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Introduction to Coalgebra Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Introduction to Coalgebra Bart Jacobs Institute for Computing and Information Sciences { Digital Security Radboud University Nijmegen EWSCS 2011: 16th Estonian Winter School in Computer Science 28 feb. { 4 march, Estonia EWSCS'11 Bart Jacobs EWSCS'11 Coalgebra Intro 1 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Outline Basics Induction and coinduction Algebras and coalgebras Bisimilarity and finality Language example Coalgebras and quantum computing Introducing QBits Combining qubits Monads for computations Walks illustrating computation types Quantum walks, coalgebraically Reversibility of computation Bart Jacobs EWSCS'11 Coalgebra Intro 2 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing What are coalgebras, intuitively • Mathematical models for state-based computation • your computer has a complicated internal state • you as user can only modify it to some extend • also, you can observe only so much • Basically only two kinds of operations: • move to a next state, somehow (deterministically, non-deterministically, probabilistically, . ) • make an observation (\measurement") about the current state • These operations can be combined (\observation has side-effect”) • Describing such state-based systems and their dynamics requires a new kind of mathematics. • related to dynamical systems (and evolution via differential equations) Bart Jacobs EWSCS'11 Coalgebra Intro 4 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing (Co)algebras, pictorially Algebras with \carrier" X are maps into X , of the form: X ··· X construct / X Coalgebras with \state space" X are maps out of X , of the form: X / X ··· X modify/observe Think of a fields & methods in an object-oriented class • Formally, the boxes are functors, usually Sets ! Sets,... • . but at this stage you may think of them as types σ(X ) containing a single type variable X Bart Jacobs EWSCS'11 Coalgebra Intro 5 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Intermezzo: type constructors for \boxes" • Identity X , and constants A • including empty set ;, often written as 0 • singleton set 1 = {∗} • products X × Y = fhx; yi j x 2 X ; y 2 Y g, with projections: hf ;gi Z −−−! X × Y π1 π2 X o X × Y / Y =============== Z −! XZ −! Y f g Q • infinite version usually written as i2I Xi • coproducts X + Y = fhx; 0i j x 2 X g [ fhy; 1i j y 2 Y g, with coprojections [f ;g] X + Y −−−! Z κ1 κ2 X / X + Y o Y =============== X −! ZY −! Z f g ` • infinite version written as i2I Xi Bart Jacobs EWSCS'11 Coalgebra Intro 6 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Algebra examples F (X ) ! X • Natural numbers [zero;succ] zero: 1 −! N 1 + N / N i.e. succ: N −! N The functor/type here is F (X ) = 1 + X . • Groups 8 0: 1 −! G [0;+;−] < 1 + (G × G) + G / G i.e. +: G × G −! G : −: G −! G • Clearly this works for many algebraic structures: for each signature Σ there is a functor: ` arity(f ) X 7−! f 2Σ X (disjoint union over all operations) whose algebras are precisely the Σ-algebras. Note: this captures algebraic operations, not equations ¨¨ Bart Jacobs EWSCS'11 Coalgebra Intro ©© 7 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Coalgebra examples X ! F (X ) • Streams (infinite lists) AN of elements of a set A: hhead;taili AN / A × AN for the functor F (X ) = A × X , where: head(α) = α(0) 0 tail(α) = α = λn 2 N: α(n + 1) • Both finite and infinite lists A1 = A? + AN possible head 1 A1 / 1 + A × A for the functor F (X ) = 1 + A × X . Note: observation with side-effect Bart Jacobs EWSCS'11 Coalgebra Intro 8 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Automata-theoretic coalgebra examples • A deterministic automaton with input actions A is a coalgebra: hstep;final?i X / X A × 2 where 2 = f0; 1g. • A non-deterministic automaton, or transition system, with input actions A is a coalgebra: succs X / P(X )A • Other functors (actually \monads") than powerset P may be used: eg. partial automata via lift, or probabilistic automata via distribution. Bart Jacobs EWSCS'11 Coalgebra Intro 9 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Markov chains For a set