# Computational Semantics with Functional Programming: A Practical Approach Based on Jan Van Eijck's Work

## Computational Semantics With Functional Programming Jan Van Eijck

Computational semantics is a branch of computational linguistics that studies the meaning of natural language expressions and how to compute them. Functional programming is a paradigm of programming that emphasizes the use of functions as the primary building blocks of programs. Jan Van Eijck is a Dutch logician, linguist, and computer scientist who has made significant contributions to both fields. In this article, we will explore what computational semantics and functional programming are, how they are related, and what Jan Van Eijck has done to advance them.

## Computational Semantics With Functional Programming Jan Van Eijck

## Introduction

Language is one of the most complex and fascinating phenomena in the world. It allows us to communicate our thoughts, feelings, intentions, and knowledge with others. But how do we understand what others mean when they speak or write? And how can we make computers understand natural language as well?

These are some of the questions that computational semantics tries to answer. Computational semantics is a subfield of computational linguistics that deals with the representation and computation of meaning in natural language. It aims to develop formal models and algorithms that can capture the semantic aspects of language, such as word meanings, sentence meanings, discourse meanings, pragmatics, and inference.

One of the tools that computational semanticists use to model and compute meaning is functional programming. Functional programming is a style of programming that treats computation as the evaluation of mathematical functions. Unlike imperative programming, which uses statements that change the state of the program, functional programming avoids side effects and mutable data. Instead, it relies on pure functions that always return the same output for the same input, and higher-order functions that can take other functions as arguments or return them as results.

Functional programming has many advantages for computational semantics, such as simplicity, elegance, expressiveness, modularity, and correctness. It also has a close connection with logic, which is the foundation of semantics. Many functional programming languages are based on or inspired by formal systems of logic, such as lambda calculus and type theory.

One of the pioneers and leaders of the field who has explored the link between computational semantics and functional programming is Jan Van Eijck. He is a professor of logic, language, and computation at the University of Amsterdam and a senior researcher at the Centrum Wiskunde & Informatica (CWI). He has written numerous papers and books on various topics related to logic, linguistics, and computer science, and has developed several software tools and educational resources for teaching and learning about them.

In this article, we will learn more about computational semantics and functional programming, and how Jan Van Eijck has contributed to their development and dissemination.

## Computational Semantics and Functional Programming

### Why use functional programming for computational semantics?

As we have seen, computational semantics is concerned with modeling and computing meaning in natural language. But how can we represent meaning formally? And how can we manipulate it computationally?

One of the most influential approaches to formal semantics is based on the idea that meaning can be represented by functions. Functions are mathematical objects that map inputs to outputs. For example, the function f(x) = x + 1 maps any number x to the number x + 1. Functions can also be applied to other functions, resulting in more complex functions. For example, the function g(f, x) = f(f(x)) applies the function f twice to the input x.

Using functions, we can model the meaning of words, phrases, and sentences in natural language. For instance, we can assign a function to a noun that maps entities to truth values, indicating whether the noun applies to them or not. For example, the noun "dog" can be assigned the function λx.dog(x), which returns true if x is a dog and false otherwise. Similarly, we can assign a function to a verb that maps entities or events to truth values, indicating whether the verb describes them or not. For example, the verb "bark" can be assigned the function λe.bark(e), which returns true if e is an event of barking and false otherwise.

Using functions, we can also model how words combine to form larger expressions. For instance, we can use function application to model how a noun and a verb form a sentence. Function application is a rule that says that if f is a function and x is an argument, then f(x) is the result of applying f to x. For example, if we apply the function λx.dog(x) to the argument "Fido", we get the result dog(Fido), which is true if Fido is a dog and false otherwise. Similarly, if we apply the function λe.bark(e) to the argument "e1", we get the result bark(e1), which is true if e1 is an event of barking and false otherwise.

Using function application, we can combine the meanings of a noun and a verb to form the meaning of a sentence. For example, if we apply the function λx.λe.bark(e) agent(e,x) to the argument "Fido", we get the function λe.bark(e) agent(e,Fido), which returns true if e is an event of Fido barking and false otherwise. This function can be seen as the meaning of the sentence "Fido barks".

As you can see, using functions and function application, we can represent and compute meaning in a systematic and compositional way. Compositional means that the meaning of a complex expression is determined by the meanings of its parts and how they are combined. This property is desirable for semantic models, as it reflects how humans understand language.

But how can we implement these functions and function application in a programming language? This is where functional programming comes in handy. Functional programming languages are designed to support and facilitate the use of functions as first-class citizens. This means that functions can be created, stored, passed, and returned just like any other data type. Functional programming languages also provide features such as recursion, pattern matching, higher-order functions, lazy evaluation, and polymorphism that make it easier to define and manipulate functions.

