Complexity of Model Checking for Modal Dependence Logic
CComplexity of Model Checking for ModalDependence Logic
Johannes Ebbing ∗ , Peter Lohmann ∗ October 29, 2018
Modal dependence logic (
MDL ) was introduced recently by V¨a¨an¨anen. It enhancesthe basic modal language by an operator =( · ). For propositional variables p , . . . , p n the atomic formula =( p , . . . , p n − , p n ) intuitively states that the value of p n isdetermined solely by those of p , . . . , p n − .We show that model checking for MDL formulae over Kripke structures is NP -complete and further consider fragments of MDL obtained by restricting the set ofallowed propositional and modal connectives. It turns out that several fragments,e.g., the one without modalities or the one without propositional connectives, remain NP -complete.We also consider the restriction of MDL where the length of each single dependenceatom is bounded by a number that is fixed for the whole logic. We show that themodel checking problem for this bounded
MDL is still NP -complete while for somefragments, e.g., the fragment with only ♦ , the complexity drops to P .We additionally extend MDL by allowing classical disjunction – introduced bySevenster – besides dependence disjunction and show that classical disjunction isalways at least as computationally bad as bounded arity dependence atoms and insome cases even worse, e.g., the fragment with nothing but the two disjunctions is NP -complete.Furthermore we almost completely classifiy the computational complexity of themodel checking problem for all restrictions of propositional and modal operators forboth unbounded as well as bounded MDL with both classical as well as dependencedisjunction.This is the second arXiv version of this paper. It extends the first version bythe investigation of the classical disjunction. A shortened variant of the first arXivversion was presented at SOFSEM 2012 [EL12].
ACM Subject Classifiers:
F.2.2 Complexity of proof procedures; F.4.1 Modal logic;D.2.4 Model checking
Keywords: dependence logic, modal logic, model checking, computational complexity ∗ Leibniz Universit¨at Hannover, Theoretical Computer Science, Appelstr. 4, 30167 Hannover, Germany, { ebbing,lohmann } @thi.uni-hannover.de c (cid:13) Johannes Ebbing and Peter Lohmann licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
This work was partially supported by the NTH Focused Research School for IT Ecosystems. a r X i v : . [ c s . L O ] J a n Introduction
Dependence among values of variables occurs everywhere in computer science (databases,software engineering, knowledge representation, AI) but also the social sciences (humanhistory, stock markets, etc.). In his monograph [V¨a¨a07] in 2007 V¨a¨an¨anen introducedfunctional dependence into the language of first-order logic.Functional dependence of the value of q from the values of p , . . . , p n means that thereexists a determinating function f with q = f ( p , . . . , p n ), i.e., the value of q is completelydetermined by the values of p , . . . , p n alone. We denote this form of dependence (ordetermination) by the dependence atom =( p , . . . , p n , q ). To examine dependence be-tween situations, plays, worlds, events or observations we consider collections of these,so called teams . For example, a database can be interpreted as a team. In this case=( p , . . . , p n , q ) means that in every record the value of the attribute q is determined bythe values of the attributes p , . . . , p n .In modal logic a team is a set of worlds in a Kripke structure. Here =( p , . . . , p n , q )means that in every world of the team the value of the atomic proposition q is determinedby the propositions p , . . . , p n , i.e., there is a fixed Boolean function f : { , } n → { , } that determines the value of q from the values of p , . . . , p n for all worlds in the team.In first-order logic =( x , . . . , x n , y ) means the same for a function f : A n → A where A is the universe of a first-order structure. Dependence logic [V¨a¨a07] is then defined bysimply adding dependence atoms to usual first-order logic and modal dependence logic ( MDL ) [V¨a¨a08, Sev09] is defined by introducing dependence atoms to modal logic.Besides the inductive semantics (which we will use here) V¨a¨an¨anen also gave twoequivalent game-theoretic semantics for
MDL [V¨a¨a08]. Sevenster showed that for sin-gleton sets of worlds there exists a translation from
MDL to plain modal logic [Sev09].Sevenster also showed that the satisfiability problem for
MDL is NEXPTIME -complete[Sev09] and Lohmann and Vollmer continued the complexity analysis of the satisfiabilityproblem for
MDL by systematically restricting the set of allowed modal and propositionaloperators and completely classifying the complexity for all fragments of
MDL definablein this way [LV10].Sevenster [Sev09] also introduced classical disjunction (which is classical in a more settheoretic way of looking at the semantics; cf. [AV09]) into the language of
MDL . In thefollowing we always think of the version that includes both classical disjunction (heredenoted by (cid:62) ) as well as dependence disjunction when we write
MDL .The method of systematically classifying the complexity of logic related problemsby restricting the set of operators allowed in formulae goes back to Lewis who usedthis method for the satisfiability problem of propositional logic [Lew79]. Recently itwas, for example, used by Hemaspaandra et al. for the satisfiability problem of modallogic [Hem05, HSS10] and by Lohmann and Vollmer for the satisfiability problem of
MDL [LV10]. The motivation for this approach is that by systematically examining allfragments of a logic one might find a fragment which allows for efficient algorithms butstill has high enough expressivity to be useful in practice. On the other hand, thissystematic approach usually leads to insights into the sources of hardness, i.e., the exact2omponents of the logic that make satisfiability, model checking etc. hard.In this paper we transfer the method from satisfiability [LV10] to model checkingand classify the model checking problem for almost all fragments of
MDL definable byrestricting the set of allowed modal ( (cid:3) , ♦ ) and propositional ( ∧ , ∨ , (cid:62) , ¬ ) operatorsto an arbitrary subset of all operators. The model checking problem asks whether agiven formula is true in a given team of a given Kripke structure. For plain modal logicthis problem is solvable in P as shown by Clarke et al. [CES86]. A detailed complexityclassification for the model checking problem over fragments of modal logic was shownby Beyersdorff et al. [BMM +
11] (who investigate the temporal logic
CTL which containsplain modal logic as a special case).In the case of
MDL it turns out that model checking is NP -complete in general andthat this still holds for several seemingly quite weak fragments of MDL , e.g., the onewithout modalities or the one where nothing except dependence atoms and ♦ is allowed(first and fourth line in Table 1). Strangely, this also holds for the case where only theboth disjunctions ∨ and (cid:62) are allowed and not even dependence atoms occur (third linein Table 1).Furthermore it seems natural to not only restrict modal and propositional operatorsbut to also impose restrictions on dependence atoms. One such restriction is to limitthe arity of dependence atoms, i.e., the number n of variables p , . . . , p n by which q hasto be determined to satisfy the formula =( p , . . . , p n , q ), to a fixed upper bound k ≥ MDL k ). For this restriction model checking remains NP -complete in general but for the fragment with only the ♦ operator allowed this does nothold any more (seventh line in Table 2). In this case either ∧ (fourth line in Table 2) or ∨ (sixth line in Table 2) is needed to still get NP -hardness.We classify the complexity of the model checking problem for fragments of MDL withunbounded as well as bounded arity dependence atoms. We are able to determinethe tractability of each fragment except the one where formulae are built from atomicpropositions and unbounded dependence atoms only by disjunction and negation (sixthline in Table 1). In each of the other cases we either show NP -completeness or show thatthe model checking problem admits an efficient (polynomial time) solution.In Table 1 we list our complexity results for the cases with unbounded arity dependenceatoms and in Table 2 for the cases with an a priori bound on the arity. In these tables a“+” means that the operator is allowed, a “-” means that the operator is forbidden anda “*” means that the operator does not affect the complexity of the problem. We will briefly present the syntax and semantics of
MDL . For a more in-depth introduc-tion we refer to V¨a¨an¨anen’s definition of
MDL [V¨a¨a08] and Sevenster’s model-theoreticand complexity analysis [Sev09] which also contains a self-contained introduction to
MDL . Definition 2.1. (Syntax of
MDL )Let AP be an arbitrary set of atomic propositions and p , . . . , p n , q ∈ AP . Then MDL perators Complexity Reference (cid:3) ♦ ∧ ∨ ¬ = (cid:62) ∗ ∗ + + ∗ + ∗ NP -complete Theorem 3.2+ ∗ ∗ + ∗ + ∗ NP -complete Theorem 3.4 ∗ ∗ ∗ + ∗ ∗ + NP -complete Theorem 5.2 ∗ + ∗ ∗ ∗ + ∗ NP -complete Theorem 3.3 ∗ + + ∗ ∗ ∗ + NP -complete Theorem 4.7,Lemma 5.1 − − − + ∗ + − in NP Proposition 3.1 ∗ ∗ − − ∗ − ∗ in P Theorem 5.3 ∗ − ∗ − ∗ ∗ ∗ in P Theorem 3.6 ∗ ∗ ∗ ∗ ∗ − − in P [CES86] + : operator present − : operator absent ∗ : complexity independent of operator Table 1: Classification of complexity for fragments of
MDL -MCis the set of all formulae built from the following rules: ϕ ::= (cid:62) | ⊥ | q | ¬ q | ϕ ∨ ϕ | ϕ (cid:62) ϕ | ϕ ∧ ϕ | (cid:3) ϕ | ♦ ϕ | =( p , . . . , p n , q ) | ¬ =( p , . . . , p n , q ) . Note that negation is only atomic, i.e., it is only defined for atomic propositions anddependence atoms. (cid:121)
We sometimes write (cid:3) k (resp. ♦ k ) for (cid:3) (cid:3) . . . (cid:3) (cid:124) (cid:123)(cid:122) (cid:125) k times (resp. ♦ ♦ . . . ♦ (cid:124) (cid:123)(cid:122) (cid:125) k times ). For a dependenceatom =( p , . . . , p n , q ) we define its arity as n , i.e., the arity of a dependence atom is thearity of the determinating function whose existence it asserts.In Section 4 we will investigate the model checking problem for the following logic. Definition 2.2. ( MDL k ) MDL k is the subset of MDL that contains all formulae which do not contain any depen-dence atoms whose arity is greater than k . (cid:121) We will classify
MDL for all fragments defined by sets of operators.
