Causal-State Splitting Reconstruction Ziba Rostamian CS 590 – Winter 2008

Probabilistic Automaton A probabilistic automaton is a finite automaton with transition probabilities which represents a distribution over the set of all strings defined over a finite alphabet. They are used in a variety of applications, including text and speech processing, image processing, and computational biology

Notations Consider, a discrete-time,discrete-valued stochastic process where value of ∈ A and A is a finite alphabet of k symbols At any time t, is future and is history. For all events A in future : does not depend on t. Problem : knowing, what is P( )?

Causal States Definition: The causal states of a process are the members of the range of a function that maps from histories to sets of histories. If μ( ) is the collection of all measurable future events, then

Recurrent, Transient and Synchronization states Recurrent: Recurrent states are returned to infinitely. Transient: Transient states visited only finitely. Synchronization: Synchronization states can never be returned to once a recurrent state has been visited.

Properties of causal states The causal states are homogeneous for future events. The causal states themselves form Markov process. The –machine is deterministic.

Causal-State Spiting Reconstruction A string w is suffix of a history if for some L. A state has been represented as a set of suffixes. The function maps a finite history to that which contains a suffix of One suffix is the child of another suffix if w = av where a is a single symbol. A suffix is a descendent of its ancestor if w = uv where u in any non-null string.

State’s Morph Each is associated with a distribution for the next observable ;i.e., is defined for each and each. They call this distribution the state’s morph.

CSSR Algorithm There are three steps in CSSR algorithm: Initialize, Homogenize and Determinize. Initialize : Start L = 0 and where.They regard suffix of any history, so that initially maps all histories to one causal state. The morph of that state is defined as: