Nikolaj Bjørner Joe Hendrix Microsoft Research & Corporation.

Linear Functional Fixed-Points

Nikolaj Bjørner Joe Hendrix

Microsoft Research & Corporation

Overview

Linear Functional Fixed-Point Logic (FFP)

Complexity results for FFP:FFP(Propositional) – PSPACE/NP

FFP(Linear/Equalities) – PSPACE

By a reduction to LTLFFP(Non-linear)– NEXPTIME hard/undecidable

Integrating FFP with an SMT solver (Z3)

A list-manipulating program

T T

F

FF

Thead

curr

data(curr) := true;curr := f(curr)

curr = head

T T

T

TT

Thead

curr

curr := head

F F

F

FF

Fhead

curr

Loop invariant: Every data element betweenhead and curr is set to true

F F

F

FF

Thead

curr

The loop invariant


T T

F

FF

T

head

curr

x [head curr] . data(x)f

invariant(head) where invariant(x) = x = curr (data(x) invariant(f(x)))

LFP Inv , x. [ x = curr (data(x) Inv(f(x))) ] (head)

Inv x [ x = curr (data(x) Inv(f(x))) ] (head)

What are practical ways of reasoning with such fixed-points?

uvf

w[Nelson 80]

Some solutions

u v wf f f f

f f

Some solutions

u v wf f f f f

uvbtwnf(u,v,w

)[Rakamarić07

+]

f

w[Nelson 80]

f

Some solutions

u vf f f f

uvbtwnf(u,v,w

)[Rakamarić07

+]

wf. Reachability[Lahiri, Qadeer

06]

f

w[Nelson 80]

B(u) = v

BSet(u)

BSet(f(u))

BSet(f(f(u)))

BSet(v)

R(u,v) u vf f f f

BSet(f(u))

BSet(f(f(u)))

From u reach vand v is the first element satisfyingBSet(v)

From u reach vand everything afteru and up to v satisfies BSet

uvbtwnf(u,v,w

)[Rakamarić07

+]


06]

f

w[Nelson 80]

Interpreted sets &

Bounded quant.[Lahiri, Qadeer

08]

Some solutions

Use first-order axioms to encode quantifier-free theory of reachability.

[LQ08] rely on SMT solver Z3 for instantiating axioms using triggers.

Required quantifier support by solver is not so off-the-shelf.

uvbtwnf(u,v,w

)[Rakamarić07

+]


06]

f

w[Nelson 80]

Interpreted sets &


08]FFP(Prop)

Lin. FFP(Eq)

FFP(Non-linear)Reachable Patterns

[Yorsh+ 06]

wSnS(finite trees) wS1S

(fin. Acyclic lists)

S1S(inf. Acyclic

lists)

SnS(inf. Trees)

wSO(f)(finite linked

lists)

SO(f)(infinite trees)

Some solutions

Many other solutions

[Immerman+ 04] First-order transitive closure

[Møller+ 05] Pointer assertion logic

[Lev-Ami+ 05] Acyclic transtive closure

[McPeak+ 05] Linked lists

[Ranise+ 05] Linked lists

[Balaban+ 07] Single parent heaps

[Bouajjani+ 06-09] Reachability + arithmetic + T

Apologies for relevant omissions.

A Quest for an SMT solver integrationExisting decision procedures for fixed-points use

- Encoding with first-order axioms-Rely on first-order instantiation engine for completeness

- Reduction to automata -Powerful combination with some theories, but flexible combination approach and “low-order” complexity results unclear to us

T T

F

FF

Thead

curr

The DPLL(T) setting for SMT

Theories

Core Theory

SAT solver

Bit-Vectors

Arithmetic

Data-types

E-matching

Arrays

Formula

Rewriting Simplification

Specialized theory solvers interoperateby exchanging learned equalities and clauses with a commoncongruence closure core

Core Theory: Equalities, asserted literalsTheory Core: Equalities, asserted literals, new clauses

T T

F

FF

Thead

curr

Back to the loop invariant


T T

F

FF

Thead

curr

x [head curr] . data(x)f

invariant(head) where invariant(x) = x = curr (data(x) invariant(f(x)))

LFP Inv , x. [ x = curr (data(x) Inv(f(x))) ] (head)


Question:

[data(x) Untilf,x x = curr] (head)

Is there a convenient propositional-like abstraction of fixed-points?

Our Approach: establish and use a connection with Linear Time Temporal Logic for linear functional fixed-points


A Until B

B [A (A Until B)]

X . B [A X]

T T

F

FF

Thead

curr

FFP Temporal Macros

[A(x) Untilf,x B(x)] (a)

R x [B(x) (A(x) R(f(x)))] (a)

[f,x A(x)] (a) [true Untilf,x

A(x)] (a)

[f,x A(x)] (a) [f,x A(x)] (a)

uvbtwnf(u,v,w

)[Rakamanic07

+]


06]

f

w[Nelson 80]

Interpreted sets &


08]FFP(Prop)

Lin. FFP(Eq)


[Yorsh+ 06]

wSnS(finite trees) wS1S

(fin. Acyclic lists)

S1S(inf. Acyclic

lists)

SnS(inf. Trees)

wSO(f)(finite linked

lists)

SO(f)(infinite trees)

Some solutions

uvbtwnf(u,v,w

)[Rakamanic07

+]


06]

f

w[Nelson 80]

Interpreted sets &


08]FFP(Prop)

Lin. FFP(Eq)


[Yorsh+ 06]

Propositional Linear Time

Temporal Logic

Our approach – a tighter sandwich

?

