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Kleene Algebra with Tests
(Part 3: Theoretical Results)
Dexter Kozen
Cornell University
Workshop on Logic & Computation
Nelson, 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  B
AB = { |   A,   B}
A* = Un0 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
0p01p12p23  n-1pn-1n
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  B
AB = A  B
A* = Un0 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 0p01  n-1pn-1n
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
{cd,cd}
dd
d
d
d
c
cc
c
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. (bc 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)}
Dynamic Model Theory
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)
Dynamic Model Theory
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
Dynamic Model Theory
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!
Dynamic Model Theory
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)
Dynamic Model Theory
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 01 and 10 complete,
respectively
Corollary There is no relatively complete deductive
system for scheme equivalence (or inequivalence)
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}
def
A is letter equivalent to B  (A) = (B)
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B
AB = {x + y | x  A, y  B}
A* = Un0 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 | 
{balanced parens} S  (S) | SS | 
{palindromes}
axb + 1  x
(x) + xx + 1  x
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 + cy
a + xy(bxy)*
a,b,...  K
x,y,... variables
K[x,y,...] is a CKA
Polynomials K[x,y,...]
(ax + by)*
1 + (ax*b*)* + bx + cy
a + xy(bxy)*
K
a,b,...  K
x,y,... variables
K[x,y,...]
K[x,y,...] is a CKA
{x,y,...}
eval
K[x,y,...] is the direct sum (coproduct) of K
and the free CKA on {x,y,...}
L
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
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
K
K[x,y,...]
{x,y,...}
K[x,y,...]

Differential Operators x

:K[x,...]  K[x,...], where
x

(x) = 1
x

(y) = 0, y  x
x

(a) = 0, a  K
x
Examples

(ax2y + bxy2 + (ax)* + 1) = axy + by2 + a(ax)*
x

(ax2y + bxy2 + (ax)* + 1) = ax2 + bxy
y
Chain Rule
For f, e  K[x],

(f(e)) =
x
f
e
(e) ·
x
x
or in more conventional notation,
f(e(x)) = f(e(x))  e(x)
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
Let K be a CKA and let f(x)  K[x]. The least
solution of f(x)  x is f(f(0))*  f(0).
Example {anbn | n  0}
f(x)  x
f(x)
f(x)
f(0)
f(f(0))*  f(0)
axb + 1  x
axb + 1
ab
1
(ab)*
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).
Example {balanced parentheses}
f(x)  x
f(x)
f(x)
f(0)
f(f(0))*  f(0)
axb + x2 + 1  x
axb + x2 + 1
ab + x
1
(ab + 1)* = (ab)*
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).
Example {palindromes}
f(x)  x
f(x)
f(x)
f(0)
f(f(0))*  f(0)
axa + bxb + a + b + 1  x
axa + bxb + a + b + 1
a2 + b2
a+b+1
(a2 + b2)* (a + b + 1)
= (a2 )* (b2)* (a + b + 1)
The 2 x 2 Case
f(x,y)  x
g(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)  x
g(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], and
e = e1,...,en,
f(e)
=
f(0) +
f
(e) ·e
x
Multivariate Taylor Theorem
For x = x1,...,xn, f = f1,...,fm  K[x], and
e = e1,...,en,
f(e)
=
f(0) +
f
(e) ·e
x
Jacobian matrix
f
(e)ij =
x
 fi
(e)
 xj
Multivariate Chain Rule
For x = x1,...,xn, f = f1,...,fm  K[x], and
e = e1,...,en,

(f(e)) =
z
e
f
(e) ·  z
x
Theorem
Let x = x1,...,xn and f = f1,...,fn  K[x].
Consider the n x n system
f(x)  x
(**)
Define
a0
=
ak+1 =
f(0)
f
(ak)* ·ak
x
For sufficiently large finite N, aN is the
least solution to (**).
How bad can N be?
How bad can N be?
N(n)  (7 ·3n - 5) / 2