=Paper=
{{Paper
|id=Vol-2362/paper23
|storemode=property
|title=Machine Learning Technique for Regular Pattern Detector Synthesis: toward Mathematical Rationale
|pdfUrl=https://ceur-ws.org/Vol-2362/paper23.pdf
|volume=Vol-2362
|authors=Grygoriy Zholtkevych,Nataliya Polyakovska
|dblpUrl=https://dblp.org/rec/conf/colins/ZholtkevychP19
}}
==Machine Learning Technique for Regular Pattern Detector Synthesis: toward Mathematical Rationale==
https://ceur-ws.org/Vol-2362/paper23.pdf
Machine Learning Technique for Regular
Pattern Detector Synthesis: toward
Mathematical Rationale
Grygoriy Zholtkevych[0000−0002−7515−2143] and Nataliya
Polyakovska[0000−0003−2855−7970]
V. N. Karazin Kharkiv National University, Kharkiv, 61022, Ukraine
g.zholtkevych@karazin.ua, natalipolyakovska@gmail.com
Abstract. In the paper, the machine learning method for synthesis reg-
ular pattern detectors proposed early by authors is studied. This method
can be used in event processing systems. This paper, in contrast to the
previous one, is focused on studying the mathematical properties of con-
cepts related to the method. The main result of the paper is proofs of
theorems grounding that the method leads to the required result some-
times. The estimation of a probability for success is not considered in
the paper.
Keywords: event· event stream processing· event pattern· pattern de-
tector· regular pattern detector· regular pattern acceptor
1 Introduction
The widespread development of cloud computing and Internet of Things (IoT)
leads to an increase in the share of cyber-physical systems of all the variety
of software systems. The distribution, heterogeneity, and the variability of the
composite of components and relationships between them are distinctive features
for systems of this class. Thus, the architecture of these systems should provide
mechanisms to plug and unplug components, reconfiguration of relationships,
and synchronisation of component behaviours.
It is known that event-driven architecture is a good choice is known that
event-driven architecture is a good choice for meeting the requirements men-
tioned above. In this context, the problem of controlling event streams is very
important and may be considered even as the main one.
Thus, a stream of events is a very important concept for cyber-physical sys-
tems and distributed software systems in whole. The validity of this statement
is caused by the practice of developing controlled asynchronously distributed
systems. This practice led to the reference architecture for event-driven systems
(see [7,2], for example). It is no coincidence that these developments are con-
ducted in the context of cloud technologies because message passing is the only
way to organise collaboration between components of cloud platforms. One can
find a detailed analysis of the event processing technology in [3].
Creating an event processing subsystem for each component of a distributed
software system lies in the focus of the system developers. Therefore, efficient
tools for specifying the valid system component behaviour and recognising vio-
lations of the validity are needed.
We need to underline that there is a contradiction between the complexity to
specify and check the behaviour requirements of the system components and the
necessity to do fast these processes. An attempt to overcome this contradiction
is discussed in the paper.
2 State of Art
The idea to use inductive methods for overcoming the mentioned above contra-
diction was firstly formulated in [10]. This idea has arisen as a result of developing
the mathematical theory of some generalisation of automata called pre-automata
and their applications in software engineering [1,12,9].
In [11], the machine learning method for the synthesis of special class event
detectors was proposed and experimentally studied.
2.1 Regular Pattern Detectors
Remind that we consider the class of machines, which we call regular pattern
detectors. A member of this class is defined as follows.
Definition 1. A regular pattern detector is a tuple R = hΣ, Q, q0 , Π, δi where
• Σ is a finite alphabet of input signals;
• Q is a finite sets of states;
• q0 ∈ Q is a fixed state called initial;
• Π is some finite alphabet whose elements mark the corresponding patterns;
• δ : Q × Σ → Π + Q is called the decision function 1 .
Below we consider non-empty words 2 over alphabet Σ as events (more precisely,
messages about event occurrences) and denote the set of events by Σ + .