Definition 2.3. ( MDL ( M ))Let M ⊆ { (cid:3) , ♦ , ∧ , ∨ , (cid:62) , ¬ , (cid:62) , ⊥ , = } . By MDL ( M ) (resp. MDL k ( M )) we denote the subsetof MDL (resp.
MDL k ) built from atomic propositions using only operators from M . Wesometimes write MDL ( op , op , . . . ) instead of MDL ( { op , op , . . . } ). (cid:121) MDL formulae are interpreted over Kripke structures.4 perators Complexity Reference (cid:3) ♦ ∧ ∨ ¬ = (cid:62) ∗ ∗ + + ∗ + ∗ NP -complete Theorem 3.2+ ∗ ∗ + ∗ + ∗ NP -complete Theorem 3.4 ∗ ∗ ∗ + ∗ ∗ + NP -complete Theorem 5.2 ∗ + + ∗ ∗ + ∗ NP -complete Theorem 4.7 ∗ + + ∗ ∗ ∗ + NP -complete Theorem 4.7,Lemma 5.1 ∗ + ∗ + ∗ + ∗ NP -complete Theorem 4.8 ∗ ∗ − − ∗ ∗ ∗ in P Theorem 5.3 ∗ − ∗ − ∗ ∗ ∗ in P Theorem 3.6 − − − ∗ ∗ ∗ − in P Theorem 4.5 ∗ ∗ ∗ ∗ ∗ − − in P [CES86] + : operator present − : operator absent ∗ : complexity independent of operator Table 2: Classification of complexity for fragments of
MDL k -MC with k ≥ Definition 2.4. (Kripke structure)An AP -Kripke structure is a tuple W = ( S, R, π ) where S is an arbitrary non-emptyset of worlds , R ⊆ S × S is the accessibility relation and π : S → P ( AP ) is the labelingfunction . (cid:121) Definition 2.5. (Semantics of
MDL )In contrast to common modal logics, truth of a
MDL formula is not defined with respectto a single world of a Kripke structure but with respect to a set (or team ) of worlds.Let AP be a set of atomic propositions and p, p , . . . , p n ∈ AP . The truth of a formula ϕ ∈ MDL in a team T ⊆ S of an AP -Kripke structure W = ( S, R, π ) is denoted by5 , T | = ϕ and is defined as follows: W, T | = (cid:62) always holds W, T | = ⊥ iff T = ∅ W, T | = p iff p ∈ π ( s ) for all s ∈ TW, T | = ¬ p iff p / ∈ π ( s ) for all s ∈ TW, T | = =( p , . . . , p n − , p n ) iff for all s , s ∈ T it holds that π ( s ) ∩{ p , . . . , p n − } (cid:54) = π ( s ) ∩{ p , . . . , p n − } or π ( s ) ∩ { p n } = π ( s ) ∩ { p n } W, T | = ¬ =( p , . . . , p n − , p n ) iff T = ∅ W, T | = ϕ ∧ ψ iff W, T | = ϕ and W, T | = ψW, T | = ϕ (cid:62) ψ iff W, T | = ϕ or W, T | = ψW, T | = ϕ ∨ ψ iff there are sets T , T with T = T ∪ T , W, T | = ϕ and W, T | = ψW, T | = (cid:3) ϕ iff W, { s (cid:48) | ∃ s ∈ T with ( s, s (cid:48) ) ∈ R } | = ϕW, T | = ♦ ϕ iff there is a set T (cid:48) ⊆ S such that W, T (cid:48) | = ϕ andfor all s ∈ T there is a s (cid:48) ∈ T (cid:48) with ( s, s (cid:48) ) ∈ R (cid:121) Note that this semantics is a conservative extension of plain modal logic semantics,i. e., it coincides with the latter for formulae which do neither contain dependence atomsnor classical disjunction. Rationales for this semantics – especially for the case of thenegative dependence atom – were given by V¨a¨an¨anen [V¨a¨a07, p. 24].In the remaining sections we will classify the complexity of the model checking problemfor fragments of
MDL and
MDL k . Definition 2.6. ( MDL -MC)Let M ⊆ { (cid:3) , ♦ , ∧ , ∨ , (cid:62) , ¬ , (cid:62) , ⊥ , = } . Then the model checking problem for MDL ( M )(resp. MDL k ( M )) over Kripke structures is defined as the canonical decision problem ofthe set MDL -MC( M )(resp. MDL k -MC( M )) := (cid:26) (cid:104) W, T, ϕ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) W = ( S, R, π ) a Kripke structure, T ⊆ S , ϕ ∈ MDL ( M ) and W, T | = ϕ (cid:27) . (cid:121) We write
MDL -MC for
MDL -MC( { (cid:3) , ♦ , ∧ , ∨ , ¬ , = , (cid:62) , (cid:62) , ⊥} ).Before we begin with the classification we state a lemma showing that it does notmatter whether we include (cid:62) , ⊥ or ¬ in a sublogic MDL ( M ) of MDL since this does notaffect the complexity of
MDL -MC( M ). Lemma 2.7.
