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[MATH] bits appears at position [MATH] We introduce as well [MATH] to pass [MATH] through the link. We also have [MATH] for all [MATH] |
to verify the equality of bit positions through links. As in the previous case, we wind up the alternation with the following transitions |
[EQUATION] for all [MATH] Notice again there are only a polynomial number of such transitions. Here, [MATH] appears on the right to verify the property at the next grid position (hence we verify all positions). We check the equality of the first [MATH] bits since these encode the column index which has to be equal. The... |
Next, we take one more step to guess the values of the bit positions being tested for equality. Thus we have transitions [EQUATION] |
for all [MATH] where [MATH] Note, the states like [MATH] are used to pass the equals check through the link as will be seen below. At the same time we fire |
[EQUATION] where [MATH] Now we do the checking. To do the bit equality checks we have similar transitions to previously. That is for all |
[MATH] [EQUATION] and to pass the checks through to the linked stacks we have [EQUATION] and once the states have been passed through the links we have |
[EQUATION] Next, to check the increment we need for each [MATH] [EQUATION] and once the state has been passed through the link we have |
[EQUATION] where the order- [MATH] states are described after we complete our description of the order- [MATH] transitions. To complete the order- [MATH] description we need a way to handle the final row of the grid. For this we use a state |
[MATH] which asserts the part of the binary number encoding the row consists of only [MATH] characters. We give its transitions below. Thus, the above states can avoid checking a stack by asserting it appears on the final row. That is |
[EQUATION] for all [MATH] It remains to check several properties the above transitions need to assert at order- [MATH] We have the states |
[MATH] for all [MATH] These states check the rightmost [MATH] appears at position [MATH] in the encoding of the row number. We first skip the leading spacer with |
[EQUATION] Then we have, for all [MATH] [EQUATION] and for all [MATH] [EQUATION] and for all [MATH] [EQUATION] Similarly, we check the rightmost [MATH] bit with |
[EQUATION] and for all [MATH] [EQUATION] It only remains to define the transitions from [MATH] For this we introduce several intermediate states |
[MATH] and the transitions [EQUATION] 6. Next we check that the first tile is [MATH] We will first introduce some order- [MATH] states for checking the tile in a stack. That is, for each |
[MATH] we have [MATH] from which we have the transitions [EQUATION] where [MATH] ranges over [MATH] Then to check the first tile we use |
[EQUATION] with the states [MATH] These transitions simply skip the leading spacer stacks and check the first stack encoding a cell holds [MATH] |
7. To check the final tile is [MATH] we use [EQUATION] which simply iterate to the final order- [MATH] stack and verify it contains |
[MATH] 8. Next we need to define transitions that check the horizonal tiling relation. We will begin with a transition reading the first spacer |
[EQUATION] where [MATH] Next we will guess the pair [MATH] that the following two order- [MATH] stacks will contain. For this we will need order- [MATH] states |
[MATH] for each [MATH] from which we verify the following two order- [MATH] stacks contain [MATH] and [MATH] respectively. In other words, the horizontal tiling relation is satisfied. The transitions we have are |
[EQUATION] where [MATH] appears on the right hand side to assert the horizontal tiling relation over subsequent pairs of order- [MATH] stacks. From each |
[MATH] we have [EQUATION] where [MATH] are intermediate states for each required [MATH] Since the horizontal tiling does not apply to the final tile in each row, we have transitions allowing the condition to be relaxed here. That is, we have, for each |
[MATH] [EQUATION] The transitions from [MATH] simply keep passing the state verifying the tiling relation along, or terminates if the bottommost stack is read. From |
