It is crucial for an intelligent agent system to be capable of representing and reasoning about high-order knowledge in the multi-agent setting. A general representative framework for these scenarios is multi-agent epistemic logics. However, many reasoning tasks in such logics are intractable, e.g., the entailment problems for and are -complete [Halpern and Moses1992].
These intractability results impede applications of multi-agent epistemic logics, e.g., multi-agent epistemic planning (MAEP) [Kominis and Geffner2015, Muise et al.2015]. An MAEP consists of a set of agents, the initial knowledge base (KB) and the goal formula that are expressed in multi-agent epistemic logics, ontic actions that change the world and epistemic actions that modify the mental attitude of agents. Two types of reasoning tasks, that are essential to solving MAEP, involve progression and entailment check. Progression updates KBs according to the effects of actions while entailment check is needed to decide if the current KB entails the goal formula and the preconditions of actions. As mentioned in [Bienvenu, Fargier, and Marquis2010], based on a normal form with efficient progression and entailment procedures, the whole planning process should also be effective.
Knowledge compilation is an effective approach to address the intractability problem [Darwiche and Marquis2002]. A basic idea is to identify a normal form such that it is a fragment of the given language and each KB can be equivalently transformed into a KB in the normal form. BieFM2010 BieFM2010 proposed a normal form, called -, for the single-agent that supports polytime bounded conjunction and forgetting. It is also applied in making the progression of actions tractable. However, many reasoning tasks of multi-agent epistemic logics, including forgetting and entailment check, is intractable in this normal form. Hence, it cannot be applied to the multi-agent case.
Some normal forms have been proposed for multi-agent epistemic logics. By using cover operators instead of standard epistemic operators, CateCMV2006 CateCMV2006 defined cover disjunctive normal forms (CDNFs) for that is a syntactic variant of . Bie2008 Bie2008 introduced prime implicate normal forms (PINFs) for . The target languages for these two compilations are tractable w.r.t. major reasoning tasks such as entailment check and forgetting. The former supports bounded conjunction while the latter does not. In the worst case, a compiled formula from CDNF has the single exponential size w.r.t. the original formula, but PINF can cause double exponential blowup. In addition, a normal form, called alternating cover disjunctive normal form (ACDNF), is proposed for the logic [Hales, French, and Davies2012]. This form prohibits direct nestings of cover operators of an agent inside those of the same agent. Recently, HuangFWL2017 HuangFWL2017 proved that polytime bounded conjunction and satisfiability check hold for ACDNFs.
To develop effective algorithms to MAEPs, we aim to develop a compilation approach for multi-agent logics such that (1) the compilation is relatively compact. That is, the compiled formula has at most single exponential size; (2) the target language is tractable for major reasoning tasks of MAEP: bounded conjunction, forgetting and entailment check; and (3) each formula can be equivalently transformed into a formulas in the normal form.
In this paper, we provide such a solution to knowledge compilation for the multi-agent epistemic logics and its extension , with the well-known introspection axioms and , by employing the theory of logical separability [Levesque1998]. Informally, we say a conjunction of formulas is logically separable if reasoning can be reduced to its conjuncts. For example, the formula is not logically separable since it logically implies a conjunct that is not derived by any single conjunct of . By conjoining with the implicit conjunct, the new formula becomes logically separable.
The main contributions of this paper are summarized as follows:
We first formulate the concept of logical separability for epistemic terms and introduce some useful properties that are desired for them. Thanks to the notion of logical separability, we are able to define two novel normal forms for , referred to as and (Section 3).
We provide an almost complete knowledge compilation map for multi-agent epistemic logics by comparing among the four normal forms: , , and from the four aspects: expressiveness, succinctness, queries and transformations. To the best of our knowledge, we are the first to construct this map for multi-agent epistemic logics (Sections 4 and 5).
We offer a tractable approach to progression and entailment checking, which are important ingredients of MAEP. To achieve this, we obtain a normal form by taking advantage of tractability of normal forms in propositional logic, e.g., and [Bryant1986] on bounded conjunction, forgetting and entailment check are tractable (Section 6).