Let M be an arbitrary set of MDL operators, i. e., M ⊆ { (cid:3) , ♦ , ∧ , ∨ , (cid:62) , ¬ , = , ⊥ , (cid:62)} . Then we have that MDL - MC( M ) ≡ pm MDL - MC( M \ {(cid:62) , ⊥ , ¬} ) . (cid:97) roof. It suffices to show ≤ pm . So let K = ( S, R, π ) a Kripke structure, T ⊆ S , ϕ ∈ MDL ( M ) and the variables of ϕ among p , . . . , p n . Let p (cid:48) , . . . , p (cid:48) n , t, f be freshpropositional variables. Then K, T | = ϕ iff K (cid:48) , T | = ϕ (cid:48) , where K (cid:48) := ( S, R, π (cid:48) ) with π (cid:48) defined by π (cid:48) ( s ) ∩ { t, f } := { t } ,π (cid:48) ( s ) ∩ { p i , p (cid:48) i } := (cid:26) { p i } if p i ∈ π ( s ) , { p (cid:48) i } if p i / ∈ π ( s ) , for all i ∈ { , . . . , n } and s ∈ S , and ϕ ∈ MDL ( M \ {(cid:62) , ⊥ , ¬} ) defined by ϕ (cid:48) := ϕ ( p (cid:48) / ¬ p )( p (cid:48) / ¬ p ) . . . ( p (cid:48) n / ¬ p n )( t/ (cid:62) )( f / ⊥ ) . (cid:4) First we will show that the most general of our problems is in NP and therefore all modelchecking problems investigated later are as well. Proposition 3.1.
Let M be an arbitrary set of MDL operators. Then
MDL - MC( M ) isin NP . And hence also MDL k - MC( M ) is in NP for every k ≥ . (cid:97) Proof.
The following non-deterministic top-down algorithm checks the truth of theformula ϕ on the Kripke structure W in the evaluation set T in polynomial time.Algorithm 1: check( W = ( S, R, π ), ϕ , T ) case ϕ when ϕ = p foreach s ∈ T i f not p ∈ π ( s ) thenreturn false return true when ϕ = ¬ p foreach s ∈ T i f p ∈ π ( s ) thenreturn false return true when ϕ = =( p , . . . , p n ) foreach ( s, s (cid:48) ) ∈ T × T i f π ( s ) ∩ { p , . . . , p n − } = π ( s (cid:48) ) ∩ { p , . . . , p n − } then // i.e., s and s (cid:48) agree on the values of the propositions p , . . . , p n − i f ( q ∈ π ( s ) and not q ∈ π ( s (cid:48) ) ) or ( not q ∈ π ( s ) and q ∈ π ( s (cid:48) ) ) thenreturn false return true hen ϕ = ¬ =( p , . . . , p n ) i f T = ∅ return true return false when ϕ = ψ ∨ θ guess two s e t s o f s t a t e s A, B ⊆ S i f not A ∪ B = T thenreturn false return ( check ( W, A, ψ ) and check ( W, B, θ ) ) when ϕ = ψ (cid:62) θ return ( check ( W, T, ψ ) or check ( W, T, θ ) ) when ϕ = ψ ∧ θ return ( check ( W, T, ψ ) and check ( W, T, θ ) ) when ϕ = (cid:3) ψT (cid:48) := ∅ foreach s (cid:48) ∈ S foreach s ∈ T i f ( s, s (cid:48) ) ∈ R then T (cid:48) := T (cid:48) ∪ { s (cid:48) } // T (cid:48) is the set of all successors of all states in T return check ( W, T (cid:48) , ψ ) when ϕ = ♦ ψ guess s e t o f s t a t e s T (cid:48) ⊆ S foreach s ∈ T i f t h e r e i s no s (cid:48) ∈ T (cid:48) with ( s, s (cid:48) ) ∈ R thenreturn false // T (cid:48) contains at least one successor of every state in T return check ( W, T (cid:48) , ψ ) Now we will see that the model checking problem is NP -hard and that this still holdswithout modalities. Theorem 3.2.
Let M ⊇ {∧ , ∨ , = } . Then MDL - MC( M ) is NP -complete. Furthermore, MDL k - MC( M ) is NP -complete for every k ≥ . (cid:97) Proof.
Membership in NP follows from Proposition 3.1. For the hardness proof wereduce from 3SAT.For this purpose let ϕ = C ∧ . . . ∧ C m be an arbitrary 3CNF formula with vari-ables x , . . . , x n . Let W be the Kripke structure ( S, R, π ) over the atomic propositions8 , . . . , r n , p , . . . , p n defined by S := { s , . . . , s m } ,R := ∅ ,π ( s i ) ∩ { r j , p j } := { r j , p j } iff x j occurs in C i positively, { r j } iff x j occurs in C i negatively, ∅ iff x j does not occur in C i .Let ψ be the MDL ( ∧ , ∨ , = ) formula n (cid:95) j =1 r j ∧ =( p j )and let T := { s , . . . , s m } the evaluation set.We will show that ϕ ∈ W, T | = ψ . Then it follows that 3SAT ≤ pm MDL -MC( M ) and therefore MDL -MC( M ) is NP -hard.Now assume ϕ ∈ θ an interpretation with θ | = ϕ . From the valuations θ ( x j )of all x j we construct subteams T , . . . , T n such that for all j ∈ { , . . . , n } it holds that W, T j | = γ j with γ j := r j ∧ =( p j ). The T j are constructed as follows T j := (cid:40) { s i ∈ S | π ( s i ) ∩ { r j , p j } = { r j , p j }} iff θ ( x j ) = 1 { s i ∈ S | π ( s i ) ∩ { r j , p j } = { r j }} iff θ ( x j ) = 0i.e., T j is the team consisting of exactly the statescorresponding to clauses satisfied by θ ( x j ) . Since every clause in ϕ is satisfied by some valuation θ ( x j ) = 1 or θ ( x j ) = 0 we havethat T ∪ . . . ∪ T n = T such that W, T | = ϕ .On the other hand, assume that W, T | = ψ , therefore we have T = T ∪ T ∪ . . . ∪ T n such that for all j ∈ { , . . . , n } it holds that T j | = γ j . Therefore π ( s i ) ∩ { p j } is constantfor all elements s i ∈ T j . From this we can construct a valid interpretation θ for ϕ .For all j let I j := { i | s i ∈ T j } . For every j ∈ { , . . . , n } we consider T j . If for everyelement s i ∈ T j it holds that π ( s i ) ∩ { p j } = { p j } then we have for all i ∈ I j that x j is aliteral in C i . In order to satisfy those C i we set θ ( x j ) = 1. If for every element s i ∈ T j it holds that π ( s i ) ∩ { p j } = ∅ then we have for every i ∈ I j that ¬ x j is a literal in C i .In order to satisfy those C i we set θ ( x j ) = 0.Since for every s i ∈ T there is a j with s i ∈ T j we have an evaluation θ that satisfiesevery clause in ϕ . Therefore we have θ | = ϕ . (cid:4) Instead of not having modalities at all we can also allow nothing but the ♦ modality,i.e., we disallow propositional connectives and the (cid:3) modality, and model checking is NP -complete as well. Theorem 3.3.
