CN113553044B - Generation method of time automaton model combining PAC learning theory and active learning - Google Patents

Generation method of time automaton model combining PAC learning theory and active learning Download PDF

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CN113553044B
CN113553044B CN202110817891.XA CN202110817891A CN113553044B CN 113553044 B CN113553044 B CN 113553044B CN 202110817891 A CN202110817891 A CN 202110817891A CN 113553044 B CN113553044 B CN 113553044B
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CN113553044A (en
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张苗苗
沈炜
安杰
詹博华
薛白
詹乃军
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Tongji University
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Abstract

The invention provides a generation method of a time automaton model combining PAC learning theory and active learning, which is used for generating a single-clock time automaton form model of a real-time system and is characterized by comprising the following steps: step S1, initializing a time observation table through the learner; step S2, processing the time observation table by the converter to satisfy closeness and consistency; step S3, constructing a hypothesis model through a learner based on the time observation table
Figure DDA0003170838510000011
Step S4, comparing the hypothesis models by the comparator
Figure DDA0003170838510000013
Quality and stability model of
Figure DDA0003170838510000012
And judging whether a counterexample is found; step S5, when step 4 judges no, the teaching device makes a pair of hypothesis models
Figure DDA0003170838510000014
Carrying out PAC approximate equivalence inquiry, judging whether a counterexample is found, stopping iteration when the counterexample is not found, and assuming a model
Figure DDA0003170838510000015
As a result model; in step S6, when the determination in step S4 or step S5 is yes, the time observation table is updated according to a counterexample, and the process returns to step S2.

Description

Generation method of time automaton model combining PAC learning theory and active learning
Technical Field
The invention belongs to the technical field of software development, and particularly relates to a generation method of a time automaton model combining a PAC (programmable automation controller) learning theory and active learning, which is used for generating a deterministic single-clock time automaton formal model of a real-time system.
Background
With the rapid development of real-time embedded systems, the reliability and security of the systems are receiving more and more attention from the industry and governments of various countries. At present, the inspection of the system is mainly realized by using technologies such as Model-based testing (Model-based testing) and Formal verification (Formal verification). Formal models are prerequisites for applying the technologies, and in reality, it is generally difficult to directly obtain the formal models due to the problems of software history left, document missing description, source code inaccessible or difficult understanding, and the like. Model learning (Model learning) is a method for automated formal modeling of a system through input-output observations of the system, by which a formal Model of the target system is learned automatically to supplement testing and verification techniques. At present, model learning is successfully applied to a plurality of industrial fields, and is becoming an efficient vulnerability searching technology.
In 1987, Angluin proposed a famous L*The method provides an Active learning (Active learning) framework of regular language. On the basis, scholars at home and abroad develop active learning algorithms for more formal models, such as Register Automata, Mealy Machine, Nondeteristic finish Automata, I/O Automata, Symbolic Automata and the like. Currently, some active learning libraries and tools, such as libalf, leanlib, Tomte, etc., have been developed by academia and industry based on the above algorithms and frameworks. However, currently, model learning, especially active learning, for real-time systems is still fresh, the library and the tool are mainly focused on systems without time information, the learned formal models lack description of the time information of the systems, however, for some real-time systems with strict time limit constraints, a time limit related error may bring significant loss, and the time is an indispensable dimension when analyzing and verifying the real-time systems, so the tool cannot be applied to such real-time embedded systems. Therefore, there is a need for a way to generate a formal model of temporal automata that incorporates active learning.
