Networks Based on Tree Semantics Approach

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1 Word Journa of Socia Science Research ISSN (Print) ISSN (Onine) Vo. 4, No. 4, Reachabiity Anaysis of Asynchronous Dynamic Pushdown Networks Based on Tree Semantics Approach Guodong Wu 1* & Junyan Qian 2 1 Schoo of Information and Computer, Anhui Agricutura University, Hefei, China 2 Guangxi Key Laboratory of Trusted Software, Guiin University of Eectronic Technoogy, Guiin , China * Guodong Wu, E-mai: @qq.com Received: October 6, 2017 Accepted: October 15, 2017 Onine Pubished: October 17, 2017 doi: /wjssr.v4n4p287 URL: Abstract ADPN (Asynchronous Dynamic Pushdown Networks) are an abstract mode for concurrent programs with recursive procedures and dynamic thread creation. Usuay, asynchronous dynamic pushdown networks are described with intereaving semantics, in which the backward anaysis is not effective. In order to improve intereaving semantics, tree semantics approach was introduced. This paper extends the tree semantics to ADPN. Because the reachabiity probem of ADPN is aso undecidabe, we address the context-bounded reachabiity probem and provide an agorithm for backward reachabiity anaysis with tree-based semantics Approach. Keywords DPN, ADPN, tree semantics, context-bounded, backward reachabiity 1. Introduction With deveopment of muti-core processor, the research on concurrent program has become the focus of programming. Usuay, the concurrent program empoys concurrent pushdown system or parae procedure ca for modeing, which two modes, however, fai to we simuate program with dynamic thread creation. Bouajjani et a. suggested in 2005 the DPN which are appicabe to modeing of concurrent program containing recursive procedures or dynamic thread creation, yet its execution semantics is intereaving semantics, in which the backward reachabiity anaysis is inefficient. Lammich et a. suggested tree semantics for DPN to mode execution of program as a tree, which more conforms to actua running of program, and they provided the reachabiity anaysis means for dynamic pushdown networks under tree semantics yet without considering intercommunication of threads. Based on dynamic pushdown networks mode, Bouajjani raised ADPN mode, whose reachabiity probem, however, is undecidabe and backward reachabiity on it is compicated. Faouzi modeed program 287

2 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 containing dynamic thread creation, presented context-bounded anaysis agorithm for this mode, and proved it is decidabe. Wenner extended dynamic pushdown networks to weighted dynamic pushdown networks to enhance modeing capacity of mode. The paper extends tree semantic anaysis into ADPN mode to make it more efficienty conduct backward reachabiity anaysis, and soved undecidabe * probem using the context-bounded approach. The backward reachabe patterns set Pr e M of this mode keeps reguarity property under tree semantics, i.e., abe to construct automator to receive a group of * backward reachabe patterns set Pr e M. The paper structures an intermediate mode to simuate execution mode of tree semantics, cacuates backward reachabe patterns set of this intermediate mode in context of finite times, and uses projection operation to derive the backward reachabe patterns set on origina mode. Finay, the compexity anaysis on this agorithm is presented. 2. Reate Knowedge ADPN is extension of DPN to simuate pushdown, new process and asynchronous communication of inter-thread using of shared memory. ADPN is a quintupe M = (G, P, Γ,, g ), where G is goba state set, P is oca state set, Γ is stack symbo, is set of oca migration rues: (a) ργ ρ 1 w 1 ; (b) ργ ρ 1 w 1 ρ 2 w 2, where ρ, ρ 1, ρ 2 P, γ Γ, w 1, w 2 Γ ; g is set of goba migration rues: (a) (g, ργ) (g, ρ 1 w 1 ); (b) (g, ργ) (g, ρ 1 w 1 ) ρ 2 w 2 ; where g, g G, ρ, ρ 1, ρ 2 P, γ Γ, w 1, w 2 Γ*. ADPN s pattern is (g, α) G (P Γ*), where g is goba state, character string α = ρ 1 w 1 ρ 2 w 2 ρ n w n, and each substring represents corresponding DPN pattern of ADPN pattern. Let C represent ADPN pattern set, migration reation is defined as: (g, ս) (g, v) is true, then ργ p1 w 1 beongs to rues set, and ս = ս 1 ρ γ ս 2, v = ս 1 ρ 1 w 1 ս 2, g = g, or ργ p1 w 1 ρ 2 w 2 beongs to rues set, or (g, ργ) (g, ρ1 w 1 ) beongs to rues set g, and ս = ս 1 ρ γ ս 2, v = ս 1 ρ 1 w 1 ս 2, or (g, ργ) (g, ρ1 w 1 ) ρ 2 w 2, and ս = ս 1 ργս 2, v = ս 1 ρ 2 w 2 ρ 1 w 1 ս 2 ; Where ս 1 (PΓ )*, ս 2 Γ (PΓ )*. Goba migration and oca migration rues sets constitute the whoe migration system. State is reachabe: There is ADPN M and patterns set S C, define post M *(S) and pre M *(S) as respectivey representing forward and backward reachabe patterns sets that start from pattern S. The forward and backward reachabiity probems can respectivey be defined as: patterns set I and F are respectivey initia and a certain given pattern sets. To judge whether the state in given pattern F is reachabe just Pubished by SCHOLINK INC. 288

