪ {χ} for every w ∈ W Note how the upgrade operation is functional: for every model M it returns one and only one model Mχ⇑+ .
The new plausibility relation is given in a PDL style: we have w ≤ u iff (1) w ≤ u and u is a χ-world, or (2) w is a ¬χ-world and w ≤ u, or (3) w u, w is a ¬χ-world and u is a χ-world. There are other possible definitions for ≤ [4; 3], and the chosen one, so-called radical upgrade, is just an example of what can be defined.
The operation preserves models in the intended class.
We extend the language to express the effect of an explicit upgrade; formulas of the form χ ⇑+ ϕ are read as “it is possible to performan explicit χ-upgrade after which ϕ holds”. There is no precondition for this action (the agent can perform an explicit upgrade whenever she wants), so the semantic interpretation is as follows.
Note how the operation puts on top those worlds that are χ-worlds in the original model, but they do not need to be χ-ones after the upgrade. The plausibility relation changes, therefore changing the truth-value of formulas containing the modalities for ≤ and/or and, in particular, changing the agent’s beliefs. This is not strange at all, and in fact it corresponds to the well-known Moore-like sentences (“p is the case and you do not know it”) in Public Announcement Logic that become false after being announced, and therefore cannot be known.
Nevertheless, the operation behaves as expected for propositional formulas. The operation does not change valuations, so if χ is purely propositional, the operation will put current χ-worlds on top, and they will still be χ-worlds after the operation so the agent will believe χ.
The validities in our new language can be axiomatized by using reduction
axioms, valid formulas that indicate how to translate a formula with the
new modality χ ⇑+ into a provably equivalent one without them. Then,
completeness follows from the completeness of the basic system. We refer
to [
Theorem 1. The axiom system of Table 1 together with axioms and rules of Table 3 (with the always true formula) provide a sound and (weakly) complete axiom system for formulas in the language L plus the explicit upgrade modality with respect to plausibility-access models.
χ+⇑ p ↔ p χ+⇑ ¬ϕ ↔ ¬ χ+⇑ ϕ χ+⇑ (ϕ ∨ ψ) ↔ χ ⇑+ ϕ ∨ χ+⇑ ψ χ+⇑ ≤ ϕ ↔ χ+⇑ ϕ ↔ From ϕ infer ≤χ ∧ χ ⇑+ ϕ ∨ ¬χ ∧ ≤
+ χϕ⇑ χ+⇑ ϕ χ+⇑ A χ ↔ χ+⇑ A ϕ ↔ A ϕ for ϕ χ+⇑ R ρ ↔ R ρ
χ χ+⇑ ϕ ∨ ¬χ ∧ (χ ∧ χ+ ⇑ ϕ)
The reduction axioms simply indicate how each kind of formula is
affected by the explicit upgrade operation. For example, χ ⇑+ p ↔ p states
that atomic propositions are not affected, and both χ ⇑+ A χ ↔ and
χ ⇑+ A ϕ ↔ A ϕ for ϕ χ together state that χ and only χ is added to the
A-sets. The interesting axiom is the one for the plausibility modality ≤ . It
is obtained with techniques from [
Retraction But there are also situations in which the agent simply retracts some explicit belief, that is, she does not acknowledge it as true anymore. This is achieved simply by removing the formula from the A-sets.
Definition 3.3 (Retraction). Let M = W, ≤, V, A, R formula in L. The PA model M−χ = W, ≤, V, A, R access set function, given for every w ∈ W as be a PA model and χ a differs from M just in the
A (w) := A(w) \ {χ}
Again, the retraction operation is functional. Moreover, it does not modify ≤, so it preserves plausibility models.
This operation is represented in the language by formulas of the form −χ ϕ, read as “it is possible to retract χ and after it ϕ holds”. Just like an upgrade, no precondition is needed.
Theorem 2. The axiom system of Table 1 together with axioms and rules of Table 4 (with ⊥ the always false formula) provide a sound and (weakly) complete axiom system for formulas in the language L plus the retraction modality with respect to plausibility-access models.
