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...on. Instead of representing states explicitly, symbolic approaches exploit advanced data structures to encode and manipulate entire sets of states at once. Paired with binary decision diagrams (BDDs) =-=[7]-=-, symbolic model checkers are able to handle enormous state spaces. At the same time, several questions remain open for BDD-based symbolic model checking. One of the most challenging ones is that the ...

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...on nor it is necessarily the most efficient way to build Xrch. Beyond state-space generation, more advanced analyses can be performed on a discrete-state model. For example, in the temporal logic CTL =-=[21, 33]-=-, the operators EX, EU, and EG are complete, that is, they can be used to express any CTL operator through complementation, conjunction, and disjunction. If A and B are the sets of states satisfying C...

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...on nor it is necessarily the most efficient way to build Xrch. Beyond state-space generation, more advanced analyses can be performed on a discrete-state model. For example, in the temporal logic CTL =-=[21, 33]-=-, the operators EX, EU, and EG are complete, that is, they can be used to express any CTL operator through complementation, conjunction, and disjunction. If A and B are the sets of states satisfying C...

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...d a path from j to i, we know we need to break the cycle (even in the sequential case) and may hurt chaining, but parallelization can further hurt chaining. This was experimentally verified in a Cilk =-=[6]-=- implementation at York University [22] running on a shared-memory multicore processor computer system. We achieved some speedup on some models on a four-core machine, but also experienced substantial...

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... computation of these more complex fixpoints, and many of the possible solutions are applicable to them. 1.2 Explicit vs. implicit techniques, which one should we parallelize? Symbolic model checking =-=[8]-=- is undoubtedly a significant breakthrough in formal verification. Instead of representing states explicitly, symbolic approaches exploit advanced data structures to encode and manipulate entire sets ...

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...licit techniques are still competitive for asynchronous systems. Such techniques take advantage of the locality and symmetry properties widely enjoyed by asynchronous systems. Partial order reduction =-=[24, 43]-=- and symmetry reduction [10] have been successfully implemented in explicit model checkers to reduce the number of states that must be explored and stored, thus the run time. These approaches explore ...

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...ls. Traditional BDD techniques can efficiently encode complex synchronous transition relations, but they are often not as compact when applied to asynchronous events, even if disjunctive partitioning =-=[9]-=- often helps. Analogously, image computation, the base of symbolic state-space generation, is often expensive for asynchronous systems. Thus, tools like SPIN [28], a model checker with advanced explic...

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...yramidal neurons. An abundance of research in the field of synaptic plasticity has demonstrated that dendritic spines display morphological plasticity in response to a myriad of extracellular stimuli =-=[1,2]-=-. These changes are thought to be cellular correlates of the plasticity seen in learning and memory [3]. Importantly, spines have repeatedly been shown to increase in both size and number following th...

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...nly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in =-=[2, 3, 32, 40]-=- and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as...

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...es exist. Traditional explicit state space generation approach, such as the one used in the model-checking tool SPIN [28], enumerates and explores each state one by one, and was first parallelized in =-=[16, 35, 40]-=-. The other approach, DD-based state-space generation, is behind all commonly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. S...

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...yramidal neurons. An abundance of research in the field of synaptic plasticity has demonstrated that dendritic spines display morphological plasticity in response to a myriad of extracellular stimuli =-=[1,2]-=-. These changes are thought to be cellular correlates of the plasticity seen in learning and memory [3]. Importantly, spines have repeatedly been shown to increase in both size and number following th...

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...wing the distribution of computation over multiple processors or cores. Parallel DD-based algorithms have been developed for a variety of platforms: shared memory multiprocessor or multi-core systems =-=[41]-=-, network of workstations (NOW) [34], distributed shared memory (DSM) architectures [37], single-instruction-multiple-data (SIMD) and multiple-instruction-multipledata (MIMD) architectures [23], and v...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...

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...ritic spines display morphological plasticity in response to a myriad of extracellular stimuli [1,2]. These changes are thought to be cellular correlates of the plasticity seen in learning and memory =-=[3]-=-. Importantly, spines have repeatedly been shown to increase in both size and number following the induction of long-term potentiation (LTP) [4-7] and to decrease in size and number following inductio...