Let M ⊇ { ♦ , = } . Then MDL - MC( M ) is NP -complete. (cid:97) roof. Membership in NP follows from Proposition 3.1 again.For hardness we again reduce from 3SAT. Let ϕ = (cid:86) mi =1 C i be an arbitrary 3CNFformula built from the variables x , . . . , x n . Let W be the Kripke structure ( S, R, π ),over the atomic propositions p , . . . , p n , q , shown in Figure 1 and formally defined by S := { c , . . . , c m , s , . . . , s n , s , . . . , s n } ,R ∩ { ( c i , s j ) , ( c i , s j ) } := { ( c i , s j ) } iff x j occurs in C i positively, { ( c i , s j ) } iff x j occurs in C i negatively, ∅ iff x j does not occur in C i , π ( c i ) := ∅ ,π ( s j ) := { p j , q } ,π ( s j ) := { p j } . c i s j p j s j p j , q Figure 1: Kripke structure part corresponding to the 3CNF fragment · · · ∧ C i ∧ . . . with C i = x j ∨ . . . .Let ψ be the MDL ( ♦ , =) formula ♦ =( p , . . . , p n , q )and let T := { c , . . . , c m } .We will show that ϕ ∈ W, T | = ψ . Hence, 3SAT ≤ pm MDL -MC( M ) and MDL -MC( M ) is NP -hard.First suppose we have an interpretation θ that satisfies ϕ . From the valuations of θ we will construct a successor team T (cid:48) of T , i.e., for all s ∈ T there is an s (cid:48) ∈ T (cid:48) s.t.( s, s (cid:48) ) ∈ R with W, T (cid:48) | = =( p , . . . , p n , q ). T (cid:48) is defined by: T (cid:48) := { s zj | θ ( x j ) = z, j ∈ { , . . . , n }} Since θ satisfies every clause C i of ϕ we have that for every C i there is an x j with θ ( x j ) = (cid:40) , iff x j ∈ C i , iff ¬ x j ∈ C i . It follows that for every s ∈ T there is an s (cid:48) ∈ T (cid:48) such that ( s, s (cid:48) ) ∈ R .By construction of T (cid:48) it is not possible to have both s j and s j in T (cid:48) . Hence forall elements s j , s j (cid:48) ∈ T (cid:48) it follows that j (cid:54) = j (cid:48) and therefore π ( s j ) ∩ { p , . . . , p n } (cid:54) = π ( s j (cid:48) ) ∩ { p , . . . , p n } . Thus W, T (cid:48) | = =( p , . . . , p n , q ).10n the other hand assume W, T | = ψ . Then there is a successor set T (cid:48) of T s.t. forevery s ∈ T there is an s (cid:48) ∈ T (cid:48) with ( s, s (cid:48) ) ∈ R and T (cid:48) | = =( p , . . . , p n , q ). We construct θ as follows: θ ( x j ) := , iff s j ∈ T (cid:48) , iff s j ∈ T (cid:48) , iff s j , s j / ∈ T (cid:48) . Note that in the latter case it does not matter if 0 or 1 is chosen.Since
W, T (cid:48) | = =( p , . . . , p n , q ) and for every j it holds that W, { s j , s j } (cid:54)| = =( p , . . . , p n ,q ) we have that for every j at most one of s j or s j is in T (cid:48) . It follows that θ is well-defined.Since for every c i ∈ T there is an s zj ∈ T (cid:48) s.t. ( c i , s zj ) ∈ R with θ ( x j ) = z , we have bycontruction of W that θ satisfies every clause C i of ϕ . From this follows ϕ ∈ (cid:4) If we disallow ♦ but allow (cid:3) instead we have to also allow ∨ to get NP -hardness. Theorem 3.4.
Let M ⊇ { (cid:3) , ∨ , = } . Then MDL - MC( M ) is NP -complete. Also, MDL k - MC( M ) is NP -complete for every k ≥ . (cid:97) Proof.
Membership in NP follows from Proposition 3.1 again. To prove hardness, wewill once again reduce 3SAT to this problem.Let ϕ = (cid:86) mi =1 C i be an arbitrary 3CNF formula over the variables x , . . . , x n . Let W be the structure ( S, R, π ), over the atomic propositions p , . . . , p n , shown in Figures 2 to7 and formally defined as follows: S := (cid:8) s i | i ∈ { , . . . , m } (cid:9) ∪ (cid:8) r jk | k ∈ { , . . . , m } , j ∈ { , . . . , n } (cid:9) ∪ (cid:8) r jk | k ∈ { , . . . , m } , j ∈ { , . . . , n } (cid:9) R ∩ (cid:83) j ∈{ ,...,n } { ( s i , r ji ) , ( s i , r ji ) } := (cid:26) { ( s i , r i ) } iff x occurs in C i (positively or negatively) (Fig. 2) { ( s i , r i ) , ( s i , r i ) } iff x does not occur in C i (Fig. 3) R ∩ (cid:83) k ∈{ ,...,n } { ( r ji , r ki ) , ( r ji , r ki ) , ( r ji , r ki ) , ( r ji , r ki ) } := { ( r ji , r j +1 i ) } iff x j and x j +1 both occur in C i (Fig. 4) { ( r ji , r j +1 i ) , ( r ji , r j +1 i ) } iff x j occurs in C i but x j +1 doesnot occur in C i (Fig. 5) { ( r ji , r j +1 i ) , ( r ji , r j +1 i ) } iff x j does not occur in C i but x j +1 does occur in C i (Fig. 6) { ( r ji , r j +1 i ) , ( r ji , r j +1 i ) } iff neither x j nor x j +1 occur in C i (Fig. 7) π ( s i ) := ∅ π ( r ji ) := (cid:40) { p j } iff x j occurs in C i positively or not at all ∅ iff x j occurs in C i negatively π ( r ji ) := ∅ i r i Figure 2: x occurs in C i . s i r i r i Figure 3: x does not occur in C i . r j +1 i r ji Figure 4: x j and x j +1 occur in C i . r ji r j +1 i r j +1 i Figure 5: x j occurs in C i but x j +1 does notoccur in C i .Let ψ be the MDL ( (cid:3) , ∨ , =) formula n (cid:95) j =1 (cid:3) j =( p j )and let T := { s , . . . , s m } .Then, as we will show, ϕ ∈ W, T | = ψ and therefore MDL -MC( (cid:3) , ∨ , =) is NP -complete. Intuitively, the direction from left to right holds because the disjunctionsplits the team { s , . . . , s m } of all starting points of chains of length n into n subsets(one for each variable) in the following way: s i is in the subset that belongs to x j iff x j r j +1 i r ji r ji Figure 6: x j does not occur in C i but x j +1 does occur in C i . r ji r ji r j +1 i r j +1 i Figure 7: x j and x j +1 do not occur in C i .12atisfies the clause C i under the variable valuation that satisfies ϕ . Then the team thatbelongs to x j collectively satisfies the disjunct (cid:3) j =( p j ) of ψ . For the reverse directionthe r ji states are needed to ensure that a state s i can only satisfy a disjunct (cid:3) j =( p j ) ifthere is a variable x j that occurs in clause C i (positively or negatively) and satisfies C i .More precisely, assume θ is a satisfying interpretation for ϕ . From θ we constructsubteams T , . . . , T n with T ∪ . . . ∪ T n = T s.t. for all j it holds that T j | = (cid:3) j =( p j ). T j is defined by T j := (cid:40) { s i | { s i } | = (cid:3) j p j } iff θ ( x j ) = 1 { s i | { s i } | = (cid:3) j ¬ p j } iff θ ( x j ) = 0for all j ∈ { , . . . , n } . Obviously, for all j it holds that T j | = (cid:3) j =( p j ). Now we will showthat for all i ∈ { , . . . , m } there is a j ∈ { , . . . , n } such that s i ∈ T j . For this purposelet i ∈ { , . . . , m } and suppose C i is satisfied by θ ( x j ) = 1 for a j ∈ { , . . . , n } . Then, bydefinition of W , π ( r ji ) = p j , hence { s i } | = (cid:3) j p j and therefore s i ∈ T j . If, on the otherhand, C i is satisfied by θ ( x j ) = 0 then we have that π ( r ji ) = ∅ , hence { s i } | = (cid:3) j ¬ p j and again it follows that s i ∈ T j . Altogether we have that for all i there is a j such that s i ∈ T j . It follows that T ∪ . . . ∪ T n = T and therefore W, T | = ψ .On the other hand assume W, T | = ψ . Therefore we have T = T ∪ . . . ∪ T n with T j | = (cid:3) j =( p j ) for all j ∈ { , . . . , n } . We define a valuation θ by θ ( x j ) := (cid:40) T j | = (cid:3) j p j T j | = (cid:3) j ¬ p j . Since every s i is contained in a T j we know that for all i ∈ { , . . . , m } there is a j ∈{ , . . . , n } with { s i } | = (cid:3) j =( p j ). From this it follows that x j occurs in C i (positively ornegatively) since otherwise, by definition of W , both r ji and r ji would be reachable from s i .It also holds that { s i } | = (cid:3) j p j or { s i } | = (cid:3) j ¬ p j . In the former case we have that π ( r ji ) = p j , hence, by definition of W , x j is a literal in C i . By construction of θ itfollows that C i is satisfied. In the latter case it holds that x j is a literal in C i . Again,by construction of θ it follows that C i is satisfied. Hence, ϕ ∈ (cid:4) The following example demonstrates the construction from the previous proof.
