DOCSLIB.ORG

Comprehending Monads

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Comprehending Monads Comprehending Monads Philip Wadler University of Glasgow Abstract Category theorists invented monads in the s to concisely express certain asp ects of universal algebra Functional programmers invented list comprehensions in the s to concisely express certain programs involving lists This pap er shows how list comprehensions may b e generalised to an arbitrary monad and howthe resulting programming feature can concisely express in a pure functional language some programs that manipulate state handle exceptions parse text or invokecon tinuations A new solution to the old problem of destructive array up date is also presented No knowledge of category theory is assumed Intro duction Is there a waytocombine the indulgences of impurity with the blessings of purity Impure strict functional languages such as Standard ML Mil HMT and Scheme RC supp ort a wide variety of features such as assigning to state handling exceptions and invoking continuations Pure lazy functional languages suchasHaskell HPW or Miranda Tur eschew such features b ecause they are incompatible with the advan tages of lazy evaluation and equational reasoning advantages that have b een describ ed at length elsewhere Hug BW Purity has its regrets and all programmers in pure functional languages will recall some moment when an impure feature has tempted them For instance if a counter is required to generate unique names then an assignable variable seems just the ticket In such cases it is always p ossible to mimic the required impure feature by straightforward or instance a counter can b e simulated by mo difying the relevant though tedious means F functions to accept an additional parameter the counters currentvalue and return an additional result the counters up dated value Miranda is a trademark of ResearchSoftware Limited Authors address Department of Computing Science University of Glasgow G QQ Scotland Elec tronic mail wadlercsglasgowacuk This pap er app eared in Mathematical Structures in Computer Science volume pp copy right Cambridge University Press This version corrects a few small errors in the published version An earlier version app eared in ACM ConferenceonLispandFunctional Programming Nice June This pap er describ es a new metho d for structuring pure programs that mimic impure features This metho d do es not completely eliminate the tension b etween purity and impurity but it do es relax it a little bit It increases the readability of the resulting programs and it eliminates the p ossibility of certain silly errors that might otherwise arise such as accidentally passing the wrong value for the counter parameter The inspiration for this technique comes from the work of Eugenio Moggi Moga Mogb His goal was to provide a way of structuring the semantic description of features such as state exceptions and continuations His discovery was that the notion of a monad from category theory suits this purp ose By dening an interpretation of calculus in an arbitrary monad he provided a framework that could describ e all these features and more It is relatively straightforward to adopt Moggis technique of structuring denotational sp ecications into a technique for structuring functional programs This pap er presents a simplied version of Moggis ideas framed in a way b etter suited to functional program mers than semanticists in particular no knowledge of category theory is assumed The pap er contains two signicant new contributions The rst contribution is a new language feature the monad comprehension This generalises the familiar notion of list comprehension Wad due originally to Burstall and Darlington and found in KRCTur Miranda Haskell and other languages Monad comprehensions are not essential to the structuring technique describ ed here but they do provide a pleasant syntax for expressing programs structured in this way The second contribution is a new solution to the old problem of destructive array een up date The solution consists of two abstract data typ es with ten op erations b etw them Under this approach the usual typing discipline eg HindleyMilner extended with abstract data typ es is sucient to guarantee that arrayupdatemay safely b e implemented byoverwriting Tomyknowledge this solution has never b een prop osed b efore and its discovery comes as a surprise considering the plethora of more elab orate solutions that have b een prop osed these include syntactic restrictions Sch runtime checks Hol abstract interpretation Huda Hudb Blo and exotic typ e systems GH Wad Wad That monads led to the discovery of this solution must countas a p oint in their favour Why has this solution not b een discovered b efore One likely reason is that the data typ es involve higherorder functions in an essential way The usual axiomatisation of arrays involves only rstorder functions index update andnewarray as describ ed in Section and so apparentlyitdidnotoccurtoanyone to search for an abstract data typ e based on higherorder functions Incidentally the higherorder nature of the solution means that it cannot b e applied in rstorder languages such as Prolog or OBJ It also casts doubt on Goguens thesis that rstorder languages are sucient for most purp oses Gog Monads and monad comprehensions help to clarify and unify some previous prop osals for incorp orating various features into functional languages exceptions Wad Spi parsers Wad Fai FL and nondeterminism HO In particular