WFST over semiring

A weighted finite state transducer (WFST) $(\Sigma,\Delta,Q,E,I,F,\lambda,\rho)$ over a semiring $(\mathbb{K},\oplus,\otimes, \bar{0}, \bar{1})$ is defined as:

$$T(x, y) = \underset{\pi\in P(I,x,y,F)}{\oplus}\lambda(p[\pi])\otimes w[\pi]\otimes\rho(n[\pi])$$

Semiring is a ring may lack negation. The two operations:

• Sum: compute the weight of a sequence (sum of the weights of the paths labeled with that sequence)
• Product: compute the weight of a path (product of the weights of constitute transitions)

Important semirings:

• Probability semiring: $(\mathbb{R}_+, +, \times, 0, 1)$
• Log semiring: $(\mathbb{R}\cup\{-\infty,+\infty\}, \oplus_{\log}, +, +\infty, 0)$
• Tropical semiring: $(\mathbb{R}\cup \{-\infty,+\infty\},\min,+, +\infty, 0)$