docs/alloy-model: add linear fee function model

In this commit, we add a model for the linear fee function we use in lnd
for fee bumping. This models can be used to reproduce the issue reported
in https://github.com/lightningnetwork/lnd/issues/8741, and can also be
shown that that bug fix resolves a counter example found by the model
checker.

docs/alloy-models/README.md

@@ -19,7 +19,7 @@ in development and understanding.
 
 Alloy is useful as when used during upfront software design (or even after the
 fact), one can create a formal model of a software system to gain better
-confidence re the _correctness_ of a system. The model can then be translated
+confidence in the _correctness_ of a system. The model can then be translated
 to code. Many times, writing the model (either before or after the code) can
 help one to better understand a given software system. Models can also be used
 to specify protocols in p2p systems such as Lightning (as it [supports temporal

docs/alloy-models/linear-fee-function/READM.md

@@ -0,0 +1,64 @@
+# Linear Fee Function
+
+This is a model of the default [Linear Fee
+Function](https://github.com/lightningnetwork/lnd/blob/b7c59b36a74975c4e710a02ea42959053735402e/sweep/fee_function.go#L66-L109)
+fee bumping mechanism in lnd.
+
+This model was inspired by a bug fix, due to an off-by-one error in the
+original code: https://github.com/lightningnetwork/lnd/issues/8741.
+
+The bug in the original code was fixed in this PR:
+https://github.com/lightningnetwork/lnd/pull/8751.
+
+
+## Model & Bug Fix Walk-through
+
+The model includes an assertion that captures the essence of the bug:
+`max_fee_rate_before_deadline`:
+```alloy
+// max_fee_rate_before_deadline is the main assertion in this model. This
+// captures a model violation for our fee function, but only if the line in
+// fee_rate_at_position is uncommented.
+//
+// In this assertion, we declare that if we have a fee function that has a conf
+// target of 4 (we want a few fee bumps), and we bump to the final block, then
+// at that point our current fee rate is the ending fee rate. In the original
+// code, assertion isn't upheld, due to an off by one error.
+assert max_fee_rate_before_deadline {
+  always req_num_blocks_to_conf[4] => bump_to_final_block => eventually (
+    all f: LinearFeeFunction | f.position = f.width.sub[1] &&
+                               f.currentFeeRate = f.endingFeeRate
+  )
+}
+```
+
+We can modify the model to find the bug described in the original issue.
+  1. First, we modify the model by forcing a `check` on the
+     `max_fee_rate_before_deadline` assertion: 
+    ```alloy
+    check max_fee_rate_before_deadline
+    ```
+
+  2. Next, we'll modify the `fee_rate_at_position` predicate to re-introduce
+     the off by one error:
+    ```alloy
+    p >= f.width => f.endingFeeRate // -- NOTE: Uncomment this to re-introduce the original bug.
+    ```
+
+If we hit `Execute` in the Alloy Analyzer, then we get a counter example:
+![Counter Example](counter-example.png)
+
+
+We can hit `Show` in the analyzer to visualize it: 
+![Counter Example Show](counter-example-show.png)
+
+We can see that even though we're one block (`position`) before the deadline
+(`width`), our fee rate isn't at the ending fee rate yet.
+
+If we modify the `fee_rate_at_position` to have the correct logic: 
+```alloy
+p >= f.width.sub[1] => f.endingFeeRate
+```
+
+Then Alloy is unable to find any counterexamples:
+![Correct Model](fixed-model.png)

docs/alloy-models/linear-fee-function/linear-fee.als

@@ -0,0 +1,311 @@
+// LinearFeeFunction is the abstract model of our fee function. We use one
+// here to indicate that only one of them exists in our model. Our model will
+// use time to drive this single instance.
+one sig LinearFeeFunction {
+  // startFeeRate is the starting fee rate of our model.
+  var startingFeeRate: Int, 
+  
+  // endingFeeRate is the max fee rate we'll be ending out once the deadline
+  // has been reached.
+  var endingFeeRate: Int,
+
+  // currentFeeRate is the current fee rate we're using to bump the
+  // transaction.
+  var currentFeeRate: Int,
+
+  // width is the number of blocks between the starting block height and the
+  // deadline height.
+  var width: Int,
+
+  // position is the number of between the current height and the starting
+  // height. This is our progress as time passes, we'll increment by a single
+  // block.
+  var position: Int, 
+
+  // deltaFeeRate is the fee rate time step in msat/kw that we'll take each
+  // time we update our progress (scaled by position). In this simplified
+  // model, this will always be 1.
+  var deltaFeeRate: Int,
+} {
+  // Here we have some _implicit_ facts attached to the definition of
+  // LinearFeeFunction. These ensure that we always get well formed traces in
+  // our model.
+
+  // position must be positive, and can never exceed our width. In our model,
+  // position is the amount of blocks that have elapsed, and width is the conf
+  // target.
+  position >= 0
+  position <= width
+
+  // startingFeeRate must be positive.
+  startingFeeRate > 0 
+
+  // currentFeeRate must be positive.
+  currentFeeRate > 0 
+
+  // endingFeeRate must be positive.
+  endingFeeRate > 0 
+
+  // deltaFeeRate always just increments by one. This simplifies our model, as
+  // Alloy doesn't support arithmetics with large numbers easily.
+  deltaFeeRate = 1
+}
+
+// ActiveFeeFunctions is a signature for the set of all active fee functions.
+// This is used to properly initialize