CN105183652A - Temporal dynamic push-down network converting method - Google Patents

Abstract

The invention discloses a time dynamic push-down network conversion method, which is used to describe real-time concurrent recursive modeling including recursion and dynamic thread creation. Firstly, a global clock that describes continuous time and a real-number clock that can describe the "age" of time-related global variables and stack characters are introduced in DPN, so that asynchronous communication based on shared memory and real-time concurrent system with dynamic thread creation can be implemented. for modeling. Secondly, for the clock equivalence technology based on integer division, an optimization technology based on clock key points is given to reduce the clock interval, thereby reducing the converted state space. Since the time dynamic pushdown network is an abstract model of a real-time concurrent recursive program, the clock equivalent optimization technology based on key points converts the model into a dynamic pushdown network, so that by confirming whether the execution of the dynamic pushdown network model will run to Error status to detect bugs or vulnerabilities in the concurrent recursive program corresponding to this model.

Description

A Transformation Method for Temporal Dynamic Pushdown Networks

technical field

The invention belongs to the field of software security and reliability research, relates to a verification method for a multi-threaded concurrent recursive program, is a reachability solution technology applicable to an abstract model of a multi-threaded concurrent recursive program containing time, and specifically relates to a time dynamic The transformation method for the pushdown network.

Background technique

With the development of multi-core technology, concurrent programs have become a hot spot in current program design research. Due to the uncertainty of concurrent execution, it is difficult for traditional testing methods to find hidden errors and loopholes in programs. Model checking is an automatic verification technology through exhaustive search, which has become an important means to ensure the safety and reliability of programs, and can be used as a supplement to testing methods. Reachability analysis is an important core technology of model checking by analyzing whether a certain state is reachable.

In recent years, researchers have introduced real-time clocks based on automata models to describe real-time system modeling and verification. In 1994, Alur proposed time automata (R.Alur, D.Dill, Atheory of timed automata [J]. TheoreticalComputerScience, 126(2), pp.183-235, 1994.) is based on automata and introduces a clock describing continuous time. , and the clock equivalent technology is given to realize the model checking time automaton (J.Bengtsson, W.Yi, Timedautomata: Semantics, algorithms and tools [C], 4thAdvancedCourseonPetriNets, Eichstaat.Germany, pp.87-124, 2004). In order to solve the modeling of real-time systems containing recursion, Trivedi (A.Trivedi, D.Wojtczak, RecursiveTimedAutomata[C], ATVA2010.LNCS, vol.6252, Springer, Heidelberg, pp.306–324.2010.) proposed a timed pushdown automata , and convert the time push-down automaton into push-down automaton through the clock equivalence technique to solve the accessibility problem of minimum time cost. In 2013, Li (LGqiang, CXjuan, O.Mizuhito, NestedTimedAutomata[R], Research report (School of Information Science, Japan Advanced Institute of Science and Technology), IS-RR-2013-004, pp.1-20, 2013) proposed a nested timed automata, using nested Set of ideas to solve recursive problems in real-time systems. But such models cannot describe the modeling of real-time concurrent systems with dynamic thread creation. Bouajjani (A.Bouajjani, M.M.Olm, T.Touili.Regularsymbolicanalysisofdynamicnetworksofpushdownsystems[C].Proceedingsofthe16thInternationalConferenceonConcurrencyTheory.LNCS3653, SanFrancisco: CiscoSyst, 2005, 473-487.) proposed a concurrent pushdown system, R.Bouskellig .Mennicke, The complexity of model checking multi-stack systems [C], Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, New Orleans, LA, USA, pp.163-72, 2013) extended model - dynamic pushdown network (DPN), to solve the dynamic creation of new threads in concurrent systems , which is suitable for modeling concurrent systems with recursion and dynamic thread creation. Based on DPN, Lammich (P.Lammich, M.M.Olm, H.Seidl, ContextualLockingforDynamicPushdownNetworks[C].StaticAnalysis.Proceedingsof20thInternationalSymposium, Seattle, WA, USA, pp.47-98, 2013.) proposed a context lock technology to solve process recursion The synchronization problem between them, and a reverse reachability analysis was carried out. Wenner (A.Wenner, Weighteddynamicpushdownnetworks[C], 19thEuropeansymposiumonprogramming, Paphos, Cyprus, pp.590-609, 2010.) introduces weights in DPN to solve the accessibility of the shortest path.

Since the above model cannot describe the interaction between threads in a real-time multi-threaded concurrent recursive system, for real-time multi-threaded concurrent recursive programs, the formal verification of such programs will cause the state space explosion problem, which brings great difficulties to the verification.

Contents of the invention

The technical problem to be solved by the present invention is to provide a conversion method of a time dynamic push-down network. The time dynamic push-down network is an abstract model of a real-time concurrent recursive program, and the model is converted based on the clock equivalent optimization technology of key points For the dynamic push-down network, by confirming whether the execution of the dynamic push-down network model will run into an error state, the error or vulnerability in the corresponding concurrent recursive program of this model is detected.

In order to solve the above problems, the present invention is achieved through the following technical solutions:

The conversion method of time dynamic push-down network includes the following steps:

Step (1) converts the real-time concurrent recursive program into a time dynamic push-down network.

