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Knowledge on Treelike Spaces

This paper presents a bimodal logic for reasoning about knowledge during knowledge acquisition. One of the modalities represents (effort during) non-deterministic time and the other represents knowledge. The semantics of this logic are tree-like spaces which are a generalization of semantics used for modeling branching time and historical necessity. A finite system of axiom schemes is shown to be canonically complete for the formentioned spaces. A characterization of the satisfaction relation implies the small model property and decidability for this system.

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1 Introduction

The notion of possible world dominates the literature in modal logic, via Kripke models, as well as in any logic dealing with the epistemic state of a reasoner. The heart of this popularity lies in the identification of an intentional state through common properties of extensional objects. Apart from genuine problems such as logical omniscience this representation suffers from, it is limited in a static description of the reasoner’s epistemic state. The “logic of knowing” is not only embodied in the representation of knowledge but also in the way knowledge is acquired. We do not refer to temporal properties but rather to methodology (though both can be intertwined).

Recently a family of logics was introduced ([MP92],[Geo94a],[Geo93],[DMP]) with the intention to fill this void. It succeeds in doing so by attaching familiar mathematical structures such as spaces of subsets, topologies and complete lattices of subsets corresponding to a natural knowledge acquisition. This paper extends this work by introducing a bimodal logic belonging to the same family of logics and establishes a correspondence between a particular epistemic process of knowledge acquisition with a space of subsets forming a tree (treelike space).

In our framework the view of a reasoner will be represented by a set of possible worlds. Each of these worlds represents an alternative state compatible with the reasoner’s knowledge of actual state. This treatment of knowledge agrees with the traditional one ([Hin62], [HM84], [PR85], [CM86]

) expressed in a variety of contexts (artificial intelligence, distributed processes, economics, etc).

We are interested in formulating a basic logical framework for reasoning about a resource-conscious acquiring of knowledge. Such a framework can be applied to many settings such as the ones involving time, computation, physical experiments or observations. In these settings an (discrete or continuous) increase of information available to us takes place and results in an increase of our knowledge. How could this simple idea be embodied in the formentioned semantical framework? An increase of knowledge can be represented with a restriction of the knower’s view, i.e. of the equivalence class of the alternative worlds. This restriction is nondeterministic (we do not know what kind of additional information will be available to us, if at all) but not arbitrary: it will always contain the actual state of the knower, i.e. it is a neighborhood restriction of the actual state. In this way, set-theoretic considerations come in.

A discrete version of our epistemic framework can arise in scientific experiments or tests. We acquire knowledge by “a step-by-step” process, each step being an experiment or test. The outcome of such an experiment or test is unknown to us beforehand, but after being known it restricts our attention to a smaller set of possibilities. A sequence of experiments, tests, or actions comprises a strategy of knowledge acquisition. This model is in many respects similar to Hintikka’s “oracle” (see [Hin86]). In Hintikka’s model the “inquirer” asks a series of questions to an external information source, called “oracle” (can be thought as a knowledge base). The oracle answers yes or no and the inquirer increases his or her knowledge by this piece of additional evidence. At any point of this process the inquirer follows a branch of a tree determined by the possible answers to his or her series of questions. Such an interrogative model is recognized by Gadamer ([Gad75]) as an important part of the epistemic process. Consider the following example:

Example: Suppose that our view, the set of possible worlds, is and our query consists of two questions , , in that order. The answer to is yes in , and no in , . The answer to is yes in , , and no in . Then the possible sequences of knowledge states comprise a tree of subsets as shown in Figure 1. The space of subsets labeling the nodes of the tree will be called a treelike space.

yesyesnonoyesno
Figure 1: A knowledge acquisition tree.

The above example shows a transition from the symbolic description of the epistemic process to a description in spatial terms. Instead of going down a proof tree, the one which entails the desired formulae, we intersect nodes of a tree labeled by subsets of a space. This transition is direct; it enables us to think in geometric terms.