Spiveys work Spi is notable for p ointing out indep endently of Moggi that monads provide a frame work for exception handling There is a translation scheme from calculus into an arbitrary monad Indeed there are twoschemes one yielding callbyvalue semantics and one yielding callbyname These can b e used to systematically transform languages with state exceptions continu ations or other features into a pure functional language Two applications are given One is to derive callbyvalue and callbyname interpretations for a simple nondeterministic language this ts the work of Hughes and ODonnell HO into the more general frame work given here The other is to apply the callbyvalue scheme in the monad of continu ations the result is the familiar continuationpassing style transformation It remains an op en question whether there is a translation scheme that corresp onds to callbyneed as opp osed to callbyname Akey feature of the monad approach is the use of typ es to indicate what parts of a program mayhave what sorts of eects In this it is similar in spirit to Giord and Lucassens eect systems GL The examples in this pap er are based on Haskell HPW though any lazy functional language incorp orating the HindleyMilner typ e system would work as well The remainder of this pap er is organised as follows Section uses list comprehensions to motivate the concept of a monad and intro duces monad comprehensions Section shows that variable binding as in let terms and control of evaluation order can b e mo delled bytwo trivial monads Section explores the use of monads to structure pro grams that manipulate state and presents the new solution to the array up date problem Two examples are considered renaming b ound variables and interpreting a simple im p erative language Section extends monad comprehensions to include lters Section intro duces the concept of monad morphism and gives a simple pro of of the equivalence of two programs Section catalogues three more monads parsers exceptions and continu ations Section giv es the translation schemes for interpreting calculus in an arbitrary monad Two examples are considered giving a semantics to a nondeterministic language and deriving continuationpassing style Comprehensions and monads Lists Let us write Mx for the data typ e of lists with elements of typ e x In Haskell this is usually written x For example MInt and a b cM Char We write map for the higherorder function that applies a function to eachelment of a list map x y Mx My In Haskell typ e variables are written with small letters eg x and y and typ e construc tors are written with capital letters eg M For example if code Char Int maps a character to its ASCI I co de then map code a b c Observethat i map id id ii map g f map g map f Here id is the identity function id x x and g f is function comp osition g f x g fx In category theory the notions of type and function are generalised to object and arrow An op erator M taking each ob ject x into an ob ject Mx combined with an op erator map taking eacharrow f x y into an arrow map f Mx My and satisfying i andii is called a functor Categorists prefer to use the same symb ol for b oth op erators and so would write Mf where we write map f The function unit converts a value into a singleton lists and the function join con catenates a list of lists into a list unit x Mx Mx Mx join M For example unit and join Observe that iii map f unit unit f iv map f join join map map f Laws iii and iv may b e derived by a systematic transformation of the p olymorphic typ es of unit and join The idea of deriving laws from typ es go es by the slogan theorems for free Wad and is a consequence of Reynolds abstraction theorem for p olymorphic lamb da calculus Rey In categorical terms unit and join are natural transformations Rather than treat unit as a single function with a p olymorphic typ e categorists treat it as a family of arrows unit unit f for anyobjects unit x Mx one for each ob ject x satisfying map f x y x x and y and any arrow f x y between them They treat join similarly
Load more

Recommended publications
  • Ambiguity and Incomplete Information in Categorical Models of Language
    Ambiguity and Incomplete Information in Categorical Models of Language Dan Marsden University of Oxford [email protected] We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied in [27, 26, 16] using Selinger’s CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn’t seem to provide a satis- factory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic information. In this work we adopt a different approach. We analyze different forms of ambiguous and in- complete information, both with and without quantitative probabilistic data. Each scheme then cor- responds to a suitable enrichment of the category in which we model language. We view different monads as encapsulating the informational behaviour of interest, by analogy with their use in mod- elling side effects in computation. Previous results of Jacobs then allow us to systematically construct suitable bases for enrichment. We show that we can freely enrich arbitrary dagger compact closed categories in order to capture all the phenomena of interest, whilst retaining the important dagger compact closed structure. This allows us to construct a model with real convex combination of binary relations that makes non-trivial use of the scalars. Finally we relate our various different enrichments, showing that finite subconvex algebra enrichment covers all the effects under consideration.