Step (1.1) Construct the abstract model of real-time concurrent recursive program, that is, the temporal dynamic pushdown network.

The constructed temporal dynamic push-down network is a quadruple T=(P, Γ, Δ, X), where: P is the state set; Γ is the stack character set; Δ=Δ _nop ∪Δ _＝ ∪Δ _├ ∪Δ _push ∪Δ _pop ∪Δ _dc is a set of migration rules, where Δ _nop means no-operation migration, Δ ₌ means clock assignment migration, Δ _├ means time lapse migration, Δ _push means stack migration, Δ _pop means stack migration, Δ _dc Represents dynamic thread creation migration; X represents the clock set, and its value function express for When the current value is θ(x), the value of the time-related global variable and the stack character "age" is also similar.

The constructed temporal dynamic push-down network pattern Indicates the state of the model at a certain moment, where: A binary group <g,θ(g)> representing the current global variable g and its "age"θ(g); p _i ∈ P represents a local state node, A binary group <ω _i ,θ(ω _i )> representing the stack content ω _i with the stack sequence i and its "age" θ(ω _i ); Represents the binary group <x,θ(x)> of the clock x and its value θ(x).

Step (1.2) describes the constructed temporal dynamic pushdown network with operational semantics.

As a model of real-time multi-threaded programs, the temporal dynamic _pushdown network _is used to describe the _migration of _{multiple pushdown systems} at the same time. The migration action gives its execution meaning;

1) When Δ=Δ _nop , op=nop, Indicates that the elements in the pattern have not changed;

2) When Δ=Δ ₌ , op=x←I, c∈I; means assign any value v within the range of I to the clock x, and the elements in other patterns do not change;

3) When Δ=Δ _├ , op=Time←c, suppose So ${\stackrel{\&OverBar;}{\mathrm{\&omega;}}}^{+v}=<a_1,x_1+v><a_2,x_2+v>...<a_{\mathrm{no}},x_{\mathrm{no}}+v>,{\stackrel{\&OverBar;}{x}}^{\&prime;}={\stackrel{\&OverBar;}{x}}^{+v};$ Indicates that all clocks in the pattern increase v, and the non-clock content in the pattern does not change;

4) When Δ=Δ _push , op=push(a, I), v ∈ I, Indicates that variable a is pushed into the top of the stack, and the corresponding clock is set as x, and its clock value is any value within the range of I;

5) When Δ=Δ _pop , op=pop(a,I), v ∈ I, Represent that the variable a whose clock value in the top of the stack is 1 range is popped up;

6) When Δ=Δ _dc , op=dc, Indicates the creation of a new thread stack content.

Step (2) Convert the temporal dynamic push-down network T=(P,Γ,Δ,X) obtained in step (1) into a dynamic push-down network M=(P ^M ,Γ ^M ,Δ ^M );

Step (2.1) transition of state P ^M : That is, the state set of T is the same as the state set of M.

Step (2.2) stack character set Transformation: If a∈{Γ,├}, then and

Step (2.3) is the conversion rule of migration relation Δ to ^ΔM .

Assuming that the current TDPN contains n stacks, for the convenience of description, only the push-down system with sequence i is described to perform stack push and pop operations, and other stack operations are similar. Assume that the stack depth of the push-down system is l, and the number of the bottom of the stack is 1, and the number of the top of the stack is l. The TDPN contains the global variable g, the clock variable x, and the stack content ω={ω ₁ …ω _i …ω _n }, where ω _i represents the stack content of the push-down system of number i, and ω _i is projected on Γ by ω _il| Γ The top character of the stack. Each represents a keypoint whose "age" and value are under clock equivalence. Thus it can be known that the equivalent domain corresponding to the current clock in M in Indicates the character at the top of the record stack in domain R _l , ├ indicates the reference clock character in domain R _l , and ├ ^· indicates the time lapse character corresponding to domain R _l .

TDPN pattern φ=(γ, op, γ′) ∈ Δ represents the pattern migration of T, and the corresponding pattern migration of M can be expressed as Among them, p and p′ are the same as the state in T, and respectively represent the state before and after the pattern migration; R _l = {R _1l ... R _il ... R _nl } represents the stack top domain, where R _il represents the stack top domain of the number i push-down system , R _l and R _l ′ represent the stack top domains before and after pattern migration respectively; the action migration set op′ corresponds to the op of T, and the following mainly describes the construction of R _l ′ according to different ops:

1) When op=nop, for If and only if there exists in M In T, the empty operation only changes the state, so in M, the pattern migration only changes the state, and the domain R _l remains unchanged.

2) When op=(x←I), for If and only if there exists in M This transition relation represents the execution of the item whose clock is x in the DPN domain R _l operation, where θ(x)′∈I, to construct the domain R _l ′. The specific implementation process is as follows:

Pop the field R _l out of the stack and get the items in R _l Reset θ(x) to θ(x)′, forming a new term

● item instead of terms in the field R _l Obtain the field R _l ′, put it into the stack, and switch to the new state p′.