[MATH] we simply dismiss the requirement if the stack is the final tile in a row. Note, we do not have similar transitions from [MATH] |
since these states represent the relation between the next tile (stack) and the previous one, and thus need to be asserted at the end of a row too. |
The state [MATH] above is an order- [MATH] state from which we verify the column index is [MATH] To implement this state we need several intermediate states |
[MATH] The first [MATH] of these states allow any binary digit, while the final [MATH] , which read the bits encoding the column index, only allow [MATH] |
digits to occur. That is, [EQUATION] 9. Finally we need to define transitions that check the vertical tiling relation. As before, we will begin with a transition reading the first spacer |
[EQUATION] where [MATH] Next we will guess the pair [MATH] that the next order- [MATH] stack and its linked-to stack respectively will contain. For this we will need order- [MATH] states |
[MATH] for each [MATH] from which we verify the next order- [MATH] stack contain [MATH] and the order- [MATH] stack linked to from the [MATH] in this stack contains |
[MATH] The transitions we have are [EQUATION] where [MATH] appears on the right hand side to assert the vertical tiling relation over subsequent pairs of order- [MATH] stacks. From each |
[MATH] and [MATH] we have [EQUATION] where [MATH] are intermediate states for each required [MATH] Recall [MATH] will pass [MATH] through the link on [MATH] |
Since the vertical tiling does not apply to the final tile in each column, we have transitions allowing the condition to be relaxed here. That is, we have, for each |
[MATH] [EQUATION] Recall [MATH] verifies the row index is [MATH] Note, as soon as the automaton reaches the first order- [MATH] stack containing a row index of [MATH] , then all subsequent stacks will contain the same row index, hence the transition to [MATH] allowing the condition to be disbanded. From |
[MATH] we simply dismiss the requirement if the stack is the final tile in a column. Note, we do not have similar transitions from |
[MATH] since these states represent the relation between the next tile (stack) and the previous one via the collapse link, and thus need to be asserted at the end of a row too. |
0.4 Conclusion We have shown that the problem of testing emptiness of stack automata is NEXPTIME-complete for collapsible pushdown stacks. In the case of annotated stacks where stack characters are augmented with further annotated stacks (instead of a link to a position elsewhere in the stack), our proof does not carry... |
# Source: arxiv 1805.11891 # Title: On the semilattice of modal operators and decompositions of the discriminator # Sections: all # Downloaded: 2026-03-03T02:36:05.668163+00:00 |
On the semilattice of modal operators and decompositions of the discriminator Abstract We investigate the join semilattice of modal operators on a Boolean algebra [MATH] . Furthermore, we consider pairs [MATH] of modal operators whose supremum is the unary discriminator on [MATH] , and study the associated bi–modal alg... |
Keywords: Algebraic logic, modal operators, unary discriminator, join semilattice, dual pseudocomplementation Dedicated to our friends Hajnal and Istvan with respect and gratitude for long lasting inspiration |
Introduction Boolean algebras with operators were introduced by Jónsson and Tarski in connection with their investigations into relation algebras. It was observed much later that the simplest case of modal algebras, that is, expansions of Boolean algebras with a single unary normal operation which preserves finite join... |
In this paper, we study the lattice theoretic properties of [MATH] , in particular, the existence and form of dual pseudocomplements. If [MATH] is a complete Boolean algebra, the problems concerning [MATH] are solved. In particular, [MATH] is dually pseudocomplemented if and only if [MATH] is complete. |
Generalizing the notion of pseudocomplement, we consider pairs [MATH] of modal operators ( companions ) whose join in [MATH] is the discriminator [MATH] . Such pairs lead to bimodal algebras [MATH] which we call discriminator decomposition algebras . These algebras give rise to a study of bimodal logics in which the mo... |
