Summary Programs as Diagrams From Categorical Computability to Computable Categories arxiv.org
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One Line
This document explores the connection between mathematicians and programmers through diagrams in categorical computability and computable categories.
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Slide Presentation (8 slides)
Key Points
- Programs as diagrams connect the worlds of mathematics and programming.
- Categorical language based on string diagrams is essential for mathematical models of computation and programming languages.
- Types and functions can be represented as strings in diagrams, and categories and monoidal categories are important in understanding them.
- The concept of propositions-as-types relates logic and type theory.
- The Church-Turing Thesis postulates that different theoretical models of computation define the same notion of computation.
- Metaprogramming involves writing programs that compute other programs.
- Computational processes involve state transitions and outputs over time.
- Program closure is a concept related to the ability of programming languages to compute functions.
Summaries
17 word summary
This document connects mathematicians and programmers by exploring programs as diagrams in categorical computability and computable categories.
63 word summary
This document explores programs as diagrams in categorical computability and computable categories, connecting mathematicians and programmers. It introduces a categorical language based on string diagrams as a compact representation of computations. The basics of computability and theoretical models of computation are explained. Categories and monoidal categories are emphasized for understanding types and functions. The research concludes that computability is intrinsic to the universe.
150 word summary
This document explores the concept of programs as diagrams in the context of categorical computability and computable categories. The author highlights the convenience of using the language of categories for mathematicians and programmers, connecting the two communities. A categorical language based on string diagrams is introduced as a compact representation of computations. The basics of math behind computer science are explained, including the concept of computability and different theoretical models of computation. Categories and monoidal categories are introduced, emphasizing their importance in understanding types and functions. The document presents a categorical language based on string diagrams as a foundational tool for understanding computability and computation. The paradigm of propositions-as-types, cartesian closed categories, arrow and string diagrams, program evaluators, and idempotent splittings are discussed. Metaprogramming, computation as a process, program-closed categories, and well-ordering in programming languages are also explored. The research concludes that computability is an intrinsic property of the universe.
548 word summary
This document explores the concept of programs as diagrams in the context of categorical computability and computable categories. The author emphasizes the convenience of using the language of categories for both mathematicians and programmers, as it connects the two communities. A categorical language based on string diagrams is introduced as a compact and basic representation of computations.
The basics of math behind computer science are explained, including the concept of computability and different theoretical models of computation. The Church-Turing Thesis states that all these models define the same notion of computation.
Types can be represented as strings in diagrams, and data services involve copying and deleting data. Monoidal functions do not necessarily preserve data services, while cartesian functions do. The concept of categories and monoidal categories is introduced, highlighting their importance in understanding types and functions.
The document presents a categorical language based on string diagrams as a foundational tool for understanding computability and computation. It connects the worlds of mathematics and programming, providing a compact and versatile representation of computations.
The author explores the paradigm of propositions-as-types, which relates logic and type theory. The historical development and adoption of this paradigm as the Curry-Howard isomorphism are discussed. The concept of cartesian closed categories and their significance in various areas of mathematics and programming is also explained.
Arrow diagrams and string diagrams are highlighted as useful tools for diagrammatic reasoning, particularly in representing complex algebraic transformations. The document introduces the concept of a computer as a universal machine, emphasizing its ability to perform any computation through programming. Program evaluators are discussed as universal parametrized functions that play a role in evaluating programs.
The document presents the idea that computable types are retracts of programming languages, with data being encoded as programs and types being checked by program evaluations. The concept of types as idempotents is introduced, ensuring the completeness of the type system.
Various concepts related to programs and computations are discussed, including idempotent splittings and their role in inducing types. The foundational view of typing and the intuitive view of type checking using program evaluations are highlighted. The correspondence between sets of expressions in different programming languages is explained, although it is not computable or computationally meaningful.
The document delves into the concepts of truth values, branching, program equality, and predicates. The concept of a monoidal computer is introduced, along with examples of programming languages as monoidal computers. Fixpoints in computable functions are discussed, as well as polymorphic quines and viruses.