3) When op=(Time←v), for If and only if there exists in M This transition relation means that except the reference clock item (├, 0) in the domain R _l , the clock values of all other items plus the time elapsed v are used to construct the domain R _l ⁺ . The specific execution steps are:

The field R _l is popped out of the stack, except for the reference clock, all add the time elapsed v, Denotes a new general term corresponding to g, ω _il , x, Each represents the corresponding new entry, Indicates the reference clock entry;

●The new item replaces the original item, obtains the field R _l ⁺ , puts it into the stack, and transitions to the new state p′.

4) When op=push(a, I), for if and only if M exists The migration relationship indicates that the push-down system for number i is pushed into the stack, and the character is a, and the value is Items of the stack field R _il are used to construct the field R _i(l+1) . The specific process is as follows:

● Get items from R _il and

● replace respectively Obtain the field R _i(l+1) and put it on the stack, and switch to the new state p′.

5) When op=pop(a, I), for if and only if M exists The transition relation indicates that the field R _il ′ is constructed by popping the stack character a in the stack domain R _il and the item θ(a)∈I. The specific steps are described as follows:

● Pop domain R _il and domain R _i(l-1) to get items in domain R _il

The clock value of all items in the domain R _(l-1) is added to θ(├ ^· ) to obtain the domain R _i(l-1) ';

● Obtain the field R _il ′ through R _il and R _i(l-1) ′, and the items of R _il ′ are: common stack character items come from field R _i(l-1) ′; common clock items and global variable items come from field R _il ; the record items all come from the field R _i(l-1) ′;

● Push the domain R _il ' into the stack, and switch to the new state p'.

6) When when, for If and only if there exists in M This migration relation means that a new thread is created to construct domain R _l '. Assuming the TDPN model, the stack number for dynamically creating a new push-down system is n+1. The specific execution steps are as follows:

● Popping domain R _l , based on the clock equivalence optimization technology of key points, ordinary character items can be obtained entry

●Put item and Add field R _l to get field R _l ′, put it on the stack, and switch to new state p′.

The research of the present invention is based on Time Dynamic Pushdown Networks (TimeDynamicPushdownNetworks, TDPN), which is used to describe real-time concurrent recursive modeling with recursion and dynamic thread creation. Firstly, a global clock that describes continuous time and a real-number clock that can describe the "age" of time-related global variables and stack characters are introduced in DPN, so that asynchronous communication based on shared memory and real-time concurrent system with dynamic thread creation can be implemented. for modeling. Secondly, for the clock equivalence technology based on integer division, an optimization technology based on clock key points is given to reduce the clock interval, thereby reducing the converted state space. In order to further reduce the state space, the continuous model TDPN is converted into the corresponding DPN by using the dynamic conversion method that only focuses on the top of the stack, and the corresponding conversion algorithm is given at the same time. Then prove the correctness of its conversion and the consistency of TDPN reachability, so that the existing DPN reachability technology can be used to solve the TDPN reachability problem.

Description of drawings

Figure 1 shows the equivalence interval of x _i and x _j clocks.

Detailed ways

The present invention proposes an accessibility analysis method for the time dynamic push-down network model. For the time dynamic push-down network model, the clock equivalence optimization technology based on key points is adopted to dynamically convert the continuous time dynamic push-down network model It is an automatic method for solving the reachability problem of the time dynamic pushdown network model for a discrete dynamic pushdown network model.

1. The stage of constructing TDPN model: TDPN model is an extension of DPN model. The basic idea is to introduce a real-time clock describing continuous time into DPN, which is used to describe real-time concurrent recursive programs with dynamic thread creation.

Based on the syntax and semantics of TMPDN, the conversion methods for converting real-time concurrent programs into TMPDN are divided into three categories: intra-stack migration conversion, inter-stack switching conversion and concurrent execution conversion.

1. Constructing an abstract model of concurrent programs - time dynamic push-down network

The TDPN model is a quaternion T=(P, Γ, Δ, X), where: P is the state set; Γ is the stack character set; Δ=Δ _nop ∪Δ ₌ ∪Δ _├ ∪Δ _push ∪Δ _pop ∪Δ _dc is a set of migration rules, wherein Δ _nop represents no-operation migration, Δ ₌ represents clock assignment migration, Δ _├ represents time lapse migration, Δ _push represents stack migration, Δ _pop represents stack migration, and Δ _dc represents dynamic thread creation migration; X represents the clock set, and its value function express for When the current value is θ(x), the value of the time-related global variable and the stack character "age" is also similar.

When the TDPN model is migrated, multiple pushdown systems are executed concurrently, that is, multiple stacks are migrated at the same time. Assuming that G is a global variable set, the TDPN structure can be expressed as: in: A binary group <g,θ(g)> representing the current global variable g and its "age"θ(g); p _i ∈ P represents a local state node, A binary group <ω _i ,θ(ω _i )> representing the stack content ω _i with the stack sequence i and its "age" θ(ω _i ); Represents the binary group <x,θ(x)> of the clock x and its value θ(x). The migration action set op of TDPN includes the empty operation nop; clock reset x←I, where x∈X, I represents the value range of the clock; time lapse Time←v, where Time represents the elapsed time, and v represents the real value of the specific elapse , the present invention uses θ+v to represent that the clock value increases v without ambiguity; push operation push (a, I) into the stack, represent the character a that pushes into the stack, and its age assignment belongs to the real number of interval I; the stack operation pop( a, I), first judge whether the "age" of the stack symbol a satisfies the interval I, if it is satisfied, perform the stack operation, otherwise do not perform the operation; dynamic thread creation operation Indicates that a new thread is dynamically created.

