A Lisp Way to Type Theory and Formal Proofs Frederic Peschanski
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Lecture 4. Higher-Order Functions Functional Programming
Lecture 4. Higher-order functions Functional Programming [Faculty of Science Information and Computing Sciences] 0 I function call and return as only control-flow primitive I no loops, break, continue, goto I (almost) unique types I no inheritance hell I high-level declarative data-structures I no explicit reference-based data structures Goal of typed purely functional programming Keep programs easy to reason about by I data-flow only through function arguments and return values I no hidden data-flow through mutable variables/state [Faculty of Science Information and Computing Sciences] 1 I (almost) unique types I no inheritance hell I high-level declarative data-structures I no explicit reference-based data structures Goal of typed purely functional programming Keep programs easy to reason about by I data-flow only through function arguments and return values I no hidden data-flow through mutable variables/state I function call and return as only control-flow primitive I no loops, break, continue, goto [Faculty of Science Information and Computing Sciences] 1 I high-level declarative data-structures I no explicit reference-based data structures Goal of typed purely functional programming Keep programs easy to reason about by I data-flow only through function arguments and return values I no hidden data-flow through mutable variables/state I function call and return as only control-flow primitive I no loops, break, continue, goto I (almost) unique types I no inheritance hell [Faculty of Science Information and Computing Sciences] 1 Goal -
Higher-Order Functions 15-150: Principles of Functional Programming – Lecture 13
Higher-order Functions 15-150: Principles of Functional Programming { Lecture 13 Giselle Reis By now you might feel like you have a pretty good idea of what is going on in functional program- ming, but in reality we have used only a fragment of the language. In this lecture we see what more we can do and what gives the name functional to this paradigm. Let's take a step back and look at ML's typing system: we have basic types (such as int, string, etc.), tuples of types (t*t' ) and functions of a type to a type (t ->t' ). In a grammar style (where α is a basic type): τ ::= α j τ ∗ τ j τ ! τ What types allowed by this grammar have we not used so far? Well, we could, for instance, have a function below a tuple. Or even a function within a function, couldn't we? The following are completely valid types: int*(int -> int) int ->(int -> int) (int -> int) -> int The first one is a pair in which the first element is an integer and the second one is a function from integers to integers. The second one is a function from integers to functions (which have type int -> int). The third type is a function from functions to integers. The two last types are examples of higher-order functions1, i.e., a function which: • receives a function as a parameter; or • returns a function. Functions can be used like any other value. They are first-class citizens. Maybe this seems strange at first, but I am sure you have used higher-order functions before without noticing it. -
Lemma Functions for Frama-C: C Programs As Proofs
Lemma Functions for Frama-C: C Programs as Proofs Grigoriy Volkov Mikhail Mandrykin Denis Efremov Faculty of Computer Science Software Engineering Department Faculty of Computer Science National Research University Ivannikov Institute for System Programming of the National Research University Higher School of Economics Russian Academy of Sciences Higher School of Economics Moscow, Russia Moscow, Russia Moscow, Russia [email protected] [email protected] [email protected] Abstract—This paper describes the development of to additionally specify loop invariants and termination an auto-active verification technique in the Frama-C expression (loop variants) in case the function under framework. We outline the lemma functions method analysis contains loops. Then the verification instrument and present the corresponding ACSL extension, its implementation in Frama-C, and evaluation on a set generates verification conditions (VCs), which can be of string-manipulating functions from the Linux kernel. separated into categories: safety and behavioral ones. We illustrate the benefits our approach can bring con- Safety VCs are responsible for runtime error checks for cerning the effort required to prove lemmas, compared known (modeled) types, while behavioral VCs represent to the approach based on interactive provers such as the functional correctness checks. All VCs need to be Coq. Current limitations of the method and its imple- mentation are discussed. discharged in order for the function to be considered fully Index Terms—formal verification, deductive verifi- proved (totally correct). This can be achieved either by cation, Frama-C, auto-active verification, lemma func- manually proving all VCs discharged with an interactive tions, Linux kernel prover (i. e., Coq, Isabelle/HOL or PVS) or with the help of automatic provers. -
