On the decidability and complexity of reasoning about only knowing

Artificial Intelligence 116 (2000) 193–215

On the decidability and complexity ofreasoning about only knowing

Riccardo Rosati1

Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Salaria 113,00198 Roma, Italy

Received 16 November 1998; received in revised form 2 July 1999

Abstract

We study reasoning in Levesque’s logic of only knowing. In particular, we first prove thatextending a decidable subset of first-order logic with the ability of reasoning about only knowingpreserves decidability of reasoning, as long asquantifying-in is not allowed in the language, anddefine a general method for reasoning about only knowing in such a case. Then, we show that theproblem of reasoning about only knowing in the propositional case lies at the second level of thepolynomial hierarchy. Thus, it is as hard as reasoning in the majority of propositional formalismsfor nonmonotonic reasoning, like default logic, circumscription, and autoepistemic logic, and itis easier than reasoning in propositional formalisms based on the minimal knowledge paradigm,which is strictly related to the notion of only knowing. Finally, we identify a syntactic restriction inwhich reasoning about only knowing is easier than in the general propositional case, and provide aspecialized deduction method for such a restricted setting. 2000 Elsevier Science B.V. All rightsreserved.

Keywords:Knowledge representation; Nonmonotonic reasoning; Epistemic logics; Computational complexity

1. Introduction

Research in the formalization of commonsense reasoning through epistemic logics[18,21,23] has pointed out the need for providing systems (agents) with the ability ofintrospecting on their own knowledge and ignorance. To this aim, anepistemic closureassumptionis generally adopted, which informally can be stated as follows: “the logicaltheory formalizing the agent is acompletespecification of the agent’s knowledge”. As a

1 Email: [email protected].

0004-3702/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0004-3702(99)00083-1

194 R. Rosati / Artificial Intelligence 116 (2000) 193–215

consequence, any fact that is not logically implied by such a theory is assumed to benotknownby the agent.2

As shown in [18], this paradigm underlies the vast majority of the logical formalizationsof nonmonotonic reasoning. Roughly speaking, there exist two different ways to embedsuch a principle into a logic:

(1) by considering anonmonotonicformalism, whose semanticsimplicitly realizes sucha “closed” interpretation of the logical theory representing the agent’s knowledge;

(2) by representing the closure assumptionexplicitly in the framework of a monotoniclogic, suitably extending its syntax and semantics.

The first approach has been pursued in the definition of severalmodalformalizations ofnonmonotonic reasoning, e.g., McDermott and Doyle’s nonmonotonic modal logics [20],Halpern and Moses’ logic of minimal epistemic states [9] and Lifschitz’s logic of minimalbelief and negation as failure [19]. On the other hand, the second approach has beenfollowed by Levesque [18] in the definition of the logic ofonly knowing.

The logic of only knowing is obtained by adding an “all-I-know” modal operatorO tomodal logicK45. Informally, such an interpretation of the modalityO is obtained througha maximization of the set of successors of each world satisfyingO-formulas.

There is a strict similarity between the interpretation of the modalityO and the semanticsof nonmonotonic modal logics. Letϕ be a modal formula specifying the knowledge of theagent: in the logic of only knowing, satisfiability of the formulaOϕ in a worldw requiresmaximization of the possible worlds connected tow and satisfyingϕ; an analogous kindof maximization is generally realized by the preference semantics of nonmonotonic modallogics, by choosing, among the models forϕ, only the models having a “maximal” setof possible worlds, where such a notion of maximality changes according to the differentproposals. In a nutshell, the logic of only knowing is a monotonic formalism, in which themodalityO allows for an explicit representation of the epistemic closure assumption at theobject level (i.e., in the language of the logic), whereas in nonmonotonic formalisms theclosure assumption is a meta-level notion.

The studies investigating the relationship between only knowing and nonmonotoniclogics [1] have stressed the analogies between the two approaches from an epistemologicalviewpoint. An analogous analysis from the computational viewpoint has not been pursuedso far. Indeed, there exist several studies concerning the computational properties ofnonmonotonic logics, in particular propositional nonmonotonic modal formalisms (e.g.,[3,5,20,22]). On the other hand, the computational properties of only knowing in thepropositional case have not been thoroughly investigated. The only related studiesappearing in the literature concern a fragment ofOL built upon a very restricted subclassof propositional formulas, for which satisfiability is tractable [15], and a computationalstudy of a framework in which only knowing is added to a formal model oflimitedreasoning [13]. Moreover, a lower bound for reasoning in the propositional fragment ofOL(6p

2) is known, due to the fact that autoepistemic logic [21] can be embedded in polynomialtime intoOL.

2 The use of the notion of logical implication here may be misleading: to be more precise, the closureassumption acts by “maximizing ignorance” (in a way that changes according to the different proposals) in eachpossibleepistemic stateof the agent.

R. Rosati / Artificial Intelligence 116 (2000) 193–215 195

The goal of the present work is to provide algorithms for computing satisfiability in thelogic of only knowing. To this end, we exploit the similarity between this formalism andnonmonotonic modal logics, in order to identify a finite characterization of the models ofa formula in the logicOL.

The main results of the paper concern both decidability and complexity of reasoningabout only knowing. Specifically, we first prove that extending a decidable subset of first-order logic (without equality) with the ability of reasoning about only knowing preservesdecidability of reasoning, as long asquantifying-in, i.e., the presence of modalitiesinside quantifiers, is not allowed. Moreover, we define a general method for computingsatisfiability in OL without quantifying-in. To the best of our knowledge, such analgorithm is the first terminating procedure for reasoning about only knowing in anydecidable fragment of first-order logic (e.g., in the full propositional fragment ofOL).

Then, we show that the problem of reasoning about only knowing in the proposi-tional case lies at the second level of the polynomial hierarchy. More precisely, satisfi-ability in the propositional fragment ofOL is a6p

2-complete problem. Thus, reasoningabout only knowing is as hard as reasoning in the majority of propositional formalismsfor nonmonotonic reasoning, like autoepistemic logic [5,22], default logic [5], circum-scription [4], and several McDermott and Doyle’s logics [20]. Moreover, reasoning aboutonly knowing is easier (unless the polynomial hierarchy collapses) than reasoning in non-monotonic modal formalisms based on theminimal knowledgeparadigm [27], like Halpernand Moses’ logic of minimal epistemic states [2], Lifschitz’s logicMBNF [24], and themoderately grounded version of autoepistemic logic [3].

We also define an interesting syntactic restriction of the propositional fragment ofOLin which deduction is easier than in the general case. Specifically, we identify a subsetof formulas inOL for which the satisfiability (and validity) problem is PNP[O(logn)]-complete, i.e., can be reduced to a logarithmic number of propositional satisfiabilityproblems. This case is particularly interesting, since it can be viewed as a generalization ofthe problem of answering epistemic queries to a propositional knowledge base [6,23].

In the following, we first briefly introduce the modal logic of only knowingOL. Then,in Section 3 we present a finite characterization of the models of a sentence inOL, whichprovides the basis for the definition of reasoning methods forOL. In Section 4 we definea deduction method for satisfiability (validity) in decidable fragments ofOL, and analyzethe computational properties ofOL in the propositional fragment ofOL; we also definea syntactic restriction ofOL, showing that reasoning in this setting is easier than in thegeneral case. Finally, in Section 5 we investigate the relationship between only knowingand the minimal knowledge paradigm, and conclude in Section 6.

2. The logicOL

In this section we briefly recall the formalization of only knowing [18]. We assume thatthe reader is familiar with the basic notions of modal logic. We recall thatK45 denotes themodal logic interpreted on Kripke structures whose accessibility relation among worlds istransitive and Euclidean, while modal logicKD45 imposes in addition that the relation be


serial; finally, modal logicS5 also imposes reflexivity on the accessibility relation (see,e.g., [10,20] for more details).

We useL to denote the language of first-order logic without equality, i.e.,L is the setof first-order sentences built in the usual way upon connectives∧,¬ (the symbols∨,⊃,≡are used as abbreviations), an existential quantifier, an infinite set of variables, an infinitesetA of propositional symbols, an infinite set of predicate symbols of every arity, and aninfinite set of function symbols.3 We assume thatA contains the symbolstrue, false. Wecall objectiveany sentence fromL.

Following [18], we interpret sentences fromL with respect to a fixed, countably infiniteinterpretation domain∆. As shown in [17], imposing a countably infinite domain doesnot influence satisfiability/validity of first-order sentences without equality, i.e., the set ofsatisfiable sentences is the same as in classical first-order logic. In the following, we callinterpretationa usual first-order interpretation forL over∆. An interpretation is also calledworld. For each interpretationw, w(true)= TRUEandw(false)= FALSE. The evaluationw(ϕ) of a sentenceϕ in an interpretationw is defined in the usual way. We say that asentenceϕ ∈ L is satisfiableif there exists an interpretationw such thatw(ϕ) = TRUE(which we also denote asw |= ϕ).

