```Kleene Algebra with Tests
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
Kleene Algebra (KA)
is the algebra of regular expressions
pq + qp
p
q
p
p*q
{pq,qp}
{q,pq,p 2q,p3q, …}
q
(p + q)* = (p*q)*p*
q
p
p,q
{all strings over p,q}
(0 + 1(01*0)*1)*
{multiples of 3 in binary}
q
(pq)*p = p(qp)*
{p,pqp,pqpqp,}
0
1
0
1
0
p
1
Standard Interpretation
Regular sets over 
A+B
AB
A*
1
0
=
=
=
=
=
AB
{xy | x  A, y  B}
Un0 An = A0  A1  A2 
{}

p   interpreted as {p}
...
Binary Relations
R, S binary relations on a set X
R+S = R  S
RS
= R ° S = {(u,v) | w (u,w)  R, (w,v)  S}
R*
= reflexive transitive closure of R
= Un0 Rn = R  R1  R2  
1
0
= identity relation = {(u,u) | u  X}
=
Applications
• Automata and formal languages
regular expressions
• Program logic and verification
Dynamic Logic
program analysis
protocol verification
compiler optimization
• Algorithms
shortest paths
connectivity
computational geometry
Prehistory
• Definition, relation to finite automata
Kleene 56
• No purely equational axiomatization
Redko 64
• Axiomatization of equational theory
Salomaa 66
• Algebraic theory Conway 71
• Equational theory PSPACE complete
(Stock+1)Meyer 74
Axioms of KA [K91]
• K is an idempotent semiring under +, ·, 0, 1
(p + q) + r = p + (q + r)
p+q=q+p
p+p=p
p+0=p
(pq)r = p(qr)
p1 = 1p = p
p0 = 0p = 0
p(q + r) = pq + pr
(p + q)r = pr + qr
• p*q = least x such that q + px  x
• qp* = least x such that q + xp  x
def
xy  x+y=y
Succinctly stated,
A Kleene algebra is an idempotent
semiring such that
• p*q is the least fixpoint of x.(q + px)
• qp* is the least fixpoint of x.(q + xp)
This is a universal Horn axiomatization
• p*q = least x such that q + px  x
q + p(p*q)  p*q
q + px  x  p*q  x
• qp* = least x such that q + xp  x
q + p(qp*)  qp*
q + px  x  qp*  x
Every system of linear inequalities
a11x1 + ... + an1xn + b1  x1
.
.
.
an1x1 + ... + annxn + bn  xn
has a unique least solution
Alternative Characterizations of *
Complete semirings (quantales)
i  I pi = supremum of {pi | i  I}
with respect to 
*-continuity
pq*r = sup pqnr
n0
• infinitary
• same equational theory Eq(KA) = Eq(KA*)
Some Useful Properties
1 + pp* = 1 + p*p = p*
p*p* = p** = p*
(pq)*p = p(qp)*
sliding
(p*q)*p* = (p + q)*
denesting
px = xq  p*x = xq*
bisimulation
qp = 0  (p + q)* = p*q*
loop distribution
qp = pq  (p + q)* = (pq)*(p* + q*)
Proof of the Sliding Rule
(ab)*a  a(ba)*
a + aba(ba)* = a(1 + ba(ba)*)
= a(ba)*
distributivity
1 + pp* = p*.
a + aba(ba)*  a(ba)*
(ab)*a  a(ba)*
q + px  x  p*q  x
The reverse inequality  is symmetric.
Completeness and Complexity
• Deductively complete for the equational
theory of regular sets of strings and relational
models [K 94]
• Complexity: Equational theory is PSPACEcomplete [Meyer+Stockmeyer 74]
• Hoare theory (Horn theory with premises p =
0) is PSPACE-complete [Cohen 93]
• Horn theory is 11-complete for starcontinuous & relational models [Hardin+K 03]
Matrices over a KA
a b
c d
a b
c d
def
0=
e f
g h
+
·
e f
g h
0 0
0 0
a b * def
=
c d
a+e b+f
c+g d+h
def
=
ae+bg af+bh
ce+dg cf+dh
def
=
def
1=
1 0
0 1
(a+bd*c)*
(a+bd*c)*bd*
(d+ca*b)*ca*
(d+ca*b)*
Matrices over a KA
a b * def
=
cd
(a+bd*c)*
(a+bd*c)*bd*
(d+ca*b)*ca*
(d+ca*b)*
b
d
a
c
Matrices over a KA
• Representation of finite automata
• Construction of regular expressions
• Solution of linear equations over a KA
• Connectivity and shortest path algorithms
Solution of Linear Inequalities
a11x1 + ... + an1xn + b1  x1
.
