Time Bounds for General Function Pointers

Robert Dockins and Aquinas Hobor(Princeton University) (NUS)

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Goal

• Use step-indexed models to prove program time bounds in the presence of certain (approximated) recursive domain equations

• Testbed: soundness of a logic of total correctness for a language with1. Function pointers2. Semantic assertions (assert truth of a (classical)

logical formula at the current program point)

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Goal

• Use step-indexed models to prove program time bounds in the presence of certain (approximated) recursive domain equations

• For example, the kinds of domains that occur in semantic models of the assertions of concurrent separation logic with first-class locks and threads.

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Goal

• Use step-indexed models to prove program time bounds in the presence of certain (approximated) recursive domain equations

• For example, the kinds of domains that occur in semantic models of the assertions of concurrent separation logic with first-class locks and threads.– But maybe this is a really hard domain to attack…

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Goal

• Use step-indexed models to prove program time bounds in the presence of certain (approximated) recursive domain equations

• Testbed: soundness of a logic of time bounds for a language with1. Function pointers2. Semantic assertions (assert truth of a (classical)

logical formula at the current program point)

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A very simple language

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Simple semantic definitions

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Simple semantic definitions

Uh oh…

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We detail our predicates/assertions informally and refer to the paper for formal construction

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We detail our predicates/assertions informally and refer to the paper for formal construction

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We detail our predicates/assertions informally and refer to the paper for formal construction

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We detail our predicates/assertions informally and refer to the paper for formal construction

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What does funptr l t [P] [Q] mean?1. The program has some code c at address l .

(This is why we want predicates to be able to judge programs.)

2. When c is called with some initial store ½, if t(½) is defined then c makes at most t(½) function calls before returning to its caller.

3. P and Q are actually functions from some function-specific type A to predicates. If t(½) is defined then for all a, if P(a) holds before the call then Q(a) will hold afterwards.

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Hoare Judgment

• Our Hoare Judgment looks a bit complex:

¡, R `n { P } c { Q }

• ¡ contains function pointer assertions and R is the postcondition of the current function

• n is an upper bound on the number of function calls c makes before it terminates

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Hoare Rules, 1

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Hoare Rules, 1

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Hoare Rules, 2

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Hoare Rules, 2

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Hoare Rules, 3

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Hoare Rules, 3

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The Call Rule

The interesting rule is for call… and it’s not too bad:

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The Call Rule

The interesting rule is for call… and it’s not too bad:

1. x must evaluate to a label l

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The Call Rule

The interesting rule is for call… and it’s not too bad:

2. l must be a pointer to a function with termination measure t, precondition Pl and postcondition Ql

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The Call Rule

The interesting rule is for call… and it’s not too bad:

3. The termination measure t must evaluate to some n on the current store

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The Call Rule

The interesting rule is for call… and it’s not too bad:

3. The termination measure t must evaluate to some n on the current store… and so this call will take no more than n+1 calls.

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The Call Rule

The interesting rule is for call… and it’s not too bad:

4. We require the (parameterized) precondition to be true at call, and guarantee the (parameterized) postcondition will be true on return.

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So, not too bad…

• Most of these rules are really nothing to get excited about… even the call rule is pretty simple when you understand the parts…

• (This is a virtue, of course…)

• But we’re not done. We’ve only shown the rule for using the funptr, not the rules for verifying that a function actually terminates

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V-rules

• We need another judgment, written ª : ¡, that says that program ª contains functions verified to “funptr” specifications in ¡.

• We will verify functions one at a time (or in the case of mutually-recursive groups, as a group).

• The “base case” is an easy rule:

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

1. We have verified part of ª to specification ¡

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

2. We want to add this specification for l

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

3. We must have verified the code for l

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

4. When the precondition P(a) holds

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

4. When the precondition P(a) holds and the termination measure is equal to some n.

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

5. This n is also an upper bound on the number of calls this function makes.

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

6. The postcondition is just ?: you can’t fall out the bottom

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

6. The postcondition is just ?: you can’t fall out the bottom, but the return condition is Q(a).

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

7. We can assume every funptr we have previously verified, and…

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The “Vsimple” rule

So this looks pretty bad, but we can take it apart:

8. We can call ourselves using a modified function specification precondition: t has decreased.

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The “Vsimple” rule

Here are the critical uses for t. If we assume the termination measure is equal to n at the functionstart, then we can make no more than n calls andcan recurse only when t < n.

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Vsimple, in a nutshell

• The Vsimple rule is fine for most functions, but it can only verify, well, simple recursive functions (as well as calls to both simple and complex functions previously verified).