Example 3.5.
Let ϕ be the 3CNF formula( ¬ x ∨ x ∨ x ) (cid:124) (cid:123)(cid:122) (cid:125) C ∧ ( x ∨ ¬ x ∨ x ) (cid:124) (cid:123)(cid:122) (cid:125) C ∧ ( x ∨ ¬ x ) (cid:124) (cid:123)(cid:122) (cid:125) C . The corresponding Kripke structure W shown in Figure 8 has levels j thlevel (corresponding to the variable x j in the formula ϕ ) is the set of nodes reachable viaexactly j transitions from the set of nodes s , s and s (corresponding to the clauses of ϕ ). In this example all non connected states (which do not play any role at all) are notshown. 13 s s s s r r r p r p r p r p r r p r r r p r p r r p r p r Figure 8: Kripke structure corresponding to ϕ = ( ¬ x ∨ x ∨ x ) ∧ ( x ∨ ¬ x ∨ x ) ∧ ( x ∨ ¬ x )The MDL formula corresponding to ϕ is ψ = (cid:3) =( p ) (cid:124) (cid:123)(cid:122) (cid:125) γ ∨ (cid:3) (cid:3) =( p ) (cid:124) (cid:123)(cid:122) (cid:125) γ ∨ (cid:3) (cid:3) (cid:3) =( p ) (cid:124) (cid:123)(cid:122) (cid:125) γ ∨ (cid:3) (cid:3) (cid:3) (cid:3) =( p ) (cid:124) (cid:123)(cid:122) (cid:125) γ . Let T = { s , s , s } with W, T | = ψ and for all j ∈ { , . . . , } let T j ⊆ T with T j | = γ j and T ∪ . . . ∪ T = T . By comparing the formulae γ j with the chains in the Kripkestructure one can easily verify that T (cid:32) { s , s } i.e., there can at most be one of s and s in T since π ( r ) ∩ p (cid:54) = π ( r ) ∩ { p } and s cannot be in T since its directsuccessors r , r do not agree on p . In this case T = { s } means that C is satisfied bysetting θ ( x ) = 0 and the fact that { s } (cid:54)| = γ corresponds to the fact that there is noway to satisfy C via x , because x does not occur in C . Analogously, T ⊆ { s , s } or T ⊆ { s } , and T (cid:32) { s , s } and T ⊆ { s } .Now, e.g., the valuation θ where x , x and x evaluate to true and x to false satisfies ϕ . From this valuation one can construct sets T , . . . , T with T ∪ · · · ∪ T = { s , s , s } such that T j | = γ j for all j = 1 , . . . , T j := { s i | x j satisfies clause C i under θ } for all j . This leads to T = T = { s } , T = { s } and T = { s } .The gray colourings indicate which chains (resp. clauses) are satisfied on which levels(resp. by which variables). ψ (resp. ϕ ) is satisfied because there is a gray coloured statein each chain. (cid:121)
14f we disallow both ♦ and ∨ the problem becomes tractable since the non-deterministicsteps in the model checking algorithm are no longer needed. Theorem 3.6.
Let M ⊆ { (cid:3) , ∧ , ¬ , = } . Then MDL - MC( M ) is in P . (cid:97) Proof.
Algorithm 1 is a non-deterministic algorithm that checks the truth of an ar-bitrary
MDL formula in a given structure in polynomial time. Since M ⊆ { (cid:3) , ∧ , ¬ , = } it holds that ♦ , ∨ / ∈ M . Therefore the non-deterministic steps are never used and thealgorithm is in fact deterministic in this case. (cid:4) Note that this deterministic polynomial time algorithm is a top-down algorithm andtherefore works in a fundamentally different way than the usual deterministic polynomialtime bottom-up algorithm for plain modal logic.Now we have seen that
MDL -MC( M ) is tractable if ∨ / ∈ M and ♦ / ∈ M since these twooperators are the only source of non-determinism. On the other hand, MDL -MC( M ) is NP -complete if = ∈ M and either ♦ ∈ M (Theorem 3.3) or ∨ , (cid:3) ∈ M (Theorem 3.4).The remaining question is what happens if only ∨ (but not (cid:3) ) is allowed. Unfortunatelythis case has to remain open for now. We will now show that
MDL -MC( {∨ , ¬ , = } ) is in P if we impose the following constrainton the dependence atoms in formulae given as part of problem instances: there is aconstant k ∈ N such that in any input formula it holds for all dependence atoms of theform =( p , . . . , p j , p ) that j ≤ k . To prove this statement we will decompose it into twosmaller propositions.First we show that even the whole {∨ , ¬ , = } fragment with unrestricted =( · ) atoms isin P as long as it is guaranteed that in every input formula at least a specific number ofdependence atoms – depending on the size of the Kripke structure – occur.We will need the following obvious lemma stating that a dependence atom is alwayssatisfied by a team containing at least half of all the worlds. Lemma 4.1.
Let W = ( S, R, π ) be a Kripke structure, ϕ := =( p , . . . , p n , q ) ( n ≥ )an atomic formula and T ⊆ S an arbitrary team. Then there is a set T (cid:48) ⊆ T such that | T (cid:48) | ≥ | T | and T (cid:48) | = ϕ . (cid:97) Proof.
Let T := { s ∈ T | q / ∈ π ( s ) } and T := { s ∈ T | q ∈ π ( s ) } . Then T ∪ T = T and T ∩ T = ∅ . Therefore there is an i ∈ { , } such that | T i | ≥ | T | . Let T (cid:48) := T i .Since q is either labeled in every state of T (cid:48) or in no one, it holds that W, T (cid:48) | = ϕ . (cid:4) We will now formalize a notion of “many dependence atoms in a formula”.
Definition 4.2.
For ϕ ∈ MDL let σ ( ϕ ) be the number of positive dependence atoms in ϕ . Let (cid:96) : N → R an arbitrary function and (cid:63) ∈ { <, ≤ , >, ≥ , = } . Then MDL -MC (cid:63) (cid:96) ( n ) ( M )(resp. MDL k -MC (cid:63)(cid:96) ( n ) ( M )) is the problem MDL -MC( M ) (resp. MDL k -MC( M )) restrictedto inputs (cid:104) W = ( S, R, π ) , T, ϕ (cid:105) that satisfy the condition σ ( ϕ ) (cid:63) (cid:96) ( | S | ). (cid:121)
15f we only allow ∨ and we are guaranteed that there are many dependence atoms ineach input formula then model checking becomes trivial – even for the case of unboundeddependence atoms. Proposition 4.3.
Let M ⊆ {∨ , ¬ , = } . Then MDL - MC > log ( n ) ( M ) is trivial, i.e., for allKripke structures W = ( S, R, π ) and all ϕ ∈ MDL ( M ) such that the number of positivedependence atoms in ϕ is greater than log ( | S | ) it holds for all T ⊆ S that W, T | = ϕ . (cid:97) Proof.
Let W = ( S, R, π ), ϕ ∈ MDL ( M ), T ⊆ S be an arbitrary instance with (cid:96) > log ( | S | ) dependence atoms in ϕ . Then either ϕ ≡ (cid:62) or ϕ ≡ (cid:96) (cid:95) i =1 =( p j i, , . . . , p j i,ki ) (cid:124) (cid:123)(cid:122) (cid:125) ψ i ∨ (cid:95) i l i , where each l i is either a (possibly negated) atomic proposition or a negated dependenceatom. Claim.
