In this paper, we continue the analysis of admissible rules with parameters (constants) in transitive modal logics satisfying certain extension properties started in Jeřábek . We recall that the first part was devoted to structural results. We introduced the class of cluster-extensible (clx) logics, encompassing the most common transitive modal logics such as , , , , , and many others. We proved that in the setting of rules with parameters, all formulas have projective approximations in any clx logic , whence -admissibility is decidable, and we can compute finite complete sets of -unifiers to any given formula. We provided semantic characterization of -admissibility in terms of certain classes of frames (called -extensible), and axiomatizations of -admissible rules by explicit bases.
The topic of this second part is the computational complexity of admissibility with parameters, and of the closely related problem of unifiability, in clx and other transitive logics. We mention that the complexity of admissibility in transitive modal logics was previously studied in : the main results were that admissibility in certain logics (called extensible) is either -complete or -complete depending on if the logic is linear (i.e., of width ) or not; the lower bound, stating that admissibility is -hard, was proved under a weak hypothesis applicable to a larger class of logics. The class of extensible logics of  is incomparable with the clx logics of : on the one hand, the condition of extensibility only constraints frames with a one-element root cluster, hence in this sense it is less restrictive; on the other hand, the definition of extensibility in  did not accommodate nonlinear logics of bounded branching. However, the principal difference is that  only considered admissibility without parameters.
As we will see, the introduction of parameters leads to a richer and more complicated landscape: we will encounter several more complexity classes than just and , and while linearity of the logic will remain an important dividing line, the complexity of the problem will also be influenced by other factors, the most important being if the logic allows clusters of unbounded size.
On the other hand, the usage of parameters makes our results on complexity more robust, and simpler to formulate. This is most clearly seen for lower bounds: first, they apply already to the special case of unifiability rather than to the full (in)admissibility problem, and second, they only require weak and easily checkable assumptions on the logics, such as being of width , or having unbounded cluster size. (In contrast, the parameter-free lower bound from  only applies to admissibility, and relies on a peculiar extensibility condition on the logic.) In general, our lower bounds will have the form that -unifiability is -hard (for a particular complexity class ) whenever certain finite frames may be embedded into -frames, or more precisely, when there exist -frames that subreduce or weakly subreduce to said finite frames; see below for definitions.
Wherever possible, we also include results on the complexity of restricted unifiability or admissibility problems in which only a constant number of parameters are allowed, though in this case the conditions on logics get more complicated, and do not seem optimal.
In contrast to lower bounds, decent upper bounds can be proved only for well-behaved logics, as random transitive modal logics may be quite wild, with already the set of tautologies being of high complexity or even undecidable. We are primarily interested in the case of clx logics, and one of the goals of this paper is to find a complete classification of the complexity of unifiability and admissibility in clx logics, both in the regime with arbitrary parameters, and with only constantly many parameters.
Additionally, we present upper bounds on the complexity of admissibility and unifiability in logics of bounded depth and width. To this end, we need first to prove a few structural results on admissibility in logics of bounded depth—in particular, semantical characterizations—since such logics are not covered by the framework from .
Similar to clx logics, logics of bounded depth and width are tame and well behaved, which allows us to prove interesting general results about their complexity. However, the class of these logics is structurally quite different from the class of clx logics. Clx logics are few and far between, they are particularly nice logics cherry-picked from the lattice of all transitive logics, both weak and strong. In contrast, logics of bounded depth and width form an ideal in : in particular, all extensions of any logic from this class are also in the class. Thus, logics of bounded depth and width may be construed as a toy model that is representative of all well-behaved logics, whereas it may easily happen that there are classes of nice logics whose properties are quite different from clx logics. In fact, as we will see in the sequel, there are reasonable classes of nice non-clx logics, including logics with a single top cluster such as , and some of these have different complexity of unifiability and admissibility from what we will encounter in this paper. These cases duly manifest already for logics of bounded depth and width, and for this reason we cannot give a complete classification of complexity of unifiability in these logics in the present paper.
We acknowledge that this guiding principle should not be taken too seriously: for example, we will also see that there is an interesting class of logics for which unifiability and admissibility are -complete, which cannot happen for logics of bounded depth and width, and likewise for all nontrivial results concerning the setting with constantly many parameters.
The main focus of the paper is on the complexity of admissibility and unifiability, but as a starting point, we also settle the complexity of derivability in clx logics. Generalizing results of Ladner , we show that nonlinear clx logics are -complete. For the lower bound, we prove -hardness for a broad class of transitive logics that includes all logics with the disjunction property, and many other logics such as : similar to our lower bounds on unifiability, the result applies to all transitive logics such that, roughly speaking, all finite trees may be embedded into -frames.
