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Applications and Techniques in Information Security. Definition 10 Trace of execution : let B be the atoms that appear in the head of the rules of a Markov k system S. We denote by T the size of the trace that is the number of elements of the sequence. As shown in Figure 3 , a Markov k system may seem non-deterministic when it is represented by a state transition diagram right part of the figure.
That is because such state transition diagram only represents 1-step transitions. But it can be Markov 2 , because all traces of size 2 of Figure 3 are consistent. Figure 3.
Eight traces of executions of the system of Example 4 left and the corresponding state transitions diagram right. Definition 13 Extended consistency : let R be a rule and I , J be a k -step interpretation transition. Let T be a sequence of state transitions, R is consistent with T if it is consistent with every k -step interpretation transitions of T. Here, we briefly summarize the essence of LFkT.
Because of the lack of space, the details of the algorithm, its pseudo-code, and the proofs of correctness are given as Supplementary Material. We refer to the pseudo-code of the appendix as follows: algo. LFkT is an algorithm that can learn the dynamics of a Markov k system from its traces of executions. LFkT takes a set of traces of executions O as input, where each trace is a sequence of state transitions. If all traces are consistent, the algorithm outputs a logic program P that realizes all transitions of O.
The learned influences can be at most k -step relations, where k is the size of the longest trace of O. Transforming the traces into pairs of interpretations allows us to use minimal specialization Ribeiro and Inoue, to iteratively learn the dynamics of the system. The idea of the algorithm is to start with the most general rules algo. The algorithm analyzes each interpretation transition one by one and revises the learned rules when they are not consistent algo. In the following, we will call an n -step rule any rule from the logic program learned from n -step transitions.
After analyzing all interpretation transitions, the programs that have been learned are merged into a unique logic program algo. This operation ensures that the rules outputted are consistent with all observations. It can be checked by comparing each rule with other logic programs. Finally, LFkT outputs a logic program that realizes all consistent traces of execution of O.
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In the previous subsections, we have illustrated step by step how the LFkT algorithm is able to learn Markov k systems. To illustrate the merits of our work, we now apply this approach to the analysis of the yeast cell cycle dataset from Spellman et al. In this paper, Li et al. The originality of their work lies in the fact they consider delayed correlations between genes.
The methodology can capture gene regulations that are delayed of k time units. The limits of the approach is that the authors only consider pairwise overlaps of expression levels shifted in time relative to each other. Another limit of the approach is that it is not able to make a distinction between a causal gene—gene regulation and the scenarios where two genes, A and B, are being co-regulated by a third gene C: do we have A that regulates B that regulates C, or is it a co-operation between A and B that regulates C?
Here, starting from a set of different traces coming from the yeast cell cycle system, we have performed various experiments where we have tuned the number of traces that have been considered on the one hand, the value of k i. Figure 4 shows the evolution of run time of learning with LFkT on the five Boolean networks of the yeast cell cycle proposed by Li et al. These fives programs are, respectively, Markov 1 to Markov 5.
In these experiments, for each Boolean network, the number of variables is 16 and the length of traces in input is five states. The five Boolean networks have been implemented as a logic program using Answer Set Programing Brewka et al. The source code of these programs is given as Supplementary Material.
Traces of executions of these programs have been computed using the answer set solver clasp Gebser et al. The main purpose of these experiments is to assess the efficiency of our approach, i. In the first table of Figure 4 , the evolution of run time from 10 to 1,, traces which is arbitrary chosen as upper bound of the scalability of the experiments shows that, in practice, learning with LFkT is linear in the number of traces when the number of variables is fixed.
Results show that the algorithm can handle more than one million of traces in less than 10 h. Since each trace is a sequence of five state transitions, when learning the Markov 5 system, each trace can be decomposed into 15 interpretation transitions one 5-step, two 4-step, three 3-step, four 2-step, and five 1-step. Learning the Markov 5 program from one million traces of executions of size five requires the processing of 15 million of interpretation transitions. Learning the Markov 4 to Markov 1 programs requires to process, respectively, 14 million, 12 million, 9 million, and 5 million of interpretation transitions.
Intuitively one could expect that learning the Markov 2 system to take significantly more time than learning the Markov 1 system. But each program is different, i. That is why run time is not always larger for a larger k : learning time also depends on the rules that are learned.
In this experiment, the best run time is obtained with the Markov 3 program. We cannot say that the rules of this program are simpler than the others, but they are simpler to learn for the algorithm. In the second table, we observe that the number of rules learned for the Markov 3 program is significantly smaller than for the others. It means that the algorithm needs to compare less rules for each traces analysis, which can explain the speed up. In this benchmark, in order to be faithful to the biological experiments presented by Li et al.
But our algorithm succeeds in processing larger memory effects. On some random dummy examples accessible at the above mentioned URL , we were able to learn Markov 7 systems with the following performances: we can learn 10 traces in 2. Even if the computation time increases, it should be kept in mind that our method is designed to allow successive refinements of a model about its memory effect.
These results show that such an approach is tractable even with a large number of input traces. In this paper, we proposed a logical method to learn such models from state transition systems. We designed an approach to learn Boolean networks with delayed influences. We have given a step by step explanation of this methodology, and illustrated its merits on a biological benchmark coming from a real-life case study.
Further works aim at adapting the approach developed in the paper to the kind of data produced by biologists. This requires connecting through various biological databases in order to extract real time series data, and subsequently explore and use them to learn gene regulatory networks. On account of the noise inherent to biological data, the ability to either perform an efficient discretization of the data or to include the notion of noise inside the modeling framework is fundamental. We will thus have to discuss the discretization procedure and the robustness of our modeling against noisy data and compare it to existing approaches, like the Bayesian ones Barker et al.
Regarding the model, we consider extending the methodology to asynchronous semantics. Garg et al. The authors focus on attractors, which are central to gene regulation. Previous studies about attractors with synchronous semantics [by Melkman et al. The benefits of the synchronous model are to be computationally tractable, while classical state space exploration algorithms fail on asynchronous ones.
Yet, the synchronous modeling relies on one quite heavy assumption: all genes can make a transition simultaneously and need an equivalent amount of time to change their expression level. Even if this is not realistic from a biological point of view, it is usually sufficient as the exact kinetics and order of transformations are generally unknown. The asynchronous semantics, however, helps to capture more realistic behaviors. That is why we plan to extend our approach to asynchronous semantics. Finally, we will also address multi-valued networks that may be useful to capture behaviors that cannot be summarized through a pure Boolean framework.
Tony Ribeiro: formalization of the problem; design, implementation, description, and pseudo-code of the algorithm; design, implementation, run, and discussion of experiments. Morgan Magnin: state of the art, introduction, biological background, case study, and conclusion. Katsumi Inoue: supervision of the work; formalization of the logic programing and learning from interpretation transition approach background.
Chiaki Sakama: formalization of the logic programing and learning from interpretation transition approach background. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. From structure to dynamics: frequency tuning in the pmdm2 network: I.
Akutsu, T. Identification of genetic networks by strategic gene disruptions and gene overexpressions under a Boolean model. Determining a singleton attractor of a Boolean network with nested canalyzing functions. Apt, K. Google Scholar. Barker, N.
Mehmet Eren Ahsen - Google Scholar Citations
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Comprehensive identification of cell cycle-regulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization. Cell 9, — Van Emden, M. The semantics of predicate logic as a programming language. Zhang, Z. Keywords: Boolean network, gene regulatory networks, delayed influences, time delay, logic programming, machine learning, state transitions.