Therefore, functional programming languages are natural choices for implementing semantic models based on functions. They allow us to write programs that closely resemble the mathematical notation and reasoning used in formal semantics. They also enable us to test and experiment with our semantic models interactively and efficiently.

### How to implement semantic models in functional programming languages?

There are many functional programming languages that can be used for implementing semantic models, such as Haskell, OCaml, Scheme, Lisp, and Prolog. Each of these languages has its own syntax, semantics, and features that make it more or less suitable for different tasks and preferences. However, they all share some common elements that are essential for computational semantics.

In this section, we will briefly introduce some of these elements and how they can be used to implement semantic models based on functions. We will use Haskell as an example language, but you can easily adapt these ideas to other languages as well.

#### Lambda calculus

Lambda calculus is a formal system of logic that was invented by Alonzo Church in the 1930s. It is based on two simple concepts: variables and functions. Variables are symbols that stand for arbitrary values, such as x, y, z. Functions are expressions that take variables as inputs and return outputs, such as λx.x + 1. The symbol λ (lambda) indicates that what follows is a function definition.

#### Type theory

Type theory is another formal system of logic that was developed by Bertrand Russell and others in the early 20th century. It is based on the idea that every expression has a type that determines what kind of values it can have and how it can be used. For example, the type Bool represents boolean values, such as True and False. The type Int represents integer values, such as 1, 2, 3. The type String represents string values, such as "hello", "world". The type (a -> b) represents function values that take an argument of type a and return a result of type b.

Type theory also has rules for how types can be combined and checked. For example, if f has the type (a -> b) and x has the type a, then f x has the type b. This rule ensures that functions are applied to arguments of the correct type. If f has the type (a -> b) and x has the type c, where c is different from a, then f x is a type error. This rule prevents functions from being applied to arguments of the wrong type.

Type theory is useful for computational semantics because it allows us to specify and verify the types of our semantic representations and computations. For example, we can assign types to our semantic functions that reflect their domain and range. For instance, we can assign the type (Entity -> Bool) to a function that maps entities to truth values, such as λx.dog(x). We can also assign types to our semantic expressions that reflect their meaning category. For example, we can assign the type Bool to a sentence that expresses a proposition, such as "Fido barks".

Using types, we can ensure that our semantic models are well-formed and consistent. We can also use types to guide our semantic analysis and implementation. For example, we can use types to determine what kind of functions we need to define or apply to obtain the meaning of a given expression.

Many functional programming languages support types and type checking. Some languages, such as Haskell and OCaml, have static typing, which means that types are checked at compile time. This helps to catch errors before running the program. Other languages, such as Scheme and Lisp, have dynamic typing, which means that types are checked at run time. This allows more flexibility and experimentation with the program.

#### Montague grammar

Montague grammar is a framework for formal semantics that was proposed by Richard Montague in the 1970s. It is based on the idea that natural language can be treated as a formal language that has a syntax and a semantics that are defined by precise rules. Montague grammar uses lambda calculus and type theory as the main tools for defining these rules.

Montague grammar consists of three components: a syntactic component, a semantic component, and an interface component. The syntactic component defines the structure and categories of natural language expressions using context-free grammars or categorial grammars. The semantic component defines the meaning and types of natural language expressions using lambda terms or logical formulas. The interface component defines how the syntactic and semantic components are related using translation functions or interpretation functions.

Using Montague grammar, we can derive the meaning of natural language expressions in a systematic and compositional way. We can also compare and contrast different languages or dialects using their syntactic and semantic rules. We can also study various linguistic phenomena, such as quantification, anaphora, modality, tense, aspect, presupposition, implicature, etc., using their formal representations and interpretations.

Many functional programming languages can be used to implement Montague grammar models. Some languages, such as Haskell and OCaml, have built-in support for context-free grammars or categorial grammars using parser combinators or monads. Other languages, such as Scheme and Lisp, have libraries or frameworks for defining and parsing grammars using macros or functions. All languages can use lambda terms or logical formulas to represent and manipulate meanings using functions or data structures.

### What are the benefits and challenges of functional programming for computational semantics?

Functional programming has many benefits for computational semantics, such as:

It allows us to write programs that closely resemble the mathematical notation and reasoning used in formal semantics.

It enables us to test and experiment with our semantic models interactively and efficiently.

It provides features such as recursion, pattern matching, higher-order functions, lazy evaluation, and polymorphism that make it easier to define and manipulate functions.

It supports types and type checking that help us to specify and verify the types of our semantic representations and computations.

It facilitates modularity and reusability of code by allowing us to compose functions from smaller functions.

It encourages clarity and elegance of code by avoiding side effects and mutable data.

