Synthesis of Mealy Machines Using DerivativesSynthesis of Mealy Machines Using Derivatives Helle...

Synthesis of Mealy Machines Using Derivatives

Helle Hvid Hansen1,2 David Costa2 Jan Rutten1,2

1Section Theorectical Computer ScienceVrije Universiteit Amsterdam (VUA)

2SEN3Centrum voor Wiskunde en Informatica (CWI)

March 25, 2006 / Coalgebraic Methods in Computer Science

Hansen, Costa, Rutten (VUA, CWI) Synthesis of Mealy Machines... CMCS 2006 1 / 25

Introduction

Introduction

Main reference:J. Rutten, Algebraic specification and coalgebraic synthesis of Mealyautomata, FACS 2005.

Mealy machines are coalgebras.Abstract implementation via final Mealy coalgebra.Symbolic computation of derivatives.

Our contributions:Implementation (Haskell),Decision procedure for deciding equivalence of expressions,Results on rational 2-adic stream functions

automaton size,complexity.


Introduction

Outline

1 Mealy Machines and Coalgebras

2 Synthesis Using Derivatives (I)Main Idea

3 Arithmetic Bitstream Specifications2-Adic Bitstream AlgebraMod-2 Bitstream Algebra

4 Synthesis Using Derivatives (II)Deciding Equivalence

5 Rational 2-Adic Functions

6 Mealy Synthesis Tool (incl. Demo)


Mealy Machines and Coalgebras Structures and Morphisms

Mealy Machines

Finite State Machine (FSM): states, I/O-transitions:

sa|b // t

In state s, reading input a, the machine produces an output b, andmoves to a new state t .Example:

b|b

a|a

b|aa|b


Mealy Machines and Coalgebras Structures and Morphisms

Mealy Coalgebras

Mealy coalgebra (input in A, output in B):Set-functor M(X ) = (B × X )A

δ : S −→ (B × S)A

s 7→ (os, ds)

o : S × A → B is the output function,d : S × A → Q is the next-state function.Mealy homomorphism h : (S, δ) → (U, ρ):

for all s, t ∈ S : sa|b // t iff h(s)

a|b // h(t) .

Mealy machine (S, δ, s0) = (finite) Mealy coalgebra + initial state.


Mealy Machines and Coalgebras Behaviour and Finality

Causal Behaviour

..., b, b, a ..., a, b, ba|b b|a

a|a

b|b

Behaviour of (S, δ, s0): deterministic transformation of inputstream to output stream (transducer).Behaviour of a state s0 in (S, δ): Beh(s0) : Aω → Bω.For σ = (a0, a1, a2, . . .) ∈ Aω, Beh(s0)(σ) = (b0, b1, b2, . . .) ∈ Bω

where

s0a0|b0 // s1

a1|b1 // . . .ak−1|bk−1// sk

ak |bk // sk+1 . . .

Beh(s0) : Aω → Bω is causal (bn depends only on a0, . . . , an).



Final Mealy Coalgebra

Let Γ = {f : Aω → Bω | f causal }.

Theorem[Rutten]: Γ carries a final Mealy coalgebra structure.

For f ∈ Γ define π(f ) = 〈of , df 〉 : A → B × Γ asinitial output of f (on input a):

of (a) := f [a] = f (a : σ)(0) = head f (a : σ)

(stream function) derivative of f (on input a):

df (a) := fa : Aω → Bω

σ 7→ f (a : σ)′ = tail f (a : σ)



Abstract Implementations

States s, t in (S, δ) are equivalent (s ∼ t) iff Beh(s) = Beh(t).Mealy machine (S, δ, s0) implements f : Aω → Bω iff Beh(s0) = f .

Generating Abstract Implementations

Let 〈f 〉 be the subcoalgebra generated by f in (Γ, π);i.e., f ∈ 〈f 〉 ⊆ Γ is minimal, derivative-closed.Then by finality of (Γ, π):

〈f 〉 implements f . (Because Beh(f ) = f .)〈f 〉 is a minimal-state implementation.If (S, s) implements f , then Beh[〈s〉] = 〈f 〉.


Synthesis Using Derivatives (I) Main Idea

Synthesis of Mealy Machines Using Derivatives

Basic Idea:Symbolic computation of 〈f 〉.

Specify causal stream functions algebraically (language L).Define output and derivatives of expressions (cf. Brzozowski).

L〈o,d〉 // (B × L)A.

θ ∈ L defines/specifies a function Beh(θ) = fθ : Aω → Bω.Compute representation of 〈fθ〉 as least fixpoint.

Questions:Specification language? Arithmetic bitstream algebra.Realisability: is 〈f 〉 finite? Rational bitstream functions.Equivalence: how to decide θ ∼ φ? Normal forms.


Arithmetic Bitstream Specifications

Outline










Semantic Domain:Bitstreams 2ω (i.e. A = B = 2 = {0, 1}),Bitstream functions f : 2ω → 2ω.