    [Show full text]
  • Comprehending Adts*
    Comprehending ADTs* Martin Erwig School of EECS Oregon State University [email protected] Abstract We show how to generalize list comprehensions to work on abstract data types. First, we make comprehension notation automatically available for any data type that is specified as a constructor/destructor-pair (bialgebra). Second, we extend comprehensions to enable the use of dif- ferent types in one comprehension and to allow to map between different types. Third, we refine the translation of comprehensions to give a reasonable behavior even for types that do not obey the monad laws, which significantly extends the scope of comprehensions. Keywords: Functional Programming, Comprehension Syntax, Abstract Data Type, Generalized Fold, Monad 1 Introduction Set comprehensions are a well-known and appreciated notation from mathematics: an expression such as fE(x) j x 2 S; P(x)g denotes the set of values given by repeatedly evaluating the expression E for all x taken from the set S for which P(x) yields true. This notation has found its way into functional languages as list comprehensions [3, 17, 18]. Assuming that s denotes a list of numbers, the following Haskell expression computes squares of the odd numbers in s. [ x ∗ x j x s; odd x ] The main difference to the mathematical set comprehensions is the use of lists in the generator and as a result. In general, a comprehension consists of an expression defining the result values and of a list of qualifiers, which are either generators for supplying variable bindings or filters restricting these. Phil Wadler has shown that the comprehension syntax can be used not only for lists and sets, but more generally for arbitrary monads [19].
    [Show full text]
  • Structure Interpretation of Text Formats
    1 1 Structure Interpretation of Text Formats 2 3 SUMIT GULWANI, Microsoft, USA 4 VU LE, Microsoft, USA 5 6 ARJUN RADHAKRISHNA, Microsoft, USA 7 IVAN RADIČEK, Microsoft, Austria 8 MOHAMMAD RAZA, Microsoft, USA 9 Data repositories often consist of text files in a wide variety of standard formats, ad-hoc formats, aswell 10 mixtures of formats where data in one format is embedded into a different format. It is therefore a significant 11 challenge to parse these files into a structured tabular form, which is important to enable any downstream 12 data processing. 13 We present Unravel, an extensible framework for structure interpretation of ad-hoc formats. Unravel 14 can automatically, with no user input, extract tabular data from a diverse range of standard, ad-hoc and 15 mixed format files. The framework is also easily extensible to add support for previously unseen formats, 16 and also supports interactivity from the user in terms of examples to guide the system when specialized data extraction is desired. Our key insight is to allow arbitrary combination of extraction and parsing techniques 17 through a concept called partial structures. Partial structures act as a common language through which the file 18 structure can be shared and refined by different techniques. This makes Unravel more powerful than applying 19 the individual techniques in parallel or sequentially. Further, with this rule-based extensible approach, we 20 introduce the novel notion of re-interpretation where the variety of techniques supported by our system can 21 be exploited to improve accuracy while optimizing for particular quality measures or restricted environments.
    [Show full text]
  • The Marriage of Effects and Monads
    The Marriage of Effects and Monads PHILIP WADLER Avaya Labs and PETER THIEMANN Universit¨at Freiburg Gifford and others proposed an effect typing discipline to delimit the scope of computational ef- fects within a program, while Moggi and others proposed monads for much the same purpose. Here we marry effects to monads, uniting two previously separate lines of research. In partic- ular, we show that the type, region, and effect system of Talpin and Jouvelot carries over di- rectly to an analogous system for monads, including a type and effect reconstruction algorithm. The same technique should allow one to transpose any effect system into a corresponding monad system. Categories and Subject Descriptors: D.3.1 [Programming Languages]: Formal Definitions and Theory; F.3.2 [Logics and Meanings of Programs]: Semantics of Programming Languages— Operational semantics General Terms: Languages, Theory Additional Key Words and Phrases: Monad, effect, type, region, type reconstruction 1. INTRODUCTION Computational effects, such as state or continuations, are powerful medicine. If taken as directed they may cure a nasty bug, but one must be wary of the side effects. For this reason, many researchers in computing seek to exploit the benefits of computational effects while delimiting their scope. Two such lines of research are the effect typing discipline, proposed by Gifford and Lucassen [Gifford and Lucassen 1986; Lucassen 1987], and pursued by Talpin and Jouvelot [Talpin and Jouvelot 1992, 1994; Talpin 1993] among others, and the use of monads, proposed by Moggi [1989, 1990], and pursued by Wadler [1990, 1992, 1993, 1995] among others. Effect systems are typically found in strict languages, such as FX [Gifford et al.
