afastalgorithmforcomputingprefixprobabilities/full.md

A Fast Algorithm for Computing Prefix Probabilities

Franz Nowak Ryan Cotterell

{franz现阶段，ryan.cotterell}@inf.ethz.ch

ETH zürich

Abstract

Multiple algorithms are known for efficiently calculating the prefix probability of a string under a probabilistic context-free grammar (PCFG). Good algorithms for the problem have a runtime cubic in the length of the input string. However, some proposed algorithms are suboptimal with respect to the size of the grammar. This paper proposes a novel speed-up of Jelinek and Lafferty's (1991) algorithm, whose original runtime is $\mathcal{O}(N^3 |\mathcal{N}|^3 +|\mathcal{N}|^4)$ , where $N$ is the input length and $|\mathcal{N}|$ is the number of non-terminals in the grammar. In contrast, our speed-up runs in $\mathcal{O}(N^2 |\mathcal{N}|^3 +N^3 |\mathcal{N}|^2)$ .

https://github.com/rycolab/ prefix-parsing

1 Introduction

Probabilistic context-free grammars (PCFGs) are an important formalism in NLP (Eisenstein, 2019, Chapter 10). One common use of PCFGs is to construct a language model. For instance, PCFGs form the backbone of many neural language models, e.g., recurrent neural network grammars (RNNGs; Dyer et al., 2016; Dyer, 2017; Kim et al., 2019). However, in order to use a PCFG as a language model, one needs to be able to compute prefix probabilities, i.e., the probability that the yield of a derivation starts with the given string. In notation, given a string $\boldsymbol{w} = w_{1}\dots w_{N}$ , we seek the probability $p(\mathrm{S} \Rightarrow \boldsymbol{w}\dots)$ where S is the distinguished start symbol of the grammar and $\Rightarrow$ is the closure over applications of derivation rules of the grammar. Our paper gives a more efficient algorithm for the simultaneous computation of the prefix probabilities of all prefixes of a string $\boldsymbol{w}$ under a PCFG.

The authors are aware of two existing efficient algorithms to compute prefix probabilities under a PCFG. The first is Jelinek and Lafferty's (1991)

algorithm which is derived from CKY (Kasami, 1965; Younger, 1967; Cocke and Schwartz, 1970) and, thus, requires the grammar to be in Chomsky normal form (CNF). Jelinek-Lafferty runs in $\mathcal{O}(N^3 |\mathcal{N}|^3 +|\mathcal{N}|^4)$ time, where $N$ is the length of the input and $\mathcal{N}$ is the number of non-terminals of the grammar, slower than the $\mathcal{O}(N^3 |\mathcal{N}|^3)$ required for parsing with CKY, when the number of non-terminals $|\mathcal{N}|$ is taken into account.

The second, due to Stolcke (1995), is derived from Earley parsing (Earley, 1970) and can parse arbitrary PCFGs, $^3$ with a runtime of $\mathcal{O}(N^3|\mathcal{N}|^3)$ . Many previous authors have improved the runtime of Earley's (Graham et al., 1980; Leermakers et al., 1992; Moore, 2000, inter alia), and Opedal et al. (2023) successfully applied this speed-up to computing prefix probabilities, achieving a runtime of $\mathcal{O}(N^3|\mathcal{G}|)$ , where $|\mathcal{G}|$ is the size of the grammar, that is, the sum of the number of symbols in all production rules.

Our paper provides a more efficient version of Jelinek and Lafferty (1991) for the computation of prefix probabilities under a PCFG in CNF. Specifically, we give an $\mathcal{O}(N^2|\mathcal{N}|^3 + N^3|\mathcal{N}|^2)$ time algorithm, which is the fastest attested in the literature for dense grammars in CNF; matching the complexity of CKY adapted for dense grammars by Eisner and Blatz (2007). We provide a full derivation and proof of correctness, as well as an open-source implementation on GitHub. We also briefly discuss how our improved algorithm can be extended to work for semiring-weighted CFGs.

2 Preliminaries

We start by introducing the necessary background on probabilistic context-free grammars.

Definition 1. A probabilistic context-free grammar (PCFG) is a five-tuple $\mathcal{G} = (\mathcal{N},\Sigma ,\mathrm{S},\mathcal{R},p)$ , made up of:

• A finite set of non-terminal symbols $\mathcal{N}$ ;
• A finite set of terminal symbols $\Sigma$ , $\Sigma \cap \mathcal{N} = \emptyset$ ;
• A distinguished start symbol $\mathrm{S} \in \mathcal{N}$ ;
• A finite set of production rules $\mathcal{R} \subset \mathcal{N} \times (\mathcal{N} \cup \Sigma)^*$ where each rule is written as $X \to \alpha$ with $X \in \mathcal{N}$ and $\alpha \in (\mathcal{N} \cup \Sigma)^*$ . Here, $*$ denotes the Kleene closure;
• A weighting function $p \colon \mathcal{R} \to [0,1]$ assigning each rule $r \in \mathcal{R}$ a probability such that $p$ is locally normalized, meaning that for all $X \in \mathcal{N}$ that appear on the left-hand side of a rule, $\sum_{X \to \alpha \in \mathcal{R}} p(X \to \alpha) = 1$ .

Note that not every locally normalized PCFG constitutes a valid distribution over $\Sigma^{}$ . Specifically, some may place probability mass on infinite trees (Chi and Geman, 1998). PCFGs that do constitute a valid distribution over $\Sigma^{}$ are referred to as tight. Furthermore, if all non-terminals of the grammar can be reached from the start non-terminal via production rules, we say the PCFG is trim.

Definition 2. A PCFG $\mathcal{G} = (\mathcal{N},\Sigma ,\mathrm{S},\mathcal{R},p)$ is in Chomsky normal form (CNF) if each production rule in $\mathcal{R}$ is in one of the following forms:

$$\begin{array}{cccc}& \mathrm{X}\to \mathrm{Y}\mathrm{Z}& & \text{(1)}\end{array}\backslash mathrm\; \{X\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\}\; \backslash tag\; \{1\}$$

$$\begin{array}{cccc}& \mathrm{X}\to a& & \text{(2)}\end{array}\backslash mathrm\; \{X\}\; \backslash rightarrow\; a\; \backslash tag\; \{2\}$$

$$\begin{array}{cccc}& \mathrm{S}\to \epsilon & & \text{(3)}\end{array}\backslash mathrm\; \{S\}\; \backslash rightarrow\; \backslash varepsilon\; \backslash tag\; \{3\}$$

where $\mathrm{X},\mathrm{Y},\mathrm{Z}\in \mathcal{N}$ such that $\mathrm{Y},\mathrm{Z}\neq \mathrm{S},a\in \Sigma ,$ and $\varepsilon$ is the empty string.6

Definition 3. A derivation step $\alpha \Rightarrow \beta$ is an application of the binary relation $\Rightarrow$ : $(\mathcal{N} \cup \Sigma)^* \times (\mathcal{N} \cup \Sigma)^*$ , which rewrites the left-most non-terminal in $\alpha$ according to a rule in $\mathcal{R}$ from the left-hand side of that rule to its right-hand side, resulting in $\beta$ . The probability of a derivation step is the probability of the applied rule: $p(\alpha X \gamma \Rightarrow \alpha \beta \gamma) \stackrel{\text{def}}{=} p(X \rightarrow \beta)$ .