Disclosure of Invention
In order to solve the problems, the invention provides a generation method of a time automaton model combining a PAC learning theory and active learning, which adopts the following technical scheme:
the invention provides a generation method of a time automaton model combining PAC learning theory and active learning, which is used for generating a single-clock time automaton form model of a real-time system and is characterized by comprising the following steps: step S1, initializing a time observation table through the learner; step S2, processing the time observation table by the converter to satisfy closeness and consistency; step S3, constructing a hypothesis model through a learner based on the time observation table
Figure GDA0003636290780000021
Step S4, comparing the hypothesis models by the comparator
Figure GDA0003636290780000022
Quality and stability model of
Figure GDA0003636290780000023
And judging whether a counterexample is found; step S5, when step 4 judges no, the teaching device makes a pair of hypothesis models
Figure GDA0003636290780000024
Carrying out PAC approximate equivalence inquiry, judging whether a counterexample is found, stopping iteration when the counterexample is not found, and assuming a model
Figure GDA0003636290780000025
As a result model; and step S6, when the step S4 or the step S5 judges that the time observation table is yes, updating the time observation table according to a counter example, and returning to the step S2, wherein the learner is used for learning and generating the single-clock time automaton model, the time observation table is stored, and the teaching aid is used for verifying the single-clock time automaton model generated by the learner.
The generation method of the time automaton model combining the PAC learning theory and the active learning provided by the invention can also have the technical characteristics that the teaching aid is provided with a target system and a sampling and inquiring part, and the target system is used for running test; the sampling inquiry unit is used for sampling data and carrying out PAC approximate equivalent inquiry by testing the sampled data on a target system.
The generation method of the time automaton model combining the PAC learning theory and the active learning provided by the invention can also have the technical characteristics that:
T＝(∑，∑t，∑r，S，R，E，f)
where Σ is a motion set, ΣtFor a set of logical time words, sigmarTo reset the set of logical time words, S is a set of prefixes, R is a set of boundaries, f is a set of suffixes, f is a classification function,
Figure GDA0003636290780000031
the method for generating the time automaton model combining the PAC learning theory and the active learning according to the present invention may further have the technical feature that the step S2 includes the following sub-steps: in step S2-1, for each action σ ∈ Σ, a logic time word γ ═ (σ,0) is first constructed, and then the corresponding reset logic time word γ is obtained by the converterr(σ,0, b) and added to the boundary set R; step S2-2, judging whether the time observation table meets the closeness, and processing the time observation table to meet the closeness when the judgment is no; and step S2-3, judging whether the time observation table meets the consistency, and processing the time observation table to meet the consistency when the judgment is no.
The generation method of the time automaton model combining the PAC learning theory and the active learning provided by the present invention may further have the technical feature that the step S3 includes the following sub-steps: step S3-1, establishing and determining a finite-state machine according to a time observation table; and step S3-2, converting the finite state machine into a single clock time automaton.
The invention provides generation of a time automaton model combining PAC learning theory and active learningThe method may further have the technical feature that the step S4 includes the following sub-steps: in step S4-1, the comparator generates a minimum time series for discriminating the hypothesis model
Figure GDA0003636290780000041
And stable hypothesis model
Figure GDA0003636290780000042
Step S4-2, a minimum time sequence is executed on the target system; step S4-3, determining whether the execution result of the time series is equal to the hypothesis model
Figure GDA0003636290780000043
If yes, stabilizing the model
Figure GDA0003636290780000044
Updating to a hypothetical model
Figure GDA0003636290780000045
When the judgment is no, the counter example is found.
The method for generating the time automaton model combining the PAC learning theory and the active learning according to the present invention may further have the technical feature that the step S5 includes the following sub-steps: step S5-1, calculating the sampling number according to the set error parameter, confidence parameter and the current equivalent query times; step S5-2, according to the sampling number, sampling to obtain a test case and testing the test case in a target system; step S5-3, judging whether all the tests pass, and if so, assuming the model
Figure GDA0003636290780000046
And (4) as a result model, and finding a counterexample when the judgment is negative.
The method for generating the time automaton model combining the PAC learning theory and the active learning according to the present invention may further have the technical feature that the step S6 includes the following sub-steps: step S6-1, converting the counterexample into a logic time word; step S6-2, sequentially reducing the logic time value of each position of the logic time word and making the logic time value as a counter example; and step S6-3, updating the time observation table according to the processed logic time word, and returning to step S2.
Action and Effect of the invention
According to the generation method of the time automaton model combining the PAC learning theory and the active learning, which is disclosed by the invention, a complete deterministic single-clock time automaton formal model can be generated by adopting an active learning method based on a time observation table and combining a plurality of existing active learning libraries and tools, and the formal model can be used for analyzing and verifying a real-time system, so that the defects of the prior art in the aspect are filled.
