kleene algebra with tests (part 3: theoretical results)

Kleene Algebra with Tests(Part 3: Theoretical Results)

Dexter KozenCornell University

Workshop on Logic & ComputationNelson, NZ, January 2004

These Lectures

1. Tutorial on KA and KAT

• model theory

• complexity, deductive completeness

• relation to Hoare logic

2. Practical applications

• compiler optimization

• scheme equivalence

• static analysis

3. Theoretical applications

• automata on guarded strings & BDDs

• algebraic version of Parikh’s theorem

• representation

• dynamic model theory

Kripke Frames over P,B

K = (K, mK)

mK : P 2K K P = atomic programs

mK : B 2K B = atomic tests

mK specifies a canonical interpretation for P,B

TP,B = {KAT terms over P,B}

Traces

K = (K, mK)

mK : P 2K K P = atomic programs

mK : B 2K B = atomic tests

A trace in K is a sequence

x = u0p0u1p1u2 … un-1pn-1un, n 0, (ui,ui+1) mK(pi)

u0p0u1 … un-1pn-1un · unpnun+1 … um-1pm-1um

= u0p0u1 … pn-1unpn … um-1pm-1um

TracesK = {traces in K} ,,... denote traces

Trace Algebras

A,B TracesK C K

A+B = A BAB = { | A, B}

A* = Un0 An 1 = K 0 = C = K − C

[[p]]K = {upv | (u,v) mK(p) }, p P

[[b]]K = mK(b), b B

extends to KAT homomorphism [[ ]]K : TP,B 2Traces(K)

TrK = { [[p]]K | p TP,B} regular trace algebra of K

Guarded Strings over P,B [Kaplan 69]

AtomsB = {atoms of free BA on B}

, , denote atoms

guarded strings 0p01p12p23 n-1pn-1n

join-irreducible elements of the free KAT on P,B

traces in Kripke frame G = (AtomsB,mG)

mG(p) = AtomsB x AtomsB

mG(b) = { | b}

TrG = {regular sets of guarded strings}

Relation Algebras

A,B K x K C idK = {(u,u) | u K}

A+B = A BAB = A B

A* = Un0 An 1 = idK 0 = C = idK − C

[p]K = mK(p) , p P

[b]K = {(u,u) | u mK(b)}, b B

extends to KAT homomorphism [ ]K : TP,B 2K x K

RelK = { [p]K | p TP,B} regular relation algebra of K

Traces and Relations

RelK is a homomorphic image of TrK

Ext(A) = {(first(),last() | A}

Ext : 2Traces(K) 2K x K

Ext : TrK RelK Ext([[p]]K) = [p]K

TrK is isomorphic to a relation algebra on TracesK

Rel(A) = {(,) | TracesK, A}

Thus Eq(REL) = Eq(TR)

Traces and Relations

g : K AtomsB

g(u) = unique such that u [[]]K

g : TracesK TracesG

g(u0p0u1 … un-1pn-1un) = g(u0)p0g(u1) … g(un-1)pn-1g(un)

g−1 : 2Traces(G) 2Traces(K)

g−1(A) = {x | g(x) A}

g−1 : TrG TrK g−1([[p]]G) = [[p]]K

Thus TrG is universal for relational and trace algebras

Automata on Guarded Strings

• ordinary finite automaton on alphabet P TB

• transitions labeled p P are action transitions

• transitions labeled b TB are test transitions

• inputs are guarded strings 0p01 n-1pn-1n

Automata on Guarded Strings

• read head always points to an atom, initially 0

• an action transition with label p is enabled if p is the next action symbol in x; advance the head past p

• a test transition with label b is enabled if b, where is the current atom in x; do not advance the head

• accept if occupying an accept state while scanning n

• ordinary NFA with -transitions is an AGS with B = {0,1}

Kleene’s Theorem for AGS

A set of guarded strings is accepted by some AGS over

P,B iff it is [[p]]G for some p TP,B

Determinization of AGS

An AGS is deterministic if

1. there is exactly one start state

2. each state is either an action state (has exiting action

transitions) or a test state (has exiting test transitions)

but not both

3. every action state has exactly one exiting action

transition for each p P (exactly one enabled)

4. the exiting test transitions of a test state are pairwise

exclusive and exhaustive (exactly one enabled)

5. every cycle contains at least one action transition

6. all accept states are action states

Determinization of AGS

Theorem Every nondeterministic AGS is equivalent to a

deterministic AGS

Proof Subset construction

State Minimization

Theorem If all possible tests are allowed, then minimal

unique deterministic AGSs exist

State Minimization

If only B and B = {b | b B} are allowed as tests, then

minimal deterministic AGSs are not unique

c c

d d d d

d d

c c c c

{cd,cd}

State Minimization

Theorem If only B and B = {b | b B} are allowed as

tests, and if the elements of B and B must be tested in

some fixed order, then unique minimal deterministic AGSs

exist

Special case unique minimal ordered BDDs

Myhill-Nerode Theorem for AGS

One can define an overlay operation ^ on prefixes of

guarded strings

Given a set A of guarded strings, define

x A y z (x^z A y^z A)

Theorem A is regular iff A has finitely many classes. The

A-classes give the minimal deterministic ordered AGS

Representation

Under what conditions is a given abstract KAT (K,B)

guaranteed to be isomorphic to a relational KAT?