In the planned third part of this series of papers, we will adapt the set-up of clx logics to logics with a single top cluster such as and , including both structural results as in , and computation complexity results as here. Among other things, this will allow us to complete the classification of the complexity of unifiability and admissibility for logics of bounded depth and width, and to obtain a complete classification of hereditary properties of transitive logics that guarantee hardness of unifiability for some complexity class. Both of these suggest that our results on complexity are in a certain sense optimal.
1.1 Overview of results
The linear organization of the paper is as follows. Section 2 contains a few preliminary definitions and facts; its Subsection 2.1 reviews the needed background in complexity theory, including a proof of completeness of certain problems for levels of the exponential hierarchy.
Section 3 deals with the complexity of derivability.
Section 4 is devoted to upper bounds on the complexity of admissibility and unifiability in certain logics: nonlinear and linear clx logics in Subsections 4.1 and 4.2 (respectively), and logics of bounded depth (and, apart from structural results, bounded width) in Subsection 4.3.
Section 5 presents lower bounds on the complexity of unifiability and admissibility. Hardness results for levels of the exponential hierarchy (, , ) appear in Subsection 5.1, except for results in the setting with a constant number of parameters, which are more complicated and are relegated to Subsection 5.2. Hardness results for and levels of the polynomial hierarchy are in Subsection 5.3.
Section 6 concludes the paper.
A summary of results on the complexity of derivability, unifiability, and admissibility in consistent clx logics is given in Table 1. Each entry in the table should be understood so that the stated problem is complete for the complexity class indicated, for every clx logic that meets the description. Notice that for a given logic, unifiability and admissibility usually have dual complexity, because unifiability is a special case of inadmissibility; in order to avoid confusing switching between dual complexity classes, we adopt in this overview the convention to indicate the complexity of unifiability and inadmissibility rather than admissibility. In accordance with this, we also indicate the complexity of nonderivability (or equivalently: local satisfiability) rather than derivability. However, detailed statements of theorems later in the paper will often mention both.
Here is a cross-reference of our complexity results sorted according to complexity classes:
Upper bounds: nonderivability in consistent linear clx logics (Thm. 3.2), unifiability with parameters in logics of bounded depth (Thm. 4.6), inadmissibility with parameters in logics of bounded depth and width (Thm. 4.19), inadmissibility without parameters in consistent linear clx logics [5, Thm. 2.6].
Lower bound: unifiability without parameters in consistent logics (Thm. 5.2).
This paper is a continuation of , and we assume the reader has access to that paper. We generally follow the same terminology and notational conventions as in , which we shall not repeat here, as it would considerably add to the length of the paper. In particular, we assume the reader is familiar with the content of [8, §2], which lays out our basic concepts and terminology. Moreover, we will rely on the definition of clx logics (Def. 4.1) and their main structural properties (§4.3); the definition of tight predecessors and extensible frames (Def. 5.1) along with the ensuing semantical characterization of admissibility in clx logics (Thm. 5.18); and the characterization of admissibility in terms of pseudoextensible models (Def. 5.21, Thm. 5.24).
We stress that unless stated otherwise, admissibility refers to admissibility with arbitrary parameters, and similarly for unification and other related notions.
All logics in this paper are normal modal logics extending , which we will not always state explicitly.
As a piece of a more obscure notation, we recall from  that ; if is a finite set of formulas (in this paper, typically ), then is the set of all assignments , and for any , we put , where , .
If is a natural number (which we take to include ), we will sometimes denote the set as just . (This in fact agrees with the common von Neumann definition of natural numbers.) Given an indexed sequence and , we extend the notation above so that . Finally, we will use this notation also for subsets
, that are identified in this context with their characteristic functions: that is,. Here, is the Iverson bracket: for any predicate ,
The set of all subsets of is denoted .
In accordance with [8, §5.1], we define a logic to have the disjunction property if it admits the rules
for all , or equivalently, for (the admissibility of being equivalent to the consistency of ).
If is a cluster or a point in a model, a cluster type, or an extension condition, let denote its reflexivity: if is irreflexive, and if it is reflexive.
Let and be general frames. Recall from [3, §9.1] that a subreduction of to is a partial map such that
for all , , and . If is a (Kripke) finite frame, condition (i) simplifies to
for all .
