In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction. == Definitions == A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of ∅ (denoting the empty set of strings), ε (denoting the singleton set containing just the empty string), a (where a is any character in Σ; denoting the singleton set just containing the single-character string a), r + s (where r and s are, in turn, simpler regular expressions; denoting their set's union) r ⋅ s (denoting the set of all possible concatenations of strings from r's and s's set), r + (denoting the set of n-fold repetitions of strings from r's set, for any n ≥ 1), or r (similarly denoting the set of n-fold repetitions, but also including the empty string, seen as 0-fold repetition). For example, using Σ = {0,1}, the regular expression (0+1+ε)⋅(0+1) denotes the set of all binary numbers with one or two digits (leading zero allowed), while 1⋅(0+1)⋅0 denotes the (infinite) set of all even binary numbers (no leading zeroes). Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given {1, 10, 100}, a "natural" description could be the regular expression 1⋅0, corresponding to the informal characterization "a 1 followed by arbitrarily many (maybe even none) 0's". However, (0+1) and 1+(1⋅0)+(1⋅0⋅0) is another regular expression, denoting the largest (assuming Σ = {0,1}) and the smallest set containing the given strings, and called the trivial overgeneralization and undergeneralization, respectively. Some approaches work in an extended setting where also a set of "negative example" strings is given; then, a regular expression is to be found that generates all of the positive, but none of the negative examples. == Lattice of automata == Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this lattice can be obtained by factoring the undergeneralized automaton by an appropriate equivalence relation. For the above example string set {1, 10, 100}, the picture shows at its bottom the undergeneralized automaton Aa,b,c,d in grey, consisting of states a, b, c, and d. On the state set {a,b,c,d}, a total of 15 equivalence relations exist, forming a lattice. Mapping each equivalence E to the corresponding quotient automaton language L(Aa,b,c,d / E) obtains the partially ordered set shown in the picture. Each node's language is denoted by a regular expression. The language may be recognized by quotient automata w.r.t. different equivalence relations, all of which are shown below the node. An arrow between two nodes indicates that the lower node's language is a proper subset of the higher node's. If both positive and negative example strings are given, Dupont et al. build the lattice from the positive examples, and then investigate the separation border between automata that generate some negative example and such that do not. Most interesting are those automata immediately below the border. In the picture, separation borders are shown for the negative example strings 11 (green), 1001 (blue), 101 (cyan), and 0 (red). Coste and Nicolas present an own search method within the lattice, which they relate to Mitchell's version space paradigm. To find the separation border, they use a graph coloring algorithm on the state inequality relation induced by the negative examples. Later, they investigate several ordering relations on the set of all possible state fusions. Kudo and Shimbo use the representation by automaton factorizations to give a unique framework for the following approaches (sketched below): k-reversible languages and the "tail clustering" follow-up approach, Successor automata and the predecessor-successor method, and pumping-based approaches (framework-integration challenged by Luzeaux, however). Each of these approaches is shown to correspond to a particular kind of equivalence relations used for factorization. == Approaches == === k-reversible languages === Angluin considers so-called "k-reversible" regular automata, that is, deterministic automata in which each state can be reached from at most one state by following a transition chain of length k. Formally, if Σ, Q, and δ denote the input alphabet, the state set, and the transition function of an automaton A, respectively, then A is called k-reversible if: ∀a0, ..., ak ∈ Σ ∀s1, s2 ∈ Q: δ(s1, a0...ak) = δ(s2, a0...ak) ⇒ s1 = s2, where δ means the homomorphic extension of δ to arbitrary words. Angluin gives a cubic algorithm for learning of the smallest k-reversible language from a given set of input words; for k = 0, the algorithm has even almost linear complexity. The required state uniqueness after k + 1 given symbols forces unifying automaton states, thus leading to a proper generalization different from the trivial undergeneralized automaton. This algorithm has been used to learn simple parts of English syntax; later, an incremental version has been provided. Another approach based on k-reversible automata is the tail clustering method. === Successor automata === From a given set of input strings, Vernadat and Richetin build a so-called successor automaton, consisting of one state for each distinct character and a transition between each two adjacent characters' states. For example, the singleton input set {aabbaabb} leads to an automaton corresponding to the regular expression (a+⋅b+). An extension of this approach is the predecessor-successor method which generalizes each character repetition immediately to a Kleene + and then includes for each character the set of its possible predecessors in its state. Successor automata can learn exactly the class of local languages. Since each regular language is the homomorphic