Mathematics – The missing ingredient in modern software development

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Mathematics The missing ingredient in modern software development Rahul Goma Phulore (@missingfaktor) 2014.11.22 XConf Bengaluru, 2014

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Software Development Today —"Agile" —"TDD" —"Object oriented" —"Composition over inheritance" —"Too many private methods is a smell"

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Throw away your buzzwords and blanket statements, and use some math!

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There’s something about the culture of software that has impeded the use of speciﬁcation. We have a wonderful way of describing things precisely that’s been developed over the last couple of millennia, called mathematics. I think that’s what we should be using as a way of thinking about what we build. -- Leslie Lamport

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What comes to your mind when I say the word "mathematics"?

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XConf Poster (See those numbers on the tree?)

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But...

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math·e·mat·ics (ˌmæθəˈmætɪks) —Mathematicians seek out patterns and use them to formulate new conjectures.

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math·e·mat·ics (ˌmæθəˈmætɪks) —Mathematicians seek out patterns and use them to formulate new conjectures. —Mathematicians resolve the truth or falsity of conjectures by mathematical proof. —When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature.

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A Motivating Example

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OpenSSL Heartbleed

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Even these rookie crackers exploited it!

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OpenSSL Heartbleed

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Language used for the example: ATS

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Why do we test our software?

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We test because: —We want our software to be reliable and correct.

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We test because: —We want our software to be reliable and correct. —Correctness matters.

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We test because: —We want our software to be reliable and correct. —Correctness matters. —It's about basic professionalism.

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What if I told you that there are better ways to verify your software?

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Formal Methods In computer science, formal methods are a particular kind of mathematically based techniques for the speciﬁcation, development and veriﬁcation of software and hardware systems.

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Type Systems A type system is a syntactic method for automatically checking the absence of certain erroneous behaviors by classifying program phrases according to the kinds of values they compute.

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What types are not —Classes —Runtime tags

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What types are —Types categorize terms, based on their properties and the kind of operations they can support —Types encode invariants/knowledge about your program —Intuition: sets

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Curry-Howard Correspondence —Types ≅ Propositions —Programs ≅ Proofs

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Not all type systems are created equal

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Java ≈ Bat

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Exhibit A

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NullPointerException!

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NullPointerException - Diagnosis null ∈ A

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NullPointerException - Diagnosis f :: A -> B —f can accept a non-null A and return a non-null B. —f can accept a null A and return a non-null B. —f can accept a non-null A and return null B. —f can accept a null A and return a null B.

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NullPointerException - Diagnosis —Check every single reference for null? Madness! —Have to rely on documentation or tribal knowledge.

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NullPointerException - Broken solutions —Null object pattern —Elvis operator (?:) —Safe navigation operator (?.)

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NullPointerException - Broken solutions

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NullPointerException - Let's apply some math! —Ditch null.

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NullPointerException - Let's apply some math! —Capture the algebra of "nullity". —Algebraic data types. data Option a = Some a | None -- inhabitants(Option a) = inhabitants(a) + 1

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NullPointerException - Let's apply some math! f :: A -> B —f can accept an A and return a B.

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NullPointerException - Let's apply some math! f :: A -> Option B —f can accept an A and optionally return a B.

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NullPointerException - Let's apply some math! —No way to mistake an A for an Option A!

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Algebraic data types + First class functions —Help in capturing constraints precisely. —Constraints propagate through program. —Patterns emerge easily. —Lead to algebraic patterns such as functors, applicatives, monads. —Better composability. Extremely high degree of code reuse.

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We have barely scratched the surface! —Dependent types —Theorem provers —Substructural types —Eﬀects —Coeﬀects —Region systems —Sequent calculus

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When you don't have Math vs When you have Math —Callbacks - Monadic futures —Go error handling - Monadic error handling (Either) —AspectJ - Higher order functions and combinators —Spring DI - ML modules, Reader —Polymer - Elm

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Other Beneﬁts Excellent design tool (See my TyDD presentation)

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Other Beneﬁts Discoverability - Hoogle

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Other Beneﬁts Equational reasoning

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Other Beneﬁts Amazingly capable tools

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Applications Reactive Extensions (Rx)

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Applications Games

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Applications Systems programming

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Applications Functional reactive programming (FRP)

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Applications Formally veriﬁed OS kernel - seL4

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Applications Facebook's Hack and FlowType

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Does this replace testing? No.

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Does this replace testing? 1 + 1 = 2 proof

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Does this replace testing? —Formalization has a cost; not always justiﬁed. You need to draw a line. —You can get a lot of mileage from the type systems of Scala, Haskell, F#. And even C# and Java. —Tests to cover what hasn't been formally veriﬁed.

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Thank You!

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References —See the source markdown. Credits —Many giants on whose shoulders us mortals stand. —Rahul Kavale, Shripad Agashe, and Priti Biyani for ideas and early feedback.