Goal Recognition Control under Network Interdiction Using a Privacy Information Metric
Abstract
:1. Introduction
 We start by introducing one gametheoretic decisionmaking framework, and then present the generative inverse pathplanning and network interdiction for goal recognition, and some information metrics for the signaling behavior.
 We adopt a minentropy based privacy information metric to quantify the privacy information leakage of the actions and states about the goal.
 We define the InfoGRC and InfoGRCT using the privacy information metric, and provide a more compact solution method for the observer to control the goal uncertainty by incorporating the information metric as additional path cost.
 We conduct some experimental evaluations to demonstrate the effectiveness of the InfoGRC and InfoGRCT model in controlling the goal recognition process under network interdiction.
2. Background and Related Work
2.1. PathPlaning and Network Interdiction
2.1.1. PathPlanning
 $\mathcal{N}$ is a nonempty set of location nodes;
 $\mathcal{E}\subseteq \mathcal{N}\times \mathcal{N}$ is a set of actions related edges between nodes;
 $\mathrm{c}:\mathcal{E}\mapsto {R}_{0}^{+}$ returns the cost of traversing each edge.
 $\mathcal{D}$ is the path planning domain;
 $s\in \mathcal{N}$ is the start location;
 $\mathcal{G}=\{{g}_{r},{g}_{0},{g}_{1},\dots \}$ is a set of candidate goals, where ${g}_{r}$ is the real goal;
 ${\mathcal{P}}_{po}\left(\mathcal{G}\right{\mathcal{O}}_{n})$ denotes the posterior probability of a goal given a sequence of observations (or last state in that sequence), which can be the model of the observer;
 $\mathsf{\Omega}=\left\{{o}_{i}i=1,\cdots ,m\right\}$ is the set of m observations that can be emitted as results of the actions and the states;
 $\mathcal{O}:(\mathcal{N}\times \mathcal{E})\to \mathsf{\Omega}$ is a manytoone observation function which maps the action taken and the next state reached to an observation in Ω.
2.1.2. Network Interdiction
 ${\mathbf{x}}^{*}$ denotes an optimal interdiction solution for the observer.
 Flowbalance constraints of variables $\mathrm{y}$, route one unit of flow from s to g, the inner minimum is a standard shortest path model with edge cost $c\left(a\right)+\alpha x\left(a\right)d\left(a\right)$.
 $c\left(a\right)$ is the nominal cost of edge a and $c\left(a\right)+d\left(a\right)$ is the interdicted cost; $d\left(a\right)$ represents the additional path cost, if sufficiently large, represents complete destruction of edge a.
 $r\left(a\right)$ is a small positive integer, representing how many resources are required to interdict edge a.
 R is the total available resource, the observer has ${C}_{R}^{\leftx\right}$ possible interdiction combinations, which will grow exponentially with R.
 y denotes a traverse path of the actor.
2.2. Goal Recognition
2.2.1. Probabilistic Goal Recognition
 $\mathcal{D}=\langle \mathcal{N},\mathcal{E},\mathrm{c}\rangle $ is a path planning domain;
 $\mathcal{G}\subseteq \mathcal{N}$ is the set of candidate goals locations;
 $s\in \mathcal{N}$ is the start location;
 $\mathcal{O}={o}_{1},\dots ,{o}_{k}$, where $k\ge 0$ and ${o}_{i}\in \mathcal{N}$ for all $i\in \{1,\dots ,k\}$, is a sequence of observation;
 ${\mathcal{P}}_{pr}$ represents the prior probabilities of the goals.
2.2.2. Goal Recognition Design
 ${\mathcal{P}}_{\mathcal{D}}$ is a planning domain formulated in STRIPS;
 ${\mathcal{G}}_{\mathcal{D}}$ is a set of possible goal;
 The output is ${\mathcal{P}}_{\mathcal{D}}^{\prime}$ such that $wcp\left({\mathcal{P}}_{\mathcal{D}}^{\prime}\right)\le wcp\left({{\mathcal{P}}_{\mathcal{D}}}^{\u2033}\right)$,
2.2.3. Trend and DualUse
2.3. Behavioral Information Metrics
3. Goal Recognition Control
3.1. Privacy Information Metrics
 initial uncertainty: ${H}_{\infty}\left(H\right)=logV\left(H\right)$
 remaining uncertainty: ${H}_{\infty}\left(H\rightL)=logV\left(H\rightL)$.
 information leakage $={H}_{\infty}\left(H\right){H}_{\infty}\left(H\rightL)$
3.2. InfoGRC and InfoGRCT
3.2.1. Accelerate and Delay
3.2.2. Control and Threshold
3.3. Dual Reformulation
Algorithm 1 The Benders decomposition based problemsolving algorithm for InfoGRCT 