However, functional programming also has some challenges for computational semantics, such as:

It requires a steep learning curve for beginners who are not familiar with the syntax and semantics of functional programming languages.

It may not be compatible with some existing tools or frameworks that are based on imperative or object-oriented programming paradigms.

It may not be optimal for some tasks or applications that require high performance, concurrency, or interactivity.

It may not capture all the aspects or nuances of natural language meaning, such as context, ambiguity, vagueness, or creativity.

Therefore, functional programming is not a silver bullet for computational semantics, but rather a powerful and elegant tool that can be used to model and compute meaning in natural language. It has its advantages and disadvantages, and it may need to be combined with other approaches or methods to achieve the best results.

## Jan Van Eijck's Contributions to Computational Semantics and Functional Programming

### His academic background and career

Jan Van Eijck was born in 1951 in the Netherlands. He studied mathematics and philosophy at the University of Amsterdam, where he obtained his PhD in 1980. His dissertation was on the topic of modal logic and its applications to natural language semantics. He then worked as a researcher and lecturer at various institutions, such as the University of Groningen, the University of Nijmegen, the CWI, the University of Amsterdam, the University of Utrecht, the University of Oxford, and the University of Stanford. He is currently a professor of logic, language, and computation at the University of Amsterdam and a senior researcher at the CWI.

His research interests include logic, linguistics, computer science, cognitive science, philosophy, and education. He has published over 200 papers and books on various topics related to these fields. He has also supervised over 20 PhD students and participated in many national and international research projects and collaborations.

### His publications and projects

Jan Van Eijck has written numerous publications and projects that demonstrate his expertise and contributions to computational semantics and functional programming. Here are some examples:

#### Logic in Action

Logic in Action is an online textbook and course on logic that Jan Van Eijck co-authored with several colleagues. It covers topics such as propositional logic, predicate logic, modal logic, dynamic logic, epistemic logic, temporal logic, game theory, and social choice theory. It also introduces functional programming using Haskell as a tool for learning and applying logic. It provides interactive exercises, examples, solutions, and feedback for students and teachers. It is available for free at https://www.logicinaction.org/.

#### Haskell for Linguists

expressiveness, modularity, and correctness.

Functional programming also has some challenges for computational semantics, such as compatibility, performance, concurrency, and interactivity.

Functional programming languages are based on or inspired by formal systems of logic, such as lambda calculus and type theory.

Functional programming languages can be used to implement semantic models based on functions and function application.

One of the most influential approaches to formal semantics is Montague grammar, which uses lambda calculus and type theory to define the syntax and semantics of natural language.

Jan Van Eijck has developed and co-developed several frameworks, languages, tools, and resources for computational semantics and functional programming.

Jan Van Eijck has also written and co-written several publications and projects that demonstrate his expertise and contributions to both fields.

Jan Van Eijck has received several awards and honors for his work, such as the Beth Prize, the ACL Lifetime Achievement Award, and the NWO Spinoza Prize.

### Call to action for the readers

If you are interested in learning more about computational semantics and functional programming, or if you want to explore Jan Van Eijck's work in more detail, here are some suggestions for further reading and action:

Visit Jan Van Eijck's personal website at https://staff.fnwi.uva.nl/j.vaneijck2/, where you can find his biography, publications, projects, courses, software, and contact information.

Read Logic in Action at https://www.logicinaction.org/, where you can learn about logic and functional programming using Haskell.

Read Haskell for Linguists at https://www.amazon.com/Haskell-Linguists-Richard-Moot/dp/3030028913, where you can learn how to implement linguistic theories and models using Haskell.

Read Computational Linguistics in the Netherlands Journal at https://clinjournal.org/, where you can find research papers on various topics related to computational linguistics.

Try out some of Jan Van Eijck's software tools and frameworks, such as Moltap, Jupyter4LA, lambda grammars, lambda dependency grammar, lambda prolog, and Haskell Curry. You can find them on his website or on GitHub.

Follow Jan Van Eijck on Twitter at @jveijck, where he posts updates on his work and other interesting topics related to logic, linguistics, and computer science.

We hope you enjoyed this article and learned something new. Thank you for reading!

## FAQs

Here are some frequently asked questions about computational semantics and functional programming:

What is the difference between computational semantics and natural language processing?

Natural language processing (NLP) is a broader field that encompasses computational semantics as well as other aspects of natural language analysis and generation. NLP also includes topics such as speech recognition, speech synthesis, machine translation, text summarization, sentiment analysis, information extraction, question answering, dialogue systems, etc. Computational semantics focuses on the meaning of natural language expressions and how to compute them.

What is the difference between functional programming and imperative programming?

imperative programming uses commands such as assignment, loop, conditional, etc. For example, x = x + 1 is an imperative statement that changes the