The Machine That Builds Itself: How the Strengths of Lisp Family
Khomtchouk et al. OPINION NOTE The Machine that Builds Itself: How the Strengths of Lisp Family Languages Facilitate Building Complex and Flexible Bioinformatic Models Bohdan B. Khomtchouk1*, Edmund Weitz2 and Claes Wahlestedt1 *Correspondence: [email protected] Abstract 1Center for Therapeutic Innovation and Department of We address the need for expanding the presence of the Lisp family of Psychiatry and Behavioral programming languages in bioinformatics and computational biology research. Sciences, University of Miami Languages of this family, like Common Lisp, Scheme, or Clojure, facilitate the Miller School of Medicine, 1120 NW 14th ST, Miami, FL, USA creation of powerful and flexible software models that are required for complex 33136 and rapidly evolving domains like biology. We will point out several important key Full list of author information is features that distinguish languages of the Lisp family from other programming available at the end of the article languages and we will explain how these features can aid researchers in becoming more productive and creating better code. We will also show how these features make these languages ideal tools for artificial intelligence and machine learning applications. We will specifically stress the advantages of domain-specific languages (DSL): languages which are specialized to a particular area and thus not only facilitate easier research problem formulation, but also aid in the establishment of standards and best programming practices as applied to the specific research field at hand. DSLs are particularly easy to build in Common Lisp, the most comprehensive Lisp dialect, which is commonly referred to as the “programmable programming language.” We are convinced that Lisp grants programmers unprecedented power to build increasingly sophisticated artificial intelligence systems that may ultimately transform machine learning and AI research in bioinformatics and computational biology. -
Better SMT Proofs for Easier Reconstruction Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr
Better SMT Proofs for Easier Reconstruction Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr To cite this version: Haniel Barbosa, Jasmin Blanchette, Mathias Fleury, Pascal Fontaine, Hans-Jörg Schurr. Better SMT Proofs for Easier Reconstruction. AITP 2019 - 4th Conference on Artificial Intelligence and Theorem Proving, Apr 2019, Obergurgl, Austria. hal-02381819 HAL Id: hal-02381819 https://hal.archives-ouvertes.fr/hal-02381819 Submitted on 26 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Better SMT Proofs for Easier Reconstruction Haniel Barbosa1, Jasmin Christian Blanchette2;3;4, Mathias Fleury4;5, Pascal Fontaine3, and Hans-J¨orgSchurr3 1 University of Iowa, 52240 Iowa City, USA [email protected] 2 Vrije Universiteit Amsterdam, 1081 HV Amsterdam, the Netherlands [email protected] 3 Universit´ede Lorraine, CNRS, Inria, LORIA, 54000 Nancy, France fjasmin.blanchette,pascal.fontaine,[email protected] 4 Max-Planck-Institut f¨urInformatik, Saarland Informatics Campus, Saarbr¨ucken, Germany fjasmin.blanchette,[email protected] 5 Saarbr¨ucken Graduate School of Computer Science, Saarland Informatics Campus, Saarbr¨ucken, Germany [email protected] Proof assistants are used in verification, formal mathematics, and other areas to provide trust- worthy, machine-checkable formal proofs of theorems. -
An Evaluation of Go and Clojure
An evaluation of Go and Clojure A thesis submitted in partial satisfaction of the requirements for the degree Bachelors of Science in Computer Science Fall 2010 Robert Stimpfling Department of Computer Science University of Colorado, Boulder Advisor: Kenneth M. Anderson, PhD Department of Computer Science University of Colorado, Boulder 1. Introduction Concurrent programming languages are not new, but they have been getting a lot of attention more recently due to their potential with multiple processors. Processors have gone from growing exponentially in terms of speed, to growing in terms of quantity. This means processes that are completely serial in execution will soon be seeing a plateau in performance gains since they can only rely on one processor. A popular approach to using these extra processors is to make programs multi-threaded. The threads can execute in parallel and use shared memory to speed up execution times. These multithreaded processes can significantly speed up performance, as long as the number of dependencies remains low. Amdahl‘s law states that these performance gains can only be relative to the amount of processing that can be parallelized [1]. However, the performance gains are significant enough to be looked into. These gains not only come from the processing being divvied up into sections that run in parallel, but from the inherent gains from sharing memory and data structures. Passing new threads a copy of a data structure can be demanding on the processor because it requires the processor to delve into memory and make an exact copy in a new location in memory. Indeed some studies have shown that the problem with optimizing concurrent threads is not in utilizing the processors optimally, but in the need for technical improvements in memory performance [2]. -