2. Operational Semantics of TDPN

For the convenience of description, when the layout is migrated, only the migration execution of one pushdown system is described, and the rest of the pushdown systems remain unchanged. The concurrent execution of multiple pushdown systems is also similar. The operational semantics of the TDPN layout migration relationship are defined as follows:

1) When Δ=Δ _nop , op=nop, Indicates that the elements in the pattern have not changed;

2) When Δ=Δ ₌ , op=x←I, c∈I; means assign any value v within the range of I to the clock x, and the elements in other patterns do not change;

3) When Δ=Δ _├ , op=Time←c, suppose So ${\stackrel{\&OverBar;}{\mathrm{\&omega;}}}^{+v}=<a_1,x_1+v><a_2,x_2+v>...<a_{\mathrm{no}},x_{\mathrm{no}}+v>,{\stackrel{\&OverBar;}{x}}^{\&prime;}={\stackrel{\&OverBar;}{x}}^{+v};$ Indicates that all clocks in the pattern increase v, and the non-clock content in the pattern does not change;

4) When Δ=Δ _push , op=push(a, I), v ∈ I, Indicates that variable a is pushed into the top of the stack, and the corresponding clock is set as x, and its clock value is any value within the range of I;

5) When Δ=Δ _pop , op=pop(a,I), v ∈ I, Represent that the variable a whose clock value in the top of the stack is 1 range is popped up;

6) When Δ=Δ _dc , op=dc, Indicates the creation of a new thread stack content.

2. Conversion method design stage: In order to analyze the accessibility of TDPN, it is necessary to abstract it and reduce its state space. The present invention is based on the clock equivalent optimization technology of key points, uses the idea of on-the-fly, and only cares about the stack The conversion of the top dynamically abstracts the continuous model into a discrete DPN model, and then uses the existing DPN reachability analysis technology to solve the reachability problem of TDPN.

In order to reduce the transformed state space, the division of the time region is firstly reduced. Compared with the integer division region, the clock equivalence optimization technology based on key points can reduce the transformed state space exponentially. Secondly, based on this technology, the variables related to continuous time in the TDPN model--global variables and their "ages", stack characters and their "ages", clock variables and their values, are converted into DPN models that describe the stack content of DPN area.

Use clock equivalent technology to convert continuous time to discrete time, and use the idea of dynamic conversion to convert pattern migration one by one until all pattern calculations are completed.

1. Clock optimization technology based on key points

In order to describe the pattern converted into a DPN system, the concept of domain R is introduced. Domain R is composed of a set of items r, and item r is composed of character set Z and key point set key.

Character set Z includes common character set Y and record character set Y ^· , described as Z=Y∪Y ^· . The common character set Y includes: (1) clock set X; (2) stack character Γ; (3) global variable set G; (4) reference clock character ├, which is used to record the passage of time, unless the stack operation is performed, otherwise It is always 0, so the common character set Y can be described as Y=X∪Γ∪G∪{├}. Record character set Y represents the time lapse ^of ordinary character Y ^, let X ＝{x ＝ |x∈X} represent ^the record global clock; Γ ＝{a ^· | ^a∈Γ } represents ^the record stack character; G ＝{ g | ^g∈G } means to record global variables; {├ ^} means to record the reference clock character set, which is mainly used to record the passage of time so that the time can be updated ^when popping the stack, so the record character set is described as Y ＝X ^{∪Γ ·} ∪G ^· ∪{├ ^· }.

For any time-related character set, its time migration is determined by some key points, and has a maximum value, so for any character, you can define a time-related maximum constant k _Max , all greater than k _Max are represented by the symbol ∞. For any character z _i ∈ Z, according to the migration of time conversion, the time key point that determines the migration can be found, so the time key point set key _{i of character z i =} {0, k _i1 ,..., k _il ,..., k _im , k _iMax , ∞}, where 1<l<m, 1<i≤|Z|. According to the above character set and key point set, the domain can be obtained $R=r_1...r_{\mathrm{no}}\&Element;{\left(2^{\underset{i=1}{\overset{\left|Z\right|}{\&Sigma;}}{z}_i\&times;{\mathrm{key}}_i}\right)}^{+}.$

The clock equivalent optimization technique is to discretize the continuous clock value, that is, the real clock value Divided into two parts: (1) key point part Indicates that the key points are taken downward for the real value x; (2) the remaining part

Suppose x ∈ X is any clock element in TDPN, denote the real value of clock x by θ(x), Indicates that the domain equivalence is performed on the clock x, and the key point part is taken. Assume that for any two clocks x _i , x _j ∈ X, when the clock value satisfies the following rules, it is an equivalent clock:

(1) θ( _xi )>k _iMax if and only if That is, the clock value of _xi is greater than the maximum value, and _xi takes infinity ∞.

