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If we know an operation has these two properties, this suggests a way to maximize the state complexity of the operation: keep adding new letters to the input DFAs, until the point where we have letters corresponding to all possible actions across all the input DFAs. The first property ensures that adding letters can on... |
Figure shows the result of applying this construction to a two-state DFA with unspecified initial and final states, to be used as input for a unary operation. The DFA has one letter for each of the four functions from [MATH] to itself. To illustrate the construction for a three-state DFA, we would need [MATH] letters! |
Since the state complexity of the output is not affected only by the actions of letters in the inputs, but also the initial states and final state sets of the inputs, we construct these witnesses for each configuration of initial and final states. Then for each configuration, it remains to solve the combinatorial probl... |
Of course, these combinatorial problems can sometimes be quite hard, and this method produces witnesses over extremely large alphabets. Nonetheless, we believe the OLPA approach is useful to know for several reasons. For one, it gives a way to compute the exact worst-case state complexity of certain operations for inpu... |
Second, using “OLPA witnesses” with one letter for each action of the input DFAs can simplify proofs and make the ideas behind them more clear. State complexity proofs using witnesses over optimal or near-optimal alphabets are often rather technical. Consider the following proof that for [MATH] , the state complexity o... |
Let [MATH] be a DFA with [MATH] states. Suppose for each function [MATH] , there is a letter [MATH] such that [MATH] Let [MATH] be an NFA for the reverse of [MATH] , where [MATH] For each [MATH] , let [MATH] be a function that maps [MATH] into [MATH] and [MATH] into [MATH] Then [MATH] , so all [MATH] subsets of [MATH] ... |
Because we are free to choose letters that do exactly what we want, this proof is very simple. It is also rather illuminating, since we can immediately extract a sufficient condition for attaining the worst-case state complexity from the proof: it suffices that there are letters which induce, for each [MATH] , a functi... |
that the state complexity is maximized if the transition monoid of [MATH] contains all functions from [MATH] to itself (in addition to strengthening the aforementioned sufficient condition from letters to words). |
For contrast, consider the proofs given by Jirásková and Šebej that the worst-case state complexity can be attained over ternary and binary alphabets. The ternary proof is about as short as the proof above, but somewhat more terse, asking the reader to compute transitions under the word [MATH] . It also does not offer ... |
Our third reason for studying the OLPA approach is that we believe it could lead to a better general understanding of state complexity of operations, and the conditions that lead to maximal blow-ups in complexity. The fact that this approach exists and applies to many of the operations studied in the state complexity l... |
, which have small alphabets and maximize the complexity of several operations simultaneously, be explained in this framework? We will also see there are operations for which the OLPA approach does not work. Our main example is the operation [MATH] , which belongs to a general class of operations called proportional re... |
If the OLPA approach fails for an operation, what does this imply about the nature and behaviour of this operation? Are the state complexity problems for these operations “harder” to solve in general? Studying operations for which the OLPA approach fails could be a fruitiful line of research. |
The purpose of this paper is to initiate a formal and general study of the OLPA approach. We define a class of regular operations, called “uniform operations”, for which the OLPA approach provably works. The class of uniform operations includes common operations such as reversal, star, power, concatenation, and all boo... |
and shuffles on trajectories The class is also closed under composition, and thus includes combined operations like “star-complement-star” |