Clojure, Given the Pun on Closure, Representing Anything Specific
dynamic, functional programming for the JVM “It (the logo) was designed by my brother, Tom Hickey. “It I wanted to involve c (c#), l (lisp) and j (java). I don't think we ever really discussed the colors Once I came up with Clojure, given the pun on closure, representing anything specific. I always vaguely the available domains and vast emptiness of the thought of them as earth and sky.” - Rich Hickey googlespace, it was an easy decision..” - Rich Hickey Mark Volkmann [email protected] Functional Programming (FP) In the spirit of saying OO is is ... encapsulation, inheritance and polymorphism ... • Pure Functions • produce results that only depend on inputs, not any global state • do not have side effects such as Real applications need some changing global state, file I/O or database updates side effects, but they should be clearly identified and isolated. • First Class Functions • can be held in variables • can be passed to and returned from other functions • Higher Order Functions • functions that do one or both of these: • accept other functions as arguments and execute them zero or more times • return another function 2 ... FP is ... Closures • main use is to pass • special functions that retain access to variables a block of code that were in their scope when the closure was created to a function • Partial Application • ability to create new functions from existing ones that take fewer arguments • Currying • transforming a function of n arguments into a chain of n one argument functions • Continuations ability to save execution state and return to it later think browser • back button 3 .. -
Topic 6: Partial Application, Function Composition and Type Classes
Recommended Exercises and Readings Topic 6: Partial Application, • From Haskell: The craft of functional programming (3rd Ed.) Function Composition and Type • Exercises: • 11.11, 11.12 Classes • 12.30, 12.31, 12.32, 12.33, 12.34, 12.35 • 13.1, 13.2, 13.3, 13.4, 13.7, 13.8, 13.9, 13.11 • If you have time: 12.37, 12.38, 12.39, 12.40, 12.41, 12.42 • Readings: • Chapter 11.3, and 11.4 • Chapter 12.5 • Chapter 13.1, 13.2, 13.3 and 13.4 1 2 Functional Forms Curried and Uncurried Forms • The parameters to a function can be viewed in two different ways • Uncurried form • As a single combined unit • Parameters are bundled into a tuple and passed as a group • All values are passed as one tuple • Can be used in Haskell • How we typically think about parameter passing in Java, C++, Python, Pascal, C#, … • Typically only when there is a specific need to do • As a sequence of values that are passed one at a time • As each value is passed, a new function is formed that requires one fewer parameters • Curried form than its predecessor • Parameters are passed to a function sequentially • How parameters are passed in Haskell • Standard form in Haskell • But it’s not a detail that we need to concentrate on except when we want to make use of it • Functions can be transformed from one form to the other 3 4 Curried and Uncurried Forms Curried and Uncurried Forms • A function in curried form • Why use curried form? • Permits partial application multiply :: Int ‐> Int ‐> Int • Standard way to define functions in Haskell multiply x y = x * y • A function of n+1 -
Proof-Assistants Using Dependent Type Systems
CHAPTER 18 Proof-Assistants Using Dependent Type Systems Henk Barendregt Herman Geuvers Contents I Proof checking 1151 2 Type-theoretic notions for proof checking 1153 2.1 Proof checking mathematical statements 1153 2.2 Propositions as types 1156 2.3 Examples of proofs as terms 1157 2.4 Intermezzo: Logical frameworks. 1160 2.5 Functions: algorithms versus graphs 1164 2.6 Subject Reduction . 1166 2.7 Conversion and Computation 1166 2.8 Equality . 1168 2.9 Connection between logic and type theory 1175 3 Type systems for proof checking 1180 3. l Higher order predicate logic . 1181 3.2 Higher order typed A-calculus . 1185 3.3 Pure Type Systems 1196 3.4 Properties of P ure Type Systems . 1199 3.5 Extensions of Pure Type Systems 1202 3.6 Products and Sums 1202 3.7 E-typcs 1204 3.8 Inductive Types 1206 4 Proof-development in type systems 1211 4.1 Tactics 1212 4.2 Examples of Proof Development 1214 4.3 Autarkic Computations 1220 5 P roof assistants 1223 5.1 Comparing proof-assistants . 1224 5.2 Applications of proof-assistants 1228 Bibliography 1230 Index 1235 Name index 1238 HANDBOOK OF AUTOMAT8D REASONING Edited by Alan Robinson and Andrei Voronkov © 2001 Elsevier Science Publishers 8.V. All rights reserved PROOF-ASSISTANTS USING DEPENDENT TYPE SYSTEMS 1151 I. Proof checking Proof checking consists of the automated verification of mathematical theories by first fully formalizing the underlying primitive notions, the definitions, the axioms and the proofs. Then the definitions are checked for their well-formedness and the proofs for their correctness, all this within a given logic. -
Can the Computer Really Help Us to Prove Theorems?