(2)k _il ≤θ(x _i )<k _i(l+1) if and only if That is, when the clock value of x _i is less than the key point k _i(l+1) and greater than or equal to k _il , x _i takes the key point k _il .

(3) assume If and only if re(θ( _xi )<re(θ(x _j )), that is, x _i takes the key point k _il , x _j takes the key point k _jl , when the clock value of _xi and the key point k _il When the difference is smaller than the difference between the clock value of x _j and the key point k _jl , it is recorded as re( _xi )<re(x _j ).

Assuming that for any two clocks _xi , x _j ∈X and key={key ₁ , . . . , key _n }, clock domain equivalence rules based on clock key points. For example (0≤θ(x _i )<k _i1 , 0≤θ(x _j )<k _j1 ) area, And re(θ(x _i ))>re(θ(x _j )); in (k _i1 ≤θ(x _i )<k _i2 , k _j1 ≤θ(x _j )<k _j2 ) area, And re(θ( _xi ))<re(θ(x _j )), then the clock values in the interval are equivalent clocks, as shown in the shaded part in Figure 1.

2. Construct conversion rules

In order to further reduce the state space, adopt the on-the-fly technology and use the idea of dynamic conversion, only focus on the domain conversion of the top of the stack and the next layer, without caring about other parts of the stack, and then give different conversion rules according to different migrations, so that The complex TDPN reachability problem can be transformed into a DPN reachability problem.

Assuming a given TDPNT=(P, Γ, Δ, X), dynamically convert T to DPNM=(P ^M , Γ ^M , Δ ^M ), the dynamic conversion rules for ^PM , Γ ^M , Δ ^M are as follows:

(I) State set P ^M : That is, P ^M =P.

(Ⅱ) Stack character set If a ∈ {Γ, ├}, then and

(Ⅲ) Construction of migration relationship Δ ^M :

Assuming that the current TDPN contains n stacks, for the convenience of description, only the push-down system with sequence i is described to perform stack push and pop operations, and other stack operations are similar. Assume that the stack depth of the push-down system is l, and the number of the bottom of the stack is 1, and the number of the top of the stack is l. The TDPN contains the global variable g, the clock variable x, and the stack content ω={ω ₁ …ω _i …ω _n }, where ω _i represents the stack content of the push-down system of number i, and ω _i is projected on Γ by ω _il| Γ The top character of the stack. Each represents a keypoint whose "age" and value are under clock equivalence. Thus it can be known that the equivalent domain corresponding to the current clock in M in Indicates the character at the top of the record stack in domain R _l , ├ indicates the reference clock character in domain R _l , and ├ ^· indicates the time lapse character corresponding to domain R _l .

TDPN pattern φ=(γ, op, γ′) ∈ Δ represents the pattern migration of T, and the corresponding pattern migration of M can be expressed as Among them, p and p′ are the same as the state in T, and respectively represent the state before and after the pattern migration; R _l = {R _1l ... R _il ... R _nl } represents the stack top domain, where R _il represents the stack top domain of the number i push-down system , R _l and R _l ′ represent the stack top domains before and after pattern migration respectively; the action migration set op′ corresponds to the op of T, and the following mainly describes the construction of R _l ′ according to different ops:

(1) When op=nop, for If and only if there exists in M In T, the empty operation only changes the state, so in M, the pattern migration only changes the state, and the domain R _l remains unchanged.

(2) When op=(x←I), for If and only if there exists in M This transition relation represents the execution of the item whose clock is x in the DPN domain R _l operation, where θ(x)′∈I, to construct the domain R _l ′. The specific implementation process is as follows:

Pop the field R _l out of the stack and get the items in R _l Reset θ(x) to θ(x)′, forming a new term

● item instead of terms in the field R _l Obtain the field R _l ′, put it into the stack, and switch to the new state p′.

(3) When op=(Time←v), for If and only if there exists in M This transition relation means that except the reference clock item (├, 0) in the domain R _l , the clock values of all other items plus the time elapsed v are used to construct the domain R _l ⁺ . The specific execution steps are:

The field R _l is popped out of the stack, except for the reference clock, all add the time elapsed v, Denotes a new general term corresponding to g, ω _il , x, Each represents the corresponding new entry, Indicates the reference clock entry;

●The new item replaces the original item, obtains the field R _l ⁺ , puts it into the stack, and transitions to the new state p′.

(4) When op=push(a,I), for if and only if M exists The migration relationship indicates that the push-down system for number i is pushed into the stack, and the character is a, and the value is Items of the stack field R _il are used to construct the field R _i(l+1) . The specific process is as follows:

● Get items from R _il and

● replace respectively Obtain the field R _i(l+1) and put it on the stack, and switch to the new state p′.