We prove that for uniform operations, the worst-case state complexity can be determined by considering just a finite set of witnesses. For an [MATH] -ary operation, if the state complexity of the [MATH] -th input is at most [MATH] , then the worst-case state complexity of the output can be determined using [MATH] diffe... |
In the main sections of this paper, we will first state our definitions and prove our results in the special case of unary operations, before moving on to the general case. While this is ultimately redundant, focusing on the unary case simplifies the notation and makes the definitions and results easier to digest. The ... |
To close the introduction, we give a history of the ideas behind the OLPA approach. As mentioned, the key insight dates back to a 1978 paper of Sakoda and Sipser |
They constructed languages wherein the alphabet letters were directed graphs representing behaviours of non-deterministic finite automata and two-way deterministic finite automata, with one letter for each possible behaviour. They used these languages to prove results on the complexity of conversions between different ... |
Perhaps the closest ancestor of our work is a 1990 paper of Ravikumar , who treated the “Sakoda-Sipser technique” as a “systematic method to prove lower bounds on the size complexity of finite automata”, and applied it to five different problems, two of which were operational state complexity problems! This is the firs... |
In 1992, building off Ravikumar’s work, Birget used this “unrefined” version of the OLPA approach to prove lower bounds on the state complexity of intersection and union. |
In 1994, Yu, Zhuang and Salomaa published their seminal paper on the state complexities of basic operations. Notably, even though Yu, Zhuang and Salomaa cited Ravikumar’s work, they did not use or mention the “Sakoda-Sipser technique” anywhere in their paper, instead using various ad-hoc methods to prove lower bounds. ... |
, who credited Sakoda and Sipser for the idea, and applied it to nondeterministic finite automata and automata on infinite words ( [MATH] -automata). Yan’s paper is frequently cited in the field of [MATH] -automata, so the idea seems to have gained some currency there. |
In the field of deterministic state complexity, the OLPA approach has made occasional past appearances. Jirásková and Okhotin and later Domaratzki and Okhotin |
used the OLPA approach to compute the exact worst-case complexity of the cyclic shift and power operations for small values. Brzozowski, Jirásková, Liu, Rajasekaran, and Szykuła |
used the OLPA approach to obtain reachability results for the state complexity of shuffle. Cho, Han, Ko and Salomaa used an OLPA-like construction to establish lower bounds on the state complexity of some “inversion” operations. Interestingly, their construction includes unnecessary extra letters; perhaps these letters... |
Outside the context of descriptional complexity, in 2002, Domaratzki, Kisman and Shallit used OLPA automata to enumerate the languages accepted by [MATH] -state automata. |
In 2018, Caron, Hamel-De le court, Luque and Patrou independently obtained many of the results in this paper using a different formalism. OLPA witnesses are called “monsters” in their work, and uniform operations are called “depictable operations”. Their paper was submitted to arXiv just ten days after the first versio... |
Preliminaries 2.1 Relations and Functions binary relation [MATH] between [MATH] and [MATH] is a subset of [MATH] . If [MATH] and [MATH] , the composition of [MATH] and [MATH] is the relation |
[EQUATION] For [MATH] and [MATH] , the image of [MATH] under [MATH] is the set [MATH] For [MATH] we define [MATH] The converse of a binary relation [MATH] is the relation [MATH] The set [MATH] is called the preimage of [MATH] under [MATH] |
function [MATH] is a binary relation [MATH] such that [MATH] for all [MATH] . Following our notation for binary relations, we write functions to the right of their arguments. Composition of functions is defined by composing the corresponding relations. Thus the order of composition is left-to-right ; in a composition [... |
2.2 Languages and Automata finite automaton (FA) is a tuple [MATH] where [MATH] is a finite set of states [MATH] is a finite set of letters called an alphabet [MATH] is a set of transitions [MATH] is a set of initial states , and [MATH] is a set of final states The triple [MATH] is called the state configuration of the... |