We extend the results of knowledge compilation for to by requiring that no consecutive epistemic operators of the same agent appears in formulas (Section 7).
2 The multi-agent modal logic
In this section, we first recall the syntax and semantics of the multi-agent epistemic logic , and then introduce two normal forms of , and major reasoning tasks in .
Syntax and semantics
Throughout this paper, we fix a set of agents and a countable set of variables.
The language is generated by the BNF:
where and .
The formula means that agent knows . The symbols , , , , and are defined as usual. We use and for agents, for sets of agents, and for variables, for finite sets of variables. For an -formula , we use for the size of (i.e., the number of occurrences of variables, logical connectives, and modalities in ), for the depth of (i.e., the maximal number of nested epistemic operators appearing in ), and for the set of variables appearing in . We say a formula is smaller than , if .
The notions of propositional literals, terms (), clauses (), disjunctive and conjunctive normal forms ( and ) are defined as usual. An -formula is in negation normal form (NNF) if the scope of contains only variables. A positive (resp. negative) epistemic literal is a formula of the form (resp. ). A formula is basic, if it is a propositional formula or epistemic literal. An epistemic term (resp. clause) is a conjunction (resp. disjunction) of basic formulas. Sometimes, we treat an epistemic term or clause as a set of formulas. For an epistemic term (resp. clause) , we use for the set of the maximal propositional formulas that are conjuncts (resp. disjuncts) of , for the set of formulas such that is a conjunct (resp. disjunct) of , and for the set of formulas such that is a conjunct (resp. disjunct) of .
A Kripke model is a tuple where
is a non-empty set of possible worlds;
where is a binary relation on ;
is a function assigning to each in a subset of .
A pointed Kripke model is a pair , where is a Kripke model and is a world of , called the actual world. For convenience, we assume that Kripke models are pointed.
Let be a Kripke model where . We interpret formulas in by induction:
if and ;
if for all , .
We say is satisfiable, if there is a model satisfying ; entails , written , if for any model satisfying , ; and are equivalent, written , if and .
Throughout this paper, we use and for sublanguages of , and and for propositional sublanguages. Throughout this paper, we assume that every propositional term and clause has a polynomial representation in and . All of the propositional sublanguages considered in [Darwiche and Marquis2002] obey with this assumption except the canonical . We say and are dual, if there is a polytime algorithm from to s.t. for any formula , , and vice verse. For example, and are dual in propositional logic.
A formula is in cover disjunctive normal form (), if it is generated by the BNF:
where is a satisfiable , are in , , and is shorthand for .
An epistemic clause111The definition of epistemic clauses in [Bienvenu2008] is slightly different from that in this paper. It is defined as a disjunction of propositional literals and epistemic literals. is an implicate of , if . An epistemic clause is a prime implicate of , if is an implicate of and for all implicate of s.t. , .
A formula is in prime implicate normal form (), if it is or , or satisfies the following:
is a conjunction of epistemic clauses where
each prime implicate of is equivalent to some conjunct ;
every is a prime implicate of s.t. (i) if is a disjunct of , then ; (ii) for ; (iii) for every , if then is in ; (iv) for every , and , we have .
Queries and transformations
For a normal form considered in knowledge compilation, it is useful if it preserves major reasoning tasks and logical constructs (also referred to as queries and transformations). In this paper, we consider those queries and transformations, discussed in [Darwiche and Marquis2002] for propositional logic. Most of them can be directly generalized to multi-agent epistemic logics except modal counting () and enumeration () since any formula generally has infinitely many distinct models.
We say a language satisfies
(resp. ), if there is a polytime algorithm deciding whether any formula is satisfiable (resp. valid).
(resp. ), if there is a polytime algorithm deciding whether any formulas satisfies the condition (resp. ).
(resp. ), if there is a polytime algorithm deciding whether (resp. ) for any formula and epistemic clause (resp. term) .
(resp. ), if there is a polytime algorithm generating a formula of equivalent to (resp. ) for every set of -formulas.
(resp. ), if there is a polytime algorithm generating a formula of equivalent to (resp. ) for any formulas .
, if there is a polytime algorithm generating a formula of equivalent to for any formula .