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...y-relevant plasticity paradigms. Activation of Rac1 and actin remodeling are critical for the clustering of AMPA receptors and changes in dendritic spine morphology that occur following LTP induction =-=[4,5,12,47]-=-. However, there are numerous Rac1 GEFs at the PSD and any of these could be affecting these changes [14]. In order to examine more specifically the Rac GEF activity of Kal7, we chose to use NPPD, whi...

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...es exist. Traditional explicit state space generation approach, such as the one used in the model-checking tool SPIN [28], enumerates and explores each state one by one, and was first parallelized in =-=[16, 35, 40]-=-. The other approach, DD-based state-space generation, is behind all commonly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. S...

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... explores each state one by one, and was first parallelized in [16, 35, 40]. The other approach, DD-based state-space generation, is behind all commonly used symbolic modelchecking tools, such as SMV =-=[30]-=- and SMART [18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods p...

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...ize Saturation. Ironically, exploring this opportunity led us to further speedup sequential Saturation first [15]. One of the reasons for the efficiency of Saturation is its extensive use of chaining =-=[39]-=-: if events α and β can be fired on the set of states X , X ∪Nα(X )∪Nβ(X ) ⊆ X ∪Nα(X )∪Nβ (X ∪Nα(X )) (for the inclusion to be strict, α must add new states that enable β ). Then, chaining was propose...

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...ection 3.1. 3.1 Distributed-memory approaches for symbolic state-space generation One way to achieve parallelization for symbolic techniques is designing parallel BDD libraries for a NOW environment. =-=[34, 38, 42]-=- discuss how to store and manipulate BDDs on a NOW. These approaches to achieve parallelism lie on low-level DD operations and, if they succeed in providing an interface similar to that provided by an...

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...tedly been shown to increase in both size and number following the induction of long-term potentiation (LTP) [4-7] and to decrease in size and number following induction of long-term depression (LTD) =-=[8,9]-=-. The ability of dendritic spines to remain labile/plastic is dependent on rearrangement of the actin cytoskeleton which forms the core of each spine [10-12]. This process is dependent on the activity...

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... rats [26-29]. We therefore employed a modified induction protocol using paired-pulse low frequency stimulation (PP-LFS), which has previously been shown to be effective in inducing LTD in older rats =-=[26,29,30]-=-. As A DC B Figure 5 Application of NPPD, an inhibitor of the Rho-GEF activity of Kal7, suppressed hippocampal LTP and LTD. A) Input-output relationship for effects of NPPD (100 μM) on fEPSPs in 4 wee...

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...licit techniques are still competitive for asynchronous systems. Such techniques take advantage of the locality and symmetry properties widely enjoyed by asynchronous systems. Partial order reduction =-=[24, 43]-=- and symmetry reduction [10] have been successfully implemented in explicit model checkers to reduce the number of states that must be explored and stored, thus the run time. These approaches explore ...

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... the actin cytoskeleton which forms the core of each spine [10-12]. This process is dependent on the activity of Rho-GTPases, which are activated by Rho-guanine nucleotide exchange factors (Rho-GEFs) =-=[13]-=-. About a dozen of the 58 RhoGEFs encoded by the mouse genome are localized to the postsynaptic density (PSD) [14]. Among the PSD-localized Rho-GEFs is Kalirin-7 (Kal7), the predominant adult splice v...

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...y-relevant plasticity paradigms. Activation of Rac1 and actin remodeling are critical for the clustering of AMPA receptors and changes in dendritic spine morphology that occur following LTP induction =-=[4,5,12,47]-=-. However, there are numerous Rac1 GEFs at the PSD and any of these could be affecting these changes [14]. In order to examine more specifically the Rac GEF activity of Kal7, we chose to use NPPD, whi...

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...tedly been shown to increase in both size and number following the induction of long-term potentiation (LTP) [4-7] and to decrease in size and number following induction of long-term depression (LTD) =-=[8,9]-=-. The ability of dendritic spines to remain labile/plastic is dependent on rearrangement of the actin cytoskeleton which forms the core of each spine [10-12]. This process is dependent on the activity...