A subreduction is cofinal if . A subreduction of onto is is a subreduction whose image is all of . A frame (cofinally) subreduces onto if there exists a (cofinal, resp.) subreduction of onto .
We will often encounter conditions concerning the occurrence of certain patterns as subframes that are oblivious to the reflexivity of individual points. In order to facilitate their formulation, we define a weak subreduction of to to be a partial map that satisfies (i), (ii), and the modified condition
for all and . We define other derived notions such as weak subreductions onto and cofinal weak subreductions similarly as above.
The reflexivization of a frame , denoted , is the frame . The following is an immediate consequence of the definitions:
þ Let and be frames, and a partial map from to .
If is a weak subreduction of to , it is a subreduction of to .
If is a subreduction of to , it is a weak subreduction of to .
If is a logic with the finite model property, and a finite set of parameters, let denote the universal -frame for parameters : it is defined as the upper part of the universal -frame of rank (denoted in [3, §8.7]), endowed with the canonical valuation of parameters from to make it a parametric frame. The double dual of is the canonical frame ([3, Cor. 8.89]); in particular, if is unprovable in , then for some .
Explicitly, we may construct as the union of the following inductively defined chain of finite parametric frames:
We start with .
The frame includes as a generated subframe. Moreover, for every , , and , where if : if has no tight -predecessor in , we include one in .
Then consists of all points such that is an -frame. We stress that if is a rooted subframe with a reflexive root cluster such that all assignments from are realized in , then no tight -predecessor of is added to , as it already has one (viz., a subset of ).
If is a clx logic, we may also describe in our terminology as the minimal locally finite -extensible parametric Kripke frame for the parameters .
It is well known that every point of is definable by a formula (see e.g. [3, Thm. 8.83]). We will need an explicit description of such formulas in the simplest case . Notice that in this case, the universal frame has no proper clusters.
For each , we define a formula by induction on the depth of as
þ Let have fmp, , and .
Proof: We prove (2) by induction on the depth of . The right-to-left implication amounts to , which is straightforward to check.
Assume that . If is irreflexive, the definition of and the induction hypothesis imply . Notice that there is no such that : this would make rooted with a reflexive root , hence no additional tight predecessor would be added to . Thus, , hence is irreflexive, i.e., it is the unique irreflexive tight predecessor of : .
If is reflexive, we have , which implies as above. Assuming for contradiction, let be maximal such that . By maximality, . Since , we have , i.e., there is such that . This again implies , hence or by maximality, i.e., or is reflexive. If , then , i.e., is a tight predecessor of other than itself, but no such tight predecessor was added into . Otherwise and is reflexive, i.e., is a reflexive tight predecessor of other than , which is again impossible.
2.1 Complexity classes
Since the topic of this paper is computational complexity, we will assume some degree of familiarity with basic computation models and complexity classes. We refer the reader to e.g. Arora and Barak  for general background on complexity theory, but for convenience, we review the definitions of classes that appear in this paper.
For any function , let (or for emphasis) denote the class of all languages (where is a finite alphabet) computable by a deterministic (multitape)Turing machine (DTM) in time at most , where is the length of input. If is a family of such functions, such as the family of all polynomially bounded functions , we put . The polynomial time and exponential time classes are then defined as
(This is exponential time with polynomial exponent; exponential classes with linear exponent, such as , will not be used in this paper.) Likewise, is the class of languages computable by a DTM using cells of memory, and
For polynomial-space languages it does not matter, but recall that in general, space usage is defined so that it only accounts for the content of work tapes, excluding the input tape (which is assumed read-only), and—if we are computing a function rather than language membership—excluding the output tape (which is write-only).
In particular, our basic notion of reduction will be many-one logarithmic-space (logspace) reductions. If is a complexity class, a language is -hard if every language is logspace-reducible to , and is -complete if additionally .
Nondeterministic Turing machines (NTM) may have a choice between multiple possible transitions in any configuration; the machine is declared to accept a given input if there exists a run of the machine that ends in an accepting state. The class of languages accepted by a NTM in time is denoted , and we put
An equivalent definition of is that it consists of languages such that membership in can be witnessed by a polynomial-size certificate whose validity can be checked in .
We could also define nondeterministic space classes, but the only example we are interested in is , which equals by Savitch’s theorem.
For any class of languages , denotes the dual class .