Now consider the following example:

Example: Suppose that a machine emits a stream of binary digits representing the output of a recursive function . After time the machine emitted the stream . The only information we have about the function being computed at this time on the basis of this (finite) observation is that

As far as our knowledge concerns, is indistinguishable from the constant function , where for all . After some additional time , i.e. spending more time and resources, might appear and thus we could be able to distinguish from . In any case, each binary stream will be an initial segment of and this initial segment is a neighborhood of . In this way, we can acquire better knowledge of the function the machine computes. The space of finite binary streams is a structure which models computation. The sets of binary streams under the initial segment ordering is an example of a treelike space.

The above example shows how the same epistemic process appears during observations of programs. Here possible worlds correspond to (total) computations and our view to observations. We can apply the same spatial reasoning to programs through the following correspondence:

Knowledge states Sets Observations

Possible worlds
Points Computations.

Therefore a common idea lies behind the knowledge-theoretic, spatial and computational framework. The connection between the last two is not new. Here is how this epistemic framework ties with previous work on establishing links between spatial reasoning and reasoning about programs.

We use two modalities for knowledge and for effort, i.e. spending of resources. Consider the formula

where is an atomic predicate and is the dual of the , i.e. . It will be clear after the presentation of semantics in Section 2.1 that if the above formula is valid, then the set which represents is an open set of the topology generated by the subsets of the treelike space as a basis. Under the reading of as “possible” and as “is known”, the above formula says that

“if is true then it is possible for to be known”,

i.e. is affirmative. Vickers defines similarly an affirmative assertion in [Vic89]

“an assertion is affirmative iff it is true precisely in the circumstances when it can be affirmed.”

Affirmative and refutative assertions are closed under infinite disjunctions and conjunctions, respectively. Smyth in [Smy83] observed first these properties in semi-decidable properties. Semi-decidable properties are those properties whose truth set is r.e. and are a particular kind of affirmative assertions. In fact, changing our power of affirming or computing we get another class of properties with a similar knowledge-theoretic character. For example, using polynomial algorithms affirmative assertions become polynomially semi-decidable, i.e. NP properties. If an object has this property then it is possible to know it with a polynomial algorithm even though it is not true we know it now.

Our approach has an independent theoretical interest. A new family of Kripke frames, called subset frames, arises. These are Kripke frames which are equivalent to sets of subsets. In particular, we have identified those which are equivalent to (complete) lattices of subsets and topologies (see [Geo93]). In this paper, we shall identify those which correspond to the above interrogative model, called treelike spaces. Treelike spaces have a particular interest; they correspond to an indeterminist’s theory of time called Ockhamism (see [Pri67]), which gives rise to branching time. We refer the reader to section 2.1 for a detailed discussion.

A family of logics for knowledge and time is studied in [HV89] and various complexity results are established. However, the framework of the above logics is restricted to distributed systems and their interpretation differs significantly from ours.

Interpreting the knowledge modal operator as a universal quantifier we present a novel way of understanding the meaning of quantifiers in varying (ordered) domains (see section 2.2 for a relevant discussion). This is one of the main difficulties in formulating a meaningful first-order system for modal logic (see [Fit93] for a discussion).

The language and semantics of our logical framework is presented in Section 2. In the same section, we present two systems which belong to the same family of logics, studied in [MP92], [Geo93] and [Geo94a]. In Section 3, we present an axiomatization, called , for our semantics and we prove completeness, small model property, and decidability.

A preliminary version of this paper has appeared in [Geo94b].

2 Two Systems: and

2.1 Language and Semantics

We follow the notation of [MP92].

We construct a bimodal propositional modal logic. Formally, we start with a countable set of atomic formulae, then the language is the least set such that and closed under the following rules:

We abbreviate, as usual, with and with . The language can be interpreted inside any spatial context as follows.

Definition 1  Let be a set and a subset of the powerset of , i.e. such that . We call the pair a subset space. A model is a triple , where is a subset space and a map from to with and called initial interpretation.

We denote the set with . For each let be the lower closed set generated by in the partial order , i.e. the set .

Definition 2  The satisfaction relation , where is the model , is a subset of defined recursively by (we write instead of )

If , for all belonging to , then is valid in , denoted by .