(5) When op=pop(a,I), for if and only if M exists The transition relation indicates that the field R _il ′ is constructed by popping the stack character a in the stack domain R _il and the item θ(a)∈I. The specific steps are described as follows:

● Pop domain R _il and domain R _i(l-1) to get items in domain R _il

The clock value of all items in the domain R _(l-1) is added to θ(├ ^· ) to obtain the domain R _i(l-1) ';

● Obtain the field R _il ′ through R _il and R _i(l-1) ′, and the items of R _il ′ are: common stack character items come from field R _i(l-1) ′; common clock items and global variable items come from field R _il ; the record items all come from the field R _i(l-1) ′;

● Push the domain R _il ' into the stack, and switch to the new state p'.

(6) when when, for If and only if there exists in M This migration relation means that a new thread is created to construct domain R _l '. Assuming the TDPN model, the stack number for dynamically creating a new push-down system is n+1. The specific execution steps are as follows:

● Popping domain R _l , based on the clock equivalence optimization technology of key points, ordinary character items can be obtained entry

●Put item and Add field R _l to get field R _l ′, put it on the stack, and switch to new state p′.

3. In the algorithm design stage: Based on the idea of clock equivalent optimization and dynamic conversion, an algorithm for converting TDPNT=(P, Γ, Δ, X) to DPNM=(P ^M , Γ ^M , Δ ^M ) is proposed, which Based on the migration relationship Δ of T, the algorithm exhaustively calculates the migration relationship Δ ^M corresponding to M through conversion rules.

Based on the idea of clock equivalent optimization and dynamic conversion, an algorithm for converting TDPNT=(P, Γ, Δ, X) to DPNM=(P ^M , Γ ^M , Δ ^M ) is proposed. The algorithm is aimed at the migration relationship Δ of T, through The conversion rules in the previous section exhaustively calculate the migration relationship Δ ^M corresponding to M. The input of the algorithm is continuous TDPNT, and the output is discrete DPNM. Suppose the initial pattern of TDPN is The content of each stack is initially empty, which corresponds to the construction of the initial field R _init of M.

If there is φ=(γ,op,γ′)∈Δ in the set of transfer relations of T, its pattern includes global variables stack string clock The current configuration of M is β=<p,R _l >, and the field R _l contains (├,0) means the common items corresponding to g, ω _il , x, ├, Indicates the corresponding entry. According to φ and R _l , the domain R _l ′ can be dynamically constructed, that is, there is a migration relationship Add this migration relation to ^ΔM .

Algorithm: TDPN into DPN algorithm

Input: TDPNT = (P, Γ, Δ, X)

Output: Corresponding DPNM=(P ^M , Γ ^M , Δ ^M )

Lines 1 and 2 in the conversion algorithm represent the initialization of the configuration and domain of the worker thread respectively. Starting from the fourth line, for the configuration migration relationship Δ of T, exhaustively calculate the migration relationship Δ ^M represented by the domain in M. Lines 8 and 9 represent no-operation migration, corresponding to M, only the state is changed, and the domain remains unchanged. Lines 10 to 12 describe the clock reset operation transition, the value of clock x in field R _l reset to The 13th to 15th lines describe the migration of time lapse. Except the reference clock item (├, 0) in domain R _l , the clock values of all other items are all added with time lapse v. Lines 16 to 18 describe the migration of the push operation, pushing the character a. Lines 19 to 22 describe the migration of the pop operation, where R _i(l-1) ′ means that all items in the domain R _i(l-1) plus the time lapse θ(├ ^· ) of the domain R _il . Lines 23 to 25 describe dynamic creation of thread migration, and the number of new threads created is n+1. For TDPNT, the algorithm is terminated, and the time complexity of the algorithm is exponential with the Cartesian product of item character set and keypoint set, and is exponential with the size of the program.

For TDPNT, the algorithm is terminated, and the time complexity of the algorithm is exponential with the Cartesian product of item character set and keypoint set, and is exponential with the size of the program.

4. Proving stage of reachability problem: By proving that the state p _F is reachable in TDPN if and only if its transition state p _F ′ is reachable in DPN, it is determined whether there is an error in the model transformation.

To transform the TDNP reachability problem into the DPN reachability problem through the clock equivalent optimization technique, it is necessary to prove the correctness of the transformation from T to M, that is, the state p _F is reachable in TDPN if and only if its transition state p _F ′ is in DPN reachable.

Definition 1 (Accessibility): Let the migration system TDPNT, is the initial configuration of T, where is the initial value of the global variable; p _init is the initial state; ε is the initial value of the stack (indicating that the stack is empty); is the initial clock (assigned to 0), the target pattern If there is pattern migration in T Then state p _F is reachable at T.

Let R=R ₀ R ₁ . . . R _n be a group of domains on the set of M-stack domains. For the two domains R ₁ and R ₂ , if R ₁ is a strict partial order relationship of R ₂ , remember If R1 is _R2 non _- strict partial ordering, remember For a domain set R, if Then R is called a correlation domain, if Then R is called a weak correlation domain. If R is a (weakly) correlated domain, then the pattern β=<p, R> is a (weakly) correlated pattern. For weak correlation domain R=R ₀ R ₁ ...R _n and domain R′=R ₀ ′R ₁ ′…R _n ′, if R _n ′=R _n , R _i ′∈R _i ⁺ (where R _i ⁺ is the time shift domain of R _i ), and Then domain R ^' is a strongly correlated domain of domain R. Given a correlation pattern β=<p, R> in M, if domain R′ is a strongly correlated domain of domain R, then pattern β′=<p, R′> is a strongly correlated pattern of β.