We now define a binary relation [MATH] for each [MATH] Define [MATH] ; in terms of maps, this is the identity map on [MATH] For [MATH] , define [MATH] For [MATH] with [MATH] , define [MATH] The relation [MATH] is called the relation induced by [MATH] or the action of [MATH] If [MATH] is a transformation of the state se... |
If [MATH] is a finite automaton such that [MATH] and [MATH] is a function for each [MATH] , we say [MATH] is deterministic We abbreviate “deterministic finite automaton” to DFA. As a result of this definition, that all DFAs we consider in this paper are complete DFAs (that is, they have exactly one transition defined f... |
Let [MATH] be an FA. A word [MATH] is accepted by [MATH] if we have [MATH] . If [MATH] is a DFA with [MATH] , this condition becomes [MATH] . The language of [MATH] , denoted [MATH] , is the set of all words it accepts. If [MATH] is the language of [MATH] , we also say that [MATH] accepts [MATH] and that [MATH] is an F... |
regular operation of arity [MATH] is a function that takes [MATH] regular languages as input and produces a regular language. DFA operation of arity [MATH] is a function that takes [MATH] DFAs as input and produces a DFA. We say a regular operation [MATH] is equivalent to a DFA operation [MATH] if both operations have ... |
In this paper we consider only DFA operations [MATH] that are alphabet-preserving in the following sense: An input [MATH] is only valid if all the DFAs have the same alphabet. |
If [MATH] is the common alphabet of the input DFAs, then [MATH] will be the alphabet of the output DFA. Furthermore, we consider only regular operations that are equivalent to an alphabet-preserving DFA operation. |
2.3 State Complexity A DFA for a regular language [MATH] is minimal if it has the minimal number of states amongst all DFAs that accept [MATH] The state complexity of a regular language is the number of states in a minimal DFA accepting the language. The state complexity of [MATH] is denoted [MATH] |
The notion of state complexity extends to regular operations. Let [MATH] be a unary regular operation. The state complexity of the operation [MATH] is the following function which takes a positive integer as input: |
[EQUATION] That is, the state complexity of [MATH] is the worst-case state complexity of the output [MATH] , expressed as a function of the maximal allowed state complexity of the input [MATH] Note that [MATH] might not exist for all [MATH] ; in such cases, the output of the function is [MATH] |
This idea generalizes to operations of higher arity. Let [MATH] be an [MATH] -ary regular operation. The state complexity of [MATH] is the following function which takes an [MATH] -tuple of positive integers as input: |
[EQUATION] The output is either a positive integer, or [MATH] if the maximum does not exist. 2.4 Morphisms Let [MATH] and [MATH] be alphabets. morphism is a function [MATH] such that [MATH] ; in other words, a morphism is just a monoid homomorphism between two free monoids. To define a morphism [MATH] , it is sufficien... |
If [MATH] is regular, then [MATH] is regular. To see this, let [MATH] be a morphism and let [MATH] be a DFA. We can construct a DFA for [MATH] as follows: let [MATH] , where [MATH] Then it is easily verified that [MATH] We call [MATH] the inverse morphism DFA of [MATH] with respect to [MATH] |
Note that [MATH] has the same state configuration as [MATH] . This will be useful for multiple reasons, but in particular it implies the following result for regular languages [MATH] and [MATH] |
Lemma 1 If [MATH] , then [MATH] Transformation Languages In this section, we formally define the witness languages that are used in the OLPA approach. |
Fix a set [MATH] and let [MATH] be a set of transformations of [MATH] For [MATH] and [MATH] , the transformation language [MATH] is the language of the DFA [MATH] , where [MATH] This DFA is called the standard DFA for the transformation language. |
The set of all transformations of a set [MATH] is called the full transformation monoid on [MATH] , and is denoted [MATH] The full transformation languages of the form [MATH] play in important role in the theory behind the OLPA approach. |