We observe that the equational class [MATH] which generalizes both [MATH] and its complementary counterpart [MATH] In the final part of the paper we address the question when [MATH] has a proper companion, i.e. a companion [MATH] with [MATH] , and give answers for several classes of modal algebras. It turns out that th... |
Notation and first definitions We regard an ordinal as the set of its predecessors, and cardinals as initial ordinals; [MATH] is the first infinite ordinal |
. As in our context no generality is lost, we shall tacitly assume that a class [MATH] of algebras is closed under isomorphic copies. If no confusion can arise, we will refer to an algebra simply by its base set. The equational class [MATH] is denoted by [MATH] |
The ternary discriminator function on an algebra [MATH] is a function [MATH] defined by [EQUATION] A ternary term [MATH] which represents the discriminator function is called a ternary discriminator for [MATH] . If [MATH] is a class of algebras with a common discriminator term, then [MATH] is called a discriminator var... |
or If [MATH] is a Boolean algebra, this can be simplified: A mapping [MATH] is a unary discriminator function , if [EQUATION] A unary term [MATH] which represents the unary discriminator function on [MATH] is called a unary discriminator for [MATH] . The next observation is well known: |
Lemma 1 A Boolean algebra [MATH] has a unary discriminator if and only if it has a ternary discriminator. frame is a pair [MATH] where [MATH] is a set and [MATH] a binary relation on [MATH] . The identity relation on [MATH] is denoted by [MATH] , or just by [MATH] if [MATH] is understood; [MATH] (or just [MATH] ) denot... |
Suppose that [MATH] is a partially ordered set and [MATH] . Then [MATH] is the downset [MATH] . If [MATH] , we just write [MATH] if no confusion can arise. If [MATH] has a smallest element [MATH] , then [MATH] , otherwise, [MATH] [MATH] is called dense (in [MATH] if for every [MATH] there is some [MATH] such that [MATH... |
In the sequel, a semilattice is assumed to be a join semilattice. Suppose that [MATH] is an upwardly bounded semilattice. A dual annihilator of [MATH] is some [MATH] such that [MATH] . Such [MATH] is proper if [MATH] [MATH] is called dually dense if its only dual annihilator is [MATH] . If [MATH] has a smallest annihil... |
Lemma 2 , Theorem 1] If [MATH] is a dually pseudocomplemented semilattice, then the set [MATH] of open elements of [MATH] is a [MATH] – subsemilattice of [MATH] and a Boolean algebra with [MATH] |
2.1 Boolean algebras Throughout, [MATH] is a nontrivial Boolean algebra (BA), usually only referred to by its universe [MATH] [MATH] is the BA with universe [MATH] , and [MATH] is the completion of [MATH] . The set of atoms of [MATH] is denoted by [MATH] , and the set of ultrafilters of [MATH] is denoted by [MATH] . Th... |
We write [MATH] , if [MATH] , and the [MATH] are nonzero and pairwise disjoint. The symmetric difference [MATH] of [MATH] is denoted by [MATH] |
[MATH] is called a finite–cofinite algebra (FC–algebra), if every element of [MATH] is a finite sum of atoms or the complement of such an element. If [MATH] is an FC–algebra, [MATH] a cardinal, and [MATH] , then [MATH] is isomorphic to the BA [MATH] which is [MATH] . If [MATH] , we let [MATH] be the ultrafilter of [MAT... |
[MATH] , and [MATH] be the ultrafilter of cofinite sets. If [MATH] is dense in [MATH] and [MATH] , then [MATH] . Moreover, there is a pairwise disjoint family [MATH] such that [MATH] |
12 , Lemma 4.9.] Recall some facts about Boolean interval algebras: Let [MATH] be a linear order with smallest element [MATH] . Suppose that [MATH] is a symbol not in [MATH] , and set [MATH] with [MATH] for all [MATH] . An interval of [MATH] is a set of the form [MATH] , where [MATH] [MATH] is the collection of all fin... |