Can the computer really help us to prove theorems? Herman Geuvers1 Radboud University Nijmegen and Eindhoven University of Technology The Netherlands ICT.Open 2011 Veldhoven, November 2011 1Thanks to Freek Wiedijk & Foundations group, RU Nijmegen Can the computer really help us to prove theorems? Can the computer really help us to prove theorems? Yes it can Can the computer really help us to prove theorems? Yes it can But it’s hard ... ◮ How does it work? ◮ Some state of the art ◮ What needs to be done Overview ◮ What are Proof Assistants? ◮ How can a computer program guarantee correctness? ◮ Challenges What are Proof Assistants – History John McCarthy (1927 – 2011) 1961, Computer Programs for Checking Mathematical Proofs What are Proof Assistants – History John McCarthy (1927 – 2011) 1961, Computer Programs for Checking Mathematical Proofs Proof-checking by computer may be as important as proof generation. It is part of the definition of formal system that proofs be machine checkable. For example, instead of trying out computer programs on test cases until they are debugged, one should prove that they have the desired properties. What are Proof Assistants – History Around 1970 five new systems / projects / ideas ◮ Automath De Bruijn (Eindhoven) ◮ Nqthm Boyer, Moore (Austin, Texas) ◮ LCF Milner (Stanford; Edinburgh) What are Proof Assistants – History Around 1970 five new systems / projects / ideas ◮ Automath De Bruijn (Eindhoven) ◮ Nqthm Boyer, Moore (Austin, Texas) ◮ LCF Milner (Stanford; Edinburgh) ◮ Mizar Trybulec (Bia lystok, Poland) -
Proceedings of the 8Th European Lisp Symposium Goldsmiths, University of London, April 20-21, 2015 Julian Padget (Ed.) Sponsors
Proceedings of the 8th European Lisp Symposium Goldsmiths, University of London, April 20-21, 2015 Julian Padget (ed.) Sponsors We gratefully acknowledge the support given to the 8th European Lisp Symposium by the following sponsors: WWWLISPWORKSCOM i Organization Programme Committee Julian Padget – University of Bath, UK (chair) Giuseppe Attardi — University of Pisa, Italy Sacha Chua — Toronto, Canada Stephen Eglen — University of Cambridge, UK Marc Feeley — University of Montreal, Canada Matthew Flatt — University of Utah, USA Rainer Joswig — Hamburg, Germany Nick Levine — RavenPack, Spain Henry Lieberman — MIT, USA Christian Queinnec — University Pierre et Marie Curie, Paris 6, France Robert Strandh — University of Bordeaux, France Edmund Weitz — University of Applied Sciences, Hamburg, Germany Local Organization Christophe Rhodes – Goldsmiths, University of London, UK (chair) Richard Lewis – Goldsmiths, University of London, UK Shivi Hotwani – Goldsmiths, University of London, UK Didier Verna – EPITA Research and Development Laboratory, France ii Contents Acknowledgments i Messages from the chairs v Invited contributions Quicklisp: On Beyond Beta 2 Zach Beane µKanren: Running the Little Things Backwards 3 Bodil Stokke Escaping the Heap 4 Ahmon Dancy Unwanted Memory Retention 5 Martin Cracauer Peer-reviewed papers Efficient Applicative Programming Environments for Computer Vision Applications 7 Benjamin Seppke and Leonie Dreschler-Fischer Keyboard? How quaint. Visual Dataflow Implemented in Lisp 15 Donald Fisk P2R: Implementation of -
Functional Programming Lecture 1: Introduction
Functional Programming Lecture 13: FP in the Real World Viliam Lisý Artificial Intelligence Center Department of Computer Science FEE, Czech Technical University in Prague [email protected] 1 Mixed paradigm languages Functional programming is great easy parallelism and concurrency referential transparency, encapsulation compact declarative code Imperative programming is great more convenient I/O better performance in certain tasks There is no reason not to combine paradigms 2 3 Source: Wikipedia 4 Scala Quite popular with industry Multi-paradigm language • simple parallelism/concurrency • able to build enterprise solutions Runs on JVM 5 Scala vs. Haskell • Adam Szlachta's slides 6 Is Java 8 a Functional Language? Based on: https://jlordiales.me/2014/11/01/overview-java-8/ Functional language first class functions higher order functions pure functions (referential transparency) recursion closures currying and partial application 7 First class functions Previously, you could pass only classes in Java File[] directories = new File(".").listFiles(new FileFilter() { @Override public boolean accept(File pathname) { return pathname.isDirectory(); } }); Java 8 has the concept of method reference File[] directories = new File(".").listFiles(File::isDirectory); 8 Lambdas Sometimes we want a single-purpose function File[] csvFiles = new File(".").listFiles(new FileFilter() { @Override public boolean accept(File pathname) { return pathname.getAbsolutePath().endsWith("csv"); } }); Java 8 has lambda functions for that File[] csvFiles = new File(".")