Metaprogramming is explored as the process of writing programs that compute other programs. Compilers and supercompilation are discussed as forms of metaprogramming that optimize program structure and improve efficiency.
The concept of computation is extended to computational processes, which take time and space to process inputs into outputs. Processes can be represented as traces or programs with sequences of instructions. The concept of program execution as a computational process is further elaborated.
The document discusses program-closed categories, well-ordering in programming languages, monotone programming, and their relation to computable functions. The upshot of the research is that a monoidal category with data services may be program-closed in at most one way, and computability is an intrinsic property of the universe.
Overall, the document provides a comprehensive overview of programs as diagrams and their relationship to categorical computability.
617 word summary
This document explores the concept of programs as diagrams in the context of categorical computability and computable categories. The author emphasizes the convenience of using the language of categories for both mathematicians and programmers, as it connects the two communities. A categorical language based on string diagrams is introduced as a compact and basic representation of computations.
The world of computation is characterized by the interaction and evolution of people and computers, making the language of categories suitable for studying this interconnectedness. The basics of math behind computer science are explained, including the concept of computability and different theoretical models of computation. The Church-Turing Thesis states that all these models define the same notion of computation.
The document discusses types as strings, data services, monoidal functions as boxes, and cartesian functions. Types can be represented as strings in diagrams, and data services involve copying and deleting data. Monoidal functions do not necessarily preserve data services, while cartesian functions do. The concept of categories and monoidal categories is introduced, highlighting their importance in understanding types and functions.
The document presents a categorical language based on string diagrams as a foundational tool for understanding computability and computation. It connects the worlds of mathematics and programming, providing a compact and versatile representation of computations.
The author explores the paradigm of propositions-as-types, which relates logic and type theory. The historical development and adoption of this paradigm as the Curry-Howard isomorphism are discussed. The concept of cartesian closed categories and their significance in various areas of mathematics and programming is also explained.
Arrow diagrams and string diagrams are highlighted as useful tools for diagrammatic reasoning, particularly in representing complex algebraic transformations. The document introduces the concept of a computer as a universal machine, emphasizing its ability to perform any computation through programming. Program evaluators are discussed as universal parametrized functions that play a role in evaluating programs. The composition of programs and the ability to compute composite functions from given programs are explored.
The document presents the idea that computable types are retracts of programming languages, with data being encoded as programs and types being checked by program evaluations. The concept of types as idempotents is introduced, ensuring the completeness of the type system.
Various concepts related to programs and computations are discussed, including idempotent splittings and their role in inducing types. The foundational view of typing and the intuitive view of type checking using program evaluations are highlighted. The correspondence between sets of expressions in different programming languages is explained, although it is not computable or computationally meaningful.
The document delves into the concepts of truth values, branching, program equality, and predicates. The concept of a monoidal computer is introduced, along with examples of programming languages as monoidal computers. Fixpoints in computable functions are discussed, as well as polymorphic quines and viruses.
Metaprogramming is explored as the process of writing programs that compute other programs. Compilers and supercompilation are discussed as forms of metaprogramming that optimize program structure and improve efficiency.
The concept of computation is extended to computational processes, which take time and space to process inputs into outputs. Processes can be represented as traces or programs with sequences of instructions. The concept of program execution as a computational process is further elaborated.
The document discusses program-closed categories, well-ordering in programming languages, monotone programming, and their relation to computable functions. The upshot of the research is that a monoidal category with data services may be program-closed in at most one way, and computability is an intrinsic property of the universe.
Overall, the document provides a comprehensive overview of programs as diagrams and their relationship to categorical computability. Key concepts and findings are highlighted in a concise and clear writing style.
1882 word summary
This document discusses the concept of programs as diagrams, exploring the relationship between categorical computability and computable categories. The author explains that while programmers use a variety of languages to program computers, mathematicians study mathematics in different languages. However, the language of categories seems to be convenient for both mathematicians and programmers, connecting the two communities. The author introduces a categorical language based on string diagrams that is essential for mathematical models of computation and programming languages. This language has evolved through years of teaching and provides a compact and basic representation of computations.
The author emphasizes that the world of computers and networks is built on numbers, and mathematics has evolved beyond the Pythagorean computing rituals. The world of computation is characterized by the interaction and evolution of people and computers, which are difficult to distinguish from each other. The author argues that the language of categories is suitable for studying this interconnectedness.
The document delves into the basics of math behind computer science, explaining that a computer runs programs and programmers program programs. The author explores the concept of computability and different theoretical models of computation, such as Turing machines and cellular automata. The Church-Turing Thesis postulates that all these models define the same notion of computation.
The author also discusses types as strings, data services, monoidal functions as boxes, and cartesian functions. They explain that types can be represented as strings in diagrams, and data services involve copying and deleting data. Monoidal functions are those that do not necessarily preserve data services, while cartesian functions do. The author introduces the concept of categories and monoidal categories, highlighting their importance in understanding types and functions.
In conclusion, this document presents a categorical language based on string diagrams as a foundational tool for understanding computability and computation. The author argues that this language connects the worlds of mathematics and programming, providing a compact and versatile representation of computations.
In this document, the author explores the concept of programs as diagrams in the context of categorical computability and computable categories. They highlight the importance of structures rather than elements in understanding types and categories. Two different types or categories may have the same elements but different structures, leading to distinct behavior and characteristics. The paradigm of propositions-as-types, which relates logic and type theory, is discussed, along with its historical development and adoption as the Curry-Howard isomorphism. The author introduces the concept of cartesian closed categories and their significance in various areas of mathematics and programming. They also discuss the use of arrow diagrams and string diagrams for diagrammatic reasoning, highlighting their benefits in representing complex algebraic transformations. The idea of a computer as a universal machine is introduced, emphasizing its ability to perform any computation through programming. The concept of program evaluators as universal parametrized functions is explained, along with their role in evaluating programs. The author discusses the composition of programs and the ability to compute composite functions from given programs. They also explore the idea that computable types are retracts of programming languages, with data being encoded as programs and types being checked by program evaluations. The concept of types as idempotents is introduced, highlighting their role in splitting idempotents and ensuring the completeness of the type system.
The text discusses various concepts related to programs and computations. It introduces the idea of idempotent splittings and how they can be used to induce types and vice versa. The text also highlights the foundational view of typing and the intuitive view of type checking using program evaluations. It explains the correspondence between sets of expressions in different programming languages, although this correspondence is not computable or computationally meaningful. However, the text mentions that there is a computable and computationally meaningful isomorphism that will be discussed later.
The text then delves into the concepts of truth values, branching, program equality, and predicates. It explains that predicates are single-valued functions that assign a truth value to each input and discusses the decidable predicates. The concept of a monoidal computer is introduced, which is a monoidal category with data services, a type of programs with a decidable equality predicate, and a program evaluator. Programming languages are given as examples of monoidal computers.
The text also discusses fixpoints in computable functions and provides examples of polymorphic quines and polymorphic viruses. It introduces the concept of ?-combinators and their classifiers, which are program transformers that can recognize and construct fixpoints. Finally, the text explains how software systems can be modeled as systems of equations and presents the method of Smullyan fixpoints for solving systems of computable equations.
Overall, the text covers a range of topics related to programs, computations, types, fixpoints, and software systems.
The concept behind the algebraic magic of programs as diagrams emerges from the pictures of functions and their fixpoints. The diagrammatic form of the construction of fixpoints follows the idea of the Fundamental Theorem. Systems of program equations can be solved by finding programs that are fixed by given computations. For any tuple of computations, there are programs that are fixed by them. Solving systems of computable equations involves defining functions and using them to construct fixpoints. Counting numbers is the process of assigning fingers or pebbles to objects and is the earliest form of computation. Numbers can be represented as sets, with 0 representing the empty set. Sequences can be represented as cartesian functions or as sequences of numbers. Counting down is achieved through induction and recursion, which are methods for computing by counting. Induction builds sequences of elements by starting with a base case and applying a step case repeatedly. Recursion builds sequences of functions by defining a base case and a recursive step. The process of evaluating entries in inductively defined or recursively defined sequences involves descending down the steps to the base case. The evaluation can be done through program evaluations within program evaluations.