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... rats [26-29]. We therefore employed a modified induction protocol using paired-pulse low frequency stimulation (PP-LFS), which has previously been shown to be effective in inducing LTD in older rats =-=[26,29,30]-=-. As A DC B Figure 5 Application of NPPD, an inhibitor of the Rho-GEF activity of Kal7, suppressed hippocampal LTP and LTD. A) Input-output relationship for effects of NPPD (100 μM) on fEPSPs in 4 wee...

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...15,16]. Kal7 has been repeatedly shown to have a profound effect on dendritic spine density in vitro, with over-expression dramatically increasing spine density and knockdown decreasing spine density =-=[17,18]-=-. More recently, we developed a mouse that cannot produce Kal7 (Kal7KO) and demonstrated that this mouse had decreased hippocampal spine density at baseline, and was unable to increase dendritic spine...

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...d algorithms have been developed for a variety of platforms: shared memory multiprocessor or multi-core systems [41], network of workstations (NOW) [34], distributed shared memory (DSM) architectures =-=[37]-=-, single-instruction-multiple-data (SIMD) and multiple-instruction-multipledata (MIMD) architectures [23], and vector processors [36]. According to the nature of the statespace generation algorithm, w...

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...y-relevant plasticity paradigms. Activation of Rac1 and actin remodeling are critical for the clustering of AMPA receptors and changes in dendritic spine morphology that occur following LTP induction =-=[4,5,12,47]-=-. However, there are numerous Rac1 GEFs at the PSD and any of these could be affecting these changes [14]. In order to examine more specifically the Rac GEF activity of Kal7, we chose to use NPPD, whi...

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... rats [26-29]. We therefore employed a modified induction protocol using paired-pulse low frequency stimulation (PP-LFS), which has previously been shown to be effective in inducing LTD in older rats =-=[26,29,30]-=-. As A DC B Figure 5 Application of NPPD, an inhibitor of the Rho-GEF activity of Kal7, suppressed hippocampal LTP and LTD. A) Input-output relationship for effects of NPPD (100 μM) on fEPSPs in 4 wee...

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...ies will explore the specific role of the NR2B subunit in the plasticity and behavioral deficits seen after Kal7 deletion. Alternative splicing generates multiple Kalirin proteins from the Kalrn gene =-=[46]-=-. Mice engineered to lack exons encoding the first GEF domain of Kalirin (KalGEF1KO) lack Kal7 as well as the larger isoforms, Kal9 and Kal12 [20]. KalGEF1KO mice show normal basal AMPA-mediated trans...

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...is a paradigm that is more relevant to the physiological range of synaptic activity [35]. We also found that low frequency stimulationinduced LTD, which requires activation of synaptic NMDA receptors =-=[36]-=-, was likewise disrupted in Kal7KO animals (Figure 3). In contrast, no genotypic deficit was seen in a non-NMDA receptor mediated form of LTP at these same synapses (Figure 4). These results suggest t...

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...y-relevant plasticity paradigms. Activation of Rac1 and actin remodeling are critical for the clustering of AMPA receptors and changes in dendritic spine morphology that occur following LTP induction =-=[4,5,12,47]-=-. However, there are numerous Rac1 GEFs at the PSD and any of these could be affecting these changes [14]. In order to examine more specifically the Rac GEF activity of Kal7, we chose to use NPPD, whi...

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...by the mouse genome are localized to the postsynaptic density (PSD) [14]. Among the PSD-localized Rho-GEFs is Kalirin-7 (Kal7), the predominant adult splice variant of the multiply spliced Kalrn gene =-=[15,16]-=-. Kal7 has been repeatedly shown to have a profound effect on dendritic spine density in vitro, with over-expression dramatically increasing spine density and knockdown decreasing spine density [17,18...

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...hat has been shown to induce a slowly developing and long-lasting form of potentiation that depends on calcium influx through voltage-gated calcium channels rather than through NMDA receptor channels =-=[31]-=-. In response to high frequency tetanic stimulation (200 Hz/2 sec) in the presence of the NMDA receptor antagonist CPP (3 μM), a transient depression was followed by a significant potentiation in WT a...