3 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 needs to judge whether post M *(I) F = φ or pre M *(F) I = φ is true. However, these two reachabiity probems are both undecidabe. K-context-bounded state is reachabe: Qadeer et a. Put forward context-bounded reachabiity anaysis. The cacuation is switched from one thread to another thread, which is defined as context switching. K-boundary means the corresponding migration sequence at most contains K contexts. K-context-bounded state being reachabe just means the set of state reachabe by ADPN in K contexts. post k,m *(S) and pre k,m *(S) are respectivey defined as set of forward and backward reachabe patterns that start from S and reachabe to K-context-boundary. The K-context-bounded state reachabiity probem is decidabe. Musuvathi et a. proved that majority of states can be detected as ong as K is set as a sma constant. 3. Tree Semantics of ADPN This section extends tree semantics to ADPN, and migration sequence is partia order, which is simiar to a tree, as shown in Figure 1. The initia state of the migration sequence ony contains a thread, and new threads are created dynamicay. The migration sequence of newy created threads correspond to execution tree on the eft of node S, and the migration sequence on the right of node S is origina thread. The form is defined as T M : = N L T M /S L T M T M /L < G,PΓ* >, where Nt corresponds to non-dynamic creation rue, and what foows it is t; S t s t corresponds to dynamic creation rue, and what foows it is a dynamicay created new thread node t s and origina thread s node t; L <g, pw> corresponds to the pattern when thread running terminates. The advantage of tree semantics is immediatey judging which progress the migration beongs to via execution mode of tree form. N 1 S 2 spawn N 4 N 3 L <g,pw> L <g,pw Figure 1. ADPN Execution Tree Definition 1 (Execution tree): The execution tree is defined as migration sequence of mode under tree semantics. Migration rue M <g, PΓ*> T M <g, Conf N > means the initia state ony contains execution tree of singe thread, where Conf N is pattern α of corresponding DPN. When the initia state Pubished by SCHOLINK INC. 289