Definition 4. A derivation under a grammar $\mathcal{G}$ is a sequence $\alpha_0, \alpha_1, \dots, \alpha_m$ , where $\alpha_0 \in \mathcal{N}, \alpha_1, \dots, \alpha_{m-1} \in (\mathcal{N} \cup \Sigma)^*$ , and $\alpha_m \in \Sigma^*$ , in which each $\alpha_{i+1}$ is formed by applying a derivation step to $\alpha_i$ . $\alpha_m = w_1 \dots w_N \in \Sigma^*$ is called the yield of the derivation. If $\alpha_0$ is not the start

symbol S, we call it a partial derivation. We define $p(\alpha \stackrel{*}{\Rightarrow} \beta)$ as the sum of probabilities of all subsequences from $\alpha$ to $\beta$ , where each subsequence probability is the product of the individual derivation step probabilities defined in Def. 3.

We represent derivations as trees whose structure corresponds to production rules, where any parent node is the non-terminal on the left-hand side of a rule and its children are the symbols from the right-hand side. The leaves of the tree, when read from left to right, form the yield. Such a tree, when rooted S, is called a derivation tree. Otherwise, it is called a derivation subtree.

Definition 5. The probability of a derivation tree (or derivation subtree) $\pmb{\tau}$ is the product of the probabilities of all its corresponding production rules:

$$\begin{array}{cccc}& p(\mathit{\tau })\prod _{(\mathit{\alpha }\to \mathit{\beta })\in \mathit{\tau }}p(\mathit{\alpha }\to \mathit{\beta })& & \text{(4)}\end{array}p\; (\backslash boldsymbol\; \{\backslash tau\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash prod\_\; \{(\backslash boldsymbol\; \{\backslash alpha\}\; \backslash rightarrow\; \backslash boldsymbol\; \{\backslash beta\})\; \backslash in\; \backslash boldsymbol\; \{\backslash tau\}\}\; p\; (\backslash boldsymbol\; \{\backslash alpha\}\; \backslash rightarrow\; \backslash boldsymbol\; \{\backslash beta\})\; \backslash tag\; \{4\}$$

Definition 6. $\mathcal{T}{\mathrm{X}}(w_i\dots w_k)$ is the set of all derivation subtrees $\pmb{\tau}$ rooted at X with yield $w{i}\dots w_{k}$

Definition 7. Given a PCFG $\mathcal{G} = (\mathcal{N},\Sigma ,\mathrm{S},\mathcal{R},p)$ a string $\boldsymbol {w} = w_{1}\dots w_{N}\in \Sigma^{*}$ , and a non-terminal $\mathrm{X}\in \mathcal{N}$ , the inside probability of X between indices $i$ and $k$ (where $0\leq i\leq k\leq N$ ) is defined as:

$$\begin{array}{c}\beta (i,\mathrm{X},k)p(\mathrm{X}{w}_{i+1}\dots w_k)(5)\\ ={\sum }_{\mathit{\tau }\in {\mathcal{T}}_{\mathrm{X}}(w_{i+1}\dots w_k)}p(\mathit{\tau })(6)\end{array}\backslash begin\{array\}\{l\}\; \backslash beta\; (i,\; \backslash mathrm\; \{X\},\; k)\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; p\; (\backslash mathrm\; \{X\}\; \backslash stackrel\; \{*\}\; \{\backslash Rightarrow\}\; w\; \_\; \{i\; +\; 1\}\; \backslash dots\; w\; \_\; \{k\})\; (5)\; \backslash \backslash \; =\; \backslash sum\_\; \{\backslash boldsymbol\; \{\backslash tau\}\; \backslash in\; \backslash mathcal\; \{T\}\; \_\; \{\backslash mathrm\; \{X\}\}\; \backslash left(w\; \_\; \{i\; +\; 1\}\; \backslash dots\; w\; \_\; \{k\}\backslash right)\}\; p\; (\backslash boldsymbol\; \{\backslash tau\})\; (6)\; \backslash \backslash \; \backslash end\{array\}$$

That is, the sum of the probabilities of all derivation trees $\pmb{\tau}$ starting at $X$ that have yield $w_{i}\dots w_{k}$

Definition 8. Given a PCFG $\mathcal{G} = (\mathcal{N},\Sigma ,\mathrm{S},\mathcal{R},p)$ a string $\boldsymbol {w} = w_{1}\dots w_{N}\in \Sigma^{*}$ , and a non-terminal $\mathrm{X}\in \mathcal{N}$ , we define the prefix probability $\pi$ , i.e., the probability of $\boldsymbol{\omega}$ being a prefix under $\mathcal{G}$ , as:

$$\begin{array}{cccc}& \pi (\mathit{w}\mid \mathrm{X})\sum _{\mathit{u}\in {\mathrm{\Sigma }}^{\ast }}p(\mathrm{X}\Rightarrow \mathit{w}\mathit{u})& & \text{(7)}\end{array}\backslash pi\; (\backslash boldsymbol\; \{w\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash boldsymbol\; \{u\}\; \backslash in\; \backslash Sigma^\; \{*\}\}\; p\; \backslash left(\backslash mathrm\; \{X\}\; \backslash Rightarrow\; \backslash boldsymbol\; \{w\}\; \backslash boldsymbol\; \{u\}\backslash right)\; \backslash tag\; \{7\}$$

In words, $\pi$ is the probability of deriving $\pmb{w}$ with an arbitrary continuation from X, that is, the sum of probabilities of deriving $\pmb{w}\pmb{u}$ from X over all possible suffixes $\pmb{u} \in \Sigma^{*}$ . In the following, we write the prefix probability of deriving prefix $\pmb{w} = w_{i+1} \cdots w_{k}$ from X as $\pi(i, X, k)$ .

Definition 9. Let $\mathcal{G}$ be a PCFG in CNF. Then for non-terminals $X, Y, Z \in \mathcal{N}$ , the left-corner expectations $\xi(Y | X)$ and $\xi(Y | Z | X)$ are defined as:

$$\begin{array}{cccc}& \xi (\mathrm{Y}\mid \mathrm{X})\sum _{\mathit{u}\in {\mathrm{\Sigma }}^{\ast }}p(\mathrm{X}\mathrm{Y}\mathit{u})& & \text{(8)}\end{array}\backslash xi\; (\backslash mathrm\; \{Y\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash boldsymbol\; \{u\}\; \backslash in\; \backslash Sigma^\; \{*\}\}\; p\; \backslash left(\backslash mathrm\; \{X\}\; \backslash stackrel\; \{*\}\; \{\backslash Rightarrow\}\; \backslash mathrm\; \{Y\}\; \backslash boldsymbol\; \{u\}\backslash right)\; \backslash tag\; \{8\}$$

$$\begin{array}{cccc}& \xi (\mathrm{Y}\mathrm{Z}\mid \mathrm{X})\sum _{{\mathrm{X}}^{\mathrm{\prime }}\in \mathcal{N}}\xi ({\mathrm{X}}^{\mathrm{\prime }}\mid \mathrm{X})\cdot p({\mathrm{X}}^{\mathrm{\prime }}\to \mathrm{Y}\mathrm{Z})& & \text{(9)}\end{array}\backslash xi\; (\backslash mathrm\; \{Y\; Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash mathrm\; \{X\}\backslash right)\; \backslash cdot\; p\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\}\backslash right)\; \backslash tag\; \{9\}$$

Algorithm 1 CKY
19: return $\beta$
1: def CKY $(\pmb {w} = w_{1}\cdot \cdot \cdot w_{N})$
2: Initialize inside probabilities
3: $\beta (\cdot ,\cdot ,\cdot)\gets 0$
4: Handle special rule, $S\rightarrow \varepsilon$
5: $\beta (0,\mathrm{S},0)\leftarrow p(\mathrm{S}\rightarrow \varepsilon)$
6: for $k\in 0,\ldots ,N - 1$ ..
7: for $\mathrm{X}\to w_{k + 1}\in \mathcal{R}$ ..
8: Handle single word tokens
9: $\beta (k,\mathrm{X},k + 1) + = p(\mathrm{X}\to w_{k + 1})$
10: is the span size
11: for $\ell \in 2,\dots ,N$ ..
12: i marks the beginning of the span
13: for $i\in 0,\dots ,N - \ell$ ..
14: k marks the end of the span
15: $k\gets i + \ell$
16: Recursively compute $\beta$
17: for $\mathrm{X}\to \mathrm{YZ}\in \mathcal{R}$ ..
18: $\beta (i,\mathrm{X},k) + = p(\mathrm{X}\to \mathrm{YZ})$ $\sum_{j = i + 1}^{k - 1}\beta (i,\mathrm{Y},j)\cdot \beta (j,\mathrm{Z},k)$