Notice that the language [MATH] has alphabet [MATH] , and the standard DFA for [MATH] has transitions [MATH] This DFA has one letter per transformation of the state set [MATH] , that is, one letter per possible action on the DFA’s states. Full transformation languages are the languages used as witnesses when applying t... |
Let [MATH] be a regular language over [MATH] recognized by a DFA [MATH] . The standard transformation morphism of [MATH] (with respect to [MATH] ), denoted by [MATH] , is defined by [MATH] The following fact is easily verified: |
Proposition 1 [MATH] Full transformation languages do not suffice as OLPA witnesses for operations of arity greater than one. When applying the OLPA approach to operations of arity [MATH] , we want to use an [MATH] -tuple [MATH] of DFAs (where [MATH] for [MATH] ) with the following property: for each [MATH] -tuple of t... |
For this purpose, we define transformation tuple languages Let [MATH] be finite sets and let [MATH] be a subset of [MATH] For [MATH] with [MATH] [MATH] , and [MATH] , the transformation tuple language |
[MATH] is the language of the DFA [MATH] where [MATH] This DFA is called the standard DFA of the transformation tuple language. The full transformation tuple languages of the form [MATH] are used as OLPA witnesses in the case of [MATH] -ary operations. |
There is a generalization of Proposition for full transformation tuple languages. Let [MATH] be an [MATH] -tuple of regular languages over [MATH] , where [MATH] is recognized by the DFA [MATH] for [MATH] The standard transformation tuple morphism of [MATH] (with respect to [MATH] ), denoted by [MATH] , is defined by [M... |
Proposition 2 We have [MATH] Proof It suffices to show for all [MATH] that [MATH] Fix [MATH] and let [MATH] be the standard DFA of [MATH] Then we have |
[EQUATION] as required. The second two-way implication may not be obvious. To see that it holds, first note that if [MATH] is empty, then [MATH] and [MATH] are both the identity map on [MATH] . Otherwise, suppose [MATH] with [MATH] We may write [MATH] , and thus [MATH] By definition, we have [MATH] for [MATH] This [MAT... |
[EQUATION] Hence the implication holds. ∎ Uniform Regular Operations Our goal in this section is to define a large class of operations for which the OLPA approach works. The approach does not work for all regular operations; it is easy to come up with rather contrived examples of operations for which OLPA fails. Consid... |
If the input DFA has one letter per action, output a DFA with no final states. Otherwise, output the input DFA. If the input DFA has one letter per action, output a DFA in which the initial state is non-final and all the actions send the initial state to a sink state. Otherwise, output the input DFA. |
The problem with this operation is that its behaviour is not “uniform” across all languages; it detects particular languages and has special behaviour for them. In the first case, the operation does not behave uniformly on states: for most DFAs it preserves the final state set, but for DFAs with one letter per action i... |
We now attempt to formally define this idea of “uniformity” for unary operations. Let [MATH] be a unary DFA operation. We say [MATH] is uniform if for every pair of DFAs [MATH] and [MATH] with the same state configuration, the image DFAs |
[MATH] and [MATH] satisfy the following conditions: 1. [MATH] 2. Whenever [MATH] for [MATH] and [MATH] , we have [MATH] We will say a unary regular operation [MATH] is uniform if there exists a uniform unary DFA operation equivalent to [MATH] |
We can interpret this definition intuitively as follows. The first condition says that the operation is uniform with respect to state configurations: if the operation is given two input DFAs with the same state configuration, it will produce two output DFAs with the same state configuration. The second condition says t... |
The definition of uniformity is heavily dependent on DFAs. Thus, it may come as a surprise that there is a simple and purely language-theoretic characterization of uniformity. A morphism [MATH] is 1-uniform if it maps letters to letters. |
Proposition 3 Let [MATH] and [MATH] be regular languages over [MATH] and [MATH] respectively. The following are equivalent: 1. The regular operation [MATH] is uniform. |