We now turn to another two important transformations: conditioning and forgetting. Conditioning is a syntactic operation defined as follows:
Let and a satisfiable propositional term. The conditioning of on , written , is the formula obtained by replacing each variable of by (resp. ) if (resp. ) is a positive (resp. negative) literal of .
A language satisfies , if there is a polytime algorithm generating a formula of equivalent to for every and satisfiable propositional term .
Intuitively, forgetting from generates the logically strongest consequence of in which any variable of does not appear. It can be applied in version control of knowledge bases and knowledge reuse. The definition of forgetting [French2006] is given as follow.
Let and . We say is a result of forgetting in , written , if
for any formula s.t. , iff .
The result of forgetting is unique up to logical equivalence [Fang, Liu, and van Ditmarsch2016]. We hereafter use to denote the result of forgetting in .
A language satisfies (resp. ), if there is a polytime algorithm generating a formula of equivalent to (resp. ) for any formula and set of variables (resp. variable ).
3 Separability-based and
In this section, based on logical separability, we introduce a general framework for defining normal forms and in .
One might define for as a disjunction of epistemic terms. However, this is not a proper definition for due to lack of some desirable properties, such as the tractability for both satisfiability check and forgetting that propositional supports. The issues distribute over disjunction, and thus the problem lies in the definition of epistemic terms as some epistemic terms are logically inseparable. Let us illustrate it in an example.
Consider the formula . The unsatisfiable formula is not derived by any single epistemic literal of . Deriving it requires reasoning about all conjuncts together. The satisfiability problem of epistemic terms cannot be decomposed into its conjuncts.
This example illustrates that the polytime check for satisfiability holds for only logically separable epistemic terms.
Let be an epistemic term. We say is logically separable, iff for every basic formula , if , then there is or is an epistemic literal that is a conjunct of s.t. .
Intuitively, logical separability requires that no logical puzzles are hidden within parts of epistemic terms.
Continued with Example 1, is logically inseparable since but no conjunct of entails . The formula , which is equivalent to , is logically separable.
Logical separable terms have the modularity property for satisfiability check and forgetting. The satisfiability problem of a logically separable epistemic term can be reduced to satisfiability subproblems of deciding whether each formula in and is satisfiable.
Let be a logically separable epistemic term. Then is satisfiable iff every formula is satisfiable.
Similarly, forgetting a set of variables in can be accomplished by individually forgetting in each formula of , and .
Let be a logically separable epistemic term and a set of variables.
To prove this property, we need a lemma.
Let be a satisfiable logically separable epistemic term. Then, the following statements hold:
For each propositional formula , iff for some ;
For each and each positive epistemic literal , iff for some ;
For each and each negative epistemic literal , iff for some .
Now we give a proof for Proposition 3.2.
For brevity, we let be the right-hand-side formula. We consider two possible cases:
Case 1. is unsatisfiable: Then is also unsatisfiable. By Proposition 3.1, there is an unsatisfiable formula , or for some , there is s.t. is unsatisfiable. Suppose that is unsatisfiable. We get that is also unsatisfiable since contains an unsatisfiable conjunct . Similarly, is unsatisfiable in the case where is unsatisfiable.
Case 2. is satisfiable: Here we only verify the only-if direction for Condition 3 of Definition 2.9: for any formula s.t. , if , then . By the De Morgan’s law, the distributive law of disjunction (resp. conjunction) over conjunction (resp. disjunction), and two transformation rules: and , every -formula can be equivalently transformed into a conjunction of epistemic clauses. So we assume w.l.o.g. that is a conjunction of epistemic clauses. Let be a conjunct of and of the form . It suffices to show that . For simplify, we let . Since , at least one of the following conditions holds.
there exist and s.t. is unsatisfiable;
there exist and s.t. is unsatisfiable.
Here, we assume that Condition 2 holds. The other cases can be proven similarly. It follows that . So . Since entails the former formula, we get that . By Lemma 3.1, there is s.t. . Since is the result of forgetting in , we have . Hence, , and . ∎
The following proposition gives the smallest logically separable epistemic term representation of an epistemic term . In this normal form, there is at most one propositional part, and at most one positive epistemic literal for each agent. Moreover, every formula inside entails the corresponding formula inside .