−χ p ↔ p −χ ¬ϕ ↔ ¬ −χ ϕ −χ (ϕ ∨ ψ) ↔ −χ ϕ ∨ −χ ψ −χ ≤ ϕ ↔ ≤ −χ ϕ −χ ϕ ↔ −χ ϕ From ϕ infer −χ ϕ −χ A χ ↔ ⊥
−χ A ϕ ↔ A ϕ for ϕ −χ R ρ ↔ R ρ χ
Here the key axioms are −χ A χ ↔ ⊥ and −χ A ϕ ↔ A ϕ for ϕ
stating that χ and only χ is removed from the A-sets. Similar reduction
axioms have been presented in [
Dynamics part two: inference on beliefs
We now turn into the main part of our work. In this section we analyze
rule-based inference on beliefs. We start by recalling the case of rule-based
inference on knowledge [
The definition of implicit and explicit knowledge are simpler than those for beliefs since they depend directly on all the worlds the agent considers possible. The agent knows ϕ implicitly when it holds in all the worlds she considers possible, and she knows ϕ explicitly when she also recognizes it as true in all such worlds. The definitions of explicit knowledge about a rule ρ is given in a similar way.
KImϕ := [ ] ϕ KImρ := [ ] tr(ρ)
KExϕ := [ ] (ϕ ∧ A ϕ)
KExρ := [ ] (tr(ρ) ∧ R ρ)
The action of inference on knowledge with rule σ is defined as an operation that adds σ’s conclusion to the A-set of those worlds where the agent knows explicitly σ and its premises (KExσ ∧ KExpm(σ)). More precisely, if M is a plausibility-access model with access set function A, then the operation of σ-inference on knowledge produces the model M →σK , differing from M just in the access set function A , which is given by
A (w) := A(w) ∪ {cn(σ)} A(w) if (M, w) otherwise
A new modality →σK is introduced to express the effects of this operation, and its semantic definition is given by
But take a closer look at the inference on knowledge operation. What it actually does is to discard all worlds where KExσ ∧ KExpm(σ) holds, and replace them with copies that are almost identical, the only difference being their A-sets that, after the operation, will have cn(σ). And this is reasonable because, under the assumption that knowledge is true information, inference based on a known (therefore truth-preserving) rule with known (therefore true) premises is simply deductive reasoning: the premises are true and the rule preserves the truth, so the conclusion should be true. In fact, inference based on a known rule with known premises is the act of recognizing two things. First, since the applied rule is truth-preserving and its premises are true, its conclusion must be true; and second, situations where the premises are true but the conclusion is not are not possible.
The case of beliefs is different, as suggested in [
Our proposal is the following. An inference on beliefs should create two copies of each world where the rule and the premises are believed: an exact copy of the original one, and another extending it by adding the rule’s conclusion to it. But not only that. The agent believes that the rule is truth-preserving and the premises are true, so the extended world should be more plausible than the ‘conclusionless’ one.
But, how to create copies of a possible world? We can use the action
models and product update of the so-called BMS approach [
Plausibility-access action models
The main idea behind action models [
This idea has been extended in two different directions: in order to
deal with plausibility models [
S, , Pre is a plausibility actionmodel [
Just as before, the plausibility relation defines an equivalence relation by putting it together with its converse: ≈ := ∪ . A pointed plausibility-access action model (A, s) has a distinguished event s ∈ S.
Examples of plausibility-access action models will be shown in Section 4.2. But first we will define the plausibility-access model that results from an action model application as well as the formula that will represent this operation and its semantic interpretation. • W := {(w, s) ∈ (W × S) | (M, w) |= Pre(s)} • (w1, s1) ≤ (w2, s2) iff s1 ≺ s2 and w1 w2 or s1 ≈ s2 and w1 ≤ w2 • V (w, s) := V(w) • A (w, s) := PosA(s, A(w)) • A (w, s) := PosR(s, R(w))
Note how the set of worlds of the new plausibility-access model is given
by the restricted cartesian product of W and S; a pair (w, s) will be a world in
the new model iff event s can be executed at world w. The new plausibility
order follows the so-called ‘Action-priority’ rule [
Now, for the valuations of the new worlds. First, a new world inherits
the atomic valuation of its static component, that is, an atom p holds at (w, s)
iff p holds at w. The cases for access sets gives us full generality: the access
set of world (w, s) is given by the function PosR with the event s and the
access set of w as parameters [
It is not hard to verify that the product update operation preserves plausibility-access models.