4 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 contains mutipe threads, definition h c M c means migration from pattern c to pattern c via execution tree h, where h = t 1 t n. The execution tree is reguar. The paper adopts hedge-automator for receiving. Hedge-automator T = (S, A 0, D), where S is set of a group of finite states, A 0 is an initia automator meeting L(A 0 ) S*. Rues set D = D L D N D S : The rue s A 1 D L simuates eaf node, rue s s DN simuates non-dynamicay created node, where, s, s, s s S, L, automator A 1 receives a patterns sets (L(A 1 ) (G, (PΓ*) + )). Execution tree h = t 1 t n T M * can be received by hedge-automator when and ony when execution tree t 1 t n can be simuated by rues set D and initia states set of execution tree respectivey correspond to initia states set s 1 s n L(A 0 ) of automator. Rue ab T S T M represents the association between hedge-automator and execution tree, as shown beow in form: [Leaf node] ab T (s, L <g, pw>) s A 1 D L <g, pw> L(A 1 ) [Non-dynamic creation] ab T (s,n t) s s D ab ( s,) t N T [Dynamic creation] ab T (s, S t s t) s s s D ab ( s,) t ab ( s, t ) s S T T s s Where ab T (s, t) means the state of execution tree t corresponds to state s of automator. It is assumed that ab T (s 1 s n,t 1 t n ) ab T (s 1,t 1 ) ab T (s n,t n ), then the anguage received by hedge-automator T is L( T ) : = { h s L( A ). ab ( s, h)} 0 T. 4. Backward Reachabe Agorithm Based on Tree Semantics It is assumed that pattern of ADPN is <g, α>, where α is corresponding DPN pattern. Receiving pattern α of automator A is structured according to iterature. The paper wi structure automator Z receiving ADPN pattern <g, α>. For any character string u 1 pu 2 L(A), automator Z receives character string w = u 1 (g,p)u 2, where u 1 (PΓ*)*, p P, u 2 Γ*(PΓ*)*. Theorem 1: ADPN M is given, and automator Z receives character string w = u 1 (g, p)u 2, based on this, the paper structures automator Z pre* as meeting L(Z pre* ) = pre*(l(z)). Time compexity for structuring Z pre* automator is O ( Q 3 ). Prove: There is ADPN M whose pattern is <g, α>, and automator A receives character string α = u 1 pu 2, automator Z = (Q, Σ, δ, q 0, F) receiving character string w = u 1 (g, p)u 2 can be easiy structured, then automator Z pre* = (Q, Σ, δ, q 0, F) is structured, wherein the rue of δ is as foows: Pubished by SCHOLINK INC. 290

5 Word Journa of Socia Science Research Vo. 4, No. 4, If (g, pγ) (g, p1 w 1 ), and q gp w 1 1 δ q, wherein q, q Q, then (qp, γ, q ) δ is true, where q p means the new state arrived after q receives character p. 2. If (g, pγ) (g, p1 w 1 ) ρ 2 w 2, and q gp w p w δ q, wherein q, q Q, then (qp, γ, q ) δ is true, wherein, q p means the new state arrived after q receives character p. As state Q is constant, structuring of automator Z pre* is aways terminabe. According to rue of migrating δ, efficient automator structuring method is adopted to structure Z pre*, with time compexity of structuring being O ( Q 3 ). Let <g, α> pre* (L(Z)), and<g, α> <g, α >, then <g, α > L(Z). Automator Z receives pattern <g, α >. It is derived from structuring rue of automator Z pre* that <g, α> L(Z pre* ). Let <g, α> L(Z pre* ), and <g, α> <g, α >, it is derived from structuring rue of Zpre* that <g, α > L(Z), and it is concuded that <g, α> pre*(l(z)). So L(Z pre* ) = pre*(l(z)) is true. q.e.d. To combine the state of hedge-automator and corresponding ADPN s oca state into a new state, and reaize execution mode of tree semantics for migration sequence, frist a specia ADPN mode M T = (G, P S, Γ, L,, g ) is structured, whose pattern is C T Conf M T, and form is as foows: C T: = {(g,(p 1, s 1 )w 1...(p n,s n )w n ) (g,p 1 w 1...p n w n ) C i.1 i n. A 1.s i A 1 D L <g,p i w i > L(A 1 )} The new pattern C T combines state in hedge-automator H with the oca state of each of its corresponding mode M, with form being (g,(p,s)γ). Goba migration rue g of ADPN mode M T is defined as foows: [Non-dynamic creation rue] ( g,( ps, ) γ ) ( g,( p, s ) w) g ( g, pγ ) ( g, pw) s s D g N is true, when and ony when is true. [Dynamic creation rue] ( g,( ps, ) γ ) ( g,( p, s ) w) ( p, s) w s s g is true, when and ony when ( g, pγ ) ( g, pw ) p w s s s D s s g s S is true. Loca migration rue is defined as foows: [Non-dynamic creation rue] ( ps, ) γ ( p, s ) w is true, when and ony when pγ pw s s D N is true. Pubished by SCHOLINK INC. 291