The smallest logically separable epistemic term representation of an epistemic term satisfies the following:
for each , ;
for each , and , .
It is trivial to prove the case where is unsatisfiable since the smallest representation of unsatisfiable formula is . We now assume that is satisfiable, and only verify Condition 1. The other two conditions can be proven similarly. On the contrary, suppose that but they are distinct. If or , then one of them is redundant, and is not the most compact form. Otherwise, and . Thus, neither nor entails . This violates Lemma 3.1. ∎
Forgetting in a logically separable epistemic term may not be tractably computed. This is because that some subformulas of may not be tractable for forgetting. To achieve polytime forgetting for logically separable epistemic terms, we need some further conditions on them. We not only require the logically separable epistemic term to be the smallest form, but also restrict the propositional part of to be in , and every formula of and to be the disjunction of formulas in this form.
An epistemic term is a separability-based term with (), if it is of the form s.t.
’s and ’s are disjunctions of ’s;
for any and .
It is natural to obtain the definition of separability-based clauses that is dual to the notion of separability-based terms.
An epistemic clause is a separability-based clause with (), if it is of the form s.t.
’s and ’s are conjunctions of ’s;
for any and .
We are ready to define separability-based and .
A formula is in separability-based disjunctive (resp. conjunctive) normal form with ( (resp. )), if is a disjunction (resp. conjunction) of ’s (resp. ’s).
It is easily verified that two existing normal forms and are sublanguages of and respectively.
In the definition of CDNF (Definition 2.4), each is an STE since (1) it is shorthand for , and (2) for each . Thus, CDNF is a fragment of SDNF.
In the definition of PINF (Definition 2.5), Conditions 2-(c)-(ii) and -(iii) correspond to the form and Condition (3) of the definition of SCL (Definition 3.3). So PINF is a fragment of SCNF. ∎
4 Expressiveness and Succinctness
In this section, we analyze the expressive power and spatial complexity of the four normal forms. Our main results include: (1) the sizes of the and for a given formula are single-exponential in the size of the given formula, and (2) we provide a full picture of the succinctness for the four normal forms , , and .
It is proven that every -formula is equivalent to a formula in (resp. ) that is at most single (resp. double) exponentially large in the given formula size. This reflects that our new normal forms have a better space complexity than and is at the same level as .
Any formula in is equivalent to a formula in (or ) that is at most single-exponentially large in the size of the original formula.
We only consider as the case of is similar.
Let . We first transform into in NNF by pushing every negation symbol into variables and eliminating double negation symbols. We then recursively transform into an equivalent formula in by induction on , the depth of nesting of epistemic operators.
Base case: If , is propositional and thus it can be equivalently transformed into a propositional DNF formula . Then we obtain a formula in by converting each disjunct of into .
Inductive case: By the distributive law, we transform the formula into a disjunction of epistemic terms . For each term , we first convert it into the form . The propositional formula is obtained by conjoining all propositional parts of , i.e., . In a similar way, we obtain the positive literal such that . For each , we obtain a negative literal where . By the inductive assumption, the subformulas and can be transformed into .
We analyze the spatial complexity of this transformation. Firstly, the NNF formula has size at most since only negation symbols are added, and there is at most one negation symbol for each occurrence of variables. Secondly, by induction on , we can show that and that is single-exponential in . ∎
In the worst case, the number of prime implicates for a formula can be double-exponential in the size of the formula [Bienvenu2009]. The following proposition shows that the size of the smallest ( and ) for a formula can be exponential in the worst case.
Every (resp. ) formula equivalent to (resp. ) has at least epistemic terms (resp. clauses).
We now turn to compare the succinctness of the four normal forms.
A language is at least as succinct as , denoted , if there is a polynomial function from to s.t. for any formula , there exists a formula s.t. and .
The following proposition indicates that the succinctness results for and can be reduced to the corresponding succinctness results in propositional logic.
iff iff .
We only prove that iff .