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...18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in =-=[4, 5, 29]-=-. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as the shared-memory approach of [22]. Section 4 discusses some p...

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...by the mouse genome are localized to the postsynaptic density (PSD) [14]. Among the PSD-localized Rho-GEFs is Kalirin-7 (Kal7), the predominant adult splice variant of the multiply spliced Kalrn gene =-=[15,16]-=-. Kal7 has been repeatedly shown to have a profound effect on dendritic spine density in vitro, with over-expression dramatically increasing spine density and knockdown decreasing spine density [17,18...

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...r Kal7-Rac1 interactions are important for the expression of hippocampal LTP, we used 1-(3-nitrophenyl)-1H-pyrrol- 2,5-dione (NPPD), an inhibitor specific for the first GEF domain of Kalirin and Trio =-=[33,34]-=-. As shown in Figure 5A, NPPD (100 μM) had no significant effect on the input-output curve in WT mice. As shown in the example traces in Figure 5B and the group time courses in Figure 5C, in response ...

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...ho-GTPases, which are activated by Rho-guanine nucleotide exchange factors (Rho-GEFs) [13]. About a dozen of the 58 RhoGEFs encoded by the mouse genome are localized to the postsynaptic density (PSD) =-=[14]-=-. Among the PSD-localized Rho-GEFs is Kalirin-7 (Kal7), the predominant adult splice variant of the multiply spliced Kalrn gene [15,16]. Kal7 has been repeatedly shown to have a profound effect on den...

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...ouse had decreased hippocampal spine density at baseline, and was unable to increase dendritic spine density in medium spiny neurons in the nucleus accumbens in response to repeated cocaine treatment =-=[17,19]-=-. Electrophysiologically, genetic deletion of Kal7 resulted in a decrease in the frequency of spontaneous excitatory postsynaptic potentials (sEPSPs) with no change in sEPSP amplitude, suggesting that...

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...decrease in sEPSP frequency was seen in cortical neurons in an animal unable to produce any of the full length Kalirin isoforms due to deletion of exons in the first GEF domain of Kalirin (KalGEF1KO) =-=[20]-=-. Interestingly, Kal7KO mice exhibited a robust decrease in LTP in the hippocampus [17] whereas KalGEF1KO mice demonstrated a small but significant decrease in hippocampal field LTP [21]. Recent bioch...

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...r Kal7-Rac1 interactions are important for the expression of hippocampal LTP, we used 1-(3-nitrophenyl)-1H-pyrrol- 2,5-dione (NPPD), an inhibitor specific for the first GEF domain of Kalirin and Trio =-=[33,34]-=-. As shown in Figure 5A, NPPD (100 μM) had no significant effect on the input-output curve in WT mice. As shown in the example traces in Figure 5B and the group time courses in Figure 5C, in response ...

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...we chose to use NPPD, which inhibits the N-terminal GEF1 domain of Kalirin (the only GEF domain in Kal7) and its ortholog Trio, but does not affect the catalytic activity of other GEF proteins tested =-=[34,48]-=-. Previous studies from our lab have shown that NPPD is effective at inhibiting the first GEF domain of Kalirin in pituitary cells [33]. LTP and LTD induction were fully blocked in the presence of NPP...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...

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...nly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in =-=[2, 3, 32, 40]-=- and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as...

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...nly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in =-=[2, 3, 32, 40]-=- and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as...

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...18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in =-=[4, 5, 29]-=-. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as the shared-memory approach of [22]. Section 4 discusses some p...

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...titive for asynchronous systems. Such techniques take advantage of the locality and symmetry properties widely enjoyed by asynchronous systems. Partial order reduction [24, 43] and symmetry reduction =-=[10]-=- have been successfully implemented in explicit model checkers to reduce the number of states that must be explored and stored, thus the run time. These approaches explore and store only a “representa...