Representation

Under what conditions is a given abstract KAT (K,B)

guaranteed to be isomorphic to a relational KAT?

1. (bc bqc = 0 bpc = 0) p q

2. pq = 0 b pb = 0 bq = 0

Theorem These conditions, together with *-continuity, are

sufficient for nonstandard representation

Proof states = ultrafilters of B

p’ = {(u,v) | b u c v bpc 0}

Dynamic Model Theory

Consider 1st-order KAT over a fixed signature

• atomic actions = assignments x := e

• atomic tests = atomic formulas R(e1,...,en)

A Kripke frame is Tarskian if it arises from a first-order

structure A

• states = valuations of variables over A

• mA(x := e) = { (u,u[x/u(e)]) | u : Var A}

• mA(R(e1,...,en)) = {u | u = R(e1,...,en)}|


Obs The equational theories of relation and trace

algebras of Tarskian frames do not coincide

x := 1; y := 2 and y := 2; x := 1 are equivalent in the

relation algebra but not in the trace algebra

Question Can we find algebras that are universal for

the Tarskian trace and relation algebras? (i.e., that

play the same role as the regular sets of guarded

strings for KAT)


Let T be a first-order theory

A quantifier-free type (qf-type) is a maximal consistent

set of quantifier-free formulas

A qf-type of T is a qf-type consistent with T

qf-types correspond to atoms in the guarded string

model


Define the frame (U,mU)

• U = {qf-types of T}

• mU(x := e) = {(,{ | [x/e] }) | U}

• mU (P(e1,...,en)) = { U | P(e1,...,en) }

Theorem TrU is universal for the equational theory of

Tarskian trace algebras over models of T:

[[p]]U = [[q]]U iff [[p]]A = [[q]]A for all models A of T

Note that U itself is not Tarskian in general!


Not true for RelU !

[P(c) P(d) ; x := c]U = [P(c) P(d) ; x := d]U

but these two programs are not equivalent in any

Tarskian frame in which c d

However they are observationally equivalent

(indistinguishable by any formulas in the language)


Theorem RelU is universal for the equational theory of

relation algebras of Tarskian frames over models of T

modulo observational equivalence; i.e.,

[p]U = [q]U iff p and q are observationally equivalent

over all models of T

Complexity of Scheme Halting and Equivalence

Theorem Let T be a recursive qf-theory. The scheme

halting and scheme equivalence problem over models

of T are 1 and 1 complete, respectively

Corollary There is no relatively complete deductive

system for scheme equivalence (or inequivalence)

00

Parikh’s Theorem [Parikh 66]

Every context-free language is letter

equivalent to a regular set

Letter equivalence: just count occurrences of

letters in strings, ignore order

Examples{ababca} {aaabbc, cbbaaa}

{anbn | n 0} (ab)*

A is letter equivalent to B

every string in A has an anagram in B and vice

versa

Parikh Map #a(x) = number of occurrences of a in x(x) = (#a1(x),...,#an(x)) Parikh vector(A) = {(x) | x A} commutative image

Examples({ababca, cbbaaa}) = {(3,2,1)}({anbn | n 0}) = ((ab)*) = {(n,n) | n 0}

A is letter equivalent to B (A) = (B)def

Parikh’s Theorem (Parikh's version)

Every context-free language is letter equivalent to a regular set.

Parikh’s Theorem (Parikh's version)

Every context-free language is letter equivalent to a regular set.

Parikh’s Theorem (our version)

Every commutative Kleene algebra is uniformly algebraically closed.

Commutative Kleene Algebra (CKA)xy = yx

A theorem of CKA but not KA(p+q)* = p*q*

() 1 + (p+q)p*q* = 1 + pp*q*+qp*q*

= 1 + pp*q*+p*qq*

p*q*

(p+q)* p*q*

Using (p+q)* = p*q* can show

Normal Form [Pilling 73]Every expression is equivalent to

y1+ ... + yn, where yi is a product of am

and (a1...ak)*.

Example(((ab)*c)* + d)* = d* + (ab)*c*cd*

Standard Model

Reg(Nn) = regular sets of Parikh vectors in Nn

A + B = A BAB = {x + y | x A, y B}

A* = Un0 An = A0 A1 A2 ...

1 = {(0,...,0)}0 =

This is the free CKA on n generators

Algebraic Closure

Every system of polynomial inequalities

f1(x1,...,xn) x1...

fn(x1,...,xn) xn

over a CKA K has a unique least solution in Kn.