Moreover, the generation method of the time automata model of the invention adopts a PAC equivalent approximate query method, so that a quantitative explanation mechanism is provided for the quality of the generated result model, and the subsequent further optimization of the result model and the application in real-time system test and verification are facilitated.
Drawings
Fig. 1 is a schematic diagram of a framework of a method for generating a time automaton model combining PAC learning theory and active learning according to an embodiment of the present invention;
FIG. 2 is a flow chart of method steps for generating a model of a time robot incorporating PAC learning theory and active learning in accordance with an embodiment of the present invention;
fig. 3 is a schematic diagram of a single clock time automaton of an embodiment of the invention.
Detailed Description
In order to make the technical means, creation features, achievement purposes and effects of the present invention easy to understand, the generation method of the time automaton model combining the PAC learning theory and the active learning of the present invention is specifically described below with reference to the embodiments and the drawings.
< example >
Fig. 1 is a schematic diagram of a method for generating a temporal automaton model combining PAC learning theory and active learning according to an embodiment of the present invention.
As shown in fig. 1, the generation method framework includes a learner 11, a tutor 12, a converter 13, and a comparator 14, wherein the converter 13 and the comparator 14 are disposed between the learner 11 and the tutor 12.
The learner 11 is for learning and generating a single-clock-time automata model, storing a time observation table T.
The teaching machine 12 is used for verifying the single-clock-time robot model generated by the learning machine 11, and the teaching machine 12 has a target system 21 and a sampling inquiry unit 22. The sampling inquiry unit 22 samples data and performs PAC approximate equivalence inquiry by testing the sampled data on the target system 21.
The converter 13 is used for converting time words.
The comparator 14 is used to compare the quality of the two models.
The generation method of the time automaton model combining the PAC learning theory and the active learning according to the embodiment of the present invention is specifically described below with reference to the above generation method framework.
Fig. 2 is a flow chart of method steps for generating a temporal automaton model incorporating PAC learning theory and active learning according to an embodiment of the present invention.
As shown in fig. 1 and 2, the single-clock time automaton model is generated by:
in step S1, the learner 11 initializes the time observation table T ═ Σ, (Σ, Σ)t,∑rS, R, E, f), where S ← { ∈ } and R ← { ∈ }.
Step S2, the time observation table is processed by the converter 13 to satisfy closeness and consistency.
Step S2 includes the following substeps:
in step S2-1, for each action σ e ∈ Σ, the learner 11 first constructs a logic time word γ ═ (σ,0), and then performs a membership query through the converter 13 to obtain a corresponding reset logic time word γ thereofr(σ,0, b) and add it to set R, with time observation table T storing the results of all previous member queries;
step S2-2, judging whether the time observation table T meets the closure, if not, the converter 13 executes corresponding operation to make the time observation table T meet the requirement of the closure;
step S2-3, judging whether the time observation table T meets the consistency, if not, the converter 13 executes corresponding operation to make the time observation table T meet the requirement of consistency;
step S3, constructing a hypothesis model by the learner 11 based on the time observation table T
Figure GDA0003636290780000076
Step S3 includes the following substeps:
step S3-1, according to the time observation table T (∑ Σ, Σ)t，∑rS, R, E, f) information to construct a deterministic finite state automaton
Figure GDA0003636290780000071
Finite state set QM＝{qrow(s)|s∈S}，
Set of finite alphabets
Figure GDA0003636290780000072
Finite set of migrations
Figure GDA0003636290780000073
Initial state
Figure GDA0003636290780000074