.
.
an1x1 + ... + annxn + bn  xn
a11 ... an1
.
.
.
an1 ... ann
x1
.
.
.
xn
+
b1
.
.
.
bn

x1
.
.
.
xn
Shortest Paths
The min,+ algebra
R+  {}
r + s = min r,s
rs
=r+s
r*
=0
0
=
1
=0

=
.9
1.4
3.2
0 1.4 3.2


0

.9
0
*
0 1.4 2.3
=


0

.9
0
Other Models
Convex polyhedra [Iwano & Steiglitz 90]
AB = {ax + by | x  A, y  B}
A
A* = convex hull of A
B
Other Models
Convex polyhedra [Iwano & Steiglitz 90]
AB = {ax + by | x  A, y  B}
A
A* = convex hull of A
B
Hoare Logic
C. A. R. “Sir Tony” Hoare, 1969
• The first formal system for verification of
well-structured programs
• Initiated the field of program correctness
• Inspiration of hundred of technical articles,
books, book chapters, surveys
• Turing Award 1980, knighted in 1999
Partial Correctness Assertions
{b}p{c}
postcondition
program
precondition
‘‘If b holds in the current state, and if p is
executed starting in the current state, then
if p halts, c will be true of the halting state.’’
Rules of Hoare Logic
{b[x/e]} x:=e {b}
{b}p{c}, {c}q{d}
{b}pq{d}
{bc}p{d}, {bc}q{d}
{c}if b then p else q{d}
{bc}p{c}
{c}while b do p{bc}
b’  b, {b}p{c}, c  c’
{b’}p{c’}
assignment rule
composition rule
conditional rule
while rule
weakening rule
Dynamic Logic [Pratt 76]
<p>b
“Starting from the current state, it
is possible for p to halt in a state
satisfying b.”
[p]b
“Starting from the current state, it
is necessary that if p halts, it does
so in a state satisfying b.”
DL subsumes HL
{b}p{c} 
b  [p]c
Propositional Dynamic Logic (PDL)
propositional modal logic + Kleene algebra
Syntax
• program operators ; + *
• propositional operators    0 1
• mixed operators <p>b [p]b b?
Semantics
• binary relations (input/output)
While programs in PDL
if b then p else q

b?;p + b?;q
while b do p

(b?;p)*;b?
• complete deductive system [Segerberg 77,
Gabbay 77, Parikh 78]
b  [p*](b  [p]b)

• EXPTIME-complete [Pratt 78]
[p*]b
For many applications in CS (simple program
manipulations, safety analysis, local
optimizations) don’t need the full power of
PDL—equational reasoning suffices
BUT
need tests to model conventional programming
constructs (if-then-else, while-do, )
Kleene Algebra with Tests (KAT)
(K, B, +, ·, *, ¯, 0, 1)
• (K, +, ·, *, 0, 1) is a Kleene algebra
• (B, +, ·, ¯, 0, 1) is a Boolean algebra
•BK
• p,q,r, range over K
• a,b,c, range over B
Kleene Algebra with Tests (KAT)
+, ·, 0, 1 serve double duty
• applied to programs, denote choice,
composition, fail, and skip, resp.
• applied to tests, denote disjunction,
conjunction, falsity, and truth, resp.
• these usages do not conflict!
bc = b  c
b+c=bc
Models
• Relational models
K = binary relations on a set X
B = subsets of the identity relation
• Trace models
K = sets of traces u0p0u1p1u2 … un-1pn-1un
B = sets of traces of length 0
• Language-theoretic models
K = regular sets of guarded strings over 
B = atoms of a finite free Boolean algebra
Guarded Strings
[Kaplan 69]
P atomic programs
B atomic tests
, , atoms (minimal nonzero elements) of the free
Boolean algebra on generators B
e.g. if B = {b1,...,b6}, then b1b2b3b4b5b6 is an atom
guarded strings
A+B
= AB
AB
A*
1
0
=
=
=
=
0p01p12p23  n-1pn-1n
{xy | x  A, y  B}
Un0 An
{atoms}

Theorem [K & Smith 96]
The family of regular sets of guarded
strings over P,B is the free KAT on
generators P,B.
Corollary
KAT is complete over relational models.