Algorithm 2 Jelinek-Lafferty
1: $P^{\prime}\gets (I - P)^{-1}$ Precompute $P^{*}$ by Eq. (14)
2: for $\mathrm{X}i,\mathrm{X}j\in \mathcal{N}$ .. Assign $\xi (\mathrm{Y}\mid X)$
3: $\xi (\mathrm{X}j\mid \mathrm{X}i)\gets P{ij}^\prime$
4: for $\mathrm{X}'\to \mathrm{YZ}\in \mathcal{R}$ .. Precompute $\xi (\mathrm{YZ}\mid X)$
5: $\xi (\mathrm{YZ}\mid \mathrm{X})\leftarrow \sum{\mathrm{X}\in \mathcal{N}}\xi (\mathrm{X}'\mid \mathrm{X})\cdot p(\mathrm{X}'\rightarrow \mathrm{YZ})$
6: def JL(w=w1...wN):
7: $\pi (\cdot ,\cdot ,\cdot)\gets 0$ Initialize prefix probabilities
8: for $k\in 0,\ldots ,N$ ..
9: for $\mathrm{X}\in \mathcal{N}$ .. Prefix probability of
10: $\pi (k,\mathrm{X},k)\gets 1$
11: $\beta \gets \mathrm{CKY}(\boldsymbol {w})$ . Compute $\beta$ with Alg. 1
12: for $k\in 0,\dots ,N - 1$ ..
13: for $\mathrm{X}\in \mathcal{N}$ .. Compute base case
14: $\pi (k,\mathrm{X},k + 1)\leftarrow \sum{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}|\mathrm{X})\cdot p(\mathrm{Y}\rightarrow w{k + 1})$
15: for $\ell \in 2\dots N$ ..
16: for $i\in 0\dots N - \ell$ ..
17: $k\gets i + \ell$
18: for $\mathrm{X,Y,Z}\in \mathcal{N}$ .. Recursively compute $\pi$
19: $\pi (i,\mathrm{X},k) + = \xi (\mathrm{YZ}\mid \mathrm{X})$ $\cdot \sum_{j = i + 1}^{k - 1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)$
20: return π

(a) $\xi (\mathrm{Y}|\mathrm{X})$
Figure 2: Visualization of left-corner expectations

Figure 1: Pseudocode for the CKY algorithm (left) and Jelinek-Lafferty (right)
(b) $\xi (\mathrm{YZ}\mid \mathrm{X})$

The left-corner expectation $\xi (\mathrm{Y}\mid \mathrm{X})$ is hence the sum of the probabilities of partial derivation subtrees rooted in X that have Y as the left-most leaf; see Fig. 2a for a visualization. Similarly, $\xi (\mathrm{YZ}\mid \mathrm{X})$ is the sum of the probabilities of partial derivation subtrees that have Y and Z as the leftmost leaves; see Fig. 2b.

3 Jelinek and Lafferty (1991)

We now give a derivation of the Jelinek-Lafferty algorithm. The first step is to derive an expression for the prefix probability in PCFG terms.

Lemma 1. Given a tight, trim PCFG in CNF and a string $\pmb{w} = w_{1}\dots w_{N}$ , the prefix probability of a

substring $w_{i + 1}\dots w_k$ of $\pmb{w}$ being derived from X can be defined recursively as follows:

$$\begin{array}{cccc}& \begin{array}{c}\pi (i,\mathrm{X},k)={\sum }_{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}\xi (\mathrm{Y}\mathrm{Z}\mid \mathrm{X})\\ \cdot {\sum }_{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)\end{array}& & \text{(10)}\end{array}\backslash begin\{array\}\{l\}\; \backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\; Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash \backslash \; \backslash cdot\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; \backslash end\{array\}\; \backslash tag\; \{10\}$$

Base case (for all $k \in 0 \dots N - 1$ and all $X \in \mathcal{N}$ ):

$$\begin{array}{cccc}& \pi (k,\mathrm{X},k+1)=\sum _{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mathrm{\mid }\mathrm{X})\cdot p(\mathrm{Y}\to w_{k+1})& & \text{(11)}\end{array}\backslash pi\; (k,\; \backslash mathrm\; \{X\},\; k\; +\; 1)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\}\; |\; \backslash mathrm\; \{X\})\; \backslash cdot\; p\; (\backslash mathrm\; \{Y\}\; \backslash rightarrow\; w\; \_\; \{k\; +\; 1\})\; \backslash tag\; \{11\}$$

Proof. A proof of Lem. 1 is given in App. A. The above formulation of the prefix probability is closely related to that of the inside probability from Baker's (1979) inside-outside algorithm, which can be efficiently computed using CKY, see Alg. 1. Next, the left-corner expectations $\xi$ as defined by Eq. (8) can be computed efficiently as follows. Let $P$ denote the square matrix of dimension $|\mathcal{N}|$ , with rows and columns indexed by the non-terminals $\mathcal{N}$ (in some fixed order), where the entry at the $i^{\text{th}}$ row and the $j^{\text{th}}$ column corresponds to $p(\mathrm{X}_i \to \mathrm{X}_j)$ , i.e., the probability of deriving $\mathrm{X}_j$ on the left corner

from $\mathrm{X}_i$ in one step (we use $_$ as a wildcard):

$$\begin{array}{cccc}& p({\mathrm{X}}_i\to {\mathrm{X}}_j)\sum _{\mathrm{Y}\in \mathcal{N}}p({\mathrm{X}}_i\to {\mathrm{X}}_j\mathrm{Y})& & \text{(12)}\end{array}p\; \backslash left(\backslash mathrm\; \{X\}\; \_\; \{i\}\; \backslash rightarrow\; \backslash mathrm\; \{X\}\; \_\; \{j\}\; \backslash quad\backslash right)\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash mathrm\; \{Y\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; \backslash left(\backslash mathrm\; \{X\}\; \_\; \{i\}\; \backslash rightarrow\; \backslash mathrm\; \{X\}\; \_\; \{j\}\; \backslash quad\; \backslash mathrm\; \{Y\}\backslash right)\; \backslash tag\; \{12\}$$

We can find the probability of getting to non-terminal $\mathrm{X}_j$ after $k$ derivation steps starting from $\mathrm{X}_i$ by multiplying $P$ with itself $k$ times:

$$\begin{array}{cccc}& p({\mathrm{X}}_i\stackrel{k}{\to }{\mathrm{X}}_j-)={\left(P^k\right)}_{ij}& & \text{(13)}\end{array}p\; \backslash left(\backslash mathrm\; \{X\}\; \_\; \{i\}\; \backslash xrightarrow\; \{k\}\; \backslash mathrm\; \{X\}\; \_\; \{j\}\; -\backslash right)\; =\; \backslash left(P\; ^\; \{k\}\backslash right)\; \_\; \{i\; j\}\; \backslash tag\; \{13\}$$