Wherein the E is equal to the S,
receiving a set of states
Figure GDA0003636290780000075
In step S3-2, the learner 11 converts the finite state automata M into a complete single clock time automata, which is the hypothetical model
Figure GDA0003636290780000081
Wherein Q is QM，
Figure GDA0003636290780000082
F＝FMC is the only clock variable, sigma is the time observation table T ═ sigmat，∑rΣ in S, R, E, f). For the
Figure GDA00036362907800000819
For the migration set Δ in (1), first
Figure GDA0003636290780000083
Figure GDA0003636290780000084
Let set Ψq，σ＝{μ|(q，(σ，μ，b)，q′)∈ΔMUsing a partition function to divide the set Ψq，σIs allocated to the interval I1，I2…IkAnd satisfies k ═ Ψq，σ|∧μi∈IkThen for each relevant migration in M (q, (σ, μ)i，bi)，q′)∈ΔMAdding the corresponding transitions (q, sigma, I) to the set of transitions deltai，bi，q′)。
In step S4, the comparator 14 determines the hypothesis model
Figure GDA0003636290780000085
Whether the quality of (2) is higher than that of the stable model
Figure GDA0003636290780000086
Wherein the comparator 14 sets a distance metric function as a metric for measuring the quality of the model, which is defined as follows:
order to
Figure GDA0003636290780000087
And
Figure GDA0003636290780000088
is divided into twoDeterministic single clock time automaton
Figure GDA0003636290780000089
And
Figure GDA00036362907800000810
the identified temporal language, then the distance metric function can be expressed as:
Figure GDA00036362907800000811
wherein n is a number
Figure GDA00036362907800000812
And
Figure GDA00036362907800000813
the length of the minimum time series.
Step S4 includes the following substeps:
in step S4-1, the comparator 14 generates a logical time series ω of minimum length for discriminating the hypothesis models
Figure GDA00036362907800000814
And a stable model
Figure GDA00036362907800000815
Step S4-2, the time series ω is tested on the target system 21 due to the hypothetical model
Figure GDA00036362907800000816
And a stable model
Figure GDA00036362907800000817
The time sequence omega generated by the comparator 14 can be directly converted into a delay time word for testing;
step S4-3, judging whether the test result is the same as the hypothesis model
Figure GDA00036362907800000818
Consistent, if the target system 21 tests results with the hypothesis model
Figure GDA0003636290780000091
If the results of the performance are not consistent, the target system 21 tests the returned ωrIs the hypothetical model found by the comparator 14
Figure GDA0003636290780000092
The counter example is directly returned to the learner 11 for reconstructing a new hypothesis model, and at this time, the learner 11 does not need to determine the target system 21 and the hypothesis model by the PAC equivalence query
Figure GDA0003636290780000093
Since counterexample has been found, if the target system 21 tests results with the hypothesis model
Figure GDA0003636290780000094
If the results of the performance are consistent, the model will be assumed
Figure GDA0003636290780000095
Set to a new stable hypothesis
Figure GDA0003636290780000096
That is, in step S4, for each newly constructed hypothesis model
Figure GDA0003636290780000097
Before the PAC equivalent query is presented to the tutor, it is associated with the current stable hypothesis
Figure GDA0003636290780000098
The comparison is performed, only by the hypothetical model of comparator 14
Figure GDA0003636290780000099
Proceed to the next PAC equivalence query such thatThe quality of the model (based on the distance metric function) for each PAC equivalent query is no worse than the previous one.
Step S5, the teaching device 12 is used to make the hypothesis model
Figure GDA00036362907800000910
Carrying out PAC approximate equivalence query, judging whether a counterexample is found, if not, stopping iteration, and assuming a model
Figure GDA00036362907800000911
As a result model. Wherein the PAC approximate equivalent query is done by a number of sampling tests, the number being defined as:
the learner 11 has learned a deterministic single-clock-time automaton model, based only on its current time of the ith equivalent query
Figure GDA00036362907800000912
Random sampling of probability distribution
Figure GDA00036362907800000913
No counter example is found after the test of the time word.