[EQUATION] together with the empty set. It is well known that [MATH] is a Boolean algebra 12 , p.10] , called the interval algebra of [MATH] Each nonzero [MATH] can be written in the form ( 2.2 ) in such a way that [MATH] [MATH] ; note that the intervals [MATH] are pairwise disjoint. The representation of [MATH] in thi... |
[EQUATION] be the set of relevant intervals of [MATH] For unexplained notation and concepts in the area of universal algebra the reader is invited to consult |
, and for Boolean algebras we refer the reader to Modal algebras An operator on [MATH] is a mapping [MATH] ; note that this is more general than the terminology of |
. If [MATH] is an operator on [MATH] , then its dual (operator) [MATH] is defined by [MATH] . We also set [MATH] , and [MATH] ; clearly, [MATH] |
A mapping [MATH] [MATH] is called normal if [MATH] , and additive if [MATH] for all [MATH] . A modal operator [MATH] on [MATH] is a normal and additive mapping. In this case, [MATH] is called a modal algebra . The class of modal algebras is denoted by [MATH] . It may be remarked that [MATH] is not locally finite, indee... |
A modal operator [MATH] on [MATH] is called completely additive , or simply complete , if for every [MATH] such that [MATH] exists, [MATH] also exists and is equal to [MATH] |
An ideal [MATH] of [MATH] is called trivial , if [MATH] proper if [MATH] , and closed , if [MATH] implies [MATH] . It is well known that there is a one–one correspondence between closed ideals and congruences on [MATH] , see e.g. , §3] , and therefore, we will call closed ideals also congruence ideals |
A modal operator [MATH] is called a closure operator or S4 operator , if it satisfies Cl [MATH] Cl [MATH] In this case, [MATH] is called a closure algebra or an S4 algebra |
If [MATH] is a frame, we define a mapping [MATH] by [MATH] . In modal logic, [MATH] is the (interpretation of) the diamond operator |
[MATH] in the frame [MATH] . The structure [MATH] is called the complex algebra of [MATH] , denoted by [MATH] Theorem 3.1 11 , Theorem 3.9] |
1. [MATH] is complete and atomic, and [MATH] is a completely additive normal operator. 2. If [MATH] is a complete atomic Boolean algebra and [MATH] is completely additive, then [MATH] is isomorphic to some [MATH] . ∎ |
The canonical structure of a modal algebra [MATH] is the frame [MATH] , where [EQUATION] [MATH] is called the canonical relation of [MATH] . If [MATH] , and [MATH] are the principal ultrafilters [MATH] , respectively, [MATH] , then |
[EQUATION] Observe that [MATH] is the universal relation on [MATH] if and only if [MATH] , and [MATH] is the empty relation if and only if [MATH] |
The following representation theorem is seminal for algebraic semantics of modal logics:. Theorem 3.2 11 , Theorem 3.10] Let [MATH] , and [MATH] be defined by [MATH] . Then, [MATH] is a MOA embedding into [MATH] ; in particular, [MATH] |
The algebra [MATH] is called the canonical extension of [MATH] , denoted by [MATH] . It is well known that [MATH] if and only if [MATH] is finite. We will sometimes assume that w.l.o.g. [MATH] is a subalgebra of [MATH] . In this case, [MATH] is denoted by [MATH] . We shall usually just write [MATH] instead of [MATH] |
For a history of and an introduction to Boolean algebras with operators, the reader is invited to consult We shall use several axioms of modal logics in their algebraic form and the corresponding property of their canonical relation: |
K. [MATH] and [MATH] [MATH] is a binary relation. T. [MATH] [MATH] is reflexive. 4. [MATH] [MATH] is transitive. B. [MATH] [MATH] is symmetric. |
Conjunctions of axioms are usually written in juxtaposition of the axioms; for example, KT means a modal logic based on the axioms and . Some abbreviations are common: |
KT4 [MATH] S4, KT4B [MATH] S5. The next result provides a convenient criterion for a modal algebra to be subdirectly irreducible: |
Theorem 3.3 (Rautenberg’s criterion 14 , p. 155] Let [MATH] be a modal algebra. Then, [MATH] is subdirectly irreducible if and only if |
[EQUATION] If [MATH] is a K4 algebra, then ( 3.3 ) is equivalent to the statement [EQUATION] and if [MATH] is an S4 algebra, i.e. if [MATH] is a closure operator, then ( 3.3 ) is equivalent to |