For all k ∈ { , . . . , (cid:96) } there is a set T k ⊆ T such that W, T k | = (cid:87) ki =1 ψ i and | T \ T k | < (cid:96) − k .The main proposition follows immediately from case k = (cid:96) of this claim: From | T \ T (cid:96) | < (cid:96) − (cid:96) = 1 follows that T = T (cid:96) and from W, T (cid:96) | = (cid:87) (cid:96)i =1 ψ i follows that W, T | = ϕ . Inductive proof of the claim.
For k = 0 we can choose T k := ∅ . For the inductive steplet the claim be true for all k (cid:48) < k . By Lemma 4.1 there is a set T (cid:48) k ⊆ T \ T k − suchthat W, T (cid:48) k | = ψ k and | T (cid:48) K | ≥ | T \ T k − | . Let T k := T k − ∪ T (cid:48) k . Since W, T k − | = (cid:87) k − i =1 ψ i itfollows by definition of the semantics of ∨ that W, T k | = (cid:87) ki =1 ψ i . Furthermore, | T \ T k | = | ( T \ T k − ) \ T (cid:48) k | = | T \ T k − | − | T (cid:48) k |≤ | T \ T k − | − | T \ T k − | = | T \ T k − | < (cid:96) − ( k − = 2 (cid:96) − k . (cid:4) Note that
MDL -MC > log ( n ) ( M ) is only trivial, i.e., all instance structures satisfy allinstance formulae, if we assume that only valid instances, i.e., where the number ofdependence atoms is guaranteed to be large enough, are given as input. However, if wehave to verify this number the problem clearly remains in P .Now we consider the case in which we have very few dependence atoms (which havebounded arity) in each formula. We use the fact that there are only a few dependenceatoms by searching through all possible determinating functions for the dependenceatoms. Note that in this case we do not need to restrict the set of allowed MDL operatorsas we have done above.
Proposition 4.4.
Let k ≥ . Then MDL k - MC ≤ log ( n ) is in P . (cid:97) Proof.
From the semantics of = it follows that =( p , . . . , p k , p ) is equivalent to ∃ f f ( p , . . . , p k ) ↔ p := ∃ f (cid:0) ( ¬ f ( p , . . . , p k ) ∨ p ) ∧ ( f ( p , . . . , p k ) ∨ ¬ p ) (cid:1) (1)16here f ( p , . . . , p k ) and ∃ f ϕ – both introduced by Sevenster [Sev09, Section 4.2] – havethe following semantics: W, T | = ∃ f ϕ iff there is a Boolean function f W such that( W, f W ) , T | = ϕ ( W, f W ) , T | = f ( p , . . . , p k ) iff for all s ∈ T and for all x , . . . , x k ∈ { , } with x i = 1 iff p i ∈ π ( s ) ( i = 1 , . . . , k ): f W ( x , . . . , x k ) = 1( W, f W ) , T | = ¬ f ( p , . . . , p k ) iff for all s ∈ T and for all x , . . . , x k ∈ { , } with x i = 1 iff p i ∈ π ( s ) ( i = 1 , . . . , k ): f W ( x , . . . , x k ) = 0Now let W = ( S, R, π ), T ⊆ S and ϕ ∈ MDL k be a problem instance. First, we countthe number (cid:96) of dependence atoms in ϕ . If (cid:96) > log ( | S | ) we reject the input instance.Otherwise we replace every dependence atom by its translation according to 1 (each timeusing a new function symbol). Since the dependence atoms in ϕ are at most k -ary we havefrom the transformation (1) that the introduced function variables f , . . . , f (cid:96) are also atmost k -ary. From this it follows that the upper bound for the number of interpretationsof each of them is 2 k . For each possible tuple of interpretations f W , . . . , f W(cid:96) for thefunction variables we obtain an ML formula ϕ ∗ by replacing each existential quantifier ∃ f i by a Boolean formula encoding of the interpretation f Wi (for example by encodingthe truth table of f i with a formula in disjunctive normal form). For each such tuple wemodel check ϕ ∗ . That is possible in polynomial time in | S | + | ϕ ∗ | as shown by Clarkeet al. [CES86]. Since the encoding of an arbitrary k -ary Boolean function has length atmost 2 k and k is constant this is a polynomial in | S | + | ϕ | .Furthermore, the number of tuples over which we have to iterate is bounded by (cid:16) k (cid:17) log ( | S | ) = 2 k · log ( | S | ) = (cid:0) log ( | S | ) (cid:1) k = | S | k ∈ | S | O(1) . (cid:4) With Proposition 4.3 and Proposition 4.4 we have shown the following theorem.
Theorem 4.5.
Let M ⊆ {∨ , ¬ , = } , k ≥ . Then MDL k - MC( M ) is in P . (cid:97) Proof.
Given a Kripke structure W = ( S, R, π ) and a
MDL k ( ∨ , ¬ , =) formula ϕ thealgorithm counts the number m of dependence atoms in ϕ . If m > log ( | S | ) the inputis accepted (because by Proposition 4.3 the formula is always fulfilled in this case).Otherwise the algorithm from the proof of Proposition 4.4 is used. (cid:4) And there is another case where we can use the exhaustive determinating functionsearch.
Theorem 4.6.
Let M ⊆ { (cid:3) , ♦ , ¬ , = } . Then MDL k - MC( M ) is in P for every k ≥ . (cid:97) roof. Let ϕ ∈ MDL k ( M ). Then there can be at most one dependence atom in ϕ because M only contains unary operators. Therefore we can once again use the algorithmfrom the proof of Proposition 4.4. (cid:4) In Theorem 3.3 we saw that
MDL -MC( ♦ , =) is NP -complete. The previous theoremincludes MDL k -MC( ♦ , =) ∈ P as a special case. Hence, the question remains which arethe minimal supersets M of { ♦ , = } such that MDL k -MC( M ) is NP -complete.We will now see that adding either ∧ (Theorem 4.7) or ∨ (Theorem 4.8) is alreadyenough to get NP -completeness again. But note that in the case of ∨ we need k ≥ k = 0 the question remains open. Theorem 4.7.
Let M ⊇ { ♦ , ∧ , = } . Then MDL k - MC( M ) is NP -complete for every k ≥ . (cid:97) Proof.