    [Show full text]
  • Lecture 10. Functors and Monads Functional Programming
    Lecture 10. Functors and monads Functional Programming [Faculty of Science Information and Computing Sciences] 0 Goals I Understand the concept of higher-kinded abstraction I Introduce two common patterns: functors and monads I Simplify code with monads Chapter 12 from Hutton’s book, except 12.2 [Faculty of Science Information and Computing Sciences] 1 Functors [Faculty of Science Information and Computing Sciences] 2 Map over lists map f xs applies f over all the elements of the list xs map :: (a -> b) -> [a] -> [b] map _ [] = [] map f (x:xs) = f x : map f xs > map (+1)[1,2,3] [2,3,4] > map even [1,2,3] [False,True,False] [Faculty of Science Information and Computing Sciences] 3 mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b [Faculty of Science Information and Computing Sciences] 4 Map over binary trees Remember binary trees with data in the inner nodes: data Tree a = Leaf | Node (Tree a) a (Tree a) deriving Show They admit a similar map operation: mapTree :: (a -> b) -> Tree a -> Tree b mapTree _ Leaf = Leaf mapTree f (Node l x r) = Node (mapTree f l) (f x) (mapTree f r) [Faculty of Science Information and Computing Sciences] 4 Map over binary trees mapTree also applies a function over all elements, but now contained in a binary tree > t = Node (Node Leaf 1 Leaf) 2 Leaf > mapTree (+1) t Node
    [Show full text]
  • Coffeescript Accelerated Javascript Development.Pdf
    Download from Wow! eBook <www.wowebook.com> What readers are saying about CoffeeScript: Accelerated JavaScript Development It’s hard to imagine a new web application today that doesn’t make heavy use of JavaScript, but if you’re used to something like Ruby, it feels like a significant step down to deal with JavaScript, more of a chore than a joy. Enter CoffeeScript: a pre-compiler that removes all the unnecessary verbosity of JavaScript and simply makes it a pleasure to write and read. Go, go, Coffee! This book is a great introduction to the world of CoffeeScript. ➤ David Heinemeier Hansson Creator, Rails Just like CoffeeScript itself, Trevor gets straight to the point and shows you the benefits of CoffeeScript and how to write concise, clear CoffeeScript code. ➤ Scott Leberknight Chief Architect, Near Infinity Though CoffeeScript is a new language, you can already find it almost everywhere. This book will show you just how powerful and fun CoffeeScript can be. ➤ Stan Angeloff Managing Director, PSP WebTech Bulgaria Download from Wow! eBook <www.wowebook.com> This book helps readers become better JavaScripters in the process of learning CoffeeScript. What’s more, it’s a blast to read, especially if you are new to Coffee- Script and ready to learn. ➤ Brendan Eich Creator, JavaScript CoffeeScript may turn out to be one of the great innovations in web application development; since I first discovered it, I’ve never had to write a line of pure JavaScript. I hope the readers of this wonderful book will be able to say the same. ➤ Dr. Nic Williams CEO/Founder, Mocra CoffeeScript: Accelerated JavaScript Development is an excellent guide to Coffee- Script from one of the community’s most esteemed members.
    [Show full text]
  • 1. Language of Operads 2. Operads As Monads
    OPERADS, APPROXIMATION, AND RECOGNITION MAXIMILIEN PEROUX´ All missing details can be found in [May72]. 1. Language of operads Let S = (S; ⊗; I) be a (closed) symmetric monoidal category. Definition 1.1. An operad O in S is a collection of object fO(j)gj≥0 in S endowed with : • a right-action of the symmetric group Σj on O(j) for each j, such that O(0) = I; • a unit map I ! O(1) in S; • composition operations that are morphisms in S : γ : O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) −! O(j1 + ··· + jk); defined for each k ≥ 0, j1; : : : ; jk ≥ 0, satisfying natural equivariance, unit and associativity relations. 0 0 A morphism of operads : O ! O is a sequence j : O(j) ! O (j) of Σj-equivariant morphisms in S compatible with the unit map and γ. Example 1.2. ⊗j Let X be an object in S. The endomorphism operad EndX is defined to be EndX (j) = HomS(X ;X), ⊗j with unit idX , and the Σj-right action is induced by permuting on X . Example 1.3. Define Assoc(j) = ` , the associative operad, where the maps γ are defined by equivariance. Let σ2Σj I Com(j) = I, the commutative operad, where γ are the canonical isomorphisms. Definition 1.4. Let O be an operad in S. An O-algebra (X; θ) in S is an object X together with a morphism of operads ⊗j θ : O ! EndX . Using adjoints, this is equivalent to a sequence of maps θj : O(j) ⊗ X ! X such that they are associative, unital and equivariant.