2. For all [MATH] -uniform morphisms [MATH] , if [MATH] then [MATH] Proof [MATH] Since [MATH] is uniform, there is a uniform DFA operation [MATH] equivalent to [MATH] Fix a 1-uniform morphism [MATH] such that [MATH] Let [MATH] be a DFA for [MATH] , and let [MATH] be a DFA for [MATH] We write [MATH] for [MATH] , and [MA... |
Note that we can choose our DFAs so that they have the same state configuration. This follows from the fact that [MATH] , and thus we can take [MATH] which has the same state configuration as [MATH] Henceforth write [MATH] [MATH] , and [MATH] |
Let [MATH] and let [MATH] Write [MATH] for [MATH] and [MATH] for [MATH] The DFA [MATH] recognizes [MATH] , and the DFA [MATH] recognizes [MATH] By the uniformity of [MATH] , we can write [MATH] [MATH] , and [MATH] |
Now, we want to show that [MATH] Since [MATH] , for all [MATH] and [MATH] we have [MATH] by definition. Thus [MATH] and [MATH] are equal as transformations of [MATH] for all [MATH] By the uniformity of [MATH] |
[MATH] and [MATH] are equal as transformations of [MATH] It follows that [MATH] and [MATH] are equal as transformations of [MATH] for all [MATH] Hence we have |
[EQUATION] This proves that [MATH] [MATH] We are given a regular operation [MATH] We want to produce a uniform DFA operation [MATH] such that for all DFAs [MATH] , we have [MATH] |
Fix an [MATH] -state DFA [MATH] and let [MATH] be its language. We define [MATH] as follows. By Proposition , we have [MATH] , where [MATH] is the standard transformation morphism of [MATH] By assumption, we then have [MATH] Let [MATH] be a minimal DFA for [MATH] and set [MATH] |
It is clear that we have [MATH] as required. To see that [MATH] is uniform, fix DFAs [MATH] and [MATH] We compute the images [MATH] |
and [MATH] Now, let [MATH] be the minimal DFA for [MATH] By definition, we have [MATH] and [MATH] So [MATH] and [MATH] both have the same state configuration as [MATH] It follows that [MATH] , as required. |
Next, fix [MATH] and [MATH] such that [MATH] We have [MATH] for all [MATH] Also, [MATH] for all [MATH] By the definition of the standard transformation morphism, we have [MATH] , since [MATH] It follows that [MATH] , as required. Thus [MATH] is uniform. |
Now that we have established the definition of uniformity and the language-theoretic characterization for unary regular operations, we turn to operations of higher arity. Let [MATH] be an [MATH] -ary DFA operation. We say [MATH] is uniform if for every pair of [MATH] -tuples of DFAs [MATH] and [MATH] where for each [MA... |
1. The image DFAs [MATH] and [MATH] have the same state configuration. 2. If there exist letters [MATH] and [MATH] such that [MATH] for each [MATH] with [MATH] , then |
[MATH] There is a corresponding language-theoretic characterization of the general definition of uniformity. Proposition 4 Let [MATH] and [MATH] be [MATH] -tuples of regular languages, where [MATH] is a language over [MATH] and [MATH] is a language over [MATH] for [MATH] The following are equivalent: |
1. The [MATH] -ary regular operation [MATH] is uniform. 2. For all [MATH] -uniform morphisms [MATH] , if [MATH] for [MATH] , then [MATH] |
The proof is very similar to the proof of Proposition , except the general definition of uniformity is used and full transformation tuple languages are used instead of full transformation languages. |
Proof [MATH] Since [MATH] is uniform, there is a uniform DFA operation [MATH] equivalent to [MATH] Fix a 1-uniform morphism [MATH] such that [MATH] for [MATH] We want to show that [MATH] |
Since [MATH] , for each [MATH] we can find a DFA [MATH] for [MATH] and a DFA [MATH] for [MATH] such that [MATH] Each pair of DFAs [MATH] and [MATH] has a common state configuration [MATH] For [MATH] , let [MATH] be the DFA for [MATH] and let [MATH] be the DFA for [MATH] By the uniformity of [MATH] , the image DFAs [MAT... |
Since [MATH] for [MATH] , for all [MATH] and [MATH] , we have [MATH] by definition. Thus [MATH] for all [MATH] and all [MATH] . By the uniformity of [MATH] , it follows that [MATH] Hence [MATH] for all [MATH] Thus we have |
[EQUATION] This proves that [MATH] [MATH] We want to produce a uniform [MATH] -ary DFA operation [MATH] such that for all tuples of DFAs [MATH] over a common alphabet, we have |