We can hence get the matrix $P^{*}$ , whose entries correspond to deriving $\mathrm{X}_j$ from $\mathrm{X}_i$ after any number of derivation steps, by summing over all the powers of the matrix $P$ :

$$\begin{array}{cccc}& \begin{array}{c}{P}^{\ast }I+P+P^2+P^3+\cdots ={\sum }_{n=0}^{\mathrm{\infty }}{P}^n\\ =I+P{\sum }_{n=0}^{\mathrm{\infty }}{P}^n=I+P{P}^{\ast }=(I-P{)}^{-1}\end{array}& & \text{(14)}\end{array}\backslash begin\{array\}\{l\}\; P\; ^\; \{*\}\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; I\; +\; P\; +\; P\; ^\; \{2\}\; +\; P\; ^\; \{3\}\; +\; \backslash dots\; =\; \backslash sum\_\; \{n\; =\; 0\}\; ^\; \{\backslash infty\}\; P\; ^\; \{n\}\; \backslash tag\; \{14\}\; \backslash \backslash \; =\; I\; +\; P\; \backslash sum\_\; \{n\; =\; 0\}\; ^\; \{\backslash infty\}\; P\; ^\; \{n\}\; =\; I\; +\; P\; P\; ^\; \{*\}\; =\; (I\; -\; P)\; ^\; \{-\; 1\}\; \backslash \backslash \; \backslash end\{array\}$$

Note that the entry at the $i^{\mathrm{th}}$ row and $j^{\mathrm{th}}$ column of $P^{*}$ is exactly the left-corner expectation $\xi (\mathrm{X}_j\mid \mathrm{X}_i)$ . Finally, we can compute the left-corner expectations $\xi (\mathrm{YZ}\mid \mathrm{X})$ using Eq. (9):

$$\xi (\mathrm{Y}\mathrm{Z}\mid \mathrm{X})\sum _{{\mathrm{X}}^{\mathrm{\prime }}\in \mathcal{N}}\xi ({\mathrm{X}}^{\mathrm{\prime }}\mid \mathrm{X})\cdot p({\mathrm{X}}^{\mathrm{\prime }}\to \mathrm{Y}\mathrm{Z})\backslash xi\; (\backslash mathrm\; \{Y\; Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash stackrel\; \{\backslash mathrm\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash cdot\; p\; (\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\})$$

Lastly, for completeness, we also compute the prefix probability of the empty string, $\varepsilon$ . For probabilistic PCFGs, this probability is simply 1 because $\varepsilon$ is a prefix of any string: $^{8}$

$$\begin{array}{cccc}& \pi (k,\mathrm{X},k)\pi (\epsilon \mathrm{\mid }\mathrm{X})=1& & \text{(15)}\end{array}\backslash pi\; (k,\; \backslash mathrm\; \{X\},\; k)\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash pi\; (\backslash varepsilon\; |\; \backslash mathrm\; \{X\})\; =\; 1\; \backslash tag\; \{15\}$$

We can now combine the quantities derived above to obtain an efficient algorithm for the computation of prefix probabilities $\pi (i,\mathrm{S},k)$ . For the full algorithm, see Alg. 2.

Proposition 1. The time complexity of the CKY algorithm as presented in Alg. 1 is $\mathcal{O}(N^3 |\mathcal{N}|^3)$ .

Proof. Clearly, the computationally critical part is in lines 11-18, where we iterate over all indices of $\pmb{w}$ for $i,j$ , and $k$ , as well as over the whole set of grammar rules, thus taking $\mathcal{O}(N^3 |\mathcal{R}|)$ . In a PCFG in CNF, with the size of $\Sigma$ taken as constant, the number of rules, $|\mathcal{R}|$ , is $\mathcal{O}(|\mathcal{N}|^3)$ , making the overall complexity of CKY $\mathcal{O}(N^3 |\mathcal{N}|^3)$ .

Proposition 2. The total time complexity of Jelinek-Lafferty is $\mathcal{O}(N^3 |\mathcal{N}|^3 +|\mathcal{N}|^4)$ :

Proof. First, we precompute any values that are independent of the input. In lines 1-3, we precompute all the left-corner expectations $\xi (\mathrm{Y}\mid \mathrm{X})$ using Eq. (14), which has the complexity of inverting the matrix $P$ , i.e., $\mathcal{O}(|\mathcal{N}|^3)$ , and move the values into a map of left-corner expectations in $\mathcal{O}(|\mathcal{N}|^2)$ (this is just for readability). In lines 4-5, we then use Eq. (9) to compute $\xi (\mathrm{YZ}\mid \mathrm{X})$ , iterating once over all non-terminals $\mathrm{X}$ for each rule, which takes $\mathcal{O}(|\mathcal{R}||\mathcal{N}|)$ , that is, $\mathcal{O}(|\mathcal{N}|^4)$ . After initializing the probabilities in lines 7-10 in $\mathcal{O}(N^2 |\mathcal{N}|)$ , we begin by pre-computing all the inside probabilities $\beta$ in line 11 of Alg. 2, which takes $\mathcal{O}(N^3 |\mathcal{N}|^3)$ by Prop. 1. Computing $\pi (k,\mathrm{X},k + 1)$ for all $\mathrm{X}\in \mathcal{N}$ by Eq. (11) in lines 12-14 takes $\mathcal{O}(N|\mathcal{N}|^2)$ as we iterate over all positions $k\in N$ and over all $\mathrm{Y}\in \mathcal{N}$ for each $\mathrm{X}\in \mathcal{N}$ . And finally, computing the $\pi$ chart in lines 15-19 takes $\mathcal{O}(N^3 |\mathcal{N}|^3)$ since we iterate over all $\ell ,i,j\leq N$ and $\mathrm{X},\mathrm{Y},\mathrm{Z}\in \mathcal{N}$ . This yields an overall time complexity of $\mathcal{O}(N^3 |\mathcal{N}|^3 +|\mathcal{N}|^4)$ .

4 Our Speed-up

We now turn to our development of a faster dynamic program to compute all prefix probabilities. The speed-up comes from a different way to factorize $\pi (i,\mathrm{X},k)$ , which allows additional memoization. Starting with the definition of the prefix probability in Eq. (16a), we first expand $\xi (\mathrm{YZ}\mid \mathrm{X})$ by Eq. (9), as seen in Eq. (16b). Then, we factor out all terms that depend on the left-corner nonterminal Y in Eq. (16c), which we store in a chart $\gamma$ , see Eq. (16e). We then do the same for all terms depending on $\mathbf{X}^{\prime}$ , factoring them out in Eq. (16d) and storing them in another chart $\delta$ , see Eq. (16f).

Our improved algorithm for computing all prefix probabilities is shown in Alg. 3.

Proposition 3. The complexity of our improved algorithm is $\mathcal{O}(N^2 |\mathcal{N}|^3 + N^3 |\mathcal{N}|^2)$ .

Proof. As before, Alg. 3 starts by precomputing and assigning the left-corner expectations in lines 1-3, which takes $\mathcal{O}(|\mathcal{N}|^3)$ and $\mathcal{O}(|\mathcal{N}|^2)$ , respectively. We then initialize the prefix probabilities and compute the inside probabilities in lines 5-8, taking $\mathcal{O}(N^2 |\mathcal{N}|)$ . As Eisner and Blatz (2007) show, one can compute $\beta$ (line 9) in $\mathcal{O}(N^2 |\mathcal{N}|^3 + N^3 |\mathcal{N}|^2)$ , thus improving the runtime of Alg. 1 for dense grammars. Pre-computing $\gamma$ and $\delta$ in lines 10-14