The deterministic and nondeterministic time classes above can be relativized: for any language , a Turing machine with oracle may query membership of words in (by writing them on a dedicated oracle query tape) at unit time cost. Then for any class , denotes the set of languages computable in polynomial-time by a NTM with oracle , and similarly for other classes. The polynomial and exponential hierarchies are defined by , and for ,
(Notice that an machine may supply exponentially long queries to the oracle, hence the oracle in the definition of should be thought of actually having the power of .) In particular, , and .
Many of the classes above can be equivalently defined using alternating Turing machines (ATM), which is a view we will favour especially when proving upper bounds (Sections 3–4). An ATM is similar to a NTM in that multiple possible transitions may be defined for any given state. However, the definition of acceptance is different. Each non-final state of an ATM is labelled as either existential or universal, and we define inductively the set of accepting configurations of the machine as the smallest set satisfying the following conditions:
A configuration in an accepting final state111In fact, we could dispense with final states altogether: an accepting (rejecting) final state is equivalent to a universal (existential, resp.) state with no possible transitions out. is accepting.
A configuration in an existential state is accepting if there exists a transition to an accepting configuration.
A configuration in a universal state is accepting if all transitions lead to accepting configurations.
(States with exactly one possible transition can be thought of as deterministic; it makes no difference whether they are labelled as existential or universal.)
If denotes the class of languages computable by an ATM in polynomial time, we have
(On a related note, , but we will not need this.)
We are especially interested in classes with bounded alternation. Here, an ATM is said to make an alternation when it transitions from an existential state to a universal state, or vice versa. For any and , denotes the class of languages computable in time by an ATM that starts in an existential state, and then makes at most alternations. The class is defined similarly, but starting from a universal state. In particular, , and . We have the following characterization for any :
We will also need convenient complete languages for our classes. The set TAUT of tautologies of is the canonical -complete language, and the dual language SAT of satisfiable classical propositional formulas is -complete. The standard -complete language222It is more transparent to think of it as an -complete language: the existential and universal quantifiers almost directly correspond to existential and universal states of an ATM. The usual textbook proof of -completeness of QSAT is for the most part actually a proof of the equality (4). is QSAT: the language of all true quantified Boolean sentences
where is a propositional formula, and each quantifies over a truth value .
Let . A quantified Boolean sentence (5) is in if the quantifier prefix may be written as at most alternating blocks, the first block consisting of existential quantifiers, the second of universal quantifiers, and so on; is defined dually (i.e., starting with a universal block). Then the language consisting of all true sentences is -complete, and the dual language is -complete. Notice that is just a notational variant of SAT: a propositional formula is satisfiable iff the corresponding existentially quantified sentence is true.
Finally, we need complete languages for (in particular, for ). Recall that in descriptive complexity, we encode words by models with domain endowed with the order relation (and possibly other arithmetical predicates, which we will not need here), and unary predicates for each symbol , such that
By Fagin’s theorem, a language is in iff there is a (i.e., existential second-order) sentence such that
More generally, languages are exactly those that are -definable. This correspondence can be generalized to the exponential hierarchy, using third-order sentences: a language is in iff it is -definable. (See Kołodziejczyk [9, Prop. 2.6], which also includes a brief historical discussion. A similar statement in Hella and Turull-Torres [4, Thm. 7] suffers from an off-by-one error.)
Since we only need complete problems rather than exact descriptions of the languages, we may simplify the sentences to a convenient form. This was already done in [5, L. 3.1] for the special case (i.e., ); here we generalize it to higher levels of the exponential hierarchy (with a more detailed proof).
þ Let . Put for odd, and for even, and let be its dual. Then the set of true sentences of the form
is a -complete language, where is given in unary, and is a Boolean combination of atomic formulas of the form or for , , and .
Proof: For ease of notation, we will assume that is odd, so that . The case of even is dual. We will denote third-order variables by capital letters (with indices etc.), second-order variables by lower-case letters , and first-order variables with Greek letters .
First, any language reduces to a language
by a simple padding argument: if, then is computable in , and is a logspace reduction of to . Thus, let us fix a -complete language .
where is a second-order formula, and all the second-order and third-order variables are unary: i.e., for a model with domain , the second-order variables range over , and third-order variables over . (Second-order variables will remain unary for the rest of the proof, but we will introduce third-order variables of higher arity during subsequent manipulations.) We may assume only uses for first-order objects.
We may write in prenex normal form, and moreover, we may assume that all second-order quantifiers precede all first-order quantifiers: this is easily accomplished by exploiting the equivalences
where stands for ; similarly for universal quantifiers. Thus, we may write
where is first-order. Next, we get rid of existential second-order quantifiers by introducing Skolem functions: is equivalent to
where each is a third-order relation variable with second-order and one first-order argument (i.e., it ranges over .) Eliminating the comprehension symbols, is equivalent to
where denotes the first-order formula
By increasing or adding dummy quantifiers if necessary, we may assume that all the tuples also have length .