Metaprogramming involves writing programs that compute other programs. This concept is represented in Figure 6.1, where a computation f is computed by a program F, and F is computed by a metaprogram F. The process of programming involves going from left to right in this diagram, while program evaluators take us from right to left. Compilers are a type of metaprogram that translate high-level human code into low-level machine code, optimizing program structure and improving efficiency. Supercompilation is another form of metaprogramming that involves partially evaluating program evaluators on code to improve efficiency. This idea was pioneered by Valentin Turchin and later expanded upon by Yoshihiko Futamura. In supercompilation, a high-level language H with a slow universal evaluator h and a low-level language L with a fast universal evaluator ? are used. A compiler is needed to translate H-programs to L-programs. However, since h and ? are universal evaluators, they can be programmed on each other, allowing for the creation of interpreters that can interpret programs in their own language. By partially evaluating these interpreters, H-programs can be compiled to L-programs. This process is known as the Futamura projections. The first projection involves partially evaluating an H-to-L-interpreter, and the second projection involves partially evaluating a specializer on an interpreter to create a compiler. These concepts are summarized in Figures 6.3 and 6.4.
The concept of computation can be extended from computable functions to computational processes. While a function produces its output as soon as it receives its input, a process takes time and space to process an input into an output. A process is defined as a pair of functions that represent the state transitions and outputs. The state transitions capture the changing state of the process, while the outputs represent the results of the computation.
Processes can be represented as traces, which capture both the state transitions and outputs. The trace starts from an initial state and is induced by a sequence of inputs. Each step in the trace consumes an input and produces an output.
Examples of processes include a multifunctional drill, where different attachments represent different states, and a general-purpose computer, where the state space represents different programs.
Processes can also be represented as programs with sequences of instructions. The state space captures the values of program parameters and tracks which instructions are ready for evaluation.
The concept of computational processes allows for a more dynamic and interactive understanding of computation, where inputs and outputs are processed over time.
This article discusses the concept of programs as diagrams in the context of categorical computability and computable categories. It begins by presenting a C-program that computes and prints the sum of user-entered inputs. The left-hand diagram in Figure 7.2 illustrates a cartesian function as a morphism between functions, while the right-hand diagram shows the same function as a process morphism from q to r. The article then explores the notion of computational processes and their relationship to programs. It explains that a function is computable when there is a program that evaluates to it, and a process is computational when there is a program whose execution simulates it. The concept of program execution as a computational process is further elaborated, with a focus on the state updates and execution traces produced. The article introduces the idea of imperative programs, which are full and faithful programs that simulate a computational process and preserve its state changes. It also discusses the properties of well-ordering in programming languages and how it relates to the universality and isomorphism of programming languages. The article concludes by highlighting the importance of program-closed categories and their role in defining computability as a property.
The concept of program closure is discussed in this document, which explores the relationship between programming languages and their ability to compute functions. The paper begins by introducing the idea of a program-closed category, which is a monoidal computer with a well-ordered programming language. It explains that in a program-closed category, any computable function can be encoded by infinitely many programs. It also highlights the importance of well-ordering in the context of programming languages.
The paper then delves into the concept of monotone programming and its relation to computable functions. It explains that any decreasing sequence of well-ordered programs is finite, and therefore, there must be an infinite increasing sequence of programs that encode a computable function. This leads to the notion that for any computation f : A → B and any bound b : P, there must always be a program F ≤ b such that F = f.
The paper further discusses the idea of program-closed categories and their relation to monotone programming. It explains that a program-closed category is a monoidal computer whose programming language is well-ordered. It also highlights the fact that any two programming languages in a program-closed category are isomorphic.
The paper concludes by discussing the upshot of the research, which is that a monoidal category with data services may be program-closed in at most one way. It emphasizes that computability is an intrinsic property of the universe and not an external structure. The paper also addresses some of the challenges and issues related to program closure, such as the existence of injections and bijections between programming languages.
Overall, the document provides a comprehensive overview of programs as diagrams and their relationship to categorical computability. It highlights key concepts and findings while maintaining a concise and clear writing style.