Step S5 includes the following substeps:
step S5-1, according to the hypothesis model
Figure GDA00036362907800000914
Calculating the required sampling number by the error parameter epsilon, the confidence parameter delta and the current equivalent query times i;
step S5-2, according to the calculated sampling number, the sampling inquiry unit 22 obtains the test case from the fixed distribution P by sampling randomly and continuously, and for each sample omega (i.e. delay time word) obtained by sampling, the target system 21 and the hypothesis model
Figure GDA0003636290780000101
Testing the same;
step S5-3, judging whether the above tests are all passed, and if the target system 21 and the hypothesis model are all passed, sampling each sample omega
Figure GDA0003636290780000102
If the test results of (a) are not consistent, a counterexample is found and returned, otherwise, if no counterexample is found for all the samples ω, the PAC approximate equivalence query returns a positive answer, at which point the learning algorithm terminates and returns to the current hypothesis model
Figure GDA0003636290780000103
As a result model, a hypothesis model
Figure GDA0003636290780000104
As PAC (∈, delta) -correct, i.e. hypothesis model
Figure GDA0003636290780000105
Considered to be approximately equivalent to the target system 21 under the current probability distribution, with a speculation error of less than e and a confidence level of greater than 1- δ.
Fig. 3 is a schematic diagram of a single clock time automaton of an embodiment of the invention.
The resulting model is in the form of a complete deterministic single clock time automaton, as shown in fig. 3.
The distribution adopted for sampling in step S5 is such that half of the total distribution is composed of time series valid for the system and the other half includes time series possibly invalid, the half of the time series being obtained by introducing random variations to the valid time series. Specifically, the length of the sample sequence is first chosen uniformly between 1 and an upper limit, and then the target system is randomly walked from the initial position, which records a series of action and logical time boundaries of the walked migration. Next, sampling logic time uniformly according to the obtained logic time boundary, and setting a constraint condition, if the local clock is not reset by the previous migration, the logic time of the current sampling must be greater than or equal to the previous logic time, and if no such logic time value exists, the current random walk path is invalid, and a new round of sampling is restarted. Also, during the sampling process, the logical time is actually sampled from the allowed region (region) of the logical time, which ensures that we can use time values up to an integer to test, otherwise, most of the sampled time values will contain fractions.
In step S6, when the counter example is found in step S4 or step S5, the time observation table T is updated according to the counter example, and the process returns to step S2 to start a new round of learning.
Step S6 includes the following substeps:
in step S6-1, after finding the counterexample (i.e. the delay time word), the counterexample is applied to the hypothesis model
Figure GDA0003636290780000111
Executing and obtaining a corresponding reset sequence and a converted logic time word;
step S6-2, performing a reduction process on the logical time value of each position of the logical time word from the beginning, when the logical time value is reduced, if the local clock is not reset after the previous time action occurs, the reduced logical time value cannot be smaller than the previous logical time, after the reduction process is completed, performing a member query on the logical time word obtained after the reduction process through the converter 13, if the logical time word after the current reduction process is still a reverse example, continuing to reduce the logical time value of the current position, otherwise, restoring the logical time value of the current position to the time value before the current reduction process, then reducing the logical time value of the next position of the logical time word, repeating the above operations until the reduction of the logical time value of the last position of the logical time word is completed, after the operation of minimizing the reverse example is completed, still in the form of a reset delay time word back to learner 11;
in step S6-3, the time observation table T is updated according to the processed logical time word (i.e. the reset delay time word), and the process returns to step S2, and the learner 11 starts a new round of learning.
By the generation method, a complete deterministic single-clock-time automaton model is generated by combining the PAC learning theory and active learning.
Examples effects and effects
According to the generation method of the time automaton model combining the PAC learning theory and the active learning provided by the embodiment, a complete deterministic single-clock time automaton formal model can be generated by adopting an active learning method based on a time observation table and combining a plurality of existing active learning libraries and tools, and the formal model can be used for analyzing and verifying a real-time system, so that the defects of the prior art in the aspect are overcome.
Moreover, the generation method of the time automaton model of the embodiment adopts the PAC equivalence approximate query method, so that a quantitative explanation mechanism is provided for the quality of the generated result model, which is beneficial to the subsequent further optimization of the result model and the application in the real-time system test and verification.
The above-described embodiments are merely illustrative of specific embodiments of the present invention, and the present invention is not limited to the description of the above-described embodiments.