$$\begin{array}{c}\pi (i,\mathrm{X},k)={\sum }_{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}\xi (\mathrm{Y},\mathrm{Z}\mid \mathrm{X})\cdot {\sum }_{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)(16a)\\ ={\sum }_{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}{\sum }_{{\mathrm{X}}^{\mathrm{\prime }}\in \mathcal{N}}\xi ({\mathrm{X}}^{\mathrm{\prime }}\mid \mathrm{X})\cdot p({\mathrm{X}}^{\mathrm{\prime }}\to \mathrm{Y}\mathrm{Z})\cdot {\sum }_{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)(16b)\\ ={\sum }_{{\mathrm{X}}^{\mathrm{\prime }},\mathrm{Z}\in \mathcal{N}}\xi ({\mathrm{X}}^{\mathrm{\prime }}\mid \mathrm{X})\cdot {\sum }_{j=i+1}^{k-1}{\gamma }_{ij}({\mathrm{X}}^{\mathrm{\prime }},\mathrm{Z})\cdot \pi (j,\mathrm{Z},k)(16c)\\ ={\sum }_{\mathrm{Z}\in \mathcal{N}}{\sum }_{j=i+1}^{k-1}{\delta }_{ij}(\mathrm{X},\mathrm{Z})\cdot \pi (j,\mathrm{Z},k)(16d)\end{array}\backslash begin\{array\}\{l\}\; \backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash cdot\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; (16a)\; \backslash \backslash \; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash sum\_\; \{\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash mathrm\; \{X\}\backslash right)\; \backslash cdot\; p\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\}\backslash right)\; \backslash cdot\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; (16b)\; \backslash \backslash \; =\; \backslash sum\_\; \{\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash mathrm\; \{X\}\backslash right)\; \backslash cdot\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash gamma\_\; \{i\; j\}\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\},\; \backslash mathrm\; \{Z\}\backslash right)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; (16c)\; \backslash \backslash \; =\; \backslash sum\_\; \{\backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash delta\_\; \{i\; j\}\; (\backslash mathrm\; \{X\},\; \backslash mathrm\; \{Z\})\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; (16d)\; \backslash \backslash \; \backslash end\{array\}$$

$$\begin{array}{cccc}& \text{w h e r e}{\gamma }_{ij}({\mathrm{X}}^{\mathrm{\prime }},\mathrm{Z})\sum _{\mathrm{Y}\in \mathcal{N}}p({\mathrm{X}}^{\mathrm{\prime }}\to \mathrm{Y}\mathrm{Z})\cdot \beta (i,\mathrm{Y},j)& & \text{(16e)}\end{array}\backslash text\; \{w\; h\; e\; r\; e\}\; \backslash gamma\_\; \{i\; j\}\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\},\; \backslash mathrm\; \{Z\}\backslash right)\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash mathrm\; \{Y\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\}\backslash right)\; \backslash cdot\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash tag\; \{16e\}$$

$$\begin{array}{cccc}& \text{a n d}{\delta }_{ij}(\mathrm{X},\mathrm{Z})\sum _{{\mathrm{X}}^{\mathrm{\prime }}\in \mathcal{N}}\xi ({\mathrm{X}}^{\mathrm{\prime }}\mid \mathrm{X})\cdot {\gamma }_{ij}({\mathrm{X}}^{\mathrm{\prime }},\mathrm{Z})& & \text{(16f)}\end{array}\backslash text\; \{a\; n\; d\}\; \backslash delta\_\; \{i\; j\}\; (\backslash mathrm\; \{X\},\; \backslash mathrm\; \{Z\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash sum\_\; \{\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\}\; \backslash mid\; \backslash mathrm\; \{X\}\backslash right)\; \backslash cdot\; \backslash gamma\_\; \{i\; j\}\; \backslash left(\backslash mathrm\; \{X\}\; ^\; \{\backslash prime\},\; \backslash mathrm\; \{Z\}\backslash right)\; \backslash tag\; \{16f\}$$

Algorithm 3 Faster prefix probability algorithm

$$\cdot p(\mathrm{Y}\to w_{k+1})\backslash cdot\; p\; (\backslash mathrm\; \{Y\}\; \backslash rightarrow\; w\; \_\; \{k\; +\; 1\})$$

1: $P^{\prime}\gets (I - P)^{-1}$ $\triangleright$ Precompute $P^{*}$ with Eq. (14)
2: for $\mathrm{X}i, \mathrm{X}j \in \mathcal{N}$ : Assign $\xi(\mathrm{Y} \mid X)$
3: $\xi (\mathrm{X}j\mid \mathrm{X}i)\gets P{ij}^{\prime}$
4: def FastJL $(\pmb {w} = w{1}\dots w{N})$
5: $\pi (\cdot ,\cdot ,\cdot)\gets 0$ $\triangleright$ Initialize prefix probabilities
6: for $k \in 0, \dots, N$ :
7: for $\mathbf{X}\in \mathcal{N}$ .. $\triangleright$ Prefix probability of $\varepsilon$
8: $\pi (k,\mathrm{X},k)\gets 1$
9: $\beta \gets \mathbf{CKY}(\boldsymbol{w})$ $\triangleright$ Compute $\beta$ with Alg. 1
10: for $i,j = 0,\dots ,N$ ..
11: for $\mathrm{X},\mathrm{Z}\in \mathcal{N}$ .. $\triangleright$ Compute $\gamma$ by Eq. (16e)
12: $\gamma{ij}(\mathrm{X},\mathrm{Z})\gets \sum_{\mathrm{Y}\in \mathcal{N}}p(\mathrm{X}\rightarrow \mathrm{YZ})\cdot \beta (i,\mathrm{Y},j)$
13: for $\mathbf{X},\mathbf{Z}\in \mathcal{N}$ .. $\triangleright$ Compute $\delta$ by Eq. (16f)
14: $\delta_{ij}(\mathrm{X},\mathrm{Z})\gets \sum_{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mid \mathrm{X})\cdot \gamma_{ij}(\mathrm{Y},\mathrm{Z})$
15: for $k \in 0, \ldots, N - 1$ :
16: for $\mathbf{X}\in \mathcal{N}$ .. $\triangleright$ Compute base case
17: $\pi (k,\mathrm{X},k + 1)\gets \sum_{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mid \mathrm{X})$
18: for $\ell \in 2\ldots N$ ..
19: for $i\in 1\dots N - \ell$ ..
20: $k\gets i + \ell$
21: for $\mathrm{X},\mathrm{Z}\in \mathcal{N}$ .. $\triangleright$ Recursively compute $\pi$
22: $\pi (i,\mathrm{X},k) + = \sum_{j = i + 1}^{k - 1}\delta_{ij}(\mathrm{X},\mathrm{Z})\cdot \pi (j,\mathrm{Z},k)$
23: return $\pi$

takes $\mathcal{O}(N^2 |\mathcal{N}|^3)$ , as we sum over non-terminals, and both charts each have two dimensions indexing $N$ and two indexing $\mathcal{N}$ . Computing the base

case $\pi (k,\mathrm{X},k + 1)$ for all non-terminals X and positions $k$ in lines 15-17 takes $\mathcal{O}(N|\mathcal{N}|^2)$ , as before. Finally, the loops computing $\pi$ in lines 18-22 take $\mathcal{O}(N^3 |\mathcal{N}|^2)$ , as we are now iterating over $\mathrm{X},\mathrm{Z}\in \mathcal{N}$ and $\ell ,i,j\leq N$ . Hence, our new overall time complexity is $\mathcal{O}(N^{2}|\mathcal{N}|^{3} + N^{3}|\mathcal{N}|^{2})$ .

5 Generalization to Semirings

It turns out that Jelinek-Lafferty, and, by extension, our improved algorithm, can be generalized to work for semiring-weighted CFGs, with the same time complexity, under the condition that the semiring is closed, i.e., it has a Kleene star. This follows from the fact that the only operations used by the algorithm are addition and multiplication if we use Lehmann's (1977) algorithm for the computation of left-corner expectations, $\xi$ . We can even relax the condition of local normalization by changing how left-corner expectations and the weight of the prefix $\varepsilon$ are computed. The relevant definitions and derivation of the adapted algorithm can be found in App. B.