Membership in NP follows from Proposition 3.1. For hardness we once againreduce 3SAT to our problem.For this purpose let ϕ := (cid:86) mi =1 C i be an arbitrary 3CNF formula built from thevariables x , . . . , x n . Let W be the Kripke structure ( S, R, π ) shown in Figure 9 andformally defined by S := { c i | i ∈ { , . . . , m }} ∪ { s j,j (cid:48) , s j,j (cid:48) | j, j (cid:48) ∈ { , . . . , n }}∪ { t j , t j | j ∈ { , . . . , n }} R := { ( c i , s ,j ) | x j ∈ C i } ∪ { ( c i , s ,j ) | x j ∈ C i }∪ { ( s k,j , s k +1 ,j ) | j ∈ { , . . . , n } , k ∈ { , . . . , n − }}∪ { ( s k,j , s k +1 ,j ) | j ∈ { , . . . , n } , k ∈ { , . . . , n − }}∪ { ( s k,j , t j ) , ( s k,j , t j ) | j ∈ { , . . . , n } , k ∈ { , . . . , n }}∪ { ( s k,j , t j ) , ( s k,j , t j ) | j ∈ { , . . . , n } , k ∈ { , . . . , n } , j (cid:54) = k } π ( c i ) := ∅ π ( s j,j (cid:48) ) := ∅ π ( s j,j (cid:48) ) := ∅ π ( t j ) := { r j , p j } π ( t j ) := { r j } . And let ψ be the MDL ( ♦ , ∧ , =) formula ♦ (cid:32) n (cid:86) j =1 ♦ j ( r j ∧ =( p j )) (cid:33) = ♦ (cid:0) ♦ ( r ∧ =( p )) ∧ ♦ ♦ ( r ∧ =( p )) ∧ . . . ∧ ♦ n ( r n ∧ =( p n )) (cid:1) . We again show that ϕ ∈ W, { c , . . . , c m } | = ψ . First assume that ϕ ∈ θ is a satisfying valuation for the variables in ϕ . Now let s j := (cid:40) s ,j if x j evaluates to true under θs ,j if x j evaluates to false under θ for all j = 1 , . . . , n . Then it holds that W, { s , . . . , s n } | = n (cid:86) j =1 ♦ j ( r j ∧ =( p j )).18 c c . . .s , s , s , s , s , . . .t r t r , p s , ... s , s , s , s , . . .s , s , s , ... s , s , . . . Figure 9: Kripke structure construction in the proof of Theorem 4.7
The underlying 3CNF formula contains the clauses C = x ∨ ¬ x , C = x ∨ x ∨ x and C = ¬ x ∨ x Furthermore, since θ satisfies ϕ it holds for all i = 1 , . . . , m that there is a j i ∈{ , . . . , n } such that ( c i , s j i ) ∈ R . Hence, W, { c , . . . , c m } | = ♦ (cid:32) n (cid:86) j =1 ♦ j ( r j ∧ =( p j )) (cid:33) .For the reverse direction assume that W, { c , . . . , c m } | = ψ . Now let T ⊆ { s , , s , ,s , , . . . , s ,n } such that T | = n (cid:86) j =1 ♦ j ( r j ∧ =( p j )) and for all i = 1 , . . . , m there is a s ∈ T with ( c i , s ) ∈ R .Since T | = ♦ j ( r j ∧ =( p j )) there is no j ∈ { , . . . , n } with s ,j ∈ T and also s ,j ∈ T .Now let θ be the valuation of x , . . . , x n defined by θ ( x j ) := (cid:40) s ,j ∈ T . Since for each i = 1 , . . . , m there is a j ∈ { , . . . , n } such that either ( c i , s ,j ) ∈ R and s ,j ∈ T or ( c i , s ,j ) ∈ R and s ,j ∈ T it follows that for each clause C i of ϕ there is a j ∈ { , . . . , n } such that x j satisfies C i under θ . (cid:4) Theorem 4.8.
Let M ⊇ { ♦ , ∨ , = } . Then MDL k - MC( M ) is NP -complete for every k ≥ . (cid:97) Proof.
As above membership in NP follows from Proposition 3.1 and for hardness wereduce 3SAT to our problem.For this purpose let ϕ := (cid:86) mi =1 C i be an arbitrary 3CNF formula built from thevariables p , . . . , p n . Let W be the Kripke structure ( S, R, π ) shown in Figure 10 and19ormally defined by S := { c i,j | i ∈ { , . . . , m } , j ∈ { , . . . , n }} ∪ { x j,j (cid:48) | j, j (cid:48) ∈ { , . . . , n } , j (cid:48) ≤ j } R := { ( c i,j , c i,j +1 ) | i ∈ { , . . . , m } , j ∈ { , . . . , n − }}∪ { ( x j,j (cid:48) , x j,j (cid:48) +1 ) | j ∈ { , . . . , n } , j (cid:48) ∈ { , . . . , j − }} π ( x j,j (cid:48) ) := (cid:26) { q, p j } iff j (cid:48) = j { q } iff j (cid:48) < jπ ( c i,j ) := { q } iff p j , ¬ p j / ∈ C i { p j } iff p j ∈ C i ∅ iff ¬ p j ∈ C i c , q c , q c , c , c , p c , qc , q c , ... c , q . . .x , qx , q, p x , qx , q, p x , q . . .x , q, p Figure 10: Kripke structure construction in the proof of Theorem 4.8
The underlying 3CNF formula contains the clauses C = ¬ p , C = p ∨ ¬ p and C = ¬ p ψ be the MDL formula n (cid:87) j =1 ♦ j − =( q, p j ) ≡ =( q, p ) ∨ ♦ =( q, p ) ∨ ♦ ♦ =( q, p ) ∨ · · · ∨ ♦ n − =( q, p n ) . Once again we show that ϕ ∈ W, { c , , . . . , c m, , x , , x , , . . . , x n, } | = ψ .First assume that ϕ ∈ θ is a satisfying valuation for the variables in ϕ .Now let P j := { c i, | C i is satisfied by p j under θ } for all j = 1 , . . . , n . Then it followsthat n (cid:83) j =1 P j = { c , , . . . , c m, } and that W, P j | = ♦ j − ( ¬ q ∧ =( p j ))for all j = 1 , . . . , n . Additionally, it holds that W, { x j, } | = ♦ j − ( q ∧ =( p j )) ( j = 1 , . . . , n ).Together it follows that W, P j ∪{ x j, } | = ♦ j − =( q, p j ) for all j = 1 , . . . , n . This implies W, n (cid:91) j =1 ( P j ∪ { x j, } ) | = n (cid:95) j =1 ♦ j − =( q, p j )which is equivalent to W, { c , , . . . , c m, , x , , x , , . . . , x n, } | = ψ. For the reverse direction assume that
W, T | = ψ with T := { c , , . . . , c m, , x , , x , ,. . . , x n, } . Let T , . . . , T n be subsets of T with T ∪ · · · ∪ T n = T such that for all j ∈ { , . . . , n } it holds that T j | = ♦ j − =( q, p j ). Then it follows that x , ∈ T since thechain starting in x , consists of only one state. From π ( x , ) = { q, p } and π ( x , ) = { q } it follows that x , / ∈ T and hence (again because of the length of the chain) x , ∈ T .Inductively, it follows that x j, ∈ T j for all j = 1 , . . . , n .Now, it follows from x j, ∈ T j that for all i ∈ { , . . . , m } with c i, ∈ T j : q / ∈ π ( c i,j )(because q, p j ∈ π ( x j,j , p j / ∈ π ( x i,j )). Since T j | = ♦ j − =( q, p j ), it then holds that T j \ { x j, } | = ♦ j − ( ¬ q ∧ =( p j )).Now let θ be the valuation of p , . . . , p n defined by θ ( p j ) := (cid:40) T j \ { x j, } | = ♦ j − ( ¬ q ∧ p j )0 if T j \ { x j, } | = ♦ j − ( ¬ q ∧ ¬ p j ) . Since for each i = 1 , . . . , m there is a j ∈ { , . . . , n } such that c i, ∈ T j it follows thatfor each clause C i of ϕ there is a j ∈ { , . . . , n } such that p j satisfies C i under θ . (cid:4) First we show that classical disjunction can substitute zero-ary dependence atoms.21 emma 5.1.
Let = , (cid:62) / ∈ M . Then MDL - MC( M ∪ { = } ) ≤ pm MDL - MC( M ∪ { (cid:62) } ) . (cid:97) Proof.
Follows immediately from the equivalence of =( p ) and p (cid:62) ¬ p together withLemma 2.7. (cid:4) The following surprising result shows that both kinds of disjunctions together arealready enough to get NP -completeness. Theorem 5.2.
Let {∨ , (cid:62) } ⊆ M . Then MDL k - MC( M ) is NP -complete for every k ≥ . (cid:97) Proof.