Uniform Algebraic Closure

Every system of polynomial inequalities

f1(x1,...,xn) x1...

fn(x1,...,xn) xn

over a CKA K has a unique least solution in Kn. The components of the solution are given by polynomials in

the coefficients of the fi.

• A context-free grammar is just a system of polynomial inequalities over the KA (*)

• The associated context-free language is its least solution in (*)

• Commutativity models letter equivalence

Examples

{anbn | n 0} S aSb | axb + 1 x

{balanced parens} S (S) | SS | (x) + xx + 1 x

{palindromes} S aSa | bSb | a | b | axa + bxb + a + b + 1 x

Previously known for

• Reg(Nn) [Pilling 73]• commutative -continuous semirings [Kuich 87]

Approach

• differential operators /x on polynomials

• Taylor’s theorem f(x+d) = f(x) + f(x+d)d

• closed form solution for n inequalities in n unknowns involving the Jacobian matrix

Polynomials K[x,y,...]

(ax + by)*1 + (ax*b*)* + bx + cya + xy(bxy)*

a,b,... Kx,y,... variables

K[x,y,...] is a CKA

Polynomials K[x,y,...]

(ax + by)*1 + (ax*b*)* + bx + cya + xy(bxy)*

a,b,... Kx,y,... variables

K[x,y,...] is a CKA

K

K[x,y,...]

{x,y,...}

L

eval

K[x,y,...] is the direct sum (coproduct) of K and

the free CKA on {x,y,...}

Differential Operators

A map D:K K is called a differential operator if for all x,y K,

• D(x+y) = Dx + Dy• D(xy) = xDy + yDx• D(x*) = x*Dx• D0 = D1 = 0


A map D:K K is called a differential operator if for all x,y K,

• D(x+y) = Dx + Dy• D(xy) = xDy + yDx• D(x*) = x*Dx• D0 = D1 = 0

K

K[x,y,...]

{x,y,...}

K[x,y,...]


:K[x,...] K[x,...], where

Examples

x

x(y) = 0, y x

x (a) = 0, a K

x (x) = 1

x (ax2y + bxy2 + (ax)* + 1) = axy + by2 + a(ax)*

y (ax2y + bxy2 + (ax)* + 1) = ax2 + bxy

x

Chain Rule

For f, e K[x],

or in more conventional notation,

f(e(x)) = f(e(x)) e(x)

f x

e x

x

(f(e)) = (e) ·

Taylor’s Theorem

For f, d K[x],

f(x+d) = f(x) + f(x+d) d

In particular, evaluating at x = 0,

f(d) = f(0) + f(d) d

Theorem

Let K be a CKA and let f(x) K[x]. The least solution of f(x) x is f(f(0))* f(0).

Theorem


Example {anbn | n 0}

f(x) x axb + 1 xf(x) axb + 1

f(x) abf(0) 1

f(f(0))* f(0) (ab)*

Theorem


Example {balanced parentheses}

f(x) x axb + x2 + 1 xf(x) axb + x2 + 1

f(x) ab + xf(0) 1

f(f(0))* f(0) (ab + 1)* = (ab)*

Theorem


Example {palindromes}

f(x) x axa + bxb + a + b + 1 xf(x) axa + bxb + a + b + 1

f(x) a2 + b2

f(0) a + b + 1

f(f(0))* f(0) (a2 + b2)* (a + b + 1)= (a2 )* (b2)* (a + b + 1)

The 2 x 2 Case

f(x,y) xg(x,y) y

Viewing K[x,y] as K[x][y], solve g(x,y) y over K[x]. Say the solution is h(x). Then solve f(x,h(x)) x over K. Say the solution is a.

Then (a,h(a)) is the least solution of (*).

(*)

The 2 x 2 Case

f(x,y) xg(x,y) y

Viewing K[x,y] as K[x][y], solve g(x,y) y over K[x]. Say the solution is h(x). Then solve f(x,h(x)) x over K. Say the solution is a.

Then (a,h(a)) is the least solution of (*).

Need uniformity: the expression f(f(0))* f(0) gives the least solution uniformly in all homomorphic images

(*)

Multivariate Taylor Theorem

For x = x1,...,xn, f = f1,...,fm K[x], ande = e1,...,en,

f(e) = f(0) + (e) ·e f x

Multivariate Taylor Theorem


f(e) = f(0) + (e) ·e f x

f x

fi

xj

(e)ij = (e)

Jacobian matrix

Multivariate Chain Rule


(f(e)) = (e) · f x

z

e z

TheoremLet x = x1,...,xn and f = f1,...,fn K[x]. Consider the n x n system

f(x) x (**)

Define

For sufficiently large finite N, aN is the least solution to (**).

a0 = f(0)

ak+1 = (ak)* ·ak

f x

How bad can N be?

How bad can N be?

N(n) (7 ·3n - 5) / 2

kleene algebra with tests (part 3: theoretical results)

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