Arithmetic Bitstream Expressions:

Constants [0] = (0, 0, 0, 0, . . .),[1] = (1, 0, 0, 0, . . .),X = (0, 1, 0, 0, . . .),X n = (0, . . . , 0︸︷︷︸

n times

, 1, 0, 0, . . .);

Bitstream variable σ;Arithmetic operations: addition, multiplication, minus, division.

2-Adic: infinitary binary arithmetic.Mod-2: infinitary modulo-2 arithmetic.



Stream Behaviour of Expressions

Mimic semantics:

Recall in final Mealy coalgebra: of (a) = f (a : σ)(0)df (a) = f (a : σ)′

– Instantiating σ with bit a ∈ 2 (later),– Stream behaviour (head, tail).

Stream behaviour of constants:

initial value(head) derivative(tail)[0](0) = 0 [0]′ = [0][1](0) = 1 [1]′ = [0]X (0) = 0 X ′ = [1]

X n(0) = 0 (X n)′ = X n−1 , n ≥ 1, (X 0 = [1])


Arithmetic Bitstream Specifications 2-Adic Bitstream Algebra

2-Adic Bitstream Algebra

2ω as the 2-adic integers:

(a0, a1, a2, . . .) ∈ 2ω ! a0 + a12 + a222 + a323 + . . . = Σ∞i=0ai2i .

2-adic operations (signature Σ2adic = {+,×,−, /}):infinitary binary addition (carry-bits propagate indefinitely):

E.g. (1, 0, 0, . . .)+(1, 1, 1, . . .) = (0, 0, 0, . . .).

stream differential equation:

(α + β)(0) = α(0)⊕ β(0)(α + β)′ = (α′ + β′) + [α(0) ∧ β(0)]





(a0, a1, a2, . . .) ∈ 2ω ! a0 + a12 + a222 + a323 + . . . = Σ∞i=0ai2i .

2-adic operations (signature Σ2adic = {+,×,−, /}):Cauchy product:

(α×β)(n) = Σni=0α(i) ∧ β(n − i) (2-adic sum).


(α× β)(0) = α(0) ∧ β(0)(α× β)′ = α′ × β + [α(0)]× β′





(a0, a1, a2, . . .) ∈ 2ω ! a0 + a12 + a222 + a323 + . . . = Σ∞i=0ai2i .

2-adic operations (signature Σ2adic = {+,×,−, /}):infinitary two’s complement:

E.g. −(1, 0, 0, . . .) = (1, 1, 1, . . .).


(−α)(0) = α(0)(−α)′ = −(α′ + [α(0)])





(a0, a1, a2, . . .) ∈ 2ω ! a0 + a12 + a222 + a323 + . . . = Σ∞i=0ai2i .

2-adic operations (signature Σ2adic = {+,×,−, /}):multiplicative inverse: β × (1/β) = [1] (well-def’d if β(0) = 1).division:

α/β = α× (1/β)


(α/β)(0) = α(0)(α/β)′ = (α′ − [α(0)]× β′)/β





(a0, a1, a2, . . .) ∈ 2ω ! a0 + a12 + a222 + a323 + . . . = Σ∞i=0ai2i .

2-adic operations (signature Σ2adic = {+,×,−, /}):(2ω,+,×,−, /, [0], [1]) is integral domain,(i.e. commutative ring with no zero divisors).Since X = (0, 1, 0, 0, . . .)! 2, we have for α ∈ 2ω:

α + α ∼ X × α,X n + X n ∼ X n+1.


Arithmetic Bitstream Specifications Mod-2 Bitstream Algebra

Mod-2 Bitstream Algebra

2ω as formal power series over Mod2-ring (2,⊕,∧, id , 0, 1):

(a0, a1, a2, . . .) ∈ 2ω ! Σ∞n=0anxn





(a0, a1, a2, . . .) ∈ 2ω ! Σ∞n=0anxn

Mod-2 operations (signature Σmod2 = {⊕,⊗,,�}):bitwise mod-2 sum: (α⊕ β)(n) = α(n)⊕ β(n),cauchy product w.r.t. mod-2 sum.minus = id: α = α (since α⊕ α = (0, 0, 0, . . .)),multiplicative inverse: 1� α

fraction: α� β = α⊗ (1� β) (well-def’d when α(0) = 1).





(a0, a1, a2, . . .) ∈ 2ω ! Σ∞n=0anxn

Mod-2 operations (signature Σmod2 = {⊕,⊗,,�}):(2ω,⊕,⊗,,�, [0], [1]) is integral domain in whichα⊕ α ∼ [0] (nilpotent).operations definable using stream differential equations.


Synthesis Using Derivatives (II)

Outline








Synthesis Using Derivatives (II)

Compute output and derivative on input a ∈ 2:

Instantiating with bit a ∈ 2:

Recall in final Mealy coalgebra: of (a) = f (a : σ)(0)df (a) = f (a : σ)′

We have for all α ∈ 2ω:

0 : α = X × α = X ⊗ α1 : α = [1] + X × α = [1]⊕ X ⊗ α

For θ(σ) ∈ Σ2adic(σ):

oθ(0) = (θ[X × σ])(0) dθ(0) = (θ[X × σ])′

oθ(1) = (θ[[1] + X × σ])(0) dθ(1) = (θ[[1] + X × σ])′

Similarly, for θ ∈ Σmod2(σ).


