Abstract
We propose new methods for goal and plan recognition based on prefix and infix probability computation in a probabilistic context-free grammar (PCFG) which are both applicable to incomplete data. We define goal recognition as a task of identifying a goal from an action sequence and plan recognition as that of discovering a plan for the goal consisting of goal-subgoal structure respectively. To achieve these tasks, in particular from incomplete data such as sentences in a PCFG that often occurs in applications, we introduce prefix and infix probability computation via parse trees in PCFGs and compute the most likely goal and plan from incomplete data by considering them as prefixes and infixes.
We applied our approach to web session logs taken from the Internet Traffic Archive whose goal and plan recognition is important to improve websites. We tackled the problem of goal recognition from incomplete logs and empirically demonstrated the superiority of our approach compared to other approaches which do not use parsing. We also showed that it is possible to estimate the most likely plans from incomplete logs. All prefix and infix probability computation together with the computation of the most likely goal and plan in this paper is carried out using logic-based modeling language PRISM.
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Notes
- 1.
In this paper, we distinguish goal recognition and plan recognition; the former is a task of identifying a goal from actions but the latter means to discover a plan consisting of goal-subgoal structure to achieve the goal.
- 2.
- 3.
\(\mathrm{expl}_0(G)\) is equivalent to G in view of the distribution semantics. When convenient, we treat \(\mathrm{expl}_0(G)\) as a bag \(\{e_1, e_2, \cdots , e_k\}\) of explanations.
- 4.
This is justified because we assume the consistency of PCFGs [16] that implies the probability of remaining nonterminals in R yielding some terminal sequences is 1.
- 5.
probf/1 is a PRISM’s built-in predicate and displays an explanation graph.
- 6.
\(\mathtt{W} = 1\) because pre_pcfg([a],[a],[]) is logically proved without involving msws.
- 7.
Clustering was done by PRISM. We used a small CFG for clustering, containing 30 rules and 12 nonterminals, because clustering by a mixture of large PCFGs tends to suffer from very high memory usage. To build this grammar, we merged similar symbols such as InternalSearch and Search in the universal session grammar shown in Table 2.
- 8.
It is conducted on a PC with Core i7 Quad 2.67Â GHz, OpenSUSE 11.4 and 72Â GB main memory.
- 9.
We applied a PCFG to prefixes by pretending them to be sentences. In this experiment, we found that the universal session grammar fails to parse at most two sequences for each data set, so we can ignore these sequences.
- 10.
We used a left-to-right HMM where the number of states is varied from 2 to 8. In Fig. 4, only the highest accuracy is plotted for each k. Since logistic regression only accepts fixed length data, we prepare 19 logistic regression models, one for each length k \((2 \le k \le 20)\).
- 11.
We used PRISM to implement a mixture of HMMs and that of PCFG and also to compute prefix probability. For the implementation of logistic regression we used the ‘nnet’ package of R.
- 12.
The entropy is defined as \(- \sum _{\tau } P(\tau )\log P(\tau )\) where \(\tau \) is a possible parse tree [2]. In our setting, a common grammar, the universal session grammar, is used for all data sets. So the entropy only depends on the parameters of a PCFG learned from the data set.
- 13.
This is to simulates a state transition of \(\mathsf{FA}\) made by a string derived from the nonterminal A using \(A \rightarrow B C\).
- 14.
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Kojima, R., Sato, T. (2015). Goal and Plan Recognition via Parse Trees Using Prefix and Infix Probability Computation. In: Davis, J., Ramon, J. (eds) Inductive Logic Programming. Lecture Notes in Computer Science(), vol 9046. Springer, Cham. https://doi.org/10.1007/978-3-319-23708-4_6
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