    [Show full text]
  • Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl
    Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl To cite this version: Wolfram Kahl. Categories of Coalgebras with Monadic Homomorphisms. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. pp.151-167, 10.1007/978-3-662-44124-4_9. hal-01408758 HAL Id: hal-01408758 https://hal.inria.fr/hal-01408758 Submitted on 5 Dec 2016 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. Distributed under a Creative Commons Attribution| 4.0 International License Categories of Coalgebras with Monadic Homomorphisms Wolfram Kahl McMaster University, Hamilton, Ontario, Canada, [email protected] Abstract. Abstract graph transformation approaches traditionally con- sider graph structures as algebras over signatures where all function sym- bols are unary. Attributed graphs, with attributes taken from (term) algebras over ar- bitrary signatures do not fit directly into this kind of transformation ap- proach, since algebras containing function symbols taking two or more arguments do not allow component-wise construction of pushouts. We show how shifting from the algebraic view to a coalgebraic view of graph structures opens up additional flexibility, and enables treat- ing term algebras over arbitrary signatures in essentially the same way as unstructured label sets.
    [Show full text]
  • Algebraic Structures Lecture 18 Thursday, April 4, 2019 1 Type
    Harvard School of Engineering and Applied Sciences — CS 152: Programming Languages Algebraic structures Lecture 18 Thursday, April 4, 2019 In abstract algebra, algebraic structures are defined by a set of elements and operations on those ele- ments that satisfy certain laws. Some of these algebraic structures have interesting and useful computa- tional interpretations. In this lecture we will consider several algebraic structures (monoids, functors, and monads), and consider the computational patterns that these algebraic structures capture. We will look at Haskell, a functional programming language named after Haskell Curry, which provides support for defin- ing and using such algebraic structures. Indeed, monads are central to practical programming in Haskell. First, however, we consider type constructors, and see two new type constructors. 1 Type constructors A type constructor allows us to create new types from existing types. We have already seen several different type constructors, including product types, sum types, reference types, and parametric types. The product type constructor × takes existing types τ1 and τ2 and constructs the product type τ1 × τ2 from them. Similarly, the sum type constructor + takes existing types τ1 and τ2 and constructs the product type τ1 + τ2 from them. We will briefly introduce list types and option types as more examples of type constructors. 1.1 Lists A list type τ list is the type of lists with elements of type τ. We write [] for the empty list, and v1 :: v2 for the list that contains value v1 as the first element, and v2 is the rest of the list. We also provide a way to check whether a list is empty (isempty? e) and to get the head and the tail of a list (head e and tail e).