Definition 2.1. We denote asLO the modal extension ofL with the modalitiesK andOinductively defined as follows:

(1) if ϕ ∈ L, thenϕ ∈ LO ;(2) if ϕ ∈ LO , thenKϕ ∈ LO ;(3) if ϕ ∈ LO , thenOϕ ∈ LO ;(4) if ϕ ∈ LO , then¬ϕ ∈LO ;(5) if ϕ1 ∈LO andϕ2 ∈LO , thenϕ1∧ ϕ2 ∈LO ;(6) nothing else belongs toLO .

Informally, the above definition does not allowquantifying-in, i.e., in all sentences fromLO , each occurrence of the modalitiesK andO lies outside the scope of quantifiers. E.g.,the sentence∀x O(p(x)) does not belong toLO , while the sentenceO(∀x p(x)) belongstoLO .

We also useLK to denote the analogous extension ofL with the only modalityK. WecallO-sentencea sentence fromLO of the formOϕ. Notice that, with respect to [18], weslightly change the language of the logic, using the modalityK instead ofB.

The semantics of a sentenceϕ ∈ LO is defined in terms of satisfiability in a structure(w,M) wherew is an interpretation (calledinitial world) andM is a set of interpretations.

Definition 2.2. Letw be an interpretation onL, and letM be a set of such interpretations.We say that a sentenceϕ ∈ LO is satisfiedin (w,M), and write (w,M) |= ϕ, iff thefollowing conditions hold:

(1) if ϕ ∈ L, then(w,M) |= ϕ iff w(ϕ)= TRUE;

3 The assumption done in [18] that constants arerigid designators(i.e., in each interpretation, each constantdenotes the same element of the interpretation domain) can be omitted here, since the case of quantifying-in isnot dealt with in this paper.


(2) if ϕ =¬ϕ1, then(w,M) |= ϕ iff (w,M) 6|= ϕ1;(3) if ϕ = ϕ1∧ ϕ2, then(w,M) |= ϕ iff (w,M) |= ϕ1 and(w,M) |= ϕ2;(4) if ϕ =Kϕ1, then(w,M) |= ϕ iff for everyw′ ∈M, (w′,M) |= ϕ1;(5) if ϕ =Oϕ1, then(w,M) |= ϕ iff for everyw′, w′ ∈M iff (w′,M) |= ϕ1.

We say thatϕ ∈LO is weaklyOL-satisfiable if there exists(w,M) such that(w,M) |=ϕ. Since the initial world does not influence satisfiability of a sentence of the formKϕ or Oϕ, we writeM |= Kϕ (respectivelyM |= Oϕ) iff (w,M) |= Kϕ (respectively(w,M) |=Oϕ) for any interpretationw.

The above semantics is not actually the one originally proposed in [18]: in additionto the above rules, a pair(w,M) must satisfy a maximality condition for the setM, asdescribed below. However, as mentioned in [7], the above, weaker notion of satisfiabilityis also meaningful.

In the following, Th(M) denotes the set of sentencesKϕ such thatϕ ∈ LK and, foreachw ∈M, (w,M) |=Kϕ. Given two sets of interpretationsM1,M2, we say thatM1 isequivalenttoM2 iff Th(M1)= Th(M2).

Definition 2.3. A set of interpretationsM is maximaliff, for each set of interpretationsM ′, if M ′ is equivalent toM thenM ′ ⊆M.

Definition 2.4. A sentenceϕ ∈ LO is OL-satisfiableiff there exists a pair(w,M) suchthat(w,M) |= ϕ andM is maximal.

Roughly speaking, the maximality condition prevents from the existence of modelswhich agree on all basic beliefs, yet disagree on what they only know [18, Section 2.2].

We say that a sentenceϕ ∈ LO is OL-valid iff ¬ϕ is notOL-satisfiable. In the nextsection we will prove that the notions ofOL-satisfiability and weakOL-satisfiabilityof a sentence fromLO coincide. Notice, however, thatOL-satisfiability and weakOL-satisfiability forinfinite theories are, in general, different (see [18, Section 2.4]).

As for reasoning inOL, we give the following definition.

Definition 2.5. A sentenceϕ ∈LO logically impliesa sentenceψ ∈LO in OL (and writeϕ |=OL ψ) iff ϕ ⊃ψ isOL-valid.

Based on the above definition, we can immediately reduce reasoning to unsatisfiabilityin OL.

Remark. An alternative definition of logical implication is given in several studies onepistemic and nonmonotonic modal logics (see, e.g., [20, Definition 7.9]). Such a notionis based on the following notion ofvalidity of a modal sentence in a model: a formulaϕis valid in a Kripke modelM iff, for each worldw in M, (w,M) |= ϕ. The notion oflogical implication is then expressed as follows: “ψ is logically implied byϕ iff ψ is validin every model in whichϕ is valid”. The two notions are in general different, and such adifference also holds for the logicOL. However, since inOL the accessibility relation ofeach interpretation structure is transitive, it can immediately be shown [20, Remark 7.11]


that the last notion of logical implication can be reduced to the one given in Definition 2.5,and hence to validity inOL. In particular,ψ is logically implied byϕ according to the lastnotion iff (ϕ ∧Kϕ)⊃ψ isOL-valid.

Notice that the above semantics strictly relates the logicOL with modal logicK45, sincethere is a precise correspondence between the pairs(w,M) used in the above definitionandK45 models. We recall that, with respect to the satisfiability problem, aK45 model canbe considered without loss of generality as a pair(w,M), wherew is a world,M is aset of worlds (possibly empty),w is connected to all the worlds inM, the worlds inMare connected with each other (i.e.,M is a universalS5 model) and no world inM isconnected tow [20] (in the case ofKD45 models,M is required to be nonempty). Thus, inthe following we will refer to a pair(w,M) as aK45 model whoseS5 component isM.Notice also that, ifΣ ∈ LK , thenΣ is OL-satisfiable if and only if it isK45-satisfiable,which is shown by the fact that, if aK45 model(w,M) satisfies such aΣ , then there existsa maximal setM ′ equivalent toM, hence(w,M ′) satisfiesΣ .

Informally, the interpretation of theO modality is obtained through the maximization ofthe set of successors of each world satisfying anO-sentence. As pointed out, e.g., in [14],the meaning of anO-sentenceOϕ such thatϕ is nonmodal is intuitive, whereas it is moredifficult to understand the semantics of anO-sentence with nested modalities.

Example 2.6. Supposeϕ ∈ L. Then,(w,M) is a model forOϕ iff M = {w: w |= ϕ}.Hence, the effect of prefixingϕ with the modalityO is that of maximizing the possibleworlds inM, which contains all the interpretations consistent withϕ.

Example 2.7. Supposeϕ ∈ L andϕ is not a tautology. Then, the sentenceOKϕ is notOL-satisfiable. In fact, supposeOKϕ is OL-satisfiable. Then, there exists(w,M) suchthat (w,M) |=OKϕ. Now, it is easy to see that, by Definition 2.2,M cannot contain anyinterpretationw′ such thatw′ 6|= ϕ. On the other hand, sinceϕ is not a tautology, thereexists such an interpretationw′; moreover,(w′,M) |=Kϕ, since the interpretation ofKϕin (w′,M) does not depend on the initial world, hence by Definition 2.2 it follows thatw′ ∈M. Contradiction. Hence,OKϕ is notOL-satisfiable. On the contrary,O(Kϕ∧ ϕ) isOL-satisfiable, under the assumption thatϕ is satisfiable.

3. CharacterizingOL-satisfiability

In this section we present a finite characterization of the models of a sentenceΣ ∈ LOwhich is based on the use of partitions of modal sentences occurring inΣ . Similartechniques are used in several methods for reasoning in nonmonotonic modal logics (e.g.,[2,3,5,20,22]): in such methods, partitions of subformulas of a modal theory are generallyused for providing a finite characterization of the epistemic states of the agent, whichcorrespond to infinite modal theories. In fact, such partitions can also be used in order toprovide a finite characterization of anS5 model. In particular, a partition satisfying certainproperties identifies a particularS5 modelM, by uniquely determining a nonmodal theory


(called theobjective knowledgeofM).M is then defined as the set of all interpretationssatisfying such objective knowledge.

Now, in order to check whether anO-sentenceOϕ is satisfied in aK45 model(w,M),we exploit the possibility of expressing, by means of an objective sentence, the objectiveknowledge of theS5 componentM of (w,M). This allows us to establish whetherϕ is“all that is known” in the set of interpretationsM.

We first introduce some preliminary definitions. Following [6], we say that an occurrenceof a sentenceψ in a sentenceϕ ∈ LO is strict if it is not in the scope of a modal operator.We also callmodal atoma sentence of the formKϕ orOϕ, with ϕ ∈ LO , and callmodalatoms ofΣ (denoted byMA(Σ)) the set of modal atoms occurring inΣ .