To define the Π-indexed family R+ = {R+ s | s ∈ Π} of event sets being
detected by the pattern detector R we extend the decision function δ up to the
partial mapping 3 δ + : Q × Σ + 99K Π + Q defined recursively as follows
δ + (q, a) ↓= δ(q, a) for any q ∈ Q and a ∈ Σ;
δ + (q, ua) ↑ if either δ + (q, u) ↓= s where s ∈ Π or δ + (q, u) ↑;
δ (q, ua) ↓= δ(q , a) if δ + (q, u) ↓= q 0 where q 0 ∈ Q.
+ 0
1
The sign “+” denotes here and below the disjoint union of sets.
2
The necessary definitions and notation are given in Appx. B.
3
The necessary definitions and notation see in Appx. A.
Definition 2. We say that an event u ∈ Σ + is detected by a regular pat-
tern detector R as a sample of the pattern s ∈ Π (symbolically, u ∈ R+
s ) if
δ + (q0 , u) ↓= s.
In other words, R+ + +
s = {u ∈ Σ | δ (q0 , u) ↓= s} for all s ∈ Π.
Thus, the Π-indexed family R+ = {R+ s | s ∈ Π} of subsets of Σ
+
is asso-
ciated with any regular pattern detector R. Properties of this family are estab-
lished by the next proposition.
Proposition 1. For any regular pattern detector R, the Π-indexed family R+
holds the following properties
1. R+ is a mutually disjoint family;
+
S +of R is a regular
2. each member set 4 ;
5
3. the set Rs is prefix-free .
s∈Π
Proof. For proving the first item let us suppose that for some u ∈ Σ + , the
statements u ∈ R+ +
s1 and u ∈ Rs2 are true where s1 , s2 ∈ Π and s1 6= s2 . By
definition of the family R , it means that δ + (q0 , u) ↓= s1 and δ + (q0 , u) ↓= s2
+
and, therefore (see Def. A1), s1 = s2 . This contradiction proves the first item.
To prove the second item let us construct a finite acceptor As = hQs , q0 , Fs , δs i
that recognises exactly words from R+ s . We define Qs = Q + Π + {⊥} where
⊥ is used to refer to the special trash-state; the initial state of the detector q0
is the initial state of the acceptor being constructed; the subset of acceptable
states Fs ⊂ Qs is the singleton {s}; the transition function δs : Qs × Σ → Qs
acts on a pair (q, a) as follows
δs (q, a) = δ(q, a) for any q ∈ Q and a ∈ Σ;
δs (s0 , a) = ⊥ for any s0 ∈ Π and a ∈ Σ;
δs (⊥, a) = ⊥ for any a ∈ Σ.
Now one can easily see that R+s coincides with the set of words being accepted
by As . S + S +
To prove the third item let us assume u ∈ Rs and uv ∈ Rs for some
s∈Π s∈Π
v ∈ Σ + . The first statement ensures that δ + (q0 , u) ↓= s for some s ∈ Π and,
therefore, for any v ∈ Σ + , we have δ + (q0 , uv) ↑. This contradiction gives the
required proof. t
u
The inverse statement is also true. We will prove this fact later using the tech-
nique presented below.
2.2 Synthesis Method
The idea of the synthesis method consists of the follows
4
See, Def. C3 of Appx. B
5
See, Def. B5 of Appx. B
• we have some finite alphabet of signals Σ and some finite alphabet of recog-
nisable pattern Π;
• we fix a Π-indexed disjoint family {Es | x ∈ Π} of finite subsets of Σ + with
+
T C ⊂ Σ of event messages
the prefix-free union and a finite subset S being
considered as undetected, moreover, E C = ∅ where E = Es ;
s∈Π
• we search a regular pattern detector R = hΣ, Q, q0 , Π, δi such that
+
s ⊂ Rs
1. E for all s ∈ Π;
T + T
2. Rs C = ∅;
s∈Π
3. the number of elements of Q is less or equal of the number of states for
any detector satisfying 1 and 2.
This idea led the authors of [11] to Algorithm1.