As above membership in NP follows from Proposition 3.1 and for hardness wereduce 3SAT to our problem – using a construction that bears some similarities with theone used in the proof of Theorem 4.8.For this purpose let ϕ := (cid:86) mi =1 C i be an arbitrary 3CNF formula built from thevariables p , . . . , p n . Let W be the Kripke structure ( S, R, π ) shown in Figure 11 andformally defined by S := { c i | i ∈ { , . . . , m }} R := ∅ π ( c i ) := { p j | p j ∈ C i } ∪ { q j | ¬ p j ∈ C i } .c p , q c p , q c q , p . . . Figure 11: Kripke structure construction in the proof of Theorem 5.2
The underlying 3CNF formula contains the clauses C = p ∨ ¬ p , C = p ∨ ¬ p and C = ¬ p ∨ p Let ψ be the MDL formula n (cid:95) j =1 ( p j (cid:62) q j ) . Once again we show that ϕ ∈ W, { c , . . . , c m } | = ψ . First assume that ϕ ∈ θ is a satisfying valuation for ϕ . Now let P j := { c i | C i is satisfied by p j under θ } for all j = 1 , . . . , n . Then it follows that (cid:83) nj =1 P j = { c , . . . , c m } and that W, P j | = p j (cid:62) q j j = 1 , . . . , n . Together it follows that W, { c , . . . , c m } | = n (cid:95) j =1 ( p j (cid:62) q j ) . For the reverse direction assume that
W, T | = ψ with T := { c , . . . , c m } . Let T , . . . , T n be subsets of T with T ∪ · · · ∪ T n = T such that for all j ∈ { , . . . , n } it holds that T j | = p j (cid:62) q j . Now let θ be the valuation of p , . . . , p n defined by θ ( p j ) := (cid:40) T j | = p j T j | = q j . Since for each i = 1 , . . . , m there is a j ∈ { , . . . , n } such that c i ∈ T j it follows thatfor each clause C i of ϕ there is a j ∈ { , . . . , n } such that p j satisfies C i under θ . (cid:4) Now we show that Theorem 4.6 still holds if we additionally allow classical disjunction.
Theorem 5.3.
Let M ⊆ { (cid:3) , ♦ , (cid:62) , ¬ , = } . Then MDL k - MC( M ) is in P for every k ≥ . (cid:97) Proof.
Let ϕ ∈ MDL ( M ). Because of the distributivity of (cid:62) with all other MDL operators there is a formula ψ equivalent to ϕ which is of the form | ϕ | (cid:54) i =1 ψ i with ψ i ∈ MDL ( M \ { (cid:62) } ) for all i ∈ { , . . . , | ϕ |} . Note that there are only linearly manyformulas ψ i because ϕ does not contain any binary operators aside from (cid:62) . Further notethat ψ can be easily computed from ϕ in polynomial time.Now it is easy to check for a given structure W and team T whether W, T | = ψ by simply checking whether W, T | = ψ i (which can be done in polynomial time byTheorem 4.6) consecutively for all i ∈ { , . . . , | ϕ |} . (cid:4) In this paper we showed that
MDL -MC is NP -complete (Theorem 3.2). Furthermore wehave systematically analyzed the complexity of model checking for fragments of MDL de-fined by restricting the set of modal and propositional operators. It turned out that thereare several fragments which stay NP -complete, e.g., the fragment obtained by restrictingthe set of operators to only (cid:3) , ∨ and = (Theorem 3.4) or only ♦ and = (Theorem 3.3).Intuitively, in the former case the NP -hardness arises from existentially guessing parti-tions of teams while evaluating disjunctions and in the latter from existentially guessingsuccessor teams while evaluating ♦ operators. Consequently, if we allow all operatorsexcept ♦ and ∨ the complexity drops to P (Theorem 3.6).23or the fragment only containing ∨ and = on the other hand we were not able todetermine whether its model checking problem is tractable. Our inability to prove either NP -hardness or containment in P led us to restrict the arity of the dependence atoms.For the aforementioned fragment the complexity drops to P in the case of bounded arity(Theorem 4.8). Furthermore, some of the cases which are known to be NP -complete forthe unbounded case drop to P in the bounded arity case as well (Theorem 4.6) whileothers remain NP -complete but require a new proof technique (Theorems 4.7 and 4.8).Most noteworthy in this context are probably the results concerning the ♦ operator.With unbounded dependence atoms this operator alone suffices to get NP -completenesswhereas with bounded dependence atoms it needs the additional expressiveness of either ∧ or ∨ to get NP -hardness.Considering the classical disjunction operator (cid:62) , we showed that the complexity of MDL k -MC( M ∪ { = } ) is never higher than the complexity of MDL k -MC( M ∪ { (cid:62) } ), i.e., (cid:62) is at least as bad as =( · ) with respect to the complexity of model-checking (in con-trast to the complexity of satisfiability; cf. [LV10]). And in the case where only ∨ isallowed we even have a higher complexity with (cid:62) (Theorem 5.2) than with = (Theo-rem 4.5). The case of MDL -MC( ∨ , (cid:62) ) is also our probably most surprising result sincethe non-determinism of the ∨ operator turned out to be powerful enough to lead to NP -completeness although neither conjunction nor dependence atoms (which also, in asense, contain some special kind of conjunction) are allowed.Interestingly, in none of our reductions to show NP -hardness the MDL formula dependson anything else but the number of propositional variables of the input 3CNF formula.The structure of the input formula is always encoded by the Kripke structure alone. Soit seems that even for a fixed formula the model checking problem could still be hard.This, however, cannot be the case since, by Theorem 4.4, model checking for a fixedformula is always in P .Another open question, apart from the unclassified unbounded arity case, is relatedto a case with bounded arity dependence atoms. In Theorem 4.8 it was only possibleto prove NP -hardness for arity at least one and it is not known what happens in thecase where the arity is zero. Additionally, it might be interesting to determine the exactcomplexity for the cases which are in P since we have not shown any lower bounds inthese cases so far. References [AV09] Samson Abramsky and Jouko V¨a¨an¨anen. From IF to BI.
Synthese ,167(2):207–230, 2009.[BMM +
11] Olaf Beyersdorff, Arne Meier, Martin Mundhenk, Thomas Schneider,Michael Thomas, and Heribert Vollmer. Model checking CTL is almostalways inherently sequential.
Logical Methods in Computer Science , 2011.[CES86] E. M. Clarke, E. A. Emerson, and A. P. Sistla. Automatic verification of24nite-state concurrent systems using temporal logic specifications.
ACMTrans. Program. Lang. Syst. , 8(2):244–263, 1986.[EL12] Johannes Ebbing and Peter Lohmann. Complexity of model checking formodal dependence logic. In
Proceedings SOFSEM 2012: Theory and Practiceof Computer Science , volume 7147 of
Lecture Notes in Computer Science ,pages 226–237. Springer Berlin / Heidelberg, 2012.[Hem05] Edith Hemaspaandra. The complexity of poor man’s logic.
CoRR ,cs.LO/9911014v2, 2005.[HSS10] Edith Hemaspaandra, Henning Schnoor, and Ilka Schnoor. Generalizedmodal satisfiability.
J. Comput. Syst. Sci. , 76(7):561–578, 2010.[Lew79] Harry Lewis. Satisfiability problems for propositional calculi.
MathematicalSystems Theory , 13:45–53, 1979.[LV10] Peter Lohmann and Heribert Vollmer. Complexity results for modal depen-dence logic. In
Proceedings 19th Conference on Computer Science Logic ,volume 6247 of
Lecture Notes in Computer Science , pages 411–425. SpringerBerlin / Heidelberg, 2010.[Sev09] Merlijn Sevenster. Model-theoretic and computational properties of modaldependence logic.
Journal of Logic and Computation , 19(6):1157–1173, 2009.[V¨a¨a07] Jouko V¨a¨an¨anen.
Dependence logic: A new approach to independencefriendly logic . Number 70 in London Mathematical Society student texts.Cambridge University Press, 2007.[V¨a¨a08] Jouko V¨a¨an¨anen. Modal dependence logic. In Krzysztof R. Apt and Robertvan Rooij, editors,
New Perspectives on Games and Interaction , volume 4of