Proposition 2. If M is a plausibility-access model and A a plausibility-access action model, then M ⊗ A is a plausibility-access model.
In order to express how product updates affect the agents’ information, we extend our language with modalities for each pointed plausibility-access action model (A, s), allowing us to build formulas of the form A, s ϕ, whose semantic interpretation is given below.
Definition 4.3. Let (M, w) be a pointed PA model and let (A, s) be a pointed PA action model with Pre its precondition function.
Pre(s) and (M ⊗ A, (w, s))
ϕ 4.2
Plausibility-access action models for basic inference The action of inference on knowledge can be represented with plausibilityaccess action models.
Definition 4.4 (Inference on knowledge). Let σ be a rule. The action of inference on knowledge is given by the pointed PA action model (A →σK , s) whose definition (left) and relevant diagram (right) are given by • S := {s} • := {(s, s)}
• Pre(s) := KExσ ∧ KExpm(σ) • PosA(s, X) := X ∪ {cn(σ)} • PosR(s, Y) := Y s
X ∪ {cn(σ)}
But now we can represent more. Following our previous discussion, here is the action model for basic inference on beliefs.
Definition 4.5 (Basic inference on beliefs). Let σ be a rule. The action of basic inference on belief is given by the pointed PA action model (A →σB , s1) whose definition is • S := {s1, s2} •
:= {(s1, s1), (s1, s2), (s2, s2)} • PPrree((ss12)) ::== PPrreeBBσσ
PreBσ := BExσ ∧ BExpm(σ)
4.3
Extended inference: an exploration
• PPoossAA((ss12,, XX)) ::== XX ∪ {cn(σ)}
• PPoossRR((ss12,, YY)) ::== YY
s2 X ∪ {cn(σ)}
Plausibility-access action models allow us to represent more than what
we have discussed. As observed in [
X ∪ {cn1(σ)} s2 s3 X ∪ {cn2(σ)}
s4 X ∪ {cn1(σ), cn2(σ)}
with more than one world, like the action model on the left that allows the agent to have cn2(σ) without having cn1(σ).
So far our examples have one characteristic in common. The new access set function is monotone, reflecting the optimism of the agent with respect to the conclusion: events that extend A-sets are always more plausible. Definition 4.6 (Plausibility-access action models for optimistic inference). Plausibility-access action models in which, for every event s1, s2, s1 s2 implies
PosA(s1, X) ⊆ PosA(s2, X) are called action models for optimistic inference.
But then we can also consider the opposite case. Models with an antimonotone new access set function reflect the pessimism of the agent with respect to the conclusion: events that extend A-sets are always less plausible. Definition 4.7 (Plausibility-access action models for pessimistic inference). Plausibility-access action models in which, for every event s1, s2, s1 s2 implies
PosA(s1, X) ⊇ PosA(s2, X) are called action models for pessimistic inference.
Of course these two classes do not cover all possibilities. Plausibility-access action models allow us to represent many different and complex inferences whose detailed study has to be left for further work. 4.4
Brief discussion on completeness
The reduction axioms of [
A, sϕ ∨ ≤
A, s ϕ
s s
s≈s
But when looking for reduction axioms for access and rule set formulas,
PosA and PosR pose a problem. The reason is that they allow the new
access and rule sets to be arbitrary sets. Compare this with other product
update definitions. The one of [
We have presented a framework for representing implicit and explicit beliefs. We have also provided representations of three actions that modify them, starting with those of explicit upgrade and retraction but, more important, discussing intuitive ideas and proposing a rich framework for representing the action of inference on beliefs.
There are parts of this work that deserve further exploration, the most
appealing being the study of the different kind of inferences that we can
represent with plausibility-access action models. We have defined those
for inference on knowledge and basic inference on beliefs, and we have
briefly explored some others, but our structures can represent much more.
Another interesting extension is to look at dynamics of rules, that is, to
look for reasonable actions that extend not only the rules the agent knows
[