6 Conclusion

In this paper, we have shown how to efficiently compute prefix probabilities for PCFGs in CNF, adapting Jelinek-Lafferty to use additional memoization, thereby reducing the time complexity from $\mathcal{O}(N^3 |\mathcal{N}|^3 +|\mathcal{N}|^4)$ to $\mathcal{O}(N^2 |\mathcal{N}|^3 +N^3 |\mathcal{N}|^2)$ . We thereby addressed one of the main limitations of the original formulation, of being slow for large grammar sizes.

Limitations

While we have improved the asymptotic running time of a classic algorithm with regard to grammar size, the time complexity of our algorithm is still cubic in the length of the input. Our result follows the tradition of dynamic programming algorithms that trade time for space by memoizing and reusing precomputed intermediate results. The usefulness of this trade-off in practice depends on the specifics of the grammar, and while the complexity is strictly better in terms of non-terminals, it will be most noticeable for denser grammars with many non-terminals.

Ethics Statement

We do not foresee any ethical issues arising from this work.

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions. We also extend our gratitude to Abra Ganz, Andreas Opedal, Anej Svete, and Tim Vieira for their valuable feedback on various versions of this paper.

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A Proof of Lem. 1

Lemma 1. Given a tight, trim PCFG in CNF and a string $\pmb{w} = w_{1}\dots w_{N}$ , the prefix probability of a substring $w_{i + 1}\dots w_{k}$ of $\pmb{w}$ being derived from X can be defined recursively as follows:

$$\begin{array}{cccc}& \pi (i,\mathrm{X},k)=\sum _{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}\xi (\mathrm{Y}\mathrm{Z}\mid \mathrm{X})\cdot \sum _{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)& & \text{(10)}\end{array}\backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\; Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash cdot\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; \backslash tag\; \{10\}$$

Base case (for all $k\in 0\dots N - 1$ and all $\mathrm{X}\in \mathcal{N}$ ):

$$\begin{array}{cccc}& \pi (k,\mathrm{X},k+1)=\sum _{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mid \mathrm{X})\cdot p(\mathrm{Y}\to w_{k+1})& & \text{(11)}\end{array}\backslash pi\; (k,\; \backslash mathrm\; \{X\},\; k\; +\; 1)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash cdot\; p\; (\backslash mathrm\; \{Y\}\; \backslash rightarrow\; w\; \_\; \{k\; +\; 1\})\; \backslash tag\; \{11\}$$

Proof. Eq. (11): The base case, $\pi(k, X, k+1)$ , is the probability of $w_{k+1}$ being the left-most terminal in the parse subtree under $X$ . It is, therefore, simply the sum of probabilities of any non-terminal $Y$ being on the left corner of the parse subtree under $X$ multiplied by the corresponding probability of $Y$ directly deriving $w_{k+1}$ .

Eq. (10): Given the PCFG is in CNF and assuming $k > i + 1$ , in order to derive the prefix $w_{i+1} \cdots w_k$ we must first apply some rule $X \to Y$ . Z, where the first part of the substring is then derived from Y and the remainder (and potentially more) from Z:

$$\begin{array}{cccc}& \pi (i,\mathrm{X},k)=\sum _{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}p(\mathrm{X}\to \mathrm{Y}\mathrm{Z})[\sum _{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)+\pi (i,\mathrm{Y},k)]& & \text{(17)}\end{array}\backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{X\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\}\; \backslash mathrm\; \{Z\})\; \backslash left[\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; +\; \backslash pi\; (i,\; \backslash mathrm\; \{Y\},\; k)\; \backslash right]\; \backslash tag\; \{17\}$$

where the last term, $\pi (i,\mathrm{Y},k)$ , handles the case where the whole prefix is derived from Y alone. This term is clearly recursively defined through Eq. (17), with X replaced by Y. Defining $R(\mathrm{Y},\mathrm{Z})\stackrel {\mathrm{def}}{=}\sum_{j = i + 1}^{k - 1}\beta (i,\mathrm{Y},j)\pi (j,\mathrm{Z},k)$ , we can rewrite Eq. (17) as:

$$\begin{array}{cccc}& \pi (i,\mathrm{X},k)=\sum _{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}p(\mathrm{X}\to \mathrm{Y}\mathrm{Z})\cdot R(\mathrm{Y},\mathrm{Z})+\sum _{\mathrm{A},\mathrm{B}\in \mathcal{N}}p(\mathrm{X}\to \mathrm{A}\mathrm{B})\cdot \pi (i,\mathrm{A},k)& & \text{(18)}\end{array}\backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{X\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\}\; \backslash mathrm\; \{Z\})\; \backslash cdot\; R\; (\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\})\; +\; \backslash sum\_\; \{\backslash mathrm\; \{A\},\; \backslash mathrm\; \{B\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{X\}\; \backslash rightarrow\; \backslash mathrm\; \{A\; B\})\; \backslash cdot\; \backslash pi\; (i,\; \backslash mathrm\; \{A\},\; k)\; \backslash tag\; \{18\}$$

After repeated substitutions ad infinitum, we get:

$$\begin{array}{cccc}& \pi (i,\mathrm{X},k)=\sum _{\mathrm{A},\mathrm{B}\in \mathcal{N}}p(\mathrm{X}\Rightarrow \mathrm{A}\mathrm{B})\sum _{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}p(\mathrm{A}\to \mathrm{Y}\mathrm{Z})\cdot R(\mathrm{Y},\mathrm{Z})& & \text{(19)}\end{array}\backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{A\},\; \backslash mathrm\; \{B\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{X\}\; \backslash Rightarrow\; \backslash mathrm\; \{A\; B\})\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{A\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\})\; \backslash cdot\; R\; (\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\})\; \backslash tag\; \{19\}$$

Note that, in the last step, infinite derivations do not carry any probability mass since we assumed the PCFG to be tight and trim. Hence, the final form of the equation is:

$$\begin{array}{cccc}& \begin{array}{c}\pi (i,\mathrm{X},k)={\sum }_{\mathrm{A},\mathrm{B}\in \mathcal{N}}p(\mathrm{X}\mathrm{A}\mathrm{B}){\sum }_{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}p(\mathrm{A}\to \mathrm{Y}\mathrm{Z})\cdot R(\mathrm{Y},\mathrm{Z})\\ ={\sum }_{\mathrm{Y},\mathrm{Z}\in \mathcal{N}}\xi (\mathrm{Y}\mathrm{Z}\mid \mathrm{X}){\sum }_{j=i+1}^{k-1}\beta (i,\mathrm{Y},j)\cdot \pi (j,\mathrm{Z},k)\end{array}& & \text{(20)}\end{array}\backslash begin\{array\}\{l\}\; \backslash pi\; (i,\; \backslash mathrm\; \{X\},\; k)\; =\; \backslash sum\_\; \{\backslash mathrm\; \{A\},\; \backslash mathrm\; \{B\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{X\}\; \backslash stackrel\; \{*\}\; \{\backslash Rightarrow\}\; \backslash mathrm\; \{A\; B\})\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; (\backslash mathrm\; \{A\}\; \backslash rightarrow\; \backslash mathrm\; \{Y\; Z\})\; \backslash cdot\; R\; (\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\})\; \backslash \backslash \; =\; \backslash sum\_\; \{\backslash mathrm\; \{Y\},\; \backslash mathrm\; \{Z\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; \backslash xi\; (\backslash mathrm\; \{Y\; Z\}\; \backslash mid\; \backslash mathrm\; \{X\})\; \backslash sum\_\; \{j\; =\; i\; +\; 1\}\; ^\; \{k\; -\; 1\}\; \backslash beta\; (i,\; \backslash mathrm\; \{Y\},\; j)\; \backslash cdot\; \backslash pi\; (j,\; \backslash mathrm\; \{Z\},\; k)\; \backslash tag\; \{20\}\; \backslash \backslash \; \backslash end\{array\}$$

B Extension of Alg. 3 to Semirings

In the following, we give the necessary background on semirings and then show how the algorithms introduced above can be framed in terms of semirings.