Algorithm 1: Synthesis Method
Data: a Π-indexed family E of finite sets of Σ + with the prefix-free union and
a finite set C of Σ + disjointed with the union of E
Result: a regular event detector R
1 build the tree-like detector hΣ, Q, q0 , Π, δi corresponding to E and denote it by
R;
2 foreach q ∈ Q and a ∈ Σ do
3 if δ(q, a) =⊥ then
4 choose randomly x from (Q + Π) \ {q, ⊥};
5 redefine δ(q, a) as x;
6 if u ∈ R+ s for some s ∈ Π and u ∈ C then
7 rollback the redefinition;
8 continue
9 end
10 minimise the modified detector R using Hoproft’s Method (see [4])
11 end
12 end
The like-tree detector hΣ, Q, q0 , Π, δi mentioned in item 1 of Algorithm 1 is
defined as follows
∗ ∗
S S
Q = u ∈ Σ | uv ∈ Es for some v ∈ Σ {⊥};
s∈Π
q0 = ;
δ(⊥, a) = ⊥ for all a ∈ Σ;
δ(u, a) = ua if ua ∈ Q;
δ(u, a) = ⊥ otherwise
followed by minimisation of Hopcroft’s Method.
3 Parallel Joint of Regular Pattern Detectors
In this section, we describe a construction that allows simplifying the problem
being studied and restricting the studying by the simple detectors called accep-
tors.
Let us assume that we have n (n > 1) regular pattern detectors denoted
by Ri = hΣ, Qi , q0,i , Πi , δi i with the same alphabet Σ of input signals for i =
1, . . . , n.
In this case, we define the regular pattern detector R = hΣ, Q, q0 , Π, δi as follows
Q = {⊥} + Q1 × . . . × Qn ;
q0 = (q0,1 , . . . , q0,n );
Π = Π1 ∪ . . . ∪ Πn ;
δ(⊥, a) = ⊥ for all a ∈ Σ;
δ((q1 , . . . , qn ), a) = (δ1 (q1 , a), . . . , δn (qn , a)) if δi (qi , a) ∈ Qi
for all i = 1, . . . , n;
δ((q1 , . . . , qn ), a) = s if δi (qi , a) ∈
/ Qi implies δi (qi , a) = s
for all i = 1, . . . , n;
δ((q1 , . . . , qn ), a) = ⊥ otherwise.
As above, the symbol ⊥ is used to refer to a trash-state, which is not a member
of any Qi .
It is easy to understand that R is really a regular pattern detector. This de-
tector is below called the parallel joint of the detectors R1 ,. . . ,Rn (symbolically,
n
R = ./ Ri ).
i=1
Below we use this definition to demonstrate that any Π-indexed family of
subsets of Σ + satisfying the conditions of Prop. 1 can be represented as R+ for
a regular pattern detector R, which is a parallel joint of primitive, in some sense,
regular pattern detectors called regular pattern acceptor.
Definition 3. A regular pattern detector A is called a regular pattern acceptor
if the correspondence set of patterns is a singleton.
It is evident that Prop. 1 ensures the following properties of the set A+ formed
by word accepted by A: this set is regular and prefix-free. The following theorem
shows that the converse fact is also true.
Theorem 1. Any regular and prefix-free subset F ⊂ Σ + is the set A+ for some
regular pattern acceptor A.
Proof. Considering F is regular let us take the minimal finite-state acceptor
M = hΣ, Q, q0 , F, δF i6 that recognises words belonging to F and only such
words. The minimality of M ensures the existence exactly one state q⊥ ∈ / F
such that
6
See App. C
1. δF (q⊥ , a) = q⊥ for all a ∈ Σ ;
∗
2. for any q ∈ Q such that q 6= q⊥ and u ∈ Σ + , δF (q0 , u) = q implies that
∗ 0 0 ∗
δF (q0 , uu ) ∈ F for some u ∈ Σ .
Further, if q ∈ F then taking into account that F is prefix-free one can conclude
that δF (q, a) = q⊥ for all a ∈ Σ . Hence, the the minimality of M ensures that
F = {qa } for some qa ∈ Q .
Let us now define Q0 = Q \ {qa } , Π = {∗} , and
∗ if δF (q, a) = qa
δ(q, a) =
δF (q, a) otherwise
Thus, A = {Σ, Π, Q0 , δ} is a regular pattern acceptor and one can easily conclude
that A+ equals F . t
u
Now we are ready to prove the statement converse to Prop. 1.