Eq(GS) = Eq(KAT) = Eq(KAT*) = Eq(REL)
Completeness and Complexity
• Deductively complete for the equational
theory of regular sets of guarded strings
and relational models [K+Smith 96]
• Complexity: Equational theory is PSPACEcomplete [K+S 96, Cohen+K+S 97]
• Hoare theory (Horn formulas with premises
of form p=0) is PSPACE-complete [K+S 96]
1
• Full Horn theory is still 1-complete
Matrices over a KAT
The n x n matrices over a KAT (K,B) forms a
KAT (K’,B’)
B’ = diagonal matrices over B
Modeling Programs
same as in PDL [Fischer & Ladner 79]
p;q

pq
if b then p else q

bp + bq
while b do p

(bp)*b
Propositional Hoare Logic (PHL)
Hoare Logic without the assignment rule
{b[x/t]} x := t {b}
Is a given rule
{b1}p1{c1}, ..., {bn}pn{cn}
{b}p{c}
• a logical consequence of the composition,
conditional, while, and weakening rules?
• relationally valid?
KAT subsumes PHL
{b}p{c} is modeled by any of the following
equivalent equations & inequalities:
• bp  pc
• bp  pc
• bp = bpc
• bpc = 0
{b}p{c}, {c}q{d}
{b}pq{d}
{bc}p{d}, {bc}q{d}
{c}if b then p else q{d}
{bc}p{c}
{c}while b do p{bc}
composition rule
conditional rule
while rule
{b}p{c}, {c}q{d}
{b}pq{d}
composition rule
 bpc = 0  cqd = 0  bpqd = 0
{bc}p{d}, {bc}q{d}
{c}if b then p else q{d}
conditional rule
 bcpd = 0  bcqd = 0  c(bp+bq)d = 0
{bc}p{c}
{c}while b do p{bc}
while rule
 bcpc = 0  c(bp)*b bc = 0
Theorem
These are all theorems of KAT
Completeness Theorem [K 99]
All relationally valid rules of the form
{b1}p1{c1}, ..., {bn}pn{cn}
{b}p{c}
are derivable in KAT (not so for PHL)
Counterexample
{c}if b then p else p{c}
{c}p{c}
is trivially unprovable in Hoare Logic, but
c(bp + bp)c = 0  cpc = 0
is easily provable in KAT
Hoare formulas
p1 = 0  p2 = 0  ...  pn = 0  q = r
Theorem
KAT is complete for the Hoare theory of
relational algebras
... not for the Horn theory!
Counterexample: p  1  p2 = p
Complexity
Theorem [Berstel 79]
It is undecidable whether a given equation holds under
a given set of commutativity conditions in all *continuous Kleene algebras
Theorem [K 99]
0
...  2 -complete ...
Horn Theories
Theorem [K 99, K + Hardin 03]
The universal Horn theories of KA*, KAT*,
1
REL are 1 -complete
Q: Is there a natural example of a sentence
in H(KA*)  H(KA)?
Q: Is H(REL) finitely axiomatizable relative to H(KA*)?
Q:
What is the complexity of H(KA)?
Complexity of KAT and PHL
Theorem [Cohen 94]
The Hoare theory of KA (Horn formulas with
premises p = 0) is PSPACE-complete
Theorem [Cohen, Kozen & Smith 96]
The Hoare theory of KAT is PSPACE-complete
Theorem
PHL is PSPACE-complete
Schematic KAT (SKAT)
x := s ; y := t
=
y := t[x/s] ; x := s
y  FV(s)
x := s ; y := t
=
x := s ; y := t[x/s]
x  FV(s)
x := s ; x := t
=
x := t[x/s]
b[x/t] ; x := t
x := x
=
1
=
x := t ; b
Special Cases
x := s ; y := t

y := t ; x := s
(x  Var(t), y  Var(s))

b ; x := t
(x  Var(b))
x := s ; x = s
(x  Var(s))
x := t ; b

x := s
x = s

x = s ; x := s
Relation to Hoare Assignment Rule
[x/t] ; x := t
=
x := t ; 
is equivalent to
{[x/t]} x := t {}
{[x/t]} x := t {}
Relation to Hoare Assignment Rule
[x/t] ; x := t
=
x := t ; 
cp = pb
is equivalent to
{[x/t]} x := t {}
cpb = 0
{[x/t]} x := t {}
cpb = 0
cp = pb

cpb + cpb = 0
Interpreted Reasoning
 = 1
where  is any property of the domain of
computation, typically an instantiated
universal first-order property
```