There are two versions of the Pumping Lemma. One is for context free grammars and one is for regular languages. This post is about the latter. The Pumping Lemma describes a property that all natural languages share. While it cannot be used by itself to prove that any given language is regular, it can be used to prove, often using proof by contradiction, that a language is not regular. In this sense the Pumping Lemma provides a necessary condition for a language to be regular but not a sufficient one.

Finite state automata (FSA), also known as finite state machines (FSM), are usually classified as being deterministic (DFA) or non-deterministic (NFA). A deterministic finite state automaton has exactly one transition from every state for each possible input. In other words, whatever state the FSA is in, if it encounters a symbol for which a transition exists, there will be just one transition and obviously as a result, one follow up state. For a given string, the path through a DFA is deterministic since there is no place along the way where the machine would have to choose between more than one transition. Given this definition it isn’t too hard to figure out what an NFA is. Unlike in DFA, it is possible for states in an NFA to have more than one transition per input symbol. Additionally, states in an NFA may have states that don’t require an input symbol at all, transitioning on the empty string ε.

Superficially it would appear that deterministic and non-deterministic finite state automata are entirely separate beasts. It turns out, however, that they are equivalent. For any language recognized by an NFA, there exists a DFA that recognizes that language and vice versa. The algorithm to make the conversion from NFA to DFA is relatively simple, even if the resulting DFA is considerably more complex than the original NFA. After the jump I will prove this equivalence and also step through a short example of converting an NFA to an equivalent DFA.

In my last post, “Kleene’s Theorem,” I provided some useful background information about strings, regular languages, regular expressions, and finite automata before introducing the eponymously named theorem that has become one of the cornerstones of artificial intelligence and more specifically, natural language processing (NLP). Kleene’s Theorem tells us that regular expressions and finite state automata are one and the same when it comes to describing regular languages. In the post I will provide a proof of this groundbreaking principle.

Stephen Cole Kleene was an American mathematician who’s groundbreaking work in the sub-field of logic known as recursion theory laid the groundwork for modern computing. While most computer programmers might not know his name or the significance of his work regarding computable functions, I am willing to bet that anyone who has ever dealt with regular expressions is intimately familiar with an indispensable operator that resulted directly from his work and even bears his name, the *****, or as it is formally known, the **Kleene star**.

While his contributions to computer science in general cannot be overstated, Kleene also authored a theorem that plays an important role in artificial intelligence, specifically the branch known as natural language processing, or NLP for short. Kleene’s Theorem relates regular languages, regular expressions, and finite state automata (FSAs). In short, he was able to prove that regular expressions and finite state automata were the same thing, just two different representations of any given regular language.

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As a computer programmer for more than a quarter of century, I don’t think I have ever thought much about strings. I knew the basics. In every language I’d worked with, strings were a data type unto themselves. Superficially they are a sequence of characters, but behind the scenes, computers store and manipulate them as arrays of one or more binary bytes. In programs, they can be stored in variables or constants, and often show up in source code as literals, ie., fixed, quoted values like “salary” or “bumfuzzle.” (That is my new favorite word, btw.) Outside of occasionally navigating the subtleties of encoding and decoding them, I never gave strings a second thought.

Even when I first dipped my toe into the waters of natural language processing, aka NLP (not to be confused with the quasi-scientific neuro linguistic programming which unfortunately shares the same acronym), I still really only worked with strings as whole entities, words or affixes, As I made my through familiarizing myself with existing NLP tools, I didn’t have to dive any deeper than that. It was only when I started programming my own tools from the ground up, did I learn about the very formal mathematics behind strings and their relationship to sets and set theory. This post will be an attempt to explain what I learned.

From Vienna Bienalle 2017, taking place this week in Austria, comes a new take on Isaac Asimov’s Three Laws of Robotics. The head of the project, Christoph Thun-Hohenstein, says the update was necessitated by:

…the need for benign intelligent robots and the necessity of cultivating a culture of quality committed to serving the common good!

That sounds a lot like Asimov’s reasoning, but the new laws are certainly worthy of consideration and debate.

Set 38 years in the future, the plot of 2002’s blockbuster film Minority Report revolves around Washington DC’s PreCrime unit, a police force who able to stop future murders from happening with the aid of three mutant human who are able to predict homicides before they happen. Minority Report managed to side step the “psychic predicts a murder” cliché storyline with its innovative use of technology: not only could precogs predict future murders, but their visions could be streamed via a neural bridge in the form of a video that the police officers could watch. Fantastical? Nope, and researchers from MIT already have a jump on the technology.

Ever since their introduction over eighty years ago, Isaac Asimov‘s Three Laws of Robotics have been the de jure rules governing the acceptable behavior of robots. Even the uninitiated and uninterested are likely to say they know of them, even if they can’t recite a single rule verbatim. When conceived, the Three Laws were nothing but a thought experiment wrapped in a science fiction story, but now, the dizzying pace of developments in the fields of robotics and ai has spurred engineers and ethicists to reinvestigate and rewrite the guidelines by which artificially intelligent entities should operate. Who better to take the lead in this initiative than Google, the company who just yesterday announced that machine learning will be at the core of everything it does.

South Korean scientists from the Department of Materials Science and Engineering at Pohang University of Science and Technology appear to have cleared the largest obstacle to the feasibility of building brain-like computers: power consumption. In their paper “Organic core-sheath nanowire artificial synapses with femtojoule energy consumption,” published in the June 17th edition of *Science Advances*, the researchers describe how they use organic nanowire (ONW) to build synaptic transistors (STs) whose power consumption is almost one-tenth of the real thing.