Definition 3.1. Let Σ ∈ LO and letP,N be sets of modal atoms such thatP ∪ N ⊇MA(Σ) andP ∩N = ∅. We denote withΣ|P,N the objective sentence obtained fromΣby substituting each strict occurrence inΣ of a sentence inP with true, and each strictoccurrence inΣ of a sentence inN with false.

Notice that only the occurrences inΣ of modal atoms which are not within the scope ofanother modality are replaced; notice also thatΣ|P,N is an objective sentence. Informally,the pair(P,N) identifies a “guess” on the modal atoms fromΣ , andΣ|P,N represents the“objective knowledge” implied byΣ under such a guess.

Definition 3.2. Let (P,N) be a partition ofMA(Σ). Then, we denote withob(P ) thefollowing objective sentence:

ob(P )=( ∧Kϕ∈P

ϕ|P,N)∧( ∧Oϕ∈P

ϕ|P,N).

Roughly speaking, the objective sentenceob(P ) represents the objective knowledgeimplied by the guess(P,N) on the modal atoms belonging toP .

Example 3.3. SupposeΣ = a ∧O(¬a ∨ Kb). Then,MA(Σ) = {O(¬a ∨ Kb),Kb}. Onepossible partition ofMA(Σ) is the following:

P = {O(¬a ∨Kb)},N = {Kb}.

Then,Σ|P,N = a ∧ true= a, andob(P )= (¬a ∨Kb)|P,N =¬a ∨ false=¬a.

Definition 3.4. Let S be a set of modal atoms. We say that a set of interpretationsM

inducesthe partition(P,N) on S if, for each modal atomKϕ ∈ S, Kϕ ∈ P iff M |=Kϕ,and for each modal atomOϕ ∈ S,Oϕ ∈ P iff M |=Oϕ.

We now define the notion of partition of a set of modal atoms induced by an objectivesentence.


Definition 3.5. Let Σ ∈ LO , ϕ ∈ L. We denote with(Pϕ(Σ),Nϕ(Σ)) the partition ofMA(Σ) induced byM = {w: w |= ϕ}.

Notice that the above definition associates a maximal set of interpretationsM with thesentenceϕ and the partition(Pϕ(Σ),Nϕ(Σ)).

In order to establish a characterization ofOL-satisfiability based on the use of partitionsof modal atoms, we prove some preliminary properties of such partitions.

Lemma 3.6. Letϕ ∈LO , letw be an interpretation, letM be a set of interpretations, andlet (P,N) be the partition induced byM on a set of modal atomsS. Then,(w,M) |= ϕ iff(w,M) |= ϕ|P,N .

Proof. Follows immediately from Definitions 3.1 and 2.2.2Lemma 3.7. LetΣ ∈ LO , ϕ ∈L. Then:

(1) each modal atomKψ of MA(Σ) belongs toPϕ(Σ) iff ϕ ⊃ψ|Pϕ(Σ),Nϕ(Σ) is a validobjective sentence;

(2) each modal atomOψ of MA(Σ) belongs toPϕ(Σ) iff ϕ ≡ψ|Pϕ(Σ),Nϕ(Σ) is a validobjective sentence.

Proof. Follows immediately from Definition 2.2.2Lemma 3.8. Let Σ ∈ LK . If Σ is K45-satisfiable andKD45-unsatisfiable, then thepartition (P,N) induced by the empty set of interpretations is such that ob(P ) isunsatisfiable.

Proof. SupposeΣ is K45-satisfiable andKD45-unsatisfiable, and let(w,M) be aK45model such that(w,M) |= Σ . Then,M = ∅. Let (P,N) be the partition ofMA(Σ)induced byM: sinceM = ∅, it follows that P = MA(Σ) (each sentence of the formKϕ is trivially satisfied byM). Now letM ′ = {w: w |= ob(P )}. We prove that(P,N) =(Pob(P )(Σ),Nob(P )(Σ)). The proof is by induction on the modal depth of the sentences inMA(Σ). First, letKϕ be a modal atom ofMA(Σ) such thatϕ ∈ L. Then, sinceKϕ ∈ P ,Definition 3.2 implies thatob(P )⊃ ϕ is a valid objective sentence, henceKϕ ∈ Pob(P )(Σ).Suppose now that(P,N) and(Pob(P )(Σ),Nob(P )(Σ)) agree on all modal atoms of modaldepth less or equal toi. Consider a modal atomKϕ of MA(Σ) of modal depthi + 1.Again, sinceKϕ ∈ P , by Definition 3.2 it follows thatob(P )⊃ ϕ|P,N is a valid objectivesentence, henceM ′ |= ϕ|P,N , and since by Definition 3.1 the value of the sentenceϕ|P,N only depends on the value of the modal atoms of modal depth less or equal toi

in (P,N), by the induction hypothesis and Lemma 3.6 it follows thatM ′ |= ϕ, henceKϕ ∈ Pob(P )(Σ). Consequently,(P,N)= (Pob(P )(Σ),Nob(P )(Σ)), which in turn impliesthat

Σ|P,N =Σ|Pob(P )(Σ),Nob(P )(Σ).

Now, since(w,M) |= Σ , by Lemma 3.6w |= Σ|P,N , hencew |= Σ|Pob(P )(Σ),Nob(P )(Σ)

and, by the same lemma,(w,M ′) |= Σ . SinceM ′ = {w: w |= ob(P )} andΣ is KD45-unsatisfiable, it follows thatM ′ is empty, henceob(P ) is unsatisfiable. 2


We say that a sentenceϕ ∈ LO hasmodal depthi if each occurrence of an objectivesentence inϕ lies within the scope of at mosti modalities, and there is an occurrence of anobjective sentence inϕ which lies within the scope of exactlyi modalities.

Lemma 3.9. LetΣ ∈ LO , ϕ ∈L. Let (P,N) be the partition of MA(Σ) built as follows:(1) start fromP =N = ∅;(2) for each modal atomKψ in MA(Σ) such thatψ|P,N ∈ L, if ϕ ⊃ ψ|P,N is a valid

objective sentence, then addKψ to P , otherwise addKψ toN ;(3) for each modal atom of the formOψ in MA(Σ) such thatψ|P,N ∈L, if ϕ ≡ψ|P,N

is a valid objective sentence, then addOψ to P , otherwise addOψ toN ;(4) iteratively apply the above rules untilP ∪N =MA(Σ).

Then,(P,N)= (Pϕ(Σ),Nϕ(Σ)).

Proof. The proof is by induction on the structure of the sentences inMA(Σ). First, fromthe fact that(Pϕ(Σ),Nϕ(Σ)) is the partition induced byM = {w: w |= ϕ}, and fromDefinition 2.2, it follows that, ifψ ∈ L, thenM |= Kψ if and only if ϕ ⊃ ψ is a validobjective sentence, andM |= Oψ if and only if ϕ ≡ ψ is a valid objective sentence.Therefore,(P,N) agrees with(Pϕ(Σ),Nϕ(Σ)) on all modal atoms of modal depth 1.Suppose now that(P,N) and(Pϕ(Σ),Nϕ(Σ)) agree on all modal atoms of modal depthless or equal toi. Consider a modal atomKψ of MA(Σ) of modal depthi + 1. FromLemma 3.7 it follows thatM |=Kψ if and only if ϕ ⊃ ψ|Pϕ(Σ),Nϕ(Σ) is a valid objectivesentence, and since by Definition 3.1 the value of the sentenceψ|Pϕ(Σ),Nϕ(Σ) only dependson the guess of the modal atoms of modal depth less or equal toi in (Pϕ(Σ),Nϕ(Σ)), bythe induction hypothesis it follows thatψ|Pϕ(Σ),Nϕ(Σ) = ψ|P,N , henceKψ belongs toPif and only if it belongs toPϕ(Σ). Analogously, it can be proven that any modal atom ofdepthi + 1 of the formOψ belongs toP if and only if it belongs toPϕ(Σ). Therefore,(P,N) and(Pϕ(Σ),Nϕ(Σ)) agree on all modal atoms of modal depthi + 1. 2

In the following, the sentenceΣ[O/false] stands for the sentence obtained fromΣ byreplacing allO-sentences occurring inΣ with false. Notice thatΣ[O/false] ∈ LK , hencesuch a sentence isOL-satisfiable if and only if it isK45-satisfiable.

We are now ready to provide a characterization of the notion of satisfiability inOL,based on the existence of a partition(P,N) of MA(Σ) which satisfies the property(P,N)= (Pob(P )(Σ),Nob(P )(Σ)).

Theorem 3.10.LetΣ ∈ LO . Then,Σ is OL-satisfiable iff at least one of the followingtwo conditions holds:

(a) Σ[O/false] is KD45-satisfiable;(b) there exists a partition(P,N) of MA(Σ) such thatΣ|P,N is satisfiable and

(P,N)= (Pob(P )(Σ),Nob(P )(Σ)).