B.1 Semirings

We start by introducing the necessary definitions and notation.

Definition 10. A monoid is a 3-tuple $\langle \mathcal{A},\circ ,\mathbf{1}\rangle$ where:

(i) $\mathcal{A}$ is a non-empty set;
(ii) $\circ$ is an associative binary operation: $\forall a, b, c \in \mathcal{A}, (a \circ b) \circ c = a \circ (b \circ c)$ ;
(iii) $\mathbf{1}$ is a left and right identity element: $\forall a\in \mathcal{A},\mathbf{1}\circ a = a\circ \mathbf{1} = a$
(iv) $\mathcal{A}$ is closed under the operation $\circ$ : $\forall a, b \in \mathcal{A}, a \circ b \in \mathcal{A}$

A monoid is commutative if $\forall a, b \in \mathcal{A}: a \circ b = b \circ a$ .

Definition 11. A semiring is a 5-tuple $\mathcal{W} = \langle \mathcal{A},\oplus ,\otimes ,\mathbf{0},\mathbf{1}\rangle$ , where

(i) $\langle \mathcal{A},\oplus ,\mathbf{0}\rangle$ is a commutative monoid over $\mathcal{A}$ with identity element 0 under the addition operation $\oplus$
(ii) $\langle \mathcal{A},\otimes ,\mathbf{1}\rangle$ is a monoid over $\mathcal{A}$ with identity element 1 under the multiplication operation $\otimes$
(iii) Multiplication is distributive over addition, that is, $\forall a,b,c\in \mathcal{A}$

$a\otimes (b\oplus c) = a\otimes b\oplus a\otimes c;$

• $(b \oplus c) \otimes a = b \otimes a \oplus c \otimes a$ .

(iv) $\mathbf{0}$ is an annihilator for $\mathcal{A}$ , that is, $\forall a\in \mathcal{A},\mathbf{0}\otimes a = a\otimes \mathbf{0} = \mathbf{0}$ .

A semiring is commutative if $\langle \mathcal{A},\otimes ,\mathbf{1}\rangle$ is a commutative monoid. A semiring is idempotent if $\forall a\in \mathcal{A}$ .. $a\oplus a = a$

Definition 12. A semiring $\mathcal{W} = \langle \mathcal{A},\oplus ,\otimes ,0,1\rangle$ is complete if it is possible to extend the addition operator $\oplus$ to infinite sums, maintaining the properties of associativity, commutativity, and distributivity from the finite case (Rozenberg and Salomaa, 1997, Chapter 9). In this case, we can define the unary operation of the Kleene star denoted by a superscript $^*$ as the infinite sum over powers of its operand, that is, $\forall a\in \mathcal{A}$ ..

$$\begin{array}{cccc}& a^{\ast }\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{a}^i& & \text{(21)}\end{array}a\; ^\; \{*\}\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash bigoplus\_\; \{i\; =\; 0\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; \backslash tag\; \{21\}$$

Analogously to Eq. (14), it then follows that:

$$\begin{array}{cccc}& a^{\ast }=\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{a}^i=a^0\oplus \underset{i=1}{\overset{\mathrm{\infty }}{⨁}}{a}^i=\mathbf{1}\oplus a\otimes \underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{a}^i=\mathbf{1}\oplus a\otimes a^{\ast }& & \text{(22)}\end{array}a\; ^\; \{*\}\; =\; \backslash bigoplus\_\; \{i\; =\; 0\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; =\; a\; ^\; \{0\}\; \backslash oplus\; \backslash bigoplus\_\; \{i\; =\; 1\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; =\; \backslash mathbf\; \{1\}\; \backslash oplus\; a\; \backslash otimes\; \backslash bigoplus\_\; \{i\; =\; 0\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; =\; \backslash mathbf\; \{1\}\; \backslash oplus\; a\; \backslash otimes\; a\; ^\; \{*\}\; \backslash tag\; \{22\}$$

and, similarly:

$$\begin{array}{cccc}& a^{\ast }=a^0\oplus \underset{i=1}{\overset{\mathrm{\infty }}{⨁}}{a}^i=\mathbf{1}\oplus \underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{a}^i\otimes a=\mathbf{1}\oplus a^{\ast }\otimes a& & \text{(23)}\end{array}a\; ^\; \{*\}\; =\; a\; ^\; \{0\}\; \backslash oplus\; \backslash bigoplus\_\; \{i\; =\; 1\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; =\; \backslash mathbf\; \{1\}\; \backslash oplus\; \backslash bigoplus\_\; \{i\; =\; 0\}\; ^\; \{\backslash infty\}\; a\; ^\; \{i\}\; \backslash otimes\; a\; =\; \backslash mathbf\; \{1\}\; \backslash oplus\; a\; ^\; \{*\}\; \backslash otimes\; a\; \backslash tag\; \{23\}$$

We now discuss how complete semirings can be lifted to square matrices. The definitions follow analogously to matrices over the real numbers.

Definition 13. We define semiring matrix addition as follows. Let $A$ and $B$ be $d \times d$ matrices whose entries are elements from a complete semiring $\mathcal{W} = \langle \mathcal{A}, \oplus, \otimes, \mathbf{0}, \mathbf{1} \rangle$ . Then the sum $('' + '')$ of $A$ and $B$ is defined as:

$$\begin{array}{cccc}& (A+B{)}_{ij}{A}_{ij}\oplus B_{ij}i,j\in 1,\dots ,d& & \text{(24)}\end{array}(A\; +\; B)\; \_\; \{i\; j\}\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; A\; \_\; \{i\; j\}\; \backslash oplus\; B\; \_\; \{i\; j\}\; \backslash quad\; i,\; j\; \backslash in\; 1,\; \backslash dots\; ,\; d\; \backslash tag\; \{24\}$$

Definition 14. We define semiring matrix multiplication as follows. Let $A$ and $B$ be $d \times d$ matrices whose entries are elements from a complete semiring $\mathcal{W} = \langle \mathcal{A}, \oplus, \otimes, \mathbf{0}, \mathbf{1} \rangle$ . Then the product of $A$ and $B$ is defined as:

$$\begin{array}{cccc}& (AB{)}_{ij}\underset{k=1}{\overset{d}{⨁}}{A}_{ik}\otimes B_{kj}i,j\in 1,\dots ,d& & \text{(25)}\end{array}(A\; B)\; \_\; \{i\; j\}\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash bigoplus\_\; \{k\; =\; 1\}\; ^\; \{d\}\; A\; \_\; \{i\; k\}\; \backslash otimes\; B\; \_\; \{k\; j\}\; \backslash quad\; i,\; j\; \backslash in\; 1,\; \backslash dots\; ,\; d\; \backslash tag\; \{25\}$$

We also define the zero matrix, $\mathbf{O}$ , over the complete semiring $\mathcal{W} = \langle \mathcal{A}, \oplus, \otimes, \mathbf{0}, \mathbf{1} \rangle$ , such that all entries are $\mathbf{0}$ , and the unit matrix $\mathbf{I}$ as $(\mathbf{I})_{ij} = \mathbf{1}$ iff $i = j$ and $\mathbf{0}$ otherwise for all indices $i, j \in 0, \ldots, d$ . It is then straightforward to show that matrix addition is associative and commutative, while matrix multiplication is associative and distributive over matrix addition. Hence, the set of square matrices of dimension $d$ over a semiring, with addition and multiplication as defined above, is itself a semiring. Furthermore, by the element-wise definition of its addition operation, it is also complete.