X , define the set of discrete distributions on X P D(X ) = f': X ! [0; 1] support(') is finite, and x '(x) = 1g Such ' 2 D(X ) is a formal convex combination: 8 < support(') = fx1;:::; xng r1x1 + ··· + rnxn where ri = '(xi ) > 0 : r1 + ··· + rn = 1 Coalgebras X !D(X ) are Markov chains, giving probabilistic transitions: ri P x −! xi with i ri = 1: They can also be used in labeled form, as X −! D(X )A or as X −! D(A × X ) Bart Jacobs EWSCS'11 Coalgebra Intro 10 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Coalgebra essentials & Plan • Generic framework for describing state-based systems X • Coinductive definition and reasoning principles Lect. II, III • Bisimilarity, as indistinguishability of states Lect. III, IV • Modal logic, also generically • Coherence between operational & denotational semantics • General trace semantics • Current own interest: coagebra & quantum computing Lect. V • Quantum language has strong coalgebraic flavour: states, transitions, observations, indistinguishability, . • Can this be made mathematically precise? Bart Jacobs EWSCS'11 Coalgebra Intro 11 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Functors, part I • Before we can proceed we need to know a little bit more about functors, for the time being only on sets • In functional programming \functoriality" is often described by \map" properties, such as list map • A functor F maps sets X to sets F (X ), and also functions to functions, in an orderly manner: • if g : X ! Y , then F (g): F (X ) ! F (Y ) id id • F preserves identity maps: F X −! X = F (X ) −! F (X ) • F preserves composition: F (h ◦ g) = F (h) ◦ F (g), in: g F (g) F (h) F X −! Y −!h Z = F (X ) −! F (Y ) −! F (Z) • Often, when the action on functions is \obvious", it is omitted from the description of a functor Bart Jacobs EWSCS'11 Coalgebra Intro 13 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Functor example I • For a set X write P(X ) for the powerset: the set of all subsets of X • This P(−) forms a functor • For a function g : X ! Y there is a suitable function P(X ) !P(Y ) • this is new function is written P(g): P(X ) !P(Y ) • it is given by image: P(g) U ⊆ X = fg(x) j x 2 Ug ⊆ Y = fy 2 Y j 9x 2 U: y = g(x)g = g(U) as it is sometimes written = g[U] also possible ` = g (U) for categorical logicians Bart Jacobs EWSCS'11 Coalgebra Intro 14 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Functor example I We check explicitly: P(id)(U) = fid(x) j x 2 Ug = U Hence P(id) = id. Similarly, P preserves composition: P(h) ◦ P(g) (U) = P(h) P(g)(U) = P(h) fy 2 Y j 9x 2 U: y = g(x)g = fz 2 Z j 9y 2 Y : 9x 2 U: y = g(x) and z = h(y)g = fz 2 Z j 9x 2 U: z = h(g(x))g = P(h ◦ g)(U) Bart Jacobs EWSCS'11 Coalgebra Intro 15 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Functor example II • For a set X write X ? for the associated set of finite lists: ? X = fhx1; x2;:::; xni j xi 2 X g • The mapping X 7−! X ? is a functor • For g : X ! Y we have g ? : X ? ! Y ? given by: ? g hx1; x2;:::; xni = hg(x1); g(x2);:::; g(xn)i • It is not hard to check the functoriality properties: id? = id and (h ◦ g)? = h? ◦ g ? Bart Jacobs EWSCS'11 Coalgebra Intro 16 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Functor examples II • Infinite lists: X 7! X N with: g N : X N ! Y N, given by: N g hx0; x1; x2;:::; i = hg(x0); g(x1); g(x2);:::i • Product with a constant X 7! X × A, with: g × A: X × A −! Y × A given by (g × A)(x; a) = (g(x); a) • Coproduct with a constant X 7! X + A, with (g + A)(κ x) = κ g(x) (g+A): X +A −! Y +A given by 1 1 (g + A)(κ2a) = κ2a Bart Jacobs EWSCS'11 Coalgebra Intro 17 / 98 Basics Induction and coinduction Bisimilarity and finality Radboud University Nijmegen Coalgebras and quantum computing Definition Let F be a functor • An algebra for F is a map F (X ) ! X • A coalgebra for F is a map X ! F (X ) Definition a b • An algebra homomorphism from F (X ) ! X to F (Y ) ! Y is a map f : X ! Y for which the rectangle on the left commutes. F (f ) F (g) F (X ) / F (Y ) F (X ) / F (Y ) O O a b c d f g X / Y X / Y c d • A coalgebra homomorphism from X ! F (X ) to Y ! F (Y ) is a map g : X ! Y for which the diagram on the right commutes.
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