Proof.If part. Suppose that either condition (a) or condition (b) of the theorem holds. Then,

there are two possible cases:


(1) Σ[O/false] is KD45-satisfiable. Then, there exists aK45 model(w,M) such thatM 6= ∅ and(w,M) |=Σ[O/false]. Let p′ be a propositional symbol not appearingin Σ : without loss of generality, we can assume thatM contains at least one worldevaluatingp′ to TRUE, and at least one world evaluatingp′ to FALSE. LetM ′ bethe maximal set equivalent toM: sinceTh(M ′) = Th(M), it follows that (w,M ′)satisfiesΣ[O/false]. Now letM ′′ be the set of interpretations obtained fromM ′by eliminating all interpretationsw′ such thatw′(p′) = FALSE. By construction,M ′′ is maximal and nonempty. Consider the model(w,M ′′): on the one hand,(w,M ′′) satisfiesΣ[O/false], sincep′ does not appear inΣ ; on the other hand,since all interpretations inM ′′ satisfyp′, M ′′ |=Kp′. Now consider a modal atomof the formOϕ in MA(Σ), and supposeM ′′ |= Oϕ. Let (P,N) be the partitioninduced byM ′′ on MA(Σ): from Lemma 3.6 it follows thatM ′′ |=Oϕ|P,N . Then,sinceM ′′ |=Kp′, from Definition 2.2 it follows thatϕ|P,N ⊃ p′ is a valid objectivesentence. Sincep′ does not occur inϕ, ϕ|P,N ⊃ p′ is valid iff ϕ|P,N is unsatisfiable,but, if we assumeϕ|P,N unsatisfiable, then by Definition 2.2M ′′ |= K false, thuscontradicting the hypothesis thatM ′′ be nonempty. Therefore, for each modal atomOϕ in MA(Σ), M ′′ 6|= Oϕ, and since(w,M ′′) satisfiesΣ[O/false], Lemma 3.6implies that(w,M ′′) satisfiesΣ . SinceM ′′ is maximal, it follows thatΣ is OL-satisfiable.

(2) There exists a partition(P,N) of MA(Σ) such thatΣ|P,N is satisfiable and(P,N) = (Pob(P )(Σ),Nob(P )(Σ)). Now, sinceΣ|P,N is satisfiable, there existsan interpretation satisfyingΣ|P,N . Let w be such an interpretation, and letM ={w′: w′ |= ob(P )}. Since(P,N) = (Pob(P )(Σ),Nob(P )(Σ)), it follows that(P,N)is the partition ofMA(Σ) induced byM. Therefore, sincew satisfiesΣ|P,N , fromLemma 3.6 it follows that(w,M) |=Σ . Moreover,M is maximal by construction,henceΣ isOL-satisfiable.

Only-if part.SupposeΣ isOL-satisfiable. Then, there exists aK45 model(w,M) suchthat(w,M) |=Σ andM is maximal. Let(P,N) be the partition induced byM onMA(Σ).There are two possible cases.

(1) All modal atoms ofMA(Σ) of the formOϕ belong toN . Then, from Lemma 3.6, itfollows that(w,M) |=Σ[O/false], i.e.,Σ[O/false] is K45-satisfiable. Now, thereare two possibilities:– Σ[O/false] is KD45-satisfiable. In this case, condition (a) of the theorem holds;– Σ[O/false] is not KD45-satisfiable. In this case, from Lemma 3.8 it follows

that ob(P ) is unsatisfiable. Moreover,M is empty, henceM = {w: w |=ob(P )}. Therefore,(P,N) coincides with the partition induced byob(P ), thatis, (P,N) = (Pob(P )(Σ),Nob(P )(Σ)). Furthermore, since(w,M) |= Σ and(P,N) is the partition induced byM on MA(Σ), from Lemma 3.6 it followsthat the interpretationw satisfiesΣ|P,N , hence condition (b) of the theoremholds.

(2) At least one modal atom ofMA(Σ) of the formOϕ belongs toP . Then, since(w,M) satisfiesΣ , it follows thatM = {w: w |= ϕ|P,N }; moreover, by definitionof ob(P ) it follows that ob(P ) is equivalent toϕ|P,N , thus (P,N) coincideswith the partition induced byob(P ), that is, (P,N) = (Pob(P )(Σ),Nob(P )(Σ)).Furthermore, since(w,M) |= Σ and (P,N) is the partition induced byM on


MA(Σ), from Lemma 3.6 it follows that the interpretationw satisfiesΣ|P,N , hencecondition (b) of the theorem holds.2

Intuitively, the above theorem provides for a characterization of the notion ofOL-satisfiability of a formulaΣ in terms of properties of partitions of the modal atoms ofΣ . Specifically, the theorem states that a formulaΣ ∈ LO is OL-satisfiable iff eitherΣ[O/false] is KD45-satisfiable, which informally corresponds to checking whether it isconsistent to assume as false everyO-sentence occurring inΣ , or there exists a partition(P,N) of MA(Σ) such thatΣ|P,N is satisfiable and(P,N) = (Pob(P )(Σ),Nob(P )(Σ)),which corresponds to checking whether there exists a guess of the modal atoms ofΣ whichis both consistent withΣ and not self-contradictory.

From the above theorem, it is easy to prove that the two notions ofOL-satisfiability andweakOL-satisfiability coincide in the case of formulas fromLO .

Theorem 3.11.LetΣ ∈LO . Then,Σ isOL-satisfiable iff it is weaklyOL-satisfiable.

Proof. Follows immediately from the fact that the proof of the only-if part of Theo-rem 3.10 holds even if the assumption thatM is maximal is discarded, since such an as-sumption is not used in the proof. This in turn implies thatΣ is weaklyOL-satisfiable iffconditions (a) and (b) of Theorem 3.10 hold. Thus, from the same theorem, it follows thatΣ is weaklyOL-satisfiable iffΣ isOL-satisfiable. 2

A property analogous to the above theorem has been proved in [8] for the propositionalfragment ofOL.

4. Reasoning inOL

In this section we study reasoning inOL. In particular, we first show that extendinga decidable fragment of first-order logic with only knowing preserves decidability ofreasoning. Then, we establish an upper bound for the satisfiability problem in thepropositional fragment ofOL, and finally analyze a restriction of the propositional case inwhich reasoning is computationally easier.

We briefly introduce the complexity classes mentioned in the following (refer, e.g.,to [11] for further details). All the classes we use reside in thepolynomial hierarchy. Inparticular, the complexity class6p

2 is the class of problems that are solved in polynomialtime by a nondeterministic Turing machine that uses an NP-oracle (i.e., that solves inconstant time any problem in NP), and5p

2 is the class of problems that are complement of aproblem in6p

2. The class PNP, also known as1p2, is the class of problems that are solved in

polynomial time by adeterministicTuring machine that uses an NP-oracle, while the classPNP[O(logn)], also known as2p

2, is the class of problems that are solved in polynomialtime by a deterministic Turing machine that makes a number of calls to an NP-oracle whichis logarithmic in the size of the input. Hence, the class PNP[O(logn)] is “mildly” harderthan the class NP, since a problem in PNP[O(logn)] can be solved by solving “few” (i.e.,a logarithmic number of) instances of problems in NP. It is generally assumed that the


polynomial hierarchy does not collapse, and that a problem in the class PNP[O(logn)] iscomputationally easier than a6p

2-hard or5p2-hard problem.

4.1. Reasoning method

As for effective methods for reasoning inOL, we recall thatOL-satisfiability inunrestrictedLO is not a decidable problem, since establishingOL-satisfiability ofobjective sentences corresponds to solving the satisfiability problem for full first-orderlogic. However, the characterization provided by Theorem 3.10 allows for the definition ofan algorithm for reasoning in subsets ofLO built upon decidable fragments of first-orderlogic. In the following, we say that a languageL′ ⊆ L is closed under boolean compositionif, for eachϕ1, ϕ2 ∈ L′, ϕ1∧ϕ2 ∈ L′ and¬ϕ1 ∈ L′. Moreover, we denote asL′O the subsetof LO built uponL′, i.e., the modal extension ofL′ obtained according to Definition 2.1.

To the aim of identifying decidable fragments ofLO , we prove the following lemma.

Lemma 4.1. LetΣ ∈ LK . Then,Σ is KD45-satisfiable iff there exists a partition(P,N)of MA(Σ) such that:

(a) Σ|P,N is satisfiable;(b) for eachKϕ ∈N , ob(P )∧¬ϕ|P,N is satisfiable;(c) ob(P ) is satisfiable.