Theorem 2. Let Σ and Π be any finite alphabets, F be a Π-indexed family of
subsets of Σ + and this family holds the conditions
1. F is a mutually disjoint family;
2. each member
S of F is a regular set;
3. the set Fs is prefix-free
s∈Π
then there exists a Π-indexed family A = {As | s ∈ Π} of regular pattern
acceptors such that R+
s = Fs for all s ∈ Π where R = ./ As .
s∈Π
Proof. Since the union of all Fs is prefix-free, each Fs is prefix-free too. Moreover,
each Fs is regular therefore Theorem 1 ensures for each s ∈ Π, the existence of
a regular pattern acceptor As such that A+ s = Fs . The condition that F is a
mutually disjoint family ensures immediately the equality R+ s = Fs for all s ∈ Π
where R = ./ As . t
u
s∈Π
Combining Prop. 1 and Theorem 2 one can easily obtain the following.
Corollary 1. For any finite alphabets Σ and Π, a Π-indexed family F of sub-
sets of Σ + is the family associated with some regular pattern detectors if and
only if the following conditions hold
1. F is a mutually disjoint family;
2. each member of F is a regular set;
3. union of all member of F is a prefix-free subset of Σ + .
4 Specification of Regular Prefix-free Sets
The results obtained above allow further us to restrict our consideration by
the class of regular pattern acceptors. Therefore, the theorem is proven in this
Algorithm 2: Synthesis Method for Acceptors
Data: a prefix-free finite set E of Σ + and a finite set C of Σ + disjointed with E
Result: a regular event acceptor A
1 build the tree-like acceptor hΣ, Q, q0 , δi corresponding to E and denote it by A;
2 foreach q ∈ Q and a ∈ Σ do
3 if δ(q, a) =⊥ then
4 choose randomly q 0 from (Q + {∗}) \ {q, ⊥};
5 redefine δ(q, a) as q 0 ;
6 if u ∈ A+ for some u ∈ C then
7 rollback the redefinition;
8 continue
9 end
10 minimise the modified detector A using Hoproft’s Method (see [4])
11 end
12 end
section is the Main Theorem of the paper. It gives a method to specify any
regular prefix-free set by two finite word subsets.
We begin with a specialised synthesis method, represented by Algorithm 1
for event pattern acceptors (see Algorithm 2 below).
The like-tree acceptor A0 = hΣ, Q, q0 , δi mentioned in item 1 of Algorithm 2
is defined as follows
Q = {u ∈ Σ ∗ | uv ∈ E for some v ∈ Σ ∗ } {⊥};
S
q0 = ;
δ(⊥, a) = ⊥ for all a ∈ Σ;
δ(u, a) = ua if ua ∈ Q;
δ(u, a) = ⊥ otherwise
followed by minimisation of Hopcroft’s Method.
Further, we need some auxiliary notion. The definition and proof of two
important properties are presented here.
Definition 4. For any alphabet Σ and a subset L ⊂ Σ + , we say that a word
u ∈ L is L-prime if for any u1 , u3 ∈ Σ ∗ and u2 ∈ Σ + , the equation u = u1 u2 u3
implies u1 u3 ∈
/ L.
Proposition 2. For any alphabet Σ and a subset L ⊂ Σ + , the subset Pm L ⊂ L
containing exactly L-prime words is prefix-free.
Proof. Indeed, if word u = u0 u00 , where u0 ∈ Pm L and u00 ∈ Σ + , belongs to
Pm L then the representation u = u0 u00 ensures that u0 = u0 ∈
/ L. The obtained
contradiction proves the proposition. t
u
Proposition 3. For any alphabet Σ and a regular subset L ⊂ Σ + , the subset
Pm L ⊂ L is finite.
Proof. According to the pumping lemma for regular languages [6, Lemma 8,
p. 119] (see also, [5, Sect. 4.6, p. 166]) there exists an integer n > 1 depending
only on L and such that any word u ∈ L of length at least n can be represented
as u1 u2 u3 , where u1 , u3 ∈ Σ ∗ , u2 ∈ Σ + , lng u1 u2 ≤ n , and u1 u3 ∈ L. Conse-
quently, all words longer than n are not L-prime. Obviously, there are a finite
set of words with length less than n, accordingly set Pm L is also finite. t
u
Now we are ready to formulate and obtain the main results of the paper.