[EQUATION] The semilattice of modal operators Let [MATH] be the set of all modal operators on a fixed Boolean algebra [MATH] . For [MATH] set [MATH] . Furthermore, let [MATH] , and |
[EQUATION] Note that [MATH] is the unary discriminator on [MATH] . In modal logics [MATH] is known as the universal modality The following observation which, is straightforward to prove, is the basis for the considerations in this section: |
Theorem 4.1 [MATH] is a bounded (join) semilattice. We remark in passing that the structure [MATH] is a bounded idempotent semiring where [MATH] is composition of functions and [MATH] is the identity. |
For each [MATH] define the relativization of [MATH] to [MATH] by [EQUATION] Clearly, each [MATH] is a modal operator on [MATH] Theorem 4.2 |
[MATH] is a complete semilattice if and only if [MATH] is complete. Proof [MATH] ”: Suppose that [MATH] . Let [MATH] , and choose some [MATH] . Our aim is to show that [MATH] is the least upper bound of [MATH] . Since [MATH] and [MATH] , it is clear that [MATH] is an upper bound of [MATH] . Next, let [MATH] be an upper... |
[MATH] ”: For [MATH] set [MATH] for each [MATH] . In this case it is in fact a complete lattice: If [MATH] , then [MATH] is a lower bound of [MATH] , and thus, [MATH] is well defined, and clearly, it is the greatest lower bound of [MATH] |
Our next topic in this section is the existence of dual pseudocomplements in [MATH] . We start with a characterization of pseudocomplements in [MATH] |
Lemma 3 Let [MATH] . If [MATH] has a dual pseudocomplement [MATH] , then [MATH] for each [MATH] . Conversely, for each [MATH] , if [MATH] exists, then [MATH] |
Proof Let [MATH] be the completion of [MATH] . If [MATH] , then [MATH] denotes the supremum of [MATH] in [MATH] . Since [MATH] is a dense subalgebra of [MATH] , all sums existing in [MATH] coincide with the sums in [MATH] ; in particular, if [MATH] , then [MATH] |
[MATH] ”: Suppose that [MATH] is a dual pseudocomplement of [MATH] , and let [MATH] [MATH] . Then, [MATH] implies [MATH] . Since [MATH] , we have [MATH] , and thus, [MATH] . It follows that [MATH] is an upper bound of [MATH] |
Assume that [MATH] for some [MATH] . Choose some [MATH] such that [EQUATION] and define [MATH] by [MATH] , and [EQUATION] Then, [MATH] , and [MATH] , if [MATH] . If [MATH] , then |
[EQUATION] Since [MATH] is the dual pseudocomplement of [MATH] , we have [MATH] , in particular, [MATH] . This contradicts [MATH] . Therefore, [MATH] exists and is equal to [MATH] for each [MATH] |
[MATH] ”: Set [MATH] , and [MATH] . Let [MATH] . If [MATH] , then [MATH] , and thus, [MATH] . It follows that [MATH] is isotone, and [MATH] |
Conversely, [EQUATION] Thus, [MATH] . Furthermore, [EQUATION] and therefore, [MATH] Next, suppose that [MATH] for some [MATH] . Let [MATH] and assume that [MATH] , i.e. [MATH] . Then, |
[EQUATION] Thus, there is some [MATH] such that [MATH] , i.e. [MATH] . This contradicts our hypothesis [MATH] , and it follows that [MATH] is the dual pseudocomplement of [MATH] |
Theorem 4.3 [MATH] is dually pseudocomplemented if and only if [MATH] is complete. Proof [MATH] ”: Suppose that [MATH] is dually pseudocomplemented, and let [MATH] . Let [MATH] be the filter of [MATH] |
[MATH] . Then, [MATH] : Since [MATH] , any lower bound of [MATH] is a lower bound of [MATH] . Conversely, let [MATH] be a lower bound of [MATH] and [MATH] . Then, there are [MATH] such that [MATH] , and therefore, [MATH] . Thus, we may assume that [MATH] is a filter of [MATH] . Let [MATH] be the identity. Then, by Lemm... |
[EQUATION] Hence, [MATH] exists and thus, [MATH] is complete. [MATH] ”: If [MATH] is complete, then [MATH] exists for each [MATH] , and the mapping defined by |
[EQUATION] is the dual pseudocomplement of [MATH] by Lemma In particular, [MATH] is dually pseudocomplemented, if [MATH] is finite. |
Let [MATH] be the set of completely additive normal operators of [MATH] . It is obvious that [MATH] is a sub–semilattice of [MATH] . If [MATH] is complete, we can say more: |
Theorem 4.4 Suppose that [MATH] is complete. 1. [MATH] is complete. 2. [MATH] for some [MATH] if and only if [MATH] is completely additive. |
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