Claims (6)

1. A method for generating a single-clock time automaton model for a real-time system, the method comprising:
step S1, initializing a time observation table through the learner;
step S2, processing the time observation table to satisfy closeness and consistency through a converter;
step S3, constructing a hypothesis model through the learner based on the time observation table
Figure FDA0003636290770000011
Step S4, comparing the hypothesis models by a comparator
Figure FDA0003636290770000012
Quality and stability ofModel (model)
Figure FDA0003636290770000013
And judging whether a counterexample is found;
step S5, when the step S4 is judged as NO, the hypothesis model is executed by the teaching device
Figure FDA0003636290770000014
Carrying out PAC approximate equivalence inquiry, judging whether a counterexample is found, stopping iteration when the counterexample is judged not to be found, and enabling the hypothesis model
Figure FDA0003636290770000015
As a result model;
a step S6, when the step S4 or the step S5 determines yes, according to the counter example, updating the time observation table, and returning to the step S2,
wherein the learner is configured to learn and generate the single clock time automaton model, the time observation table being stored,
the learner is configured to validate the single-clock-time automaton model generated by the learner,
the time observation table is as follows:
T＝(Σ,∑t,∑r,S,R,E,f)
where Σ is the set of actions, ΣtFor a set of logical time words, sigmarTo reset the set of logical time words, S is the set of prefixes, R is the set of boundaries, E is the set of suffixes, f is the classification function,
Figure FDA0003636290770000021
the step S2 includes the following sub-steps:
step S2-1, for each action σ ∈ Σ, construct a logical time word γ ═ (σ,0), and then obtain, through the converter, the corresponding reset logical time word γr(σ,0, b) and add it to the boundary set R;
step S2-2, judging whether the time observation table meets the closeness, and processing the time observation table to meet the closeness when the judgment is no;
and step S2-3, judging whether the time observation table meets the consistency, and processing the time observation table to meet the consistency when the judgment is no.
2. The method of generating a temporal automaton model combining PAC learning theory and active learning according to claim 1, wherein:
wherein the teaching tool comprises a target system and a sampling inquiry part,
the sampling inquiry unit is configured to sample data and perform the PAC proximity equivalence inquiry by testing the sampled data on the target system.
3. The method of generating a time automaton model combining PAC learning theory and active learning according to claim 1, wherein:
wherein the step S3 includes the following sub-steps:
step S3-1, constructing and determining a finite state machine according to the time observation table;
step S3-2, converting the finite state machine into a single clock time automaton, wherein the single clock time automaton is an assumed model
Figure FDA0003636290770000031
4. The method of generating a time automaton model combining PAC learning theory and active learning according to claim 2, wherein:
wherein the step S4 includes the following sub-steps:
step S4-1, the comparator generates a minimum time series for distinguishing the hypothesis models
Figure FDA0003636290770000032
And the stable model
Figure FDA0003636290770000033
Step S4-2, the minimum time series being executed on the target system;
step S4-3, judging whether the execution result is the same as the hypothesis model
Figure FDA0003636290770000034
If yes, the stable model is used
Figure FDA0003636290770000035
Updating to the hypothesis model
Figure FDA0003636290770000036
And when the judgment is no, finding the counter example.
5. The method of generating a temporal automaton model combining PAC learning theory and active learning according to claim 2, wherein:
wherein the step S5 includes the following sub-steps:
step S5-1, calculating the sampling number according to the set error parameter, confidence parameter and the current equivalent query times;
s5-2, sampling according to the sampling number to obtain test cases, and for each test case, in the target system and the hypothesis model
Figure FDA0003636290770000037
Testing the same;
step S5-3, judging whether the tests are all passed, if so, then the hypothesis model
Figure FDA0003636290770000038
For the result model, when judged no,then the counterexample is found.
6. The method of generating a temporal automaton model combining PAC learning theory and active learning according to claim 1, wherein:
wherein the step S6 includes the following sub-steps:
step S6-1, converting the counter example into the logic time word;
step S6-2, sequentially reducing the logic time value of each position of the logic time word and making the logic time value be the counter example;
and step S6-3, updating the time observation table according to the processed logic time word, and returning to the step S2.
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