The case for atomic formulae shows that we deal with analytic sentences, i.e. sentences which do not change their truth value. If a formula does not contain then it has the same interpretation as . This has also the consequence that the universal substitution rule does not hold. Thus, time does not affect the semantic value of sentences but rather the knowledge we have of them. This difference makes the modality not collapsing to a temporal modality but being closer to necessity.

We abbreviate and with and respectively. We have that

Definition 3  A treelike space is a subset space where for all , either , or , or . A model induced by a tree space will be called a treelike model.

It is clear that in the countable case the set of subsets of a treelike space forms a tree under the subset ordering.

Example: Let

Now, let

where are natural numbers, and

Then it is easily verified, using definition 2.1, that is a treelike space.

Now let be a predicate with

Then the formula

which translates to “it will never be known that appears infinitely often”, is valid in the treelike model . This comes with no surprise, since the knowledge of “infinitely often” requires an infinite amount of resources. This formula is an example of a refutative assertion (see introduction).

Treelike spaces get their name from treelike frames (see [Pri67]). A treelike frame is a pair , where is a nonempty set and is a transitive ordering on such that if and then either or or . Treelike frames have appeared as semantics for the Ockhamist’s concept of non-deterministic time and been used for treating historical necessity and conditionals (see [Tho84] and [VF81]). The validity on these frames is called Ockhamist validity. A treelike space is a special form of a treelike frame where the temporal instants of the frame are labeled by subsets of a space and whenever instants are incomparable the respective subsets are disjoint. It can be easily seen that the ordering among subsets is a treelike frame. The similarities do not end here. Let be a treelike frame and, for each , the set of maximal linear ordered subsets of containing , i.e. the branches intersecting . Then is a treelike space. The difference lies on the interpretation of atomic formulae. We interpret atomic formulae on branches while an Ockhamist assignment interprets atomic formulae on temporal instances. This bring up another dimension of our logic. Our logic is not conservative over a logic which interprets as (the “future” modality) for if contains no occurrences of then is valid in a treelike space exactly when is. We adopt the indeterminist’s view of necessity (knowledge). Although may be true in our world, may be false. This is because there is no special world in our view which deserves to be called actual. Setting apart Ockhamist validity, treelike spaces are more general than treelike frames (and their derivative frames) due to the fact that we do not assume an overall temporal ordering. In this sense treelike spaces are closer to a more general structure, first introduced by Kamp and subsequently called Kamp frames, where worlds do not participate in the same temporal structure (for definition and discussion see [Tho84]). In fact, it is easily seen that treelike spaces are equivalent to Ockhamist frames introduced by Zanardo in [Zan85] for the completeness of strong Ockhamist validity. At any rate, our work seems to have more than superficial links with work in historical necessity and questions such as what the connections between the two notions of validity are should be the subject of a more systematic investigation.

2.2 and

We saw that the semantics of the bimodal language is interpreted in any pair . What happens when we allow to be any class of sets of subsets? If is an arbitrary set of subsets then the system is complete for such subset spaces. The axiom system consists of axiom schemes 1 through 10 and rules of Table 1 (see page 1) and appeared first in [MP92].

The following was proved in [MP92].

Theorem 4

The axioms and rules of are sound and complete with respect to subset spaces.

If is a complete lattice under set-theoretic union and intersection then the system is canonically complete for this class of subset spaces. The axiom system consists of the axiom schemes and rules of plus the following two additional axiom schemes:

and

The first axiom is a well-known formula which characterizes incestual frames, i.e. if two points and in a frame can be accessed by a common point then there is a point which can be accessed by both and . The second characterizes union.

The following was proved in [Geo93].

Theorem 5

The axioms and rules of are sound and canonically complete with respect to subset spaces, which are complete lattices.

The proof of the above theorem was later shortened and improved through an elegant embedding of (and therefore intuitionistic logic via the Gödel translation) by Dabrowski, Moss and Parikh in [DMP]. This translation reveals that truth in intuitionistic logic coincides with “possibility of knowing” in our system. It also reveals a connection with another line of work, that of Fischer Servi. In [FS80] and [FS84] the semantics and syntax of the family -IC of intuitionistic modal logics is studied. This family is is naturally embedded via the Gödel translation to the family (-) of bimodal logics, where is always one of the coordinates (like in our case). However, the semantics called double model structures (birelational modal frames) deviate from our space theoretic framework; a fact that declares itself on the presence of different connecting axioms, i.e. axioms involving both modalities.