6 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 [Dynamic creation rue] ( ps, ) γ ( p, s ) w ( p, s) w s s g is true, when and ony when pγ pw pw s s s D s s g s S is true. Definition 2 (backward reachabe pattern under tree semantics). There is given ADPN M, whose pattern C = (g, α), C Conf M, where α is corresponding DPN pattern. As pattern α can be received by automator A, the form can aso be C = (g, A). It is known that execution tree H T M *, and backward reachabe patterns set based on tree semantics is defined as foows: pre [ H ]( C ) : = {( g, α) Conf ( g, α ) C, h H.( g, α) M ( g, α )} M M h where H is reguar execution tree. As C is reguar set, so pre M [H](C ) is aso kept reguar, and pre M [H](C ) can be received by structuring automator. However, reachabiity probem of ADPN under tree sementics remain undecidabe, and context-bounded technoogy is adopted to sove this probem. Definition pre K,M [H](C ) represents the K-context-bounded backward reachabe patterns set that starts from pattern C under mode M. Cacuation of pre K,M [H](C ) can be divided into K steps: first cacuate pre 1,M [H](C ), i.e., cacuate backward patterns set starting from pattern C and with context switching not occurring. For a patterns set pre 1,M [H](C ) cacuated from previous context, and for a g G, if (g, u ) pre 1,M [H](C ), then cacuate pre 1,M [H](g, u ). Cacuate recursivey ike this unti finishing cacuating K contexts, to get K-context-bounded backward reachabe patterns set. Foowing is anaysis on how to cacuate pre 1,M [H](g, u). As DPN automator A receives character string u 1 (p, s)u 2, automator Z receiving a character string u 1 (g,(p,s))u 2 can be structured, where u 1 (p, s)u 2 L(A), u 1 ((P, S)Γ*)*,p P,u 2 Γ*((P, S)Γ*)*. According to theorum 1, the automator Z pre* can be structured, hence, foowing is derived: pre 1,M [H](g,u) = + ( g,{ w ( PΓ *) : w = upu u( g, p) u L( Z )}) g G pre* The patterns set pre 1,M [H](C ) cacuated each time is of (G,((P, S)Γ*)*) form. In oca state, S represents execution tree represented by hedge-automator in cacuating reachabe pattern. To get the patterns set of origina mode, a projection operation proj T : 2 ConfM T 2 ConfM is defined. proj T (C) = {(g,p 1 w 1... p n w n ) s 1,...s n S. s 1,...s n L(A 0 ) (g,(p 1, s 1 )w 1...(p n, s n )w n ) C } A backward reachabe agorithm can be derived from above anaysis, as shown in Figure 2. Pubished by SCHOLINK INC. 292

