How does TensorFlow name tensors?

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Your observations on Tensor naming are absolutely correct: the name of a Tensor is the concatenation of

  1. the name of the operation that produced it,
  2. a colon (:), and
  3. the index of that tensor in the outputs of the operation that produced it.

Therefore the tensor named "foo:2" is the output of the op named "foo" at position 2 (with indices starting from zero).

The naming of tf.Variable objects is slightly strange. Every tf.Variable contains a mutable tensor object that holds the state of the variable (and a few other tensors). A "Variable" op (which has the name "variable_name" in your example) “produces” this mutable tensor each time it is run as its 0th output, so the name of the mutable tensor is "variable_name:0".

Since a tf.Variable is mostly indistinguishable from a tf.Tensor—in that it can be used in the same places—we took the decision to make variable names resemble tensor names, so the Variable.name property returns the name of the mutable tensor. (This contrasts with tf.QueueBase and tf.ReaderBase objects, which are not usable directly as tensors (instead you have to call methods on them to create ops that operate on their state), so these do not have a tensor-like name.)

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