Synthesis Using Derivatives (II) Deciding Equivalence

Deciding Equivalence of Expressions

Normal forms in integral domains:

Normal form of θ ∈ ExprΣ(σ) is NF(θ) = pq where p, q in distributive

polynomial normal form (PNF).

Reducing Constant Coefficients:2-Adic: Use numeric interpretation: signed, binary expansion. E.g.

X × ([1]− X 2) ! 2(1− 4) = −6NF (X × ([1]− X 2)) = −(X + X 2).

Mod-2: Use ring laws and nilpotency: E.g.

X ⊗ ([1] X 2)⊕ X = X X 3 ⊕ X = X 3.




Normal forms in integral domains:

Normal form of θ ∈ ExprΣ(σ) is NF(θ) = pq where p, q in distributive

polynomial normal form (PNF).

Example

θ =[1]

[1] + X+ σ + [1]

([1] + [1] + X ) + ([1] + X ) · σ[1] + X

normal form in 2-adic algebra:X 2 + ([1] + X )× σ

[1] + X

normal form in mod-2 algebra:X ⊕ ([1]⊕ X )⊗ σ

[1] + X




Equivalence in integral domains:

Given two expressions in normal form, θ1 = p1q1

and θ2 = p2q2

,

θ1 ∼ θ2 iff p1q1∼ p2

q2

iff PNF (p1 × q2) = PNF (p2 × q1)

Complexity:

computing PNF(θ) is exponential: 2O(|θ|).deciding θ ∼ φ is exponential: 2O(|θ|+|φ|).



Synthesis Example: θ(σ) = 6×σ9

θ(σ) =(X + X 2)× σ

[1] + X 3 !6× σ

9

θ = 6σ9 = (X+X 2)×(σ)

1+X 3

θ1 = 3+6σ9 = (1+X)+(X+X 2)×(σ)

1+X 3

θ10 = −3+6σ9 = −(1+X)+(X+X 2)×(σ)

1+X 3

θ100 = −6+6σ9 = −(X+X 2)+(X+X 2)×(σ)

1+X 3

θ1001 = 6+6σ9 = (X+X 2)+(X+X 2)×(σ)

1+X 3


Rational 2-Adic Functions


Rational 2-adic function:f (σ) =

nm× σ

where n, m are constant, polynomial (i.e. integers), m odd.Example specification:

θ(σ) =1− X 2

1− X + X 3 × σ

(−3

7× σ

)Note: In NF (θ) = p×σ

q , denominator q is constantall derivatives have denominator q,((α/β)′ = (α′ − [α(0)]× β′)/β),equivalence of derivatives in linear time,(syntactic equality of numerators).



Rational 2-Adic Automaton Size and Complexity

Theorem:

For rational 2-adic stream function f (σ) =nm× σ,

AutSize( n

m× σ

)≤

|n|+ |m|gcd(n, m)

− 1 ifnm

> 0

|n|+ |m|gcd(n, m)

ifnm≤ 0

Conjecture (based on experiments): Upper bound is also lower bound.Corollary: A Mealy machine implementing a rational 2-adicspecification θ(σ) =

nm× σ can be constructed in time 2O(|θ|).


Mealy Synthesis Tool (incl. Demo)

Mealy Synthesis Tool (Haskell)

Input specifications θ in:2-adic arithmeticmod-2 arithmetic

Output:LaTeX-document: automaton states and transitions,DOT-file: graph representation of automaton

Source code, executable, documentation:http://www.cwi.nl/˜costa/diffcal


http://www.cwi.nl/~costa/diffcal

Mealy Synthesis Tool (incl. Demo)

Tool Demo


Summary

Related Work

Derivative ConstructionsRaney (1958): stream function derivative (semantics only).Brzozowski (1964): DFA from regex.Antimirov (1996): NFA from regex (partial derivatives).Redziejowski (1999): ω-DFA from ω-regex (related to Safra).Sakarovitch (2005): Generalises [Antimirov] to formal powerseries.


Summary

Related Work

Logic Synthesis

Buchi&Landweber (1969): S1S.Pnueli&Rosner (1989): LTL.Kupferman&Vardi (2004):

distributed synthesis from LTL and CTL specs.imperfect information.

Uses theory of automata, alternation, games:determinization,constructive non-emptiness test.


Summary

Concluding

SummaryMealy synthesis using derivatives: direct symbolic construction ofdeterministic automaton.2-Adic and Mod-2 equivalence is decidable.Synthesis for rational (2-adic) functions:– EXPTIME in size |θ| of specification θ,– Automaton size.Haskell tool available.

Future work:Extend approach to more general types of transducers andstream functions,Closer investigation of logic vs coalgebraic synthesis,Develop coinductive stream languages.


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