FFP(Propositional Logic): basic results

[f,x P(f(x))](a) [ f,x P(x)](b) [Q(x) Until f,x

P(f(x))](b)

- Distinguished function f - Unary predicate symbols, P, Q, R- At most one bound variable in

scope at any time [Q(x) Until f,x [P(f(x)) Until f,y R(y)]](b)

FFP(PL): basic results

From LTL to FFP(PL)

P f,x f,x P(f(x))(anchor)

From FFP(PL) to LTL

f,x P(f(x))(a) f,x P(x)(b) Pa Pb

Complexity(FFP(PL)) = Complexity(pLTL)

FFP(Equalities): propositions and equalities

fu v u v

f f f f

f

[True Until f,x x = v](u)

f,x (x = v)(u)

FFP(E): propositions and equalities

fu v w

u v wf f f f

f f

[x w Until f,x x = v](u)


btwnf(u,v,w)

[x w Until f,x x = v](u) f,x (x = w)(v)

u v wf f f f f f


u vf f f f

B(u) = v

BSet(u)

BSet(f(u))

BSet(f(f(u)))

BSet(v)

R(u,v) u vf f f f

BSet(f(u))

BSet(f(f(u)))

[BSet(f(x)) Until f,x x = v](u)

[BSet(x) Until f,x x = v](u) BSet(v)


[ f,x x c](b)

[g,x P(g(x))](a) [f,x P(f(x))](a) [x fff(x) Until f,x x = a](b) [g,x g(g(x)) = x](c)

- Distinguished functions f, g - As long as f and g are separate

- Unary predicate symbols, P, Q, R- At most one bound variable in scope

at any time

FFP(E): A litmus test. Closure under updates.

wp(f(u) := v, [A Until f,x B](w)) f’ := x. if x = u then v

else f(x)= [A Until f,x B](w)[f f’]

A’ := A[f f’], B’ := B[f f’]

= [A’ Until f’,x B’](w)

= ….

= [A’’ Until f,x B’’](w)

A’’ := A’ u x B’’ := B’ (u = x [(u x A’) Until

f,x B’](v))

FFP(E) : reduction to LTL?

From LTL to FFP(E)


From FFP(E) to LTL?

[f,x x = c f,x P(x)](a) a and b reach c

[f,x x = c f,x P(x)](b) after that there is a

common P state.

FFP(E) : reduction to LTL?

From LTL to FFP(E)


From FFP(E) to LTL

[f,x (T(x) U(x)) f(x) = b](a)

[f,x (T(x) U(x)) f(x) = c](b)

[f,x (T(x) U(x)) f(x) = a](c)

T b

T

cT

aU

UUObstacle: f is a function.-

The Temporal Next operator does not encode functionality by itself.

FFP(E) encoding forcing functionality

Tableau()

F – acc. cond

PTL PTL*

Normalize f Erasure Functionality axioms

Functionality axioms

FFP(E) encoding forcing functionality

Tableau()

F – acc. cond

PTL PTL*

Normalize f Erasure Functionality axioms

Proposition: Validity for FFP(E) is PSPACE complete

Size of PTL* is quadratic in

Pure pLTL formula

FFP(E) extensions

FFP(NL) – more than one variable in nested bound context

[ f,x [ f,y f(x) y](x)] (a)

NEXPTIME hard FFP(NL) MSO(f)

2FFP(E) – allow nested use of functions f g: [ f,x g(f(x)) = f(g(x))] (a)

2FFP(E) is undecidable

af f f f f f f

af f f f

f f f f

g g g g

ggg g

SMT solver Integration

Most SMT solvers use a DPLL(T) architecture

SAT Equality Core Theories

SAT Equality Core Theories

Literal assignmentsEqualities

Literal assignments

Literal assignmentsEqualitiesLiteral assignments

Lemmas (Conflict Clauses)

SMT solver Integration (Theory)

Property: FFP(E) is stably infiniteIf FFP(E) formula has a model, it has a model of size N, it has a model of size N+1

Theorem: Let T be stably infinite, decidable, and have disjoint signature from f, g, Then quantifier-free formulas over FFP(E) + T are decidable

SMT solver Integration (Incremental)

pLTL Equality Core Theories

pLTL Equality Core Theories

Trace of Literal assignments

EqualitiesLiteral assignments

EqualitiesLiteral assignmentsInvariants

Safety properties

Summary

Linear Functional Fixed-Point Logic (FFP)

Complexity results for FFP:FFP(Propositional) – PSPACE/NP

FFP(Linear/Equalities) – PSPACE

By a reduction to LTLFFP(Non-linear)– NEXPTIME hard/undecidable

Integrating FFP with the SMT solver

Conclusions

We established a sandwich link between

Linear Functional Fixed-Point Logic andPropositional Linear Time Temporal LogicMore sandwiched links plausible, but open.

From DPLL(T) to SMC(T)We show how to integrate a solver based on LTL with an SMT Solver A prototype using CUDD and shows signs of life

Nikolaj Bjørner Joe Hendrix Microsoft Research & Corporation.

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