    [Show full text]
  • Pattern Matching
    Functional Programming Steven Lau March 2015 before function programming... https://www.youtube.com/watch?v=92WHN-pAFCs Models of computation ● Turing machine ○ invented by Alan Turing in 1936 ● Lambda calculus ○ invented by Alonzo Church in 1930 ● more... Turing machine ● A machine operates on an infinite tape (memory) and execute a program stored ● It may ○ read a symbol ○ write a symbol ○ move to the left cell ○ move to the right cell ○ change the machine’s state ○ halt Turing machine Have some fun http://www.google.com/logos/2012/turing-doodle-static.html http://www.ioi2012.org/wp-content/uploads/2011/12/Odometer.pdf http://wcipeg.com/problems/desc/ioi1211 Turing machine incrementer state symbol action next_state _____ state 0 __1__ state 1 0 _ or 0 write 1 1 _10__ state 2 __1__ state 1 0 1 write 0 2 _10__ state 0 __1__ state 0 _00__ state 2 1 _ left 0 __0__ state 2 _00__ state 0 __0__ state 0 1 0 or 1 right 1 100__ state 1 _10__ state 1 2 0 left 0 100__ state 1 _10__ state 1 100__ state 1 _10__ state 1 100__ state 1 _10__ state 0 100__ state 0 _11__ state 1 101__ state 1 _11__ state 1 101__ state 1 _11__ state 0 101__ state 0 λ-calculus Beware! ● think mathematical, not C++/Pascal ● (parentheses) are for grouping ● variables cannot be mutated ○ x = 1 OK ○ x = 2 NO ○ x = x + 1 NO λ-calculus Simplification 1 of 2: ● Only anonymous functions are used ○ f(x) = x2+1 f(1) = 12+1 = 2 is written as ○ (λx.x2+1)(1) = 12+1 = 2 note that f = λx.x2+1 λ-calculus Simplification 2 of 2: ● Only unary functions are used ○ a binary function can be written as a unary function that return another unary function ○ (λ(x,y).x+y)(1,2) = 1+2 = 3 is written as [(λx.(λy.x+y))(1)](2) = [(λy.1+y)](2) = 1+2 = 3 ○ this technique is known as Currying Haskell Curry λ-calculus ● A lambda term has 3 forms: ○ x ○ λx.A ○ AB where x is a variable, A and B are lambda terms.
    [Show full text]
  • A Critique of Abelson and Sussman Why Calculating Is Better Than
    A critique of Abelson and Sussman - or - Why calculating is better than scheming Philip Wadler Programming Research Group 11 Keble Road Oxford, OX1 3QD Abelson and Sussman is taught [Abelson and Sussman 1985a, b]. Instead of emphasizing a particular programming language, they emphasize standard engineering techniques as they apply to programming. Still, their textbook is intimately tied to the Scheme dialect of Lisp. I believe that the same approach used in their text, if applied to a language such as KRC or Miranda, would result in an even better introduction to programming as an engineering discipline. My belief has strengthened as my experience in teaching with Scheme and with KRC has increased. - This paper contrasts teaching in Scheme to teaching in KRC and Miranda, particularly with reference to Abelson and Sussman's text. Scheme is a "~dly-scoped dialect of Lisp [Steele and Sussman 19781; languages in a similar style are T [Rees and Adams 19821 and Common Lisp [Steele 19821. KRC is a functional language in an equational style [Turner 19811; its successor is Miranda [Turner 1985k languages in a similar style are SASL [Turner 1976, Richards 19841 LML [Augustsson 19841, and Orwell [Wadler 1984bl. (Only readers who know that KRC stands for "Kent Recursive Calculator" will have understood the title of this There are four language features absent in Scheme and present in KRC/Miranda that are important: 1. Pattern-matching. 2. A syntax close to traditional mathematical notation. 3. A static type discipline and user-defined types. 4. Lazy evaluation. KRC and SASL do not have a type discipline, so point 3 applies only to Miranda, Philip Wadhr Why Calculating is Better than Scheming 2 b LML,and Orwell.
    [Show full text]
  • Richard S. Uhler
    Smten and the Art of Satisfiability-Based Search by Richard S. Uhler B.S., University of California, Los Angeles (2008) S.M., Massachusetts Institute of Technology (2010) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2014 c Massachusetts Institute of Technology 2014. All rights reserved. Author.............................................................. Department of Electrical Engineering and Computer Science August 22, 2014 Certified by. Jack B. Dennis Professor of Computer Science and Engineering Emeritus Thesis Supervisor Accepted by . Leslie A. Kolodziejski Chairman, Committee for Graduate Students 2 Smten and the Art of Satisfiability-Based Search by Richard S. Uhler Submitted to the Department of Electrical Engineering and Computer Science on August 22, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract Satisfiability (SAT) and Satisfiability Modulo Theories (SMT) have been leveraged in solving a wide variety of important and challenging combinatorial search prob- lems, including automatic test generation, logic synthesis, model checking, program synthesis, and software verification. Though in principle SAT and SMT solvers sim- plify the task of developing practical solutions to these hard combinatorial search problems, in practice developing an application that effectively uses a SAT or SMT solver can be a daunting task. SAT and SMT solvers have limited support, if any, for queries described using the wide array of features developers rely on for describing their programs: procedure calls, loops, recursion, user-defined data types, recursively defined data types, and other mechanisms for abstraction, encapsulation, and modu- larity.
    [Show full text]