Symbolic Encoding of Neural Networks using Communicating Automata with Applications to Verification of Neural Network Based Controllers* Li Su, Howard Bowman and Brad Wyble Centre for Cognitive Neuroscience and Cognitive Systems, University of Kent, Canterbury, Kent, CT2 7NF, UK *To Appear in Neural-Symbolic Learning and Reasoning Workshop at Nineteenth International Joint Conference on Artificial Intelligence, EDINBURGH, SCOTLAND, 2005.

Outline Background: –Symbolic Computation –Sub-symbolic Computation Motivation for integrating Symbolic and Sub-symbolic Computation –Cognitive Viewpoint –Application Viewpoint Formal Methods –Model Checking –Specification –Properties –Result Summary

Background 1: Symbolic Computation Traditional symbolic computation: –Systems have explicit elements that correspond to symbols organised in systematic ways, representing information in the external world. –Programmes or rules can manipulate these symbolic representations. –Key characteristic: symbol manipulation.

Background 2: Sub-symbolic Computation Connectionism/neural networks are computational models inspired by neuron physiology, which can be regarded as sub-symbolic computation: –Aims at massively parallel simple and uniform processing elements, which are interconnected. –Representations are distributed throughout processing elements.

Motivation 1: Cognitive Viewpoint It has been argued that cognition/mind can be regarded as symbolic computation. (E.g. SOAR, ACT-R and EPIC) Sub-symbolic (neural network) architectures constitute abstract model of the human brain.

Motivation 1: Cognitive Viewpoint (cont.) Combining symbolic and sub-symbolic techniques to specify and justify behaviour of complex cognitive architectures in an abstract and suitable form. –Concurrent, Distributed Control, Hierarchical Decomposition –How do high-level cognitive properties emerge from interactions between low-level neuron components? Our approach is to encode and reason about cognitive systems or neural networks in symbolic form. –E.g. Formal Methods. –Automatic mathematical analysis can be applied.

Motivation 2: Application Viewpoint Connectionist networks can be applied to extending traditional controllers in order to handle: –Catastrophic changes –Gradual degradation –Complex and highly non-linear systems –E.g. aircraft, spacecraft or robots Reliability/Stability of adaptive systems (neural networks) needs to be guaranteed in safety/mission critical domains. However, connectionist models rarely provide an indication of the accuracy or reliability of their predictions.

Formal Methods: Model Checking Automatic analysis technique, which can be applied at system design stage. Checking whether a formal specification satisfies a set of properties, which are expressed in a requirements language. Model Checker Inputs: Yes +Witness / No + Counter-example specification properties Output(s):

An Example of a Neural Network Specification I1 I2 H1 H2 O1 Environment Tester Input Layer Hidden Layer Output Layer NeuralNet Note: this is not a realistic model of controller, but a “toy” model to evaluate the ability of model checking neural networks.

Neuron Automaton Input Middle Output k: identify of neuron;t: local clock;: activation of neuron i; i: pre-synaptic neuron identity; : speed of update;: activation of neuron k; j: post-synaptic neuron identity; : sigmoid function;: error; : net input; : weight;: learning rate.

Requirements Language

Requirements Language (cont.) Reachability Properties: –E.g. Safety Properties: –E.g. – Liveness Properties: –E.g. Note: the state formula success is true when SSE is less than a specified value.

Result The network satisfies the following properties and is guaranteed to learn XOR according to the required timing constraints using BP learning. It also guarantees the learning process is eventually stabilised. – deadline success ……

Summary Formal methods are justifiable techniques to represent low- level neural networks. They can also help to understand how high-level cognitive properties emerge from interactions between low-level neuron components. Formal methods may allow neural networks within engineering applications to be specified and justified at the system design stage. Verifications may give theoretically well-founded ways to evaluate and justify learning methods. Some pproperties can be hard to justify by simulation. –Simulations can only test that something occurs, but are unable to test that something can never occur without explicit mathematical analysis. (An open issue.)