7 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 Input: ADPN M,tupe M = (g, A) and integer k and pattern C Output: Backward reachabe patterns set pre*(c) 1. Structure hedge-automator T according to the rue as stated in section 3; 2. M M T; //Structure a new mode according to mode M and automator T of M s execution tree. 3. reachabe φ, eve = 0; //initiaize reachabe set and context-boundary identifier. 4. reachabe pre 1,M[H](g, A); //Cacuate a reachabe patterns in the first context. 5. C0 pre1, M[H](g, A); //intermediate resut is temporariy saved in C0 set. 6. eve++; 7. A = oca (C0 (g, (PΓ*)+)); //Update oca state pattern. 8. whie(eve k) // Time of context-boundary is K. 10. {for a g G. 11. {A = oca (Ceve -1 (g, (PΓ*)+)); //Update oca state pattern. 12. Ceve pre1, M[H](g, A); } //intermediate resut is temporariy saved in Ceve set. 13. reachabe Ceve; //Put a reachabe patterns in each context into reachabe. 14. eve++;}. 15. projt(reachabe); //Conduct projection operation for patterns set. Figure 2. Reverse Reachabiity Agorithm Based on Tree Semantics The first ine simuates execution mode of tree semantics for migration sequence by structuring an intermediate mode M. whie cycicay contros K context-boundaries. oca() represents the operation of projecting ADPN patterns to DPN pattern. C eve represents cacuating reachabe pattern under the eveth context. pre 1,M [H]() operation represents a possibe reachabe patterns sets in each context. As the patterns sets from reach abe sets are patterns set of intermediate mode M, the pattern of reach abe pattern set reachabe is subjected to projection to get reachabe patterns set under mode M. 5. Anaysis on Agorithm The compexity of agorithm suggested herein is O( G k-1 Q 3 ). The compexity of new mode structured via given mode M and execution tree H is inear compexity. According to theorem 1, the time compexity for structuring of automator Z pre* each time is Q 3, as the time for context-boundary is k, the execution sequence g 1 g k-1 G k-1 for for oop in the worst case has G k-1 possibiities. proj T () function is inear compexity, hence the compexity of agorithm is O( G k-1 Q 3 ), which has certain advantages over the agorithms such as k-deimited DPN reachabiity anaysis, etc. 6. Concusion The tree semantics simuates the running of concurrent program more accuratey than intereaving semantics and extends modeing capacity of mode (such as being abe to show which thread the migration beongs to), and can more efficienty conduct backward reachabiity anaysis than intereaving semantics. To make the reachabiity probem decidabe, the agorithm adopts context-bounded approach to make backward reachabiity anaysis on ADPN, and structures an intermediate mode to reaize Pubished by SCHOLINK INC. 293

8 Word Journa of Socia Science Research Vo. 4, No. 4, 2017 execution mode of semantics. Besides, it combines the state of hedge-automator and corresponding mode state into a new state to cacuate reachabe pattern of this intermediate mode. Lasty, projection operation is conducted to get the patterns set on origina mode, and compexity of agorithm is presented to we sove the reachabiity probem under this mode. The key content in future research wi be foowing two aspects: 1) Combine summarization technoogy and optimize the agorithm by dint of summary course being abstract and abe to be reused for many times; 2) As the agorithm herein can ony be used in abstract modes with imited goba variabes, next it is considered to extend it to abstract modes with infinite goba variabes. References Abdua, P. A. et a. (2014). Budget-bounded mode-checking pushdown systems. Forma Methods in System Design, 2014, Atig, M. F., Abdua, P. A., Kumar, K. N., & Saivasan, P. (2012). Linear time mode-checking for mutithread programs under scope-bounding//proceedings of the 10th internationa symposium on automated technoogy for verification and anaysis. Thiruvananthapuram, 2012, Bouajjani, A., Esparza, J., Schwoon, S., & Strejcek, J. (2005). Reachabiity anaysis of mutithreaded software with asynchronous communication. In Proc. of FSTTCS (Vo. 2005, pp ). Hyderabad: Springer. Bouajjani, A., Müer-Om, M., & Touii, T. (2005). Reguar symboic anaysis of dynamic networks of pushdown systems. In Proceedings of the 16th Internationa Conference on Concurrency Theory (pp ). Emmi, M. et a. (2015). Anaysis of Asynchronous Programs with Event-Based Synchronization. In Programming Languages and Systems (pp ). Springer Berin Heideberg. Faouzi Atig, M., Bouajjani, A., & Qadeer, S. (2009). Context-bounded anaysis for concurrent programs with dynamic creation of threads. In Proceedings of the 15th Internationa Conference on TACAS, LNCS 5505 (Vo. 2009, pp ). Univ York: European Assoc. Gawitza, T. M., Lammich, P., Müer-Om, M., Seid, H., & Wenner, A. (2011). Join-Lock-Sensitive forward reachabiity anaysis for concurrent programs with dynamic process creation. In Proceeding of the 12th internationa conference on verification, mode checking, LNCS 6538 (pp ). Irigoin, F., Jouveot, P., & Trioet, R. (2014). Semantica interprocedura paraeization: An overview of the PIPS project. In 25th Anniversary Internationa Conference on Supercomputing Anniversary (Vo. 2014, pp ). Pubished by SCHOLINK INC. 294

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