Theorem 3. Any regular pattern acceptor A = hΣ, Q, q0 , δi can be reached by
using Synthesis Method specified by Algorithm 2 with set E equal to Pm A+ .
Proof. Since Pm A+ is prefix-free and regular according to Propositions 2 and
3, it can be used as set E for Synthesis Method.
Taking into account that after each iteration of the loop 2–12 in Algorithm 2
1. the acceptor A has at most one inefficient state
2. the number of its states does not increases,
one can use breadth-first search method in the graph of the target acceptor
for the sequential redefining transition function of the like-tree acceptor built
in accordance with item 1 of Algorithm 2. This process leads evidently to the
target acceptor. t
u
Note that in the proof of Theorem 3 we assigned C = ∅. Therefore, we do not
have any guarantees that Algorithm 2 halts after reaching the target acceptor.
However, we prove that there exists some finite subset C ⊂ Σ + disjointed to
Pm A that provide the correct termination of the algorithm. We use methods of
general topology to choose C. All necessary definitions and facts can be found
in any textbook (for example, in [8]).
Theorem 4. For any regular event pattern acceptor A, a finite set C ⊂ Σ + that
allows to exactly restore acceptor A with the set Pm A+ as E using Algorithm 2
exists.
Proof. Let us consider the set Σ ω formed by infinite sequences of elements of Σ.
It is evident that the family {u · Σ ω ⊂ Σ ω | u ∈ Σ + }, where u · Σ ω is formed
by sequences with prefix u, is a base of a topology. This topology is Tikhonov
topology on Σ ω considered as the countable power of Σ in the assumption that
the last is equipped with the discrete topology. Tikhonov Theorem guarantees
that the space under consideration is compact. It is this fact that we need in the
proof.
Further, let ⊥ denotes the unique inefficientS
state for acceptor A. One can divide
Σ ω into two disjoint components Σ ω = C1 C2 as follows
• a sequence s ∈ Σ ω belongs to C1 if all and only if for some its prefix u ∈ Σ +
the equation δ + (q0 , u) =⊥ is fulfilled;
• a sequence s ∈ Σ ω belongs to C2 if all and only if for any its prefix u ∈ Σ +
either δ + (q0 , u) ↑ or the equation δ + (q0 , u) ↓= x implies x 6=⊥.
u · Σ ω where A⊥ is formed by such u that
S
It is evident that C1 =
u∈A⊥
1. δ + (q0 , u) ↓=⊥ and
2. for any proper prefix u0 of u the equation δ + (q0 , u0 ) ↓=⊥ is not fulfilled.
Further, one can easily conclude that C2 is open in Tikhonov topology. Indeed,
if a sequence s ∈ C2 then there exists some prefix u of s and u0 ∈ Σ ∗ such that
δ + (q0 , uu0 ) ↓= ∗. But in this case, any s0 ∈ uu0 · Σ ω belongs to C2 and, therefore
uu0 · Σ ω ⊂ C2 .
Openness of C2 ensures closedness of C1 . Moreover, the family {u · Σ ω | u ∈ A⊥ }
is an open covering of C1 . Thus, compactness of Σ ω ensures the finiteness of A⊥
and, therefore, we can assign C = A⊥ for Algorithm 2.
It is evident that this choice ensures reaching exactly A under using Algorithm 2.
t
u
Theorem 1, Theorem 2, Theorem 3, and Theorem 4 ensure the following fact.
Main Theorem. For any regular pattern acceptor A, the finite prefix-free set
Pm A+ and the finite set A⊥ (see proof of the previous theorem) determine
uniquely the language A+ by using Algorithm 2.
5 Conclusion
The paper examines the method of machine learning proposed earlier by the
authors, designed to synthesise regular pattern detectors based on training sets.
In the paper, the method for decomposing an arbitrary regular pattern de-
tector into prime ones has been grounded. These prime regular pattern detec-
tors have been called regular pattern acceptors and characterised as detectors
with the single output. It has been established that any recognition problem for
regular pattern detectors can be reduced to the recognition problems for the
corresponding acceptors (Theorems 1 and 2).