[MATH] Fix a tuple [MATH] of DFAs over [MATH] , where [MATH] has state configuration [MATH] , and let [MATH] for [MATH] We define the image [MATH] as follows. By Proposition we have [MATH] , where [MATH] is the standard transformation tuple morphism of [MATH] with respect to [MATH] , and [MATH] Let [MATH] be a minimal ... |
We claim that [MATH] Indeed, since [MATH] we have [EQUATION] It follows that [EQUATION] as required. To see that [MATH] is uniform, fix [MATH] -tuples of DFAs [MATH] and [MATH] such that for [MATH] , the DFAs [MATH] and [MATH] have the same state configuration [MATH] , the DFA [MATH] has alphabet [MATH] and transition ... |
and [MATH] Let [MATH] be a minimal DFA for [MATH] used in the definition of [MATH] Then [MATH] and [MATH] are both inverse morphism DFAs constructed from [MATH] , so they both have the same state configuration as [MATH] Write [MATH] for this common state configuration. |
It remains to show that whenever we have [MATH] and [MATH] such that [MATH] for [MATH] , it follows that [MATH] Fix [MATH] and [MATH] with this property. Write [MATH] as shorthand for [MATH] , and write [MATH] for [MATH] By definition, we have [MATH] and [MATH] Let [MATH] be the transition set of [MATH] Then for [MATH]... |
[MATH] and [MATH] But we are assuming that [MATH] for [MATH] , so in fact [MATH] It then follows that [MATH] , as required. This proves that [MATH] is uniform, and thus [MATH] is uniform, since it is equivalent to a uniform DFA operation. ∎ |
The Main Theorem The goal of this section is to prove that the OLPA approach works for all uniform operations. Thanks to Proposition |
and its generalization in Proposition this is not especially difficult. First we consider unary operations. The following lemma formalizes a “weak” version of the OLPA approach for unary operations. The short proof contains all the essential ideas, but the expression it gives for the state complexity function is not pr... |
Lemma 2 Let [MATH] be a uniform unary regular operation. Let [MATH] be a regular language recognized by a DFA [MATH] . Then [MATH] . In particular, the state complexity of [MATH] is given by the following function: |
[EQUATION] Proof Fix [MATH] , and recall that [MATH] , where [MATH] is the standard transformation morphism of [MATH] Since [MATH] is uniform, we have [MATH] by Proposition By Lemma , we have [MATH] as required. ∎ |
Now, we show that to compute the state complexity function, it suffices to just consider the set [MATH] instead of all sets of size [MATH] . Furthermore, we may assume that [MATH] , and that [MATH] is either [MATH] or [MATH] for some [MATH] . Thus it suffices to just check [MATH] OLPA witnesses. Write [MATH] for [MATH]... |
Theorem 5.1 Let [MATH] be a uniform unary regular operation. Let [MATH] be a regular language recognized by a DFA [MATH] Then [MATH] , where [MATH] [MATH] , and [MATH] is defined to be [MATH] if [MATH] and [MATH] if [MATH] The state complexity of [MATH] is the following function: |
[EQUATION] Proof We know from Lemma that [MATH] Let us prove that [MATH] , with [MATH] [MATH] and [MATH] defined as in the statement of the theorem. |
By Lemma and the uniformity of [MATH] , it suffices to exhibit a morphism [MATH] such that [MATH] To define [MATH] , first we define a bijection [MATH] We take [MATH] to be a bijection with the following properties: [MATH] and [MATH] The remaining elements of [MATH] are mapped to the remaining elements of [MATH] arbitr... |
We show that [MATH] if and only if [MATH] Let [MATH] for [MATH] [EQUATION] Thus [MATH] , as required. This completes the proof. ∎ |
We now consider uniform operations of arbitrary arity. The proof strategies in this case are much the same, except full transformation tuple languages are used as witnesses, rather than full transformation languages. |
Lemma 3 Let [MATH] be a uniform [MATH] -ary regular operation. Let [MATH] be regular languages, where [MATH] is recognized by a DFA [MATH] Let [MATH] Then [MATH] |
Proof Fix [MATH] , and recall from Proposition that [MATH] , where the morphism [MATH] is the standard transformation tuple morphism of [MATH] Since [MATH] is uniform, we have |
[EQUATION] by Proposition Then by Lemma , we have [EQUATION] as required. ∎ As before, it suffices to only check a finite number of witnesses. Recall that we defined [MATH] , and for [MATH] we defined [MATH] and [MATH] |
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