(): If , then there exists a mapping from to satisfying two conditions: (1) for any formula and (2) there is a polynomial s.t. for each formula , .
We inductively construct a mapping from to as follows:
, if ;
, if and .
It is easily verified that is a formula in such that and .
(): On the contrary, assume that . Let s.t. no equivalent formula in satisfying the condition: for any polynomial . Obviously, is in . The smallest representation of propositional formula is an -formula. Hence, . ∎
Table 1 summarizes the results of succinctness for the four normal forms. The symbol (or ) in the cell of row and column of Table 1 means that “the normal form given at column is at least as succinct as given at column (under the condition that in the case of )”. The symbol means that “ is not at least as succinct as ”.
We make three observations from Table 1. First of all, (resp. ) we propose are strictly more succinct than the existing normal form (resp. ). In addition, and are incomparable w.r.t. succinctness. This incomparability relation also holds for the other three pairs of normal forms: (, ), (, ), and (, ). Finally, and are not at least as succinct as the other normal forms.
The results in Table 1 hold.
We only prove that . The other statements can be seen by Propositions 3.4 - 4.3, the corresponding results for [ten Cate et al.2006] and [Bienvenu2009], and the assumption that every term and clause has polynomial representation in and .
We define a class of formulas as follows:
Here and are propositional atoms. The size of is linear in , more precisely, . Let be a polynomial s.t. any clause has a representation in with size at most . Each has a polynomial representation in with size . The smallest representation in equivalent to is . This formula has size single-exponential in . ∎
5 Queries and Transformations
In this section, we mainly discuss against the class of queries and transformations, and identify conditions of under which some useful properties hold in . In particular, we give a tractable and modular algorithm for verifying the satisfiability of formulas in . More importantly, we provide an almost complete picture for tractability of the four normal forms. These results, together with the results on succinctness, show that is the normal form most suitable for MAEP.
It is well-known that the satisfiability problem of is tractable. This positive result is still valid for if allows polytime satisfiability check. Based on a given subprocedure for the satisfiability of , Algorithm 1 is the whole procedure that recursively decides if a formula is satisfiable via repeated application of the subprocedure. Due to the modularity property (cf. Proposition 3.1), a logically separable epistemic term is satisfiable iff all of and ’s are satisfiable. Hence, the subprocedure is polytime, so is Algorithm 1. Interestingly, even if the satisfiability problem of is NP-Complete, the upper bound of the time complexity of Algorithm 1 falls into since the number of propositional subformulas in is at most , and this algorithm only calls for the subprocedure at most times.
If satisfies , then satisfies .
The negative results about other queries also carry forward from to .
does not satisfy , , , or unless .
: Let be a . For each , there exists s.t. and for some polynomial . Clearly, is in . If we can decide whether this disjunction is valid in polytime, then the validity of can be tractably accomplished. However, the latter problem is -complete. A contradiction.
and : Since implies , does not satisfy . Similarly, fails to satisfy .
and : Let be an epistemic literal where is propositional. Clearly, is in and is an epistemic term. We get that iff is valid. The validity problem of propositional logic is -complete, and so is the problem that decides if . Hence, does not satisfy . Similarly, fails to satisfy . ∎
Unlike , even if satisfies the polytime clause entailment check (), does not possess such a property. Actually, it is impossible to propose a normal form permitting such a check. In the following, we will show that supports a restricted polytime clausal entailment check after showing that satisfies polytime bounded conjunction ().
By Definition 3.4, it is obvious that the disjunction of formulas can be generated efficiently.
satisfies , and hence .
Similarly to , is not closed under conjunction and negation.
does not satisfy or .
Nonetheless, it supports bounded conjunction.
If satisfies , then satisfies .
By assumption, there exists a polytime algorithm for generating an -formula equivalent to for each pair of formulas . Let be its time complexity, the degree of , and the sum of the coefficients of . So .
Given , we construct a formula in that is equivalent to by simply taking the disjunction of all epistemic terms where for each disjunct of and each disjunct of . If for every pair and , then .
It remains to prove that . Let and . W.l.o.g., assume that . We construct a formula , where , , and . It is easy to verify that