In order to simplify each third-order quantifier block to a single unary variable, we pass from to a larger model , defined as follows. The domain of is (i.e., disjoint copies of the domain of ). For each , includes a unary predicate that selects the th copy of (i.e., is satisfied by elements of ), and a binary predicate that defines the equivalence relation
(i.e., the kernel of the projection ). The original relations of are included on the th copy . Clearly, the mapping is still logspace computable. We will translate to a formula such that
We may represent elements by elements of that satisfy , and subsets by subsets of ; however, we will actually need to quantify over the copies of in each as well. We represent an -tuple of third-order objects by a single third-order object defined as
Moreover, if we are in addition to given a tuple , where , we represent together by
With this representation in mind, will have the form
where each is an -tuple , the formula is a conjunction of (10) and (11) below, and is constructed from as follows. We replace first-order quantifiers and by and , respectively. Atomic subformulas of that only mention first-order objects are left intact, atomic formulas are replaced with , and atomic formulas are replaced with .
The first conjunct of ,
ensures that the sets are correctly formed: i.e., , and for fixed , all the are copies of the same set . The second conjunct is a translation of (8), which can be written as
Expanding the definition of , we arrive at
By construction, (9) holds. Notice that , specifically (11), contains a second-order quantifier. When we bring to prenex normal form, we obtain a sentence of the form (dropping the decoration from variables)
where is first-order. (This is still a single, constant-size sentence that only depends on , not on .)
As a final step, we transform for any given word to a sentence that embeds the structure of as follows. Using constants for elements (i.e., ), we expand each first-order quantifier to a disjunction , and to . Then we evaluate each atomic formula that does not involve higher-order variables, and replace it with or according to its truth-value.
The resulting formula has size polynomial in (the exponent being roughly the nesting depth of first-order quantifiers in ), and it is easy to see that it is logspace computable. It has the form (7), and we have
Before we embark on our main quest for the complexity of admissibility and unifiability in clx logics, let us first settle a more basic question: what is the complexity of tautologicity or derivability in these logics.
þ Notice that (single-conclusion) derivability has the same complexity as tautologicity for any transitive logic , as
gives a logspace reduction.
Recall that any clx logic is -definable on finite frames [8, Thm. 4.29], and as a consequence, finite -frames are recognizable in polynomial time (in fact, in , a subclass of uniform ).
þ For any consistent linear clx logic , derivability in is -complete.
Proof: Since is a conservative extension of classical propositional logic, is -hard.
On the other hand, if is a formula of size such that , then there exists an -model of depth and cluster size at most , hence size at most , by [8, Thm. 4.38]. (In fact, it is easy to show that size is enough.) Since finite -frames are polynomial-time recognizable by þ3.1, this shows that is in .
The complexity of nonlinear clx logics is a bit more difficult to establish (although it is just a variant of standard -completeness results for modal logics starting with Ladner ). For this reason, we state the upper and lower bounds separately. We begin with the former.
Recall from [8, Def. 4.20] that is a finite set of extension conditions that determines the shape of -frames.
þ For any clx logic , -derivability is in .
Proof: Recall from [8, Thm. 4.38] that if , then is falsified in a rooted -model which is a tree of depth of clusters of size . Similarly to the standard case of and other common logics, we may search for such a tree in polynomial space—or as we prefer to think about it, in alternating polynomial time—by exploring one branch at a time.
In more detail, let be a formula whose provability in we want to determine. Since variables and parameters work the same way with respect to derivability, we may assume contains no parameters. Put , , , and . For any and , let denote the Boolean assignment to modal formulas that agrees with on variables, and that makes true for , and false for .
We will describe a recursive algorithm that computes the predicate for , defined as
We may then express (un)provability of as
We will also consider auxiliary predicates and for , where . They are defined as follows:
for and .
þ For any ,
Proof: Left-to-right: Fix a finite rooted model such that
holds in , and such that it does not hold in any point strictly above the root cluster. Assume that is of type , and fix such that . Put
We claim that : if , let be the assignment of variables in the (unique) root , i.e., . Since , we have , and it is readily seen that
thus witnesses that holds. If is reflexive, then , thus for any , , where . If , fix a (not necessarily injective) enumeration ; then is witnessed by