Theorem 1: For any pattern γ of T, through the equivalent transformation of the clock, there is a corresponding pattern β in M.

Proof: Suppose a pattern of M β=<p, R>, a pattern of T in Assuming that S is the variable set of the migration system T at the moment, S is equivalently transformed into the domain R in M through the clock domain. set up R=R ₀ R ₁ ... R _n and the value θ of S (that is, θ _|=S ), the following expression is established:

●p'=p

●

●

Then γ|= _S β, that is, for any pattern γ of T, there is a corresponding pattern β in M after it is converted by clock domain coding.

To prove accessibility, the following two laws must be introduced first:

Law 1: For any regular reachable pattern β belonging to M, the strongly correlated pattern β′=<p, R′> of β, S is the set of variables at the moment of T, and there must be a corresponding pattern γ in T, there is γ |= _S β and

Law 2: For any pattern γ belonging to T, there must be a corresponding pattern β in M, there is at least one strongly correlated pattern β′=<p, R′> of β, and there is a transformation set S of domain R′, then There exists γ|= _S β and

Theorem 2: The state p _F is reachable in TDPNT if and only if p _F ′ is reachable in DPNM.

Proof: Proof of Sufficiency: State p _F is reachable in TDPN Its transition state p _F ' is reachable in DPN.

If the target state p _F ′ is reachable in M, then there is a regular reachable pattern β (p _F ′ is the state of pattern β). Since all the reachable patterns of DPNM are weakly correlated, that is, the reachable pattern β is a weakly correlated pattern, so there is at least one corresponding strongly correlated pattern β′=<p, R′>. It can be seen from Law 1 that in a regular pattern β of the migration system M, there is a strongly correlated pattern β′ and a set S transformed into R′, there must be a corresponding pattern γ in T, there exists γ|= _S β and That is, the state p _F (p _F is the state of the pattern γ) is reachable at T.

Re-certification necessity: state p _F is reachable in TDPN Its transition state p _F ' is reachable in DPN.

If the target state p _F ′ is reachable in T, it can be seen from Theorem 1 that there must be a corresponding pattern β in M (p _F ′ is the state of pattern β), so there is at least one strongly correlated pattern β′=<p, R'>. It can be seen from Law 2 that in a pattern γ of the migration system T, there exists a strongly correlated pattern β′ and a field set S transformed into R′, and there must be a corresponding pattern β in M, there exists γ|= _S β and That is, the state p _F ′ (p _F ′ is the state of pattern β) is reachable in M.

Therefore, state p _F is reachable in TDPN if and only if its transition state _pF ' is reachable in DPN.

Design errors or loopholes existing in the concurrent recursive program can be found through the above method steps to ensure the reliability and correctness of the program. This method is an automatic reachability solution method, which can realize the decidable solution to the time dynamic pushdown network reachability problem without too much participation of users, and the reachability pattern calculation process is simple and effective.

Claims (1)

1. to push net under Time dynamic the conversion method of network, it is characterized in that, comprise the steps:

Step (1) is pushed net under described Real-time and Concurrent recursive program is converted to a Time dynamic network;

To push net under the abstract model of step (1.1) structure Real-time and Concurrent recursive program and Time dynamic network;

Network of pushing net under the Time dynamic constructed is a four-tuple T=(P, Γ, Δ, X), and wherein P is state set; Γ is stack character set; Δ=Δ _nop∪ Δ ₌∪ Δ _├∪ Δ _push∪ Δ _pop∪ Δ _dcmigration rules set, wherein Δ _noprepresent blank operation migration, Δ ₌represent the migration of clock assignment, Δ _├represent migration time lapse, Δ _pushrepresent stacked migration, Δ _poprepresent migration of popping, Δ _dcrepresent that dynamic thread creation moves; X represents clock collection, its value function represent for be θ (x) in current value;

Construct Time dynamic under push away network situation represent described model state at a time, wherein: represent the two tuple <g at current global variable g and its age θ (g), θ (g) >; p _i∈ P represents local state node, represent that stack sequence is the stack contents ω of i _iwith its age θ (ω _i) two tuple < ω _i, θ (ω _i) >; represent the two tuple <x of clock x and its value θ (x), θ (x) >;

Network operational semantics of pushing net under constructed Time dynamic is described by step (1.2);

Pushing net under Time dynamic the model of network as real-time multithread program, producing migration for describing multiple lower pushing system, its transition relationship Δ=Δ simultaneously _nop∪ Δ ₌∪ Δ _├∪ Δ _push∪ Δ _pop∪ Δ _dcprovide it according to different migration actions below and perform implication;

1) Δ=Δ _noptime, op=nop, represent that general layout interior element does not change;

2) Δ=Δ ₌time, op=x ← I, c ∈ I; Represent to the arbitrary value v within the scope of clock x assigned I, other general layout interior element does not change;