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...usly more efficient than breadth-first iterations, we should take this approach to parallelize Saturation. Ironically, exploring this opportunity led us to further speedup sequential Saturation first =-=[15]-=-. One of the reasons for the efficiency of Saturation is its extensive use of chaining [39]: if events α and β can be fired on the set of states X , X ∪Nα(X )∪Nβ(X ) ⊆ X ∪Nα(X )∪Nβ (X ∪Nα(X )) (for th...

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...ective, our confidence in the appropriateness of symbolic techniques mainly comes from an even better algorithm: Saturation. Initially defined for asynchronous models satisfying Kronecker-consistency =-=[17]-=-, where, for each event α , Nα is the conjunction of L “local” functions Nα ,k : Xk → 2Xk , for L≥ k≥ 1, Saturation has later been extended to a fully general setting where each Nα is the conjunction ...

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...state one by one, and was first parallelized in [16, 35, 40]. The other approach, DD-based state-space generation, is behind all commonly used symbolic modelchecking tools, such as SMV [30] and SMART =-=[18]-=-. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4,...

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... methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as the shared-memory approach of =-=[22]-=-. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 Parallelizing explicit state-space generation Mo...

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...ystems [41], network of workstations (NOW) [34], distributed shared memory (DSM) architectures [37], single-instruction-multiple-data (SIMD) and multiple-instruction-multipledata (MIMD) architectures =-=[23]-=-, and vector processors [36]. According to the nature of the statespace generation algorithm, we can generally classify the implementation platform into two categories: distributed-memory architecture...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...

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...ents, even if disjunctive partitioning [9] often helps. Analogously, image computation, the base of symbolic state-space generation, is often expensive for asynchronous systems. Thus, tools like SPIN =-=[28]-=-, a model checker with advanced explicit techniques, have achieved wide acceptance and success in industrial protocol verification. Furthermore, as shown in the following sections, the parallelization...

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...18]. The remainder of this paper is organized as follows. Section 2 focuses on the explicit distributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in =-=[4, 5, 29]-=-. Section 3 surveys parallel symbolic approaches for distributed-memory architectures [1, 11, 12, 25, 26, 27, 31, 41, 42, 46], as well as the shared-memory approach of [22]. Section 4 discusses some p...

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...es exist. Traditional explicit state space generation approach, such as the one used in the model-checking tool SPIN [28], enumerates and explores each state one by one, and was first parallelized in =-=[16, 35, 40]-=-. The other approach, DD-based state-space generation, is behind all commonly used symbolic modelchecking tools, such as SMV [30] and SMART [18]. The remainder of this paper is organized as follows. S...

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...stations (NOW) [34], distributed shared memory (DSM) architectures [37], single-instruction-multiple-data (SIMD) and multiple-instruction-multipledata (MIMD) architectures [23], and vector processors =-=[36]-=-. According to the nature of the statespace generation algorithm, we can generally classify the implementation platform into two categories: distributed-memory architecture vs. shared-memory architect...

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... symbolic breadth-first iterations. Indeed, we have applied the Saturation approach also to CTL model checking [47], distance function computation [19], and timed reachability in integer-timed models =-=[44]-=-, but here we will limit our discussion to state-space generation. G. Ciardo, Y. Zhao, X. Jin 5 1.3 Classification of previous work on parallel state-space analysis Modern technology offers new parall...

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...nd Z , and relations, i.e., N , are encoded using BDDs (if the local state variable xk is not boolean, we can use dlog2 nke boolean variables for it, or we can use MDDs, in particular extensible MDDs =-=[45]-=- if nk is not known a priori; in the following, we simply use the term DD). This is essentially a symbolic breadth-first exploration, where iteration d discovers all states at distance d from Xinit . ...

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...ributedmemory approaches proposed in [2, 3, 32, 40] and the explicit shared-memory methods presented in [4, 5, 29]. Section 3 surveys parallel symbolic approaches for distributed-memory architectures =-=[1, 11, 12, 25, 26, 27, 31, 41, 42, 46]-=-, as well as the shared-memory approach of [22]. Section 4 discusses some promising directions for further research on optimizing the parallelization of symbolic state-space generation algorithms. 2 P...