Proof.If part. Let (P,N) be a partition ofMA(Σ) satisfying conditions (a), (b), and (c) of the

lemma, and letM = {w: w |= ob(P )}. Condition (c) impliesM 6= ∅. Moreover, sinceΣ|P,N is satisfiable, there exists an interpretationw such thatw |= Σ|P,N . Now weprove, by induction on the modal depth of the modal atoms inMA(Σ), that (P,N) =(Pob(P )(Σ),Nob(P )(Σ)). First, from Lemma 3.7 and condition (b), it follows that eachmodal atomKϕ in N of modal depth 1 (i.e., such thatϕ ∈ L) also belongs toNob(P )(Σ);moreover, Definition 3.2 and Lemma 3.7 imply that each modal atomKϕ in P of modaldepth 1 belongs toPob(P )(Σ). Now suppose that(P,N) and (Pob(P )(Σ),Nob(P )(Σ))

agree on all modal atoms of modal depth less or equal toi. Consider a modal atomKϕ ofMA(Σ) of modal depthi+ 1. Since by Definition 3.1 the value of the sentenceϕ|P,N onlydepends on the value of the modal atoms of modal depth less or equal toi in (P,N), bythe induction hypothesis it follows that

ϕ|P,N = ϕ|Pob(P )(Σ),Nob(P )(Σ),

hence condition (b) and Lemma 3.7 imply that, ifKϕ ∈ N , thenKϕ ∈ Nob(P )(Σ), whileDefinition 3.2 and Lemma 3.7 imply that, ifKϕ ∈ P , thenKϕ ∈ Pob(P )(Σ). Therefore,(P,N)= (Pob(P )(Σ),Nob(P )(Σ)), and sincew |=Σ|P,N and(Pob(P )(Σ),Nob(P )(Σ)) isthe partition ofMA(Σ) induced byM, from Lemma 3.6 it follows that(w,M) |=Σ , whichproves thatΣ is KD45-satisfiable.

Only-if part. SupposeΣ is KD45-satisfiable. Then, there exists a model(w,M) suchthat (w,M) |= Σ andM 6= ∅. Let (P,N) be the partition ofMA(Σ) induced byM.Then, from Lemma 3.6 it follows thatw |= Σ|P,N , hence condition (a) holds. We nowprove, by induction on the modal depth of the modal atoms inMA(Σ), that (P,N) =


(Pob(P )(Σ),Nob(P )(Σ)). Let M ′ = {w: w |= ob(P )}. First, by Definition 3.2 it followsthat, for eachKϕ ∈ MA(Σ) such thatϕ ∈ L, M |= Kϕ iff M ′ |= Kϕ, henceKϕ ∈ Piff Kϕ ∈ Pob(P )(Σ). Now suppose that(P,N) and(Pob(P )(Σ),Nob(P )(Σ)) agree on allmodal atoms of modal depth less or equal toi. Consider a modal atomKϕ of MA(Σ) ofmodal depthi + 1. SinceKϕ ∈ P , by Definition 3.2 it follows thatob(P ) ⊃ ϕ|P,N is avalid objective sentence, henceM ′′ |= ϕ|P,N , and since by Definition 3.1 the value of thesentenceϕ|P,N only depends on the value of the modal atoms of modal depth less or equalto i in (P,N), by the induction hypothesis and Lemma 3.6 it follows thatM ′ |= ϕ, henceKϕ ∈ Pob(P )(Σ). Consequently,(P,N)= (Pob(P )(Σ),Nob(P )(Σ)), and by Lemma 3.7 itfollows that condition (b) holds. Finally, sinceM 6= ∅, it follows thatob(P ) is satisfiable,hence condition (c) holds.2

We are now ready to prove decidability ofOL-satisfiability for subsets ofLO built upondecidable subsets of the first-order languageL.

Theorem 4.2. LetL′ ⊂ L. If L′ is closed under boolean composition and satisfiability inL′ is decidable, thenOL-satisfiability inL′O is decidable.

Proof. Let Σ ∈ L′O . Theorem 3.10 implies thatOL-satisfiability ofΣ can be decidedthrough the following steps.

(1) First, checkingKD45-satisfiability ofΣ[O/false]. From Lemma 4.1, this can beaccomplished by verifying the existence of a partition(P,N) of MA(Σ[O/false])such that:(a) Σ[O/false]|P,N is satisfiable. SinceL′ is closed under boolean composition, it

follows thatΣ[O/false]|P,N ∈ L′, and since satisfiability inL′ is decidable, thischeck is decidable.

(b) For eachKϕ ∈N , ob(P )∧¬ϕ|P,N is satisfiable. Again, sinceL′ is closed underboolean composition, it follows that, for eachKϕ ∈ N , ob(P ) ∧ ¬ϕ|P,N ∈ L′,hence this check is decidable.

(c) ob(P ) is satisfiable. Again, sinceob(P ) ∈L′, this check is decidable.(2) Verifying the existence of a partition(P,N) of MA(Σ) such thatΣ|P,N is satisfiable

and (P,N) = (Pob(P )(Σ),Nob(P )(Σ)). Again, sinceL′ is closed under booleancomposition,Σ|P,N ∈ L′, hence verifying satisfiability ofΣ|P,N is decidable.Moreover, sinceL′ is closed under boolean composition, and since satisfiabilityin L′ is decidable, Lemma 3.9 provides an effective method to build the partition(Pob(P )(Σ),Nob(P )(Σ)) in a finite amount of time, hence checking whether(Pob(P )(Σ),Nob(P )(Σ)) is equal to(P,N) is decidable. 2

Therefore, the above theorem states that reasoning about only knowing in the modalextension (without quantifying-in) of a decidable fragment of first-order logic closed underboolean composition is decidable.

In Fig. 1 we present the algorithmOL-Sat for computing satisfiability in any fragmentL′O of LO satisfying the conditions of Theorem 4.2. The algorithm is based onTheorem 3.10, and relies on both Lemma 4.1, which provides a method for computingKD45-satisfiability inL′K by using a procedure for computing satisfiability inL′, and


Algorithm OL-Sat(Σ )Input: sentenceΣ ∈L′

O;

Output: true if Σ isOL-satisfiable,false otherwise.beginif Σ[O/false] is KD45-satisfiablethen return trueelse if there existspartition(P,N) of MA(Σ) such that

(a)Σ |P,N is satisfiableand(b) (P,N)= (Pob(P )(Σ),Nob(P )(Σ))

then return trueelse return falseend

Fig. 1. AlgorithmOL-Sat.

Lemma 3.9, which provides a constructive way to build the partition(Pϕ(Σ),Nϕ(Σ))

starting from the sentencesΣ and ϕ, again using a procedure for satisfiability inL′.Therefore, the algorithm computesOL-satisfiability inL′O by reducing such a problem toa number of satisfiability problems inL′. Correctness of the algorithm follows immediatelyfrom Theorem 3.10.

Informally, the algorithm first checks whether it is possible to satisfyΣ by assuming asfalse allO-sentences occurring inΣ , that is, by making no closure assumptions about whatis known. If in this way it is not possible to satisfyΣ , that is, the sentenceΣ[O/false] isnot KD45-satisfiable, then the algorithm checks whether there exists a partition(P,N) ofMA(Σ) satisfying certain conditions. Intuitively, the partition must be consistent withΣ

(condition (a)) and cannot be self-contradictory (condition (b)). In particular, the condition(P,N)= (Pob(P )(Σ),Nob(P )(Σ)) establishes that the objective knowledge implied by thepartition (P,N) (that is, the sentenceob(P )) identifies a set of interpretations whichinduces the same partition(P,N) on MA(Σ). We illustrate the algorithm through thefollowing simple example.

Example 4.3. Let us consider the sentenceΣ defined in Example 3.3. Then,

Σ[O/false] = a ∧ false= false,

henceΣ[O/false] is notKD45-satisfiable. Now, consider the partition(P,N)= ({O(¬a∨Kb)}, {Kb}) of MA(Σ). As shown in Example 3.3,Σ|P,N = a, hence(P,N) satisfiescondition (a) of the algorithm. Now, sinceob(P ) = ¬a, and (¬a ∨ Kb)|P,N = ¬a, itfollows thatO(¬a ∨ Kb) ∈ Pob(P )(Σ). Moreover, the objective sentence¬a ⊃ b is notvalid, henceKb∈Nob(P )(Σ). Therefore,

(P,N)= (Pob(P )(Σ),Nob(P )(Σ)),

i.e., condition (b) of the algorithm is satisfied. Consequently,OL-Sat(Σ) returnstrue. Infact, the partition(P,N) identifies the set of interpretationsM = {w: w |= ¬a}. Moreover,sinceΣ|P,N is satisfiable, it follows that there exists an interpretationw satisfyingΣ|P,N ,which implies that(w,M) |=Σ .


4.2. Propositional case: Complexity

We now study the complexity of reasoning about only knowing in the propositional case.To this aim, we analyze the complexity of the algorithmOL-Sat, reported in Fig. 1, underthe restriction thatΣ is a modal propositional formula. To the best of our knowledge, suchan algorithm is the first terminating method for reasoning about only knowing in the fullpropositional case.4 In the following, we denote asLp the propositional fragment ofL,and withLp

O the propositional fragment ofLO .Observe that, ifΣ ∈ Lp

O , then all formulas involved in the conditions reported inthe algorithm are propositional, hence all such conditions can be checked by solvingpropositional satisfiability/validity problems. In particular:

– Satisfiability in propositionalKD45 can be computed in nondeterministic polynomialtime, since such a problem is NP-complete [10]. Membership in NP is also animmediate consequence of Lemma 4.1.