B.2 Semiring-weighted prefix algorithm

We now consider a semiring-weighted CFG $\mathcal{G} = \langle \mathcal{N}, \Sigma, \mathrm{S}, \mathcal{R}, p, \mathcal{W} \rangle$ , where $\mathcal{N}, \Sigma, \mathrm{S}, \mathcal{R}$ are defined as before but the weighting function $p: \mathcal{R} \to \mathcal{W}$ now maps rules to elements of a commutative semiring $\mathcal{W}$ . Note that we no longer require the rule weights to sum to one. As before, we define the matrix $P$ as the square matrix of dimension $|\mathcal{N}|$ whose rows and columns are indexed by the non-terminals $\mathcal{N}$ in some fixed order so that the entry $P_{ij}$ corresponds to the weight of getting the non-terminal $\mathrm{X_j}$ on the left after one rule application to $\mathrm{X_i}$ . Since the rule weights are no longer locally normalized, however, we need to include the treesum under the right non-terminal of each rule as an additional term:

$$\begin{array}{cccc}& P_{ij}=p({\mathrm{X}}_i\to {\mathrm{X}}_j-)\underset{\mathrm{Y}\in \mathcal{N}}{⨁}p({\mathrm{X}}_i\to {\mathrm{X}}_j\mathrm{Y})\cdot \mathrm{treesum}(\mathrm{Y})& & \text{(26)}\end{array}P\; \_\; \{i\; j\}\; =\; p\; \backslash left(\backslash mathrm\; \{X\}\; \_\; \{i\}\; \backslash rightarrow\; \backslash mathrm\; \{X\}\; \_\; \{j\}\; -\backslash right)\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash bigoplus\_\; \{\backslash mathrm\; \{Y\}\; \backslash in\; \backslash mathcal\; \{N\}\}\; p\; \backslash left(\backslash mathrm\; \{X\}\; \_\; \{i\}\; \backslash rightarrow\; \backslash mathrm\; \{X\}\; \_\; \{j\}\; \backslash mathrm\; \{Y\}\backslash right)\; \backslash cdot\; \backslash operatorname\; \{t\; r\; e\; e\; s\; u\; m\}\; (\backslash mathrm\; \{Y\})\; \backslash tag\; \{26\}$$

We can then calculate the weight of getting $\mathrm{X}j$ from $\mathrm{X}i$ at the leftmost non-terminal after exactly $k$ derivation steps as $(P^k){ij}$ , where $P^k \stackrel{\mathrm{def}}{=} \underbrace{P \cdots P}{k \text{ times}}$ . Finally, to get the left-corner expectations, we then

need to calculate the Kleene closure over the matrix $P$ ,¹¹ that is, we want to find $P^{*} = \sum_{k=0}^{\infty} P^{k}$ . To compute the Kleene closure over the transition matrix we can use an efficient algorithm by Lehmann (1977) which is a generalization of the well-known shortest-path algorithm usually attributed to Floyd (1962) and Warshall (1962), but introduced previously by Roy (1959).¹² The algorithm works under the condition that the Kleene closure of all individual matrix entries from semiring $\mathcal{W}$ exists, which is true for our case since we assumed $\mathcal{W}$ to be complete. The algorithm is shown in Alg. 4.

Algorithm 4 Lehmann's algorithm for computing the Kleene closure over a transition matrix
1: def Lehmann(M):
2: $d \gets \dim(M)$
3: $M^{(0)} \gets M$
4: for $j = 1, \ldots, d$ :
5: for $i = 1, \ldots, d$ :
6: for $k = 1, \ldots, d$ :
7: $M_{ik}^{(j)} \gets M_{ik}^{(j-1)} \oplus M_{ij}^{(j-1)} \otimes (M_{jj}^{(j-1)})^* \otimes M_{jk}^{(j-1)}$
8: return $\mathbf{I} + M^{(d)}$

Lastly, since we no longer have normalized rule weights, we need to set the prefix weight of $\varepsilon$ under any non-terminal $X \in \mathcal{N}$ to the treesum under $X$ . With this, we can now generalize our prefix weight algorithm to semirings, as shown in Alg. 5.

Algorithm 5 Faster prefix algorithm over semirings
1: $P^{\prime}\gets \mathrm{Lehmann}(P)$ ▷Precompute $P^{*}$ with Alg.4
2: for $\mathrm{X}i,\mathrm{X}j\in \mathcal{N}$ .. Assign $\xi (\mathrm{X}j\mid \mathrm{X}i)$
3: $\xi (\mathrm{X}j\mid \mathrm{X}i)\leftarrow P^{\prime}{ij}$
4: def FastSemiringJL $(w = w{1}\cdot \cdot \cdot w{N},\mathcal{G})$ .
5: $\pi (\cdot ,\cdot ,\cdot)\gets 0$ ▷ Initialize prefix probabilities
6: for $k\in 0,\ldots ,N$ ..
7: for $\mathrm{X}\in \mathcal{N}$ .. Prefix weight of $\varepsilon$
8: $\pi (k,\mathrm{X},k)\gets \mathrm{treesum}(\mathrm{X})$
9: $\beta \gets \mathrm{CKY}(\pmb {w})$ ▷ Compute $\beta$ with Alg.1
10: for $i,j = 0,\dots ,N$ ..
11: for X,Z E N: ▷Compute $\gamma$ by Eq. (16e)
12: $\gamma{ij}(\mathrm{X},\mathrm{Z})\leftarrow \bigoplus{\mathrm{Y}\in \mathcal{N}}p(\mathrm{X}\rightarrow \mathrm{YZ})\otimes \beta (i,\mathrm{Y},j)$
13: for X,Z E N: ▷Compute $\delta$ by Eq. (16f)
14: $\delta{ij}(\mathrm{X},\mathrm{Z})\leftarrow \bigoplus_{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mid \mathrm{X})\otimes \gamma_{ij}(\mathrm{Y},\mathrm{Z})$
15: for $k\in 0,\ldots ,N - 1$ ..
16: for X E N: ▷Compute base case
17: $\pi (k,\mathrm{X},k + 1)\leftarrow \bigoplus_{\mathrm{Y}\in \mathcal{N}}\xi (\mathrm{Y}\mid \mathrm{X})\otimes p(\mathrm{Y}\rightarrow w_{k + 1})$
18: for $\ell \in 2\dots N$ ..
19: for $i\in 1\dots N - \ell$ ..
20: $k\gets i + \ell$
21: for X,Z E N: ▷Recursively compute $\pi$
22: $\pi (i,\mathrm{X},k) + = \bigoplus_{j = i + 1}^{k - 1}\delta_{ij}(\mathrm{X},\mathrm{Z})\otimes \pi (j,\mathrm{Z},k)$
23: return π

Proposition 4. The semiring-weighted version of our algorithm runs in $\mathcal{O}(N^2 |\mathcal{N}|^3 + N^3 |\mathcal{N}|^2)$ .

Proof. Lehmann's algorithm, as presented in Alg. 4, has three nested for loops of $d$ iterations each, where $d$ is the dimension of the input matrix. In our case, $d$ is the number of non-terminals, $|\mathcal{N}|$ . Assuming the Kleene closure of elements in $\mathcal{W}$ can be evaluated in $\mathcal{O}(1)$ , this means that computing the left corner expectations in line 1 of Alg. 5 takes $\mathcal{O}(|\mathcal{N}|^3)$ , as before. Hence, the complexity of the overall algorithm remains unchanged, that is, we can compute the prefix probabilities under a semiring-weighted, locally normalized CFG $\mathcal{G}$ in $\mathcal{O}(N^2 |\mathcal{N}|^3 + N^3 |\mathcal{N}|^2)$ .