This paper introduces an end-to-end probabilistic sequence transduction system based on recurrent neural networks (RNNs) that can transform any input sequence into any finite, discrete output sequence. The system addresses the challenge of sequence transduction, where the alignment between input and output sequences is unknown. It extends the Connectionist Temporal Classification (CTC) approach by defining a distribution over output sequences of all lengths and jointly modeling input-output and output-output dependencies. The system uses two RNNs: a transcription network that processes the input sequence and a prediction network that processes the output sequence. The transcription network is a bidirectional RNN that processes the input sequence in both directions, while the prediction network is a unidirectional RNN that processes the output sequence. The system defines a conditional distribution over output sequences and uses a forward-backward algorithm to compute the probability of the output sequence given the input sequence. The system is trained to minimize the log-loss of the target sequence. Experimental results on the TIMIT speech corpus show that the system achieves a phoneme error rate among the lowest recorded on TIMIT, demonstrating its effectiveness in sequence transduction tasks. The system is also compared to standalone next-step prediction RNNs and CTC RNNs, showing that it performs better in integrating acoustic and linguistic information. The system is applicable to a wide range of sequence transduction problems, including text-to-speech and machine translation. The paper concludes that the system has the potential to improve sequence transduction tasks and suggests future work in this area.This paper introduces an end-to-end probabilistic sequence transduction system based on recurrent neural networks (RNNs) that can transform any input sequence into any finite, discrete output sequence. The system addresses the challenge of sequence transduction, where the alignment between input and output sequences is unknown. It extends the Connectionist Temporal Classification (CTC) approach by defining a distribution over output sequences of all lengths and jointly modeling input-output and output-output dependencies. The system uses two RNNs: a transcription network that processes the input sequence and a prediction network that processes the output sequence. The transcription network is a bidirectional RNN that processes the input sequence in both directions, while the prediction network is a unidirectional RNN that processes the output sequence. The system defines a conditional distribution over output sequences and uses a forward-backward algorithm to compute the probability of the output sequence given the input sequence. The system is trained to minimize the log-loss of the target sequence. Experimental results on the TIMIT speech corpus show that the system achieves a phoneme error rate among the lowest recorded on TIMIT, demonstrating its effectiveness in sequence transduction tasks. The system is also compared to standalone next-step prediction RNNs and CTC RNNs, showing that it performs better in integrating acoustic and linguistic information. The system is applicable to a wide range of sequence transduction problems, including text-to-speech and machine translation. The paper concludes that the system has the potential to improve sequence transduction tasks and suggests future work in this area.