– Condition (a) can be checked in time linear with respect to the size ofΣ .– Given(P,N), the formulaob(P ) can be computed in time linear with respect to the

size ofP . Moreover, by Lemma 3.9 it follows that, sinceMA(Σ) has size linear withrespect to the size ofΣ , construction of the partition(Pob(P )(Σ),Nob(P )(Σ)) canbe performed through a linear number (with respect to the size ofΣ) of calls to anNP-oracle for propositional satisfiability. Therefore, condition (b) can be checked inlinear time (with respect to the size ofΣ) using an NP-oracle.

Now, since the guess of the partition(P,N) of MA(Σ) requires a nondeterministicchoice, it follows that the algorithmOL-Sat, if considered as a nondeterministic procedure,is able to establish satisfiability of a formulaΣ ∈ Lp

O in nondeterministic polynomial time(with respect to the size ofΣ), using an NP-oracle for propositional satisfiability. Thus, weobtain an upper bound of6p

2 for the problem.

Lemma 4.4. LetΣ ∈ LpO . The problem of establishingOL-satisfiability ofΣ is in6p

2.

As for the lower bound of the satisfiability problem in propositionalOL, we recall thatreasoning in Moore’s autoepistemic logic (AEL) [21] can be reduced to reasoning inOL.We now briefly recall the notion of stable expansion in AEL. In order to keep notation to aminimum, we change the language ofAEL, using the modalityK instead ofL: thus, in thefollowing a formula ofAEL is a formula belonging to the propositional fragment ofLK(denoted asLp

K ).A set of formulasT fromLp

K is astable expansionfor a formulaΣ ∈ LpK if T satisfies

the equation

T =Cn({Σ} ∪ {Kϕ: ϕ ∈ T } ∪ {¬Kϕ: ϕ /∈ T }),

whereCn is the logical consequence operator of propositional logic.

Proposition 4.5 [18, Theorem 3.9].Letϕ ∈LpK . Then,Oϕ isOL-satisfiable iff there exists

a stable expansion forϕ.

4 In fact, Levesque’s axiomatization of the propositional fragment ofOL [18] does not directly imply theexistence of a terminating procedure for reasoning in the propositional fragment ofOL.


Lemma 4.6. LetΣ ∈LpO . The problem of establishingOL-satisfiability ofΣ is6p

2-hard.

Proof. Let ϕ ∈ LpK . By Proposition 4.5,Oϕ is OL-satisfiable iff there exists a stable

expansion forϕ. And since the problem of establishing whether a formulaϕ ∈ LpK admits

a stable expansion is6p2-hard [5, Theorem 4.3], this proves the thesis.2

The last two lemmas imply the following property.

Theorem 4.7. Let Σ ∈ LpO . The problem of establishingOL-satisfiability ofΣ is 6p

2-complete.

The previous theorem implies that validity in propositionalOL is 5p2-complete, and

that the logical implication problemϕ |=OL ψ is 5p2-complete as well (with respect to

the size ofϕ ∧ ψ). Consequently, the algorithmOL-Sat is “optimal” with respect to thecomplexity of satisfiability in propositionalOL, in the sense that it matches the lowerbound of the problem.

4.3. Propositional case: Restrictions

We now define an interesting subset of propositionalOL in which the modalityO canbe used in a restricted way. We prove that reasoning in such a fragment ofOL is easierthan in the general propositional case.

First, from Lemma 4.6 it follows that if we impose no restrictions on formulas which liewithin the scope of the operatorO , thenOL-satisfiability is a6p

2-hard problem. Hence, inorder to find a fragment ofLp

O for which satisfiability is computationally easier, we needto impose some restrictions on the structure ofO-subformulas.

The first significant restriction corresponds to the case of formulas of the formOϕ ∧¬Kψ in whichϕ ∈ Lp, ψ ∈Lp

K . Satisfiability of this kind of formulas inOL is analogousto a reasoning problem which has been analyzed in several different settings (e.g., [5,18,23]), and corresponds to posing an epistemic queryψ ∈Lp

K to the propositional knowledgebaseϕ, interpreting queries under the following intuitive epistemic closure assumption:

– for any ξ ∈ Lp, if ξ is logically implied (in propositional logic) byϕ thenKξ isimplied byϕ, otherwise¬Kξ is implied byϕ;

– the interpretation of an epistemic queryψ with nested occurrences of the modaloperator is obtained by iteratively checking all modal subformulasKξ such thatξ ∈ Lp, then substituting all such subformulas withtrue or false in ψ accordingly, thusobtaining new modal subformulas inψ without nested occurrences of the modality;when all modal subformulas inψ have been replaced in this way, it can be checkedwhetherψ is implied byϕ.

It can be shown (see [18, Corollary 3.13]) thatOϕ∧¬Kψ is satisfiable if and only ifψis not implied byϕ under the above semantics for epistemic queries. Moreover, it is knownthat the problem is PNP[O(logn)]-complete [6], that is, it can be solved in polynomial timethrough a number of calls to the NP-oracle which is logarithmic in the size of the formulaϕ ∧ ψ . Therefore, satisfiability inOL of a formula of the formOϕ ∧ ¬Kψ in which


ϕ ∈Lp, ψ ∈ LpK , is PNP[O(logn)]-complete as well, hence it is easier than the problem of

OL-satisfiability of a generic formula inLpO .

We now define a large superclass of the above set of formulas, and show that satisfiabilityin OL for such kind of formulas is still easier than in the general case.

Definition 4.8. Let LSO denote the set of formulas belonging toLpO in which each

propositional symbol lies within the scope of a modality. Then, we denote withL−O theset of formulas belonging toLp

O in which eachO-subformulaOϕ is such thatϕ is of theform f ∧ψ or f ∨ψ , with f ∈ Lp, ψ ∈ LSO .

Notice that the only restriction imposed by the above definition is on the form ofO-subformulas: roughly speaking, in eachO-subformulaϕ it must be possible to isolate an“objective” (i.e., belonging toLp) and a “subjective” (i.e., belonging toLSO ) subformula.For instance, the formulaΣ = a ∧O(¬a ∨Kb) defined in Example 3.3 belongs to the setL−O , since theO-subformulaO(¬a∨Kb) has an objective subformula¬a and a subjectivesubformulaKb. Conversely, the formulaa ∧O(¬a ∨ (Kb∧ c)) does not belong to the setL−O , since the subformulaKb∧ c is neither objective nor subjective.

The languageL−O allows for a nice formalization of a generalization of the abovementioned setting of epistemic queries, in which one can express queries regarding theepistemic state of a number (sayn) of propositional knowledge bases. A multimodallanguage withn operatorsK1, . . . ,Kn can be used for expressing the epistemic stateof each of the knowledge bases. Given a set ofn propositional knowledge basesK ={KB1, . . . ,KBn}, in which eachKBi is a formula fromLp, we define an epistemic query toK as a boolean combination of epistemic queries to the single knowledge bases. E.g., wecan pose a query of the form

Q=K1ϕ ∧ (¬K2ψ ∨K3ξ)

such thatϕ is an epistemic query toKB1 (i.e., in which the only modalityK1 is used),ψ isan epistemic query toKB2, andξ is an epistemic query toKB3. Q is implied byK if andonly if ϕ is implied byKB1 and eitherψ is not implied byKB2 or ξ is implied byKB3. Itis immediate to see that the evaluation of such forms of epistemic queries can be reducedto checking validity of formulas inL−O . In the case of the above example,Q is implied byK iff the formula

(OKB1⊃Kϕ′)∧((OKB2⊃¬Kψ ′)∨ (OKB3⊃Kξ ′)

)is OL-valid, whereϕ′ is obtained fromϕ by substituting each occurrence ofK1 with K,andψ ′, ξ ′ are obtained in a similar way fromψ andξ .

We now prove thatOL-satisfiability for a formulaΣ belonging toL−O is easier than forgeneric formulas inLp

O . Informally, the key point is that the syntactic restriction satisfiedby a formula inL−O allows for easily identifying a “small” (i.e., linear in the size ofΣ)number of possible sets of propositional interpretations, each one represented in terms ofa propositional formula. In particular, givenΣ ∈ L−O , there is a finite number (sayn) ofoccurrences ofO-subformulas inΣ , and each of such formulas is of the formfi ∧ ψi orfi ∨ψi , with fi ∈ Lp andψi ∈LSO , for i = 1, . . . , n: it is then possible to show that a model


Algorithm L−O

-Sat(Σ )Input: formulaΣ ∈ L−

O;

Output: true if Σ isOL-satisfiable,false otherwise.beginif Σ[O/false] is KD45-satisfiablethen return trueelse if there existsϕ in SΣ = {f1, f2, . . . , fn, true, false} such that

Σ |Pϕ(Σ),Nϕ(Σ) is satisfiablethen return trueelse return false

end

Fig. 2. AlgorithmL−O

-Sat.