image of a local language, grammars from the former class can be learned by lifting, if an appropriate (depending on the intended application) homomorphism is provided. In particular, there is such a homomorphism for the class of languages learnable by the predecessor-successor method. The learnability of local languages can be reduced to that of k-reversible languages. === Early approaches === Chomsky and Miller (1957) used the pumping lemma: they guess a part v of an input string uvw and try to build a corresponding cycle into the automaton to be learned; using membership queries they ask, for appropriate k, which of the strings uw, uvvw, uvvvw, ..., uvkw also belongs to the language to be learned, thereby refining the structure of their automaton. In 1959, Solomonoff generalized this approach to context-free languages, which also obey a pumping lemma. === Cover automata === Câmpeanu et al. learn a finite automaton as a compact representation of a large finite language. Given such a language F, they search a so-called cover automaton A such that its language L(A) covers F in the following sense: L(A) ∩ Σ≤ l = F, where l is the length of the longest string in F, and Σ≤ l denotes the set of all strings not longer than l. If such a cover automaton exists, F is uniquely determined by A and l. For example, F = {ad, read, reread } has l = 6 and a cover automaton corresponding to the regular expression (r⋅e)⋅a⋅d. For two strings x and y, Câmpeanu et al. define x ~ y if xz ∈ F ⇔ yz ∈ F for all strings z of a length such that both xz and yz are not longer than l. Based on this relation, whose lack of transitivity causes considerable technical problems, they give an O(n4) algorithm to construct from F a cover automaton A of minimal state count. Moreover, for union, intersection, and difference of two finite languages they provide corresponding operations on their cover automata. Păun et al. improve the time complexity to O(n2). === Residual automata === For a set S of strings and a string u, the Brzozowski derivative u−1S is defined as the set of all rest-strings obtainable from a string in S by cutting off its prefix u (if possible), formally: u−1S = {v ∈ Σ: uv ∈ S}, cf. picture. Denis et al. define a
Retrieval-augmented generation
Retrieval-augmented generation (RAG) is a technique that enables large language models (LLMs) to retrieve and incorporate new information from external data sources. With RAG, LLMs first refer to a specified set of documents, then respond to user queries. These documents supplement information from the LLM's pre-existing training data. This allows LLMs to use domain-specific and/or updated information that is not available in the training data. For example, this enables LLM-based chatbots to access internal company data or generate responses based on authoritative sources. RAG improves LLMs by incorporating information retrieval before generating responses. Unlike LLMs that rely on static training data, RAG pulls relevant text from databases, uploaded documents, or web sources. According to Ars Technica, "RAG is a way of improving LLM performance, in essence by blending the LLM process with a web search or other document look-up process to help LLMs stick to the facts." This method helps reduce AI hallucinations, which have caused chatbots to describe policies that don't exist, or recommend nonexistent legal cases to lawyers that are looking for citations to support their arguments. RAG also reduces the need to retrain LLMs with new data, saving on computational and financial costs. Beyond efficiency gains, RAG also allows LLMs to include sources in their responses, so users can verify the cited sources. This provides greater transparency, as users can cross-check retrieved content to ensure accuracy and relevance. The term retrieval-augmented generation (RAG) was introduced in a 2020 paper that described combining a parametric language model with a non-parametric external memory accessed through retrieval at inference time. == RAG and LLM limitations == LLMs can provide incorrect information. For example, when Google first demonstrated its LLM tool "Google Bard" (later re-branded to Gemini), the LLM provided incorrect information about the James Webb Space Telescope. This error contributed to a $100 billion decline in Google's stock value. RAG is used to prevent these errors, but it does not solve all the problems. For example, LLMs can generate misinformation even when pulling from factually correct sources if they misinterpret the context. MIT Technology Review gives the example of an AI-generated response stating, "The United States has had one Muslim president, Barack Hussein Obama." The model retrieved this from an academic book rhetorically titled Barack Hussein Obama: America's First Muslim President? The LLM did not "know" or "understand" the context of the title, generating a false statement. LLMs with RAG are programmed to prioritize new information. This technique has been called "prompt stuffing." Without prompt stuffing, the LLM's input is generated by a user; with prompt stuffing, additional relevant context is added to this input to guide the model's response. This approach provides the LLM with key information early in the prompt, encouraging it to prioritize the supplied data over pre-existing training knowledge. == Process == Retrieval-augmented generation (RAG) enhances large language models (LLMs) by incorporating an information-retrieval mechanism that allows models to access and utilize additional data beyond their original training set. Ars Technica notes that "when new information becomes available, rather than having to retrain the model, all that's needed is to augment the model's external knowledge base with the updated information" ("augmentation"). IBM states that "in the generative phase, the LLM draws from the augmented prompt and its internal representation of its training data to synthesize" an answer. === RAG key stages === Typically, the data to be referenced is converted into LLM embeddings, numerical representations in the form of a large vector space. RAG can be used on unstructured (usually text), semi-structured, or structured data (for example knowledge graphs). These embeddings are then stored in a vector database to allow for document retrieval. Given a user query, a document retriever is first called to select the most relevant documents that will be used to augment the query. This comparison can be done using a variety of methods, which depend in part on the type of indexing used. The model feeds this relevant retrieved information into the LLM via prompt engineering of the user's original query. Newer implementations (as of 2023) can also incorporate specific augmentation modules with abilities such as expanding queries into multiple domains and using memory and self-improvement to learn from previous retrievals. Finally, the LLM can generate output based on both the query and the retrieved documents. Some models incorporate extra steps to improve output, such as the re-ranking of retrieved information, context selection, and fine-tuning. == Applications == Retrieval-augmented generation is used in applications where generated responses need to be grounded in external or frequently updated information. Commonly cited use cases include search engines, question-answering systems, customer support chatbots, enterprise knowledge assistants, content generation, recommendation systems, retail and e-commerce, and industrial or manufacturing workflows. In healthcare, RAG has been studied as a way to ground large language model outputs in external medical knowledge sources, although reviews have noted continuing challenges around evaluation, ethics, and clinical reliability. == Improvements == Improvements to the basic process above can be applied at different stages in the RAG flow. === Encoder === These methods focus on the encoding of text as either dense or sparse vectors. Sparse vectors, which encode the identity of a word, are typically dictionary-length and contain mostly zeros. Dense vectors, which encode meaning, are more compact and contain fewer zeros. Various enhancements can improve the way similarities are calculated in the vector stores (databases). Performance improves by optimizing how vector similarities are calculated. Dot products enhance similarity scoring, while approximate nearest neighbor (ANN) searches improve retrieval efficiency over K-nearest neighbors (KNN) searches. Accuracy may be improved with Late Interactions, which allow the system to compare words more precisely after retrieval. This helps refine document ranking and improve search relevance. Hybrid vector approaches may be used to combine dense vector representations with sparse one-hot vectors, taking advantage of the computational efficiency of sparse dot products over dense vector operations. Other retrieval techniques focus on improving accuracy by refining how documents are selected. Some retrieval methods combine sparse representations, such as SPLADE, with query expansion strategies to improve search accuracy and recall. === Retriever-centric methods === These methods aim to enhance the quality of document retrieval in vector databases: Pre-training the retriever using the Inverse Cloze Task (ICT), a technique that helps the model learn retrieval patterns by predicting masked text within documents. Supervised retriever optimization aligns retrieval probabilities with the generator model's likelihood distribution. This involves retrieving the top-k vectors for a given prompt, scoring the generated response's perplexity, and minimizing KL divergence between the retriever's selections and the model's likelihoods to refine retrieval. Reranking techniques can refine retriever performance by prioritizing the most relevant retrieved documents during training. === Language model === By redesigning the language model with the retriever in mind, a 25-time smaller network can get comparable perplexity as its much larger counterparts. Because it is trained from scratch, this method (Retro) incurs the high cost of training runs that the original RAG scheme avoided. The hypothesis is that by giving domain knowledge during training, Retro needs less focus on the domain and can devote its smaller weight resources only to language semantics. The redesigned language model is shown here. It has been reported that Retro is not reproducible, so modifications were made to make it so. The more reproducible version is called Retro++ and includes in-context RAG. === Chunking === Chunking involves various strategies for breaking up the data into vectors so the retriever can find details in it. Three types of chunking strategies are: Fixed length with overlap. This is fast and easy. Overlapping consecutive chunks helps to maintain semantic context across chunks. Syntax-based chunks can break the document up into sentences. Libraries such as spaCy or NLTK can also help. File format-based chunking. Certain file types have natural chunks built in, and it's best to respect them. For example, code files are best chunked and vectorized as whole functions or classes. HTML files should leave