3 The system

We add the axioms 11 and 12 to form the system for the purpose of axiomatizing treelike spaces.

Axioms

  1. All propositional tautologies

  2. , for

Rules

Table 1: Axioms and Rules of .

A word about the axioms (most of the following facts can be found in any introductory book about modal logic, e.g. [Che80] or [Gol87].) Axiom 2 expresses the fact that the truth of atomic formulae is independent of the choice of subset and depends only on the choice of point. Axioms 3 through 5 and Axioms 6 through 9 are used to axiomatize the normal modal logics S4 and S5 respectively. The former group of axioms expresses the fact that the passage from one subset to its restriction is done in a constructive way, as actually happens in an increase of information or a spending of resources (the classical interpretation of necessity in intuitionistic logic is axiomatized in the same way). The latter group is generally used for axiomatizing logics of knowledge.

Axiom 10 expresses the fact that if a formula holds in arbitrary subsets is going to hold as well in the ones which are neighborhoods of a point. The converse of this axiom is not sound.

Axiom 11 is a well-known axiom which characterizes reflexive, transitive and connected frames, i.e. if two points and in a frame can be accessed by a common point then either accesses or accesses (or both).

Soundness of Axioms 1 through 10 has already been established for arbitrary subset spaces (see [MP92]). The soundness of Axiom 11 is easy to see, since the subset frame (see [Geo93]), i.e. the birelational modal frame, of a tree model is connected.

Proposition 6

The axiom 12 is sound.

Proof. We shall show soundness for the equivalent formula

Let . Then there exists such that . This implies that there exists such that . Now, observe that . For, if and then there are two cases. Either and , since , and we are done, or and so we have , since the subsets containing are linearly ordered. In this case, we have , since . By our assumption , we have . So . Now, and imply together .  

Note that Axiom 10 follows from Axiom 12 (substitute with ). Axiom 10 has a particular interest; if we replace with the universal quantifier it becomes the well-known Barcan formula

Our system (and therefore and , since this formula belongs to their axiomatization) can be thought as a propositional analogue of a first order modal system interpreted over varying restricting domains (see [Fit93]).

3.1 Completeness

Our proof of completeness is based on a construction of a treelike model which is (strongly) equivalent to each generated canonical submodel of the canonical model of .

The canonical model of is the structure

where

along with the usual satisfaction relation (defined inductively):

We write , if for all .

A canonical model exists for all consistent bimodal systems with the normal axiom scheme for each modality (as ). We have the following well known theorems (see [Che80], or [Gol87]).

Theorem 7 (Truth Theorem)

For all and ,

Theorem 8 (Completeness Theorem)

For all ,

We shall now prove some properties of .

Proposition 9
  1. The canonical frame is reflexive, transitive and connected with respect to the relation .

  2. The relation is an equivalence relation.

  3. For all , if then there exists such that .

  4. For all , if and then .

  5. The relation is antisymmetric.

Proof. For Part a, Axioms 3 through 5 and Axiom 11 characterize reflexive, transitive and connected frames (these axioms comprise the system ).

For Part b, is axiomatized with the axioms.

Part c is an immediate consequence of Axiom 10.

To show Part d, let

for all pairs such that and .

We shall prove by induction on the complexity of that, for all pairs such that and , belongs to some if and only if belongs to s. This shows that . Further, we have , since . Therefore .

If is an atomic formula and , for some , then . Therefore, by axiom 2, . Hence, .

The cases of negation and conjunction are straightforward.

If , let , for some . In particular, and by induction hypothesis, . Suppose, towards a contradiction, that . Then there exists such that and . Since the frame is connected, and imply that either or . If then which is contradiction, since and . If then, by induction hypothesis, which is a contradiction, since and is consistent. Hence .

If , let for some . Suppose, towards a contradiction, that . Then there exists such that and . We have , since . Since , there exists, by Part c, such that . We have , since . Since , there exists, by Part c, such that . Notice that and , and so . By our previous assumption, we have and . By induction hypothesis on , both and should belong to which is a contradiction to its consistency.