(w,M) forΣ mustbe such thatM is one of the maximal sets of interpretations representedby one of thefi ’s (plus the formulastrue, false). This property simplifies the problem offinding a model forΣ , since in this case the search can be restricted to a linear number ofcandidate sets of interpretations, while in the general case there is an exponential numberof such candidate sets (represented in the algorithmOL-Sat by all the possible partitionsof the modal atoms ofΣ).

In Fig. 2 we present the algorithmL−O -Sat for computingOL-satisfiability of formulasin L−O . In the algorithm, we assume without loss of generality that the setMA(Σ) containsn> 0 modal atoms prefixed by the operatorO , of the formO(fi ∧ψi) orO(fi ∨ψi), fori = 1, . . . , n, with fi ∈ Lp, ψi ∈LSO .

Example 4.9. Let us again consider the formulaΣ = a ∧ O(¬a ∨ Kb) defined inExample 3.3. As shown before,Σ[O/false] is not KD45-satisfiable. Moreover,SΣ ={¬a, true, false}. Now let ϕ = ¬a. As shown before,Pϕ(Σ) = {O(¬a ∨ Kb)}, Nϕ(Σ) ={Kb}, andΣ|Pϕ(Σ),Nϕ(Σ) = a ∧ true is satisfiable. Therefore,L−O -Sat(Σ) returnstrue.

Correctness of the algorithm is established by the following theorem.

Theorem 4.10.LetΣ ∈L−O . Then,Σ isOL-satisfiable iffL−O -Sat(Σ) returnstrue.

Proof.If part. SupposeL−O -Sat(Σ) returnstrue. Then, there are two possible cases:(1) Σ[O/false] is KD45-satisfiable. As shown in the proof of Theorem 3.10, this implies

thatΣ isOL-satisfiable.(2) There exists a formulaϕ in the set SΣ = {f1, . . . , fn, true, false} such that

Σ|Pϕ(Σ),Nϕ(Σ) is satisfiable. Now letM = {w′: w′ |= ϕ}; moreover, letw be aninterpretation satisfying the satisfiable propositional formulaΣ|Pϕ(Σ),Nϕ(Σ). Fromthe definition of(Pϕ(Σ),Nϕ(Σ)) it follows that each modal atom inMA(Σ) issatisfied by(w,M) iff it belongs toPϕ(Σ). Therefore,(w,M) satisfiesΣ , andsinceM is maximal by construction, it follows thatΣ isOL-satisfiable.


Only-if part.SupposeΣ isOL-satisfiable. Then, there is a model(w,M) satisfyingΣand such thatM is maximal. Let(P,N) be the partition induced byM on MA(Σ). Then,there are two possible cases.

(1) There is no modal atom of the formOϕ in P . Then,Σ[O/false] ∈ LpK is K45-

satisfiable, since(w,M) satisfies it. Now, there are two possibilities:– Σ[O/false] is KD45-satisfiable. In this case, the algorithmL−O -Sat(Σ) returns

true.– Σ[O/false] is not KD45-satisfiable. In this case, from Lemma 3.8 it follows

that ob(P ) is unsatisfiable, henceob(P ) is equivalent tofalse. Moreover,Mis empty, henceM = {w: w |= false}. Therefore,(P,N) coincides with thepartition induced byfalse, that is,(P,N) = (Pfalse(Σ),Nfalse(Σ)). Furthermore,since (w,M) |= Σ and (P,N) is the partition induced byM on MA(Σ),from Lemma 3.6 it follows that the interpretationw satisfiesΣ|P,N , hence thealgorithmL−O -Sat(Σ) returnstrue.

(2) There existsOϕ ∈ P . Then, since(w,M) satisfiesΣ andM is maximal, it followsthatM = {w′: w′ |= ϕ|P,N }. Now, since by hypothesisΣ ∈ L−O , Oϕ is of the formfi ∧ψ or fi ∨ψ , with 16 i 6 n andψ ∈LSO , thereforeψ|P,N is equivalent eitherto true or to false, which implies thatϕ|P,N is equivalent to one of the formulasin the setSΣ = {f1, . . . , fn, true, false}. Consequently,(P,N) is induced by one ofthe formulas inSΣ , and since the formulaΣ|P,N is satisfiable, it follows that thealgorithmL−O -Sat(Σ) returnstrue. 2

We now analyze the complexity of the algorithmL−O -Sat reported in Fig. 2. As noticedabove, the partition(Pϕ(Σ),Nϕ(Σ)) can be computed through a linear number (withrespect to the size ofΣ) of calls to an NP-oracle for propositional satisfiability. Now,since the cardinality ofMA(Σ) (and hence the number of formulas in the setSΣ ) is alsolinearly bounded by the size ofΣ , it follows that the algorithmL−O -Sat is able to establishOL-satisfiability of a formulaΣ ∈L−O in deterministicpolynomial time (in the size ofΣ),using an NP-oracle for propositional satisfiability.

More precisely, it can be shown that the problem ofOL-satisfiability of a formulaΣin L−O is PNP[O(logn)]-complete, namely it can be computed in polynomial time by alogarithmic (in the size ofΣ) number of calls to an NP-oracle. To this aim, we recallthe decision problem TREES(SAT) and the notion of NP-tree [6]. An NP-tree is a triple〈Var,G,Fr 〉 in which:

– Var is a set of propositional variablesv1, . . . , vn (called thelinking variables);– G = (V ,E) is a directed tree, with edges directed from the leaves to the root. Each

element of the set of nodesV = {F1, . . . ,Fn} contains a propositional formulaFibuilt upon a set of private propositional symbols (i.e., symbols which do not appearin any other node) and the linking variablesvj such that(Fj ,Fi) ∈E;

– Fr is a distinguished node, called terminal node.The truth assignmentσ to the propositional variables in an NP-tree is defined as follows:

σ(vi) = TRUE iff the formula F ′i is satisfiable, whereF ′i stands for for the formulaobtained fromFi by replacing each propositional linking variablevj occurring inFi withtrue if σ(vj )= TRUE, and withfalse if σ(vj )= FALSE. The result value of an NP-tree isthe valueσ(vr ).


The decision problem TREES(SAT) is the problem of establishing, given an NP-treeT , whether the result value ofT is TRUE. It has been shown [6, Theorem 4.5] thatTREES(SAT) is PNP[O(logn)]-complete.

Theorem 4.11. LetΣ ∈ L−O . Then, the problem of establishingOL-satisfiability ofΣ isPNP[O(logn)]-complete.

Proof. Hardness follows from the aforementioned fact that satisfiability inOL of aformula of the formOϕ∧¬Kψ such thatϕ ∈ Lp, ψ ∈Lp

K , corresponds to verify whetherthe epistemic queryψ is not implied byϕ. In turn, this last problem corresponds (see [6])to check nonmembership of the formulaKψ in the stable set identified by the formulaϕ,which is a PNP[O(logn)]-complete problem [6, Theorem 5.3.6].

As for membership in PNP[O(logn)], we show that the condition expressed in theinnermostif-then-else statement in the algorithmL−O -Sat, namely the existence of aformula ϕ among{f1, f2, . . . , fn, true, false} such thatΣ|Pϕ(Σ),Nϕ(Σ) is satisfiable, canbe computed in PNP[O(logn)]. To this aim, we reduce, in time polynomial in the size ofΣ , the problem of checking whether such a statement returnstrue to TREES(SAT). Weconstruct the NP-treeT (Σ) as follows.5 First, letm= n+ 2, fn+1 = true, fn+2 = false.We start from the following tree:

– Fr = v1∨ v2∨ · · · ∨ vm;– Fi =Σ , for i = 1, . . . ,m;– E = {(Fi,Fr): i = 1, . . . ,m}.

Then, we obtainT (Σ) by expanding each nodeFi (i = 1, . . . ,m) of the above tree asfollows. LetF be the nodeFi or any successor node ofFi . Now:

– for each strict occurrence of a formulaKϕ in F , create a new nodeFj =¬(fi ⊃ ϕ),replace such an occurrence ofKϕ in F with the linking variablevj , and add the edge(Fj ,F ) toE;

– for each strict occurrence of a formulaOϕ in F , create a new nodeFj =¬(fi ≡ ϕ),replace such an occurrence ofOϕ in F with the linking variablevj , and add the edge(Fj ,F ) toE.

We repeat the above expansion until there are no nodes inT (Σ) which contain modaloperators. Now, letvj be a linking variable occurring in a nodeFi or in a successor nodeof Fi , and letϕ be the modal atom that generatesvj in the above construction. FromLemma 3.7 it immediately follows thatσ(vj ) = TRUE iff ϕ ∈ Pfi (Σ). Consequently,for each nodeFi such that 16 i 6 m, Fi is satisfiable iffΣ|Pϕ(Σ),Nϕ(Σ) is satisfiable.And sinceFr = v1 ∨ · · · ∨ vm, it follows that the result value ofT (Σ) is TRUE iff thereexistsϕ ∈ {f1, . . . , fm} such thatΣ|Pϕ(Σ),Nϕ(Σ) is satisfiable. Moreover, it is immediateto verify thatT (Σ) can be constructed in time polynomial in the size ofΣ . Therefore,the condition expressed by the innermostif-then-elsestatement of the algorithmL−O -Satcan be computed in PNP[O(logn)]. And since satisfiability in propositionalKD45 can becomputed in nondeterministic polynomial time, Theorem 4.10 implies that the satisfiabilityproblem for formulas inL−O is in PNP[O(logn)]. 2

5 The construction trivially extends the technique employed in [6] in the case of Carnap’s modal logic.


5. Only knowing versus minimal knowledge

As mentioned in the introduction, only knowing is strictly related to theminimalknowledgeprinciple. We now compare these two notions from the computationalviewpoint.