4. Experiments
4.1. Experimental Setup
4.2. Experimental Scenarios
4.3. Goal Recognition Control under Network Interdiction
5. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
Abbreviations
GR  Goal recognition 
GRD  Goal recognition design 
GRC  Goal recognition control 
CIP  Critical infrastructure protection 
PAIR  Plan activity and intent recognition 
HAIP  Human–AI planning 
XAIP  Explainable planning 
HMI  Human–machine interaction 
COA  Course Of action 
HTN  Hierarchical task network 
BLMIP  Bilevel mixedinteger programming 
$wcd$  worstcase distinctiveness 
MXFI  Maximumflow network interdiction 
SPNI  Shortest path network interdiction 
MXSP  Maximizing the shortest path 
KKT  Karush–Kuhn–Tucker 
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Parameters  Meaning 

Sets and indices  
$\mathcal{G}=(\mathcal{N},\mathcal{E})$  Road graph network with nodes $\mathcal{N}$ and edges $\mathcal{E}$ 
$i\in \mathcal{N}$  Node i in $\mathcal{G}$ 
$a=(i,j)\in \mathcal{E}$  Edge $(i,j)$ in $\mathcal{G}$ 
$s\in \mathcal{N}$  Start node s 
$g\in \mathcal{N}$  Goal node g 
$\mathrm{In}\left(\mathrm{i}\right)/\mathrm{Out}\left(\mathrm{i}\right)$  Edges set directed into or out of node i 
Data  
$0\le c\left(a\right)<\infty $  Cost of edge a, vector form $\mathbf{c}$ 
$0<d\left(a\right)<\infty $  Interdiction increment if edge a is interdicted, vector form $\mathbf{d}$ 
${\mathcal{I}}_{A}\left(a\right)$  The privacy information metric of action a 
$r\left(a\right)>0$  Resource required to interdict edge a, vector form $\mathbf{r}$ 
R  Total amount of interdiction resource available 
$\tilde{\theta}>0$  Threshold of the shortest path 
$\overline{\theta}>0$  Upper bound with full interdiction 
$\underline{\theta}>0$  Lower bound without interdiction 
Decision Variables  
$x\left(a\right)$  Observer’s interdiction resource allocation, $x\left(a\right)=1$ if edge a is interdicted 
$y\left(a\right)$  Actor’s traveling edge, $y\left(a\right)=1$ if edge a is traveled by the actor 
$\mathit{E}\left(\mathit{e}\right)$$(\%)$  Scenario 1  Scenario 2  Scenario 3 

InfoGRC  65.4/63.7  62.7/78.4  88.7/77.5/90.8 
InfoGRCT  65.1/63.3  62.1/78.1  88.2/77.2/90.4 
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Luo, J.; Ji, X.; Gao, W.; Zhang, W.; Chen, S. Goal Recognition Control under Network Interdiction Using a Privacy Information Metric. Symmetry 2019, 11, 1059. https://doi.org/10.3390/sym11081059
Luo J, Ji X, Gao W, Zhang W, Chen S. Goal Recognition Control under Network Interdiction Using a Privacy Information Metric. Symmetry. 2019; 11(8):1059. https://doi.org/10.3390/sym11081059
Chicago/Turabian StyleLuo, Junren, Xiang Ji, Wei Gao, Wanpeng Zhang, and Shaofei Chen. 2019. "Goal Recognition Control under Network Interdiction Using a Privacy Information Metric" Symmetry 11, no. 8: 1059. https://doi.org/10.3390/sym11081059