For Part e, we shall prove by induction on the structure of that, for all such that , if and only .

The cases of atomic formula, negation, conjunction and are straightforward. We shall show the step. Let , and suppose towards a contradiction. Then there exists such that and . Since , there exists such that . Also, , since . Now, since there exists such that . This implies and . Therefore, by Part d, . Thus we have with and which is a contradiction to the induction hypothesis.  

The canonical model is not a (model corresponding to a) treelike model. A counterexample will appear later on (see Figure 2). However, by defining a number of equivalence relations, we shall be able to construct a treelike model equivalent to each generated part of the canonical model.

For all , let , i.e. the equivalence class under where belongs. Let . We define the following relation on .

Proposition 10

The relation is a partial order.

Proof. Since , we have and reflexivity follows.

For antisymmetry, let and for some . Then there exist such that , , and . Since , there exists such that . So we have and which implies, by Proposition 9(d), . Therefore , by ’s antisymmetry. Hence .

For transitivity, let for some . Then there exist , , and such that and . Since , there exists such that . So , and therefore .

 

A subset of , the domain of the canonical model , is called -closed whenever

The intersection of -closed sets is still -closed, therefore we can define the smallest -closed containing , for all . We shall denote this set by . Fix . We define the model

where , and are the restrictions of , and to and respectively. We shall call this model the submodel of generated by .

Observe that if we restrict the partial order to then is the greatest element under .

For each generated submodel of the canonical model, we shall construct a treelike model which is equivalent to it.

For each , let

Notice that .

For each , we define the following relation on

Proposition 11

For all , the relation is an equivalence relation.

Proof. This is because inherits the properties of .  

We denote the equivalence class of under with . We have .

Let be the subset space where

and

It is clear that .

Lemma 12

If and then .

Proof. Immediate from the definition of .  

To elaborate the above process, we present the following simple example.

Example: A part of the canonical model appears in Figure 2.

Figure 2: A generated submodel of the canonical model.

(Horizontal and downward arrows correspond to and , respectively.) We would like to make subsets of a treelike space correspond to equivalence classes under . Canonical model worlds related with will be represented by a single point. However, this model is not a treelike model: and should make two distinct points. To remedy that, we “trace back” each equivalence class under to the uppermost one. For instance, is traced back to . The latter forms . Next, we split into equivalence classes under , i.e. and , since and . Finally, we replace with as many copies as these equivalence classes (see Figure 3).

Figure 3: The treelike model corresponding to Figure 2

The infinite case is taken care of by Lemma 14. The resulting space (of Figure 3) is a treelike space. Note that we could have replaced this procedure by one that employs maximal branches but we find the present one simpler.

Proposition 13

The subset space is a treelike space.

Proof. Suppose . Let . We have either or , since , , for some and , and the canonical frame is connected. The former implies . Thus, by Lemma 12, . Similarly, the latter implies .  

Let be the treelike model where and are as above, and where is the initial interpretation restricted on .

An element of can have more than one representation. In order to prove the semantical equivalence we are opting for, we shall choose a canonical representation. So, given a pair , its canonical representation is where is such that . Its existence is assured by the definition of and uniqueness by Proposition 9(d). From now on, we shall use the canonical representation wherever is possible.

Lemma 14

Let and such that . Then for all there exists such that and , i.e. .

Proof. Let

be the linear order of all members of under such that .

Now, let

is consistent. For if not, then there would be as above with and such that

Thus

We shall prove that the negation of the above formula belongs to and reach a contradiction. Since , we have . Hence

Observe that so, by applying axiom 12, we have

Since , we have

Also, and

So, by axiom 12,

Since , we have

Also, and

So, by axiom 12,

i.e.

Arguing this way and by repeated applications of axiom 12 we have

Since , we have

which is the negation of the formula that proves. Therefore is consistent. Let be a maximal extension of .

We shall show that is the required theory of the lemma. We begin by showing that if then , for some , i.e. . So suppose that . If we are done. If not, let

is consistent. For if not, then there would be and as above such that

Since and , there exists such that . Let