The principle of minimal knowledge is a very general notion which can be phrased asfollows: “In each possible epistemic state, the agent hasminimal objective knowledge”,that is, the agent has as much ignorance as possible about the current state of the world. Asa consequence, there exists no epistemic state whose objective knowledge logically impliesthe objective knowledge of another epistemic state.

There are several proposals in the literature based on the minimal knowledge paradigm(see, e.g., [9,12,19,26]). Among them, the first attempt is due to Halpern and Moses [9]and is the most similar to the notion of only knowing. Informally, Halpern and Mosesapply minimal knowledge to modal logicS5: thus, they define a preference semantics [27]over S5, by considering as intended models of a modal theoryΣ only thoseS5 modelssatisfyingΣ in which the set of possible worlds is maximal with respect to set containment.Hence, in this case the notion of maximization lies at the semantic level.

Recently, it has been proven [2] that reasoning in Halpern and Moses’ version ofS5(also known as ground nonmonotonic modal logicS5G) lies at the third level of thepolynomial hierarchy. In particular, logical implication inS5G is a5p

3-complete problem.Moreover, many other formalisms based on the minimal knowledge paradigm share thesame computational properties ofS5G [2,3,24]. Hence, we can conclude that (unlessthe polynomial hierarchy collapses) minimal knowledge is computationally harder thanonly knowing. In particular,S5G cannot be “polynomially embedded” intoOL. This isa surprising result, since the logic of only knowing is generally considered as a veryexpressive formalism, due to its powerful ability of explicitly expressing minimization ofknowledge.

On the other hand, it can be shown that the reason whyS5G (and more generally alllogics based onS5G) is computationally harder thanOL (and all major propositionalnonmonotonic formalisms) is thatS5G allows for expressing minimal knowledge statesin a more compact form thanOL. See [25] for a detailed study of the epistemologicalproperties ofS5G.

6. Conclusions

In this paper we have defined a general method for reasoning about only knowing basedon deduction techniques developed for nonmonotonic modal logics, which proves the strictsimilarity between these logics and Levesque’s monotonic formalism. Based on such areasoning method, we have investigated the computational properties of the propositionalfragment of Levesque’s modal logic of only knowing. Our analysis shows that the problemof reasoning about only knowing in the propositional case lies at the second level ofthe polynomial hierarchy, just like reasoning in most of the propositional formalisms fornonmonotonic reasoning. We have also studied syntactic restrictions in which reasoningabout only knowing is easier than in the general case, and have shown the connections


between such a restricted setting and the framework of epistemic queries to “classical”knowledge bases [23].

We remark that a computational analysis of reasoning about only knowing is interestingnot only from a theoretical perspective, but also for the development of automatedreasoning procedures in the setting of reasoning about actions, where the logic of onlyknowing has been recently applied [14,16].

One further development of the present work is towards the analysis of reasoning aboutonly knowing in the presence of quantifying-in: in particular, it should be interesting to seewhether it is possible to extend the techniques presented here for fragments of such a moreexpressive case. This analysis may also take advantage of recent results on reasoning withquantifying-in in standard modal logics [28].

Furthermore, the problem of embedding only knowing into nonmonotonic formalisms(as autoepistemic logic or the logicS5G) is very interesting from the theoretical viewpoint,in order to establish further relationships between reasoning about only knowing and otherforms of nonmonotonic reasoning.

Acknowledgements

We gratefully thank Daniele Nardi, for his comments on an earlier version of thispaper. We are also thankful to the anonymous referees, for their invaluable remarks andobservations.

References

[1] J. Chen, The logic of only knowing as a unified framework for nonmonotonic reasoning, FundamentaInformaticae 21 (1994) 205–220.

[2] F.M. Donini, D. Nardi, R. Rosati, Ground nonmonotonic modal logics, J. Logic and Computation 7 (4)(1997) 523–548.

[3] T. Eiter, G. Gottlob, Reasoning with parsimonious and moderately grounded expansions, FundamentaInformaticae 17 (1, 2) (1992) 31–54.

[4] T. Eiter, G. Gottlob, Propositional circumscription and extended closed world reasoning are5p2-complete,

Theoret. Comput. Sci. 114 (1993) 231–245.[5] G. Gottlob, Complexity results for nonmonotonic logics, J. Logic and Computation 2 (1992) 397–425.[6] G. Gottlob, NP trees and Carnap’s modal logic, J. ACM 42 (2) (1995) 421–457.[7] J.Y. Halpern, G. Lakemeyer, Levesque’s axiomatization of only knowing is incomplete, Artificial

Intelligence 74 (2) (1995) 381–387.[8] J.Y. Halpern, G. Lakemeyer, Multi-agent only knowing, in: Proc. 6th Conference on Theoretical Aspects of

Reasoning about Knowledge (TARK-96), Morgan Kaufmann, San Mateo, CA, 1996.[9] J.Y. Halpern, Y. Moses, Towards a theory of knowledge and ignorance: Preliminary report, in: K. Apt (Ed.),

Logic and Models of Concurrent Systems, Springer, Berlin, 1985.[10] J.Y. Halpern, Y. Moses, A guide to completeness and complexity for modal logics of knowledge and belief,

Artificial Intelligence 54 (1992) 319–379.[11] D.S. Johnson, A catalog of complexity classes, in: J. van Leuven (Ed.), Handbook of Theoretical Computer

Science, Vol. A, Elsevier Science, North-Holland, Amsterdam, 1990, Chapter 2.[12] M. Kaminski, Embedding a default system into nonmonotonic logics, Fundamenta Informaticae 14 (1991)

345–354.[13] G. Lakemeyer, Limited reasoning in first-order knowledge bases with full introspection, Artificial

Intelligence 84 (1996) 209–255.


[14] G. Lakemeyer, Only knowing in the situation calculus, in: L.C. Aiello, J. Doyle, S.C. Shapiro (Eds.),Proc. 5th International Conference on the Principles of Knowledge Representation and Reasoning (KR-96),Morgan Kaufmann, Los Altos, CA, 1996, pp. 14–25.

[15] G. Lakemeyer, H.J. Levesque, A tractable knowledge representation service with full introspection, in: Proc.2nd Conference on Theoretical Aspects of Reasoning about Knowledge (TARK-88), Morgan Kaufmann,Los Altos, CA, 1988, pp. 145–159.

[16] G. Lakemeyer, H.J. Levesque, AOL: A logic of acting, sensing, knowing, and only knowing, in: Proc. 6thInternational Conference on the Principles of Knowledge Representation and Reasoning (KR-98), MorganKaufmann, Los Altos, CA, 1998, pp. 316–327.

[17] H.J. Levesque, Foundations of a functional approach to knowledge representation, Artificial Intelligence 23(1984) 155–212.

[18] H.J. Levesque, All I know: A study in autoepistemic logic, Artificial Intelligence 42 (1990) 263–310.[19] V. Lifschitz, Minimal belief and negation as failure, Artificial Intelligence 70 (1994) 53–72.[20] W. Marek, M. Truszczynski, Nonmonotonic Logics—Context-Dependent Reasoning, Springer, Berlin,

1993.[21] R.C. Moore, Semantical considerations on nonmonotonic logic, Artificial Intelligence 25 (1985) 75–94.[22] I. Niemeläa, On the decidability and complexity of autoepistemic reasoning, Fundamenta Informati-

cae 17 (1, 2) (1992) 117–156.[23] R. Reiter, What should a database know?, J. Logic Programming 14 (1990) 127–153.[24] R. Rosati, Reasoning with minimal belief and negation as failure: Algorithms and complexity, in: Proc.

AAAI-97, Providence, RI, 1997, pp. 430–435.[25] R. Rosati, Reasoning about minimal knowledge in nonmonotonic modal logics, J. Logic, Language and

Information 8 (2) (1999) 187–203.[26] G. Schwarz, M. Truszczynski, Minimal knowledge problem: A new approach, Artificial Intelligence 67

(1994) 113–141.[27] Y. Shoham, Nonmonotonic logics: Meaning and utility, in: Proc. IJCAI-87, Milan, Italy, 1987, pp. 388–392.[28] F. Wolter, M. Zakharyaschev, Decidable fragments of first-order modal logics. Available at

www.informatik.uni-leipzig.de/ ∼wolter , 1999.

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On the decidability and complexity of reasoning about only knowing