3) Δ=Δ _├time, op=Time ← c, suppose $\stackrel{\&OverBar;}{\mathrm{\&omega;}}=<a_1,x_1><a_2,x_2>...<a_n,x_n>,$ So ${\stackrel{\&OverBar;}{\mathrm{\&omega;}}}^{+v}=<a_1,x_1+v><a_2,x_2+v>...<a_n,x_n+v>,$ ${\stackrel{\&OverBar;}{x}}^{\&prime;}={\stackrel{\&OverBar;}{x}}^{+v},$ Represent that in general layout, all clocks increase v, in general layout, non-clock contents does not change;

4) Δ=Δ _pushtime, op=push (a, I), v ∈ I, represent and variable a is pressed into stack top, and to set corresponding clock be x, its clock value is the arbitrary value within the scope of I;

5) Δ=Δ _poptime, op=pop (a, I), v ∈ I, stack top internal clock value is that the variable a of I scope ejects by expression;

6) Δ=Δ _dctime, op=dc, represent and create new thread stack contents;

Push net under the Time dynamic that step (1) obtains by step (2) network T=(P, Γ, Δ, X), is converted to by following conversion method the network M=(P that to push net dynamically ^m, Γ ^m, Δ ^m);

Step (2.1) state P ^mconversion: namely the state set of T is identical with the state set of M;

Step (2.2) stack character set conversion: if a ∈ { Γ, ├ }, then and

Step (2.3) transition relationship Δ is to Δ ^mtransformation rule;

If the stack level of this lower pushing system is l, and is numbered 1 at the bottom of stack, stack top is numbered l; This TDPN contains global variable g, clock variable x, stack contents ω={ ω ₁ω _iω _n, wherein ω _irepresent the stack contents of No. i lower pushing system, use ω _il| _Γrepresent ω _ibe projected in the stack top character of Γ; its age of each expression and the key point of value under clock equivalence; Thus known correspondence is in the present clock of M territory of equal value wherein representative domain R _lrecord stack top character, ├ representative domain R _lreference clock character, ├ ^.representative domain R _lcorresponding character time lapse;

TDPN general layout φ=(γ, op, γ ') ∈ Δ represents the general layout migration of T, and the general layout migration of corresponding M can be expressed as wherein p with p ' is identical with state in T, represents the state before and after general layout migration respectively; R _l={ R _1lr _ilr _nlrepresent stack top territory, wherein R _ilrepresent the stack top territory of No. i lower pushing system, R _land R _l' represent that general layout moves the stack top territory of front and back respectively; Action migration collection op ', corresponding to the op of T, the following describes and constructs R according to different op _l':

1) as op=nop, for exist in and if only if M only state is changed, so general layout migration also only changes state, territory R in M at T hollow operations _lremain unchanged;

2) as op=(x ← I), for exist in and if only if M this transition relationship represents DPN territory R _lmiddle clock is that the item of x performs operation, wherein θ (x) ' ∈ I, carrys out structural domain R _l'; Concrete implementation is as follows:

Territory R _lpop, obtain R _lin item resetting θ (x) is θ (x) ', forms new item

? replace territory R _lin item obtain territory R _l', and stacked, be transformed into new state p ';

3) as op=(Time ← v), for exist in and if only if M this transition relationship representative domain R _lin except reference clock item (├, 0), the clock value of all the other all items adds v time lapse, carrys out structural domain R _l ⁺; Concrete execution step is:

Territory R _lpop, except reference clock, all add v time lapse, represent and correspond to g, ω _il, x new general term, the corresponding new record item of each expression, represent reference clock entry;

New item replaces original item, obtains territory R _l ⁺, and stacked, be transformed into new state p ';

4) as op=push (a, I), for and if only if, and M exists this transition relationship represents, carrying out stack-incoming operation to No. i lower pushing system, is a by character, is worth to be the stacked territory R of item _il, carry out structural domain R _{i (l+1)}; Detailed process is as follows:

From R _ilobtain item with

replace respectively obtain territory R _{i (l+1)}, and stacked, be transformed into new state p ';

5) as op=pop (a, I), for and if only if, and M exists this transition relationship represents, pop territory R _ilmiddle stack character is a, and the item of θ (a) ∈ I, carry out structural domain R _il'; Concrete steps are described below:

Pop territory R _ilwith territory R _{i (l-1)}, obtain territory R _ilin item

Territory R _(l-1)in the clock value of all items add θ (├ ^.), obtain territory R _{i (l-1)}';

Pass through R _iland R _{i (l-1)}' obtain territory R _il', R _il' item is respectively: common stack character item is from territory R _{i (l-1)}'; Ordinary clock item, global variable item are from territory R _il; Entry is all from territory R _{i (l-1)}';

Stacked territory R _il', be transformed into new state p ';

6) when time, for exist in and if only if M this transition relationship represents, creates new thread and carrys out structural domain R _l'; Suppose TDPN model, dynamic creation newly descends the stack of pushing system to be numbered n+1; Concrete execution step is as follows:

Pop territory R _l, the optimisation technique of equal value of the clock based on key point, can obtain general character item entry

Item with join domain R _l, obtain territory R _l', and stacked, be transformed into new state p '.