• If we want “the goodies” (mutual recursion, polymorphic termination arguments, etc.) then we need to call in Vsimple’s big brother…

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This is moderately horrible. We have (at least):

The “Vfull” rule

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This is moderately horrible. We have (at least):

1. A set © of funptr specifications in the goal as well as the assumptions

The “Vfull” rule

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This is moderately horrible. We have (at least):

2. The same basic termination measure trick

The “Vfull” rule

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This is moderately horrible. We have (at least):

3. A parameter b used for higher-order functions

The “Vfull” rule

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This is moderately horrible. We have (at least):

3. There can be a strong dependency between b and the other factors (e.g., the type of a)

The “Vfull” rule

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The Vsimple rule is just a special case of Vfull.

The “Vfull” rule

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Using Vfull

• We are going to verify the simplest nontrivial higher order function we can, “apply”.

• It takes as an argument a pair of a function f and an argument v and just applies f to v.

• The “interesting” part is how the polymorphism is verified as opposed to the function behavior.

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Defining a calling convention

• To specify the “apply” function, we must define a calling convention for the sub-function

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Defining a calling convention

• The registers r1 – r4 are callee-saves; registers from r5 ! 1 registers are caller-saves

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Defining a calling convention

• A stdfun’s precondition, postcondition, and termination measure only depend on r0.

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Apply’s precondition, postcondition, and termination measure

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Apply’s precondition, postcondition, and termination measure

• These look “obvious” until you realize that P, Q, and t seem free in the definition. We will see how the Vfull rule “pipes” these in.

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The code is simple, although the verification isa bit cluttered…

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Notice how we track the termination measure

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The verification obligation from Vfull

This is somewhat large (thank goodness for machine-checking!). There are a few points of particular interest.

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The verification obligation from Vfull

1. We can check this function first, — i.e., ¡ = >That is, we verify this function before we verify thefunctions we will pass to it.

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The verification obligation from Vfull

2. The Vfull rule “pipes in” t, P, and Q – in fact, thetriple (t,P,Q) is the “b” from Vfull.

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The verification obligation from Vfull

3. We thread the termination argument into the verification required.

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“Stacking” apply

• As it happens, the apply function, itself, is a standard function (r1 – r4 are preserved, argument passed in r0, return in r0).

• In fact, we can pass “apply” to itself without difficulty. The rules will prevent us from unbounded recursion. We can “apply (apply (apply … (apply (f))))” but the function f “at the bottom” must terminate “on its own”.

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The key technical hurdle for soundness

All our key definitions (Hoare tuple, funptr, etc.) are defined in terms of “halt”, which is not obviously hereditary/continuous/ monotonic/downward closed.

A. What is hereditary?B. Why is halting not hereditary?

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The key technical hurdle for soundness

P hereditary ´ (w ² P Æ wÃw’) ! w’ ² P

Here à is the “age” or “approximate” operation on step-indexed worlds.

We can only approximate a finite number of times before we “hit bottom”. The step relation approximates w during function call.

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The key technical hurdle for soundness

The standard definition:

w ² halts(¾) ´ 9 wh. (w,¾) * (wh, [])

Let w à w’. The problem is that the relation approximates the world (at function call), so w’ might not have enough “time” left to actually reach the halting state [].

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The key technical hurdle for soundness

Our definition:

w ² haltsn(¾) ´

|w| > n ! (9 wh. (|w|-|wh| · n) Æ

(w,¾) * (wh, []) )

This is actually very similar to the basic definition.

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The key technical hurdle for soundness

Our definition:

w ² haltsn(¾) ´

|w| > n ! (9 wh. (|w|-|wh| · n) Æ

(w,¾) * (wh, []) )

Here is the exists and requirement that we step to the halt state.

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The key technical hurdle for soundness

Our definition:

w ² haltsn(¾) ´

|w| > n ! (9 wh. (|w|-|wh| · n) Æ

(w,¾) * (wh, []) )

This is the key “trick” – if the amount of time left in w is not enough, then we become true (and thus hereditary) trivially.

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The key technical hurdle for soundness

Our definition:

w ² haltsn(¾) ´

|w| > n ! (9 wh. (|w|-|wh| · n) Æ

(w,¾) * (wh, []) )

We must be sure that n is really a bound on the number of steps required to halt.

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The key technical hurdle for soundness

Our definition:

w ² haltsn(¾) ´

|w| > n ! (9 wh. (|w|-|wh| · n) Æ

(w,¾) * (wh, []) )

So, really not that bad. This is the “fundamental” trick that makes everything else possible.

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The key technical hurdle for soundness

See the paper (and/or Coq development) for how to use this definition to build up the Hoare tuple, etc.

Our top-level “erased” theorems are in standard form: if a program is verified in our logic, then it terminates.

All our proofs are machine checked in Coq.

Our core proofs are quite compact.

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Compact Proofs

• The main file is 826 lines long, and contains:A. Semantic model of the terminating function

pointer, Hoare judgment, whole-program verification judgment

B. 10+ Hoare rules and soundness proofsC. 3 whole-program rules and soundness proofsD. Top-level theorems (if P is verified, then it halts)

• Really quite short…