The use of these facts allowed to prove the main result of the work, justifying
to reach the required acceptor using the previously proposed Synthesis Method
specified for the case of acceptors (Theorems 3 and 4).
Thus, we have proven that the Synthesis Method based on Machine Learning
Technique proposed in [10,11] leads sometimes to the required result. The next
step of studying is to estimate the probability of success of the synthesis process
and understand whether a good probability for success gotten experimentally in
[11] is grounded.
Appendix A Partial Mappings
This section contains the definitions and notation that are relating to the concept
of a partial mapping and used above.
Definition A1. A partial mapping f from a set X into a set Y (symbolically,
f : X 99K Y ) is the triple hX, Y, Γf i where Γf ⊂ X × Y if this triple satisfies the
condition
for all x ∈ X and y 0 , y 00 ∈ Y, hx, y 0 i ∈ Γf and hx, y 00 i ∈ Γf imply y 0 = y 00 .
Notation A1. Let X and Y be sets, and f : X 99K Y then for any x ∈ X,
f (x) ↑ means that hx, yi ∈
/ Γf for any y ∈ Y ;
f (x) ↓ means that hx, yi ∈ Γf for some y ∈ Y ;
f (x) ↓= y means that hx, yi ∈ Γf where y ∈ Y .
We use the symbol to refer to any partial mapping of the form hX, Y, ∅i.
Appendix B Alphabets and Words
This section contains the definitions, notation, and facts that are relating to the
concept of a word and used above.
Definition B1. An alphabet is a finite set whose elements called tokens or sym-
bols.
Definition B2. A word over an alphabet Σ is a partial mapping u : N 99K Σ
such that
there exists n ∈ N such that u(n) ↑;
for any m, n ∈ N such that m ≤ n, u(n) ↓ implies u(m) ↓ .
Notation B1. The set of words over an alphabet Σ is denoted by Σ ∗ , and the
set Σ ∗ \ {} is denoted by Σ + .
If u ∈ Σ + and n ∈ N such that u(n) ↓ then u[n] is the token from Σ satisfying
the condition u(n) ↓= u[n].
Definition B3. The length of a word u ∈ Σ ∗ is defined as follows
lng u = min{n ∈ N | u(n) ↑}.
Definition B4. For any alphabet Σ and u, v ∈ Σ ∗ , the word uv is defined as
follows
(uv)(n) ↓= u[n] whenever 0 ≤ n < lng u;
(uv)(n) ↓= v[n − lng u] whenever lng u ≤ n < lng u + lng v;
(uv)(n) ↑ whenever n ≥ lng u + lng v.
This word is called concatenation of u and v.
Definition B5. For any alphabet Σ, a subset L ⊂ Σ ∗ is called prefix-free if
uv ∈ L ensures v = for any u ∈ L and v ∈ Σ ∗ .
Appendix C Finite-State Acceptors and Regular Sets of
Words
Definition C1. A finite-state acceptor is a pentacle M = hΣ, Q, q0 , F, δi where
Σ is a finite input alphabet;
Q is a finite set of states;
q0 ∈ Q is some fixed state called initial;
F ⊂Q is some fixed subset of states that are called acceptable;
δ : Q × Σ → Q is a mapping called a transition function.
For a finite-state acceptor M = hΣ, Q, q0 , F, δi , one can define the following
extension δ ∗ : Q × Σ ∗ → Q of the mapping δ
δ ∗ (q, ) = q for any q ∈ Q ;
δ ∗ (q, ua) = δ(δ ∗ (q, u), a) for any q ∈ Q , u ∈ Σ ∗ , and a ∈ Σ .
Definition C2. A finite state acceptor M = hΣ, Q, q0 , F, δi recognises a word
u ∈ Σ ∗ if δ ∗ (q0 , u) ∈ F .
Definition C3. For any alphabet Σ, a subset L ⊂ Σ ∗ is called regular if there
is a finite-state acceptor that recognises exactly words from L.
More detailed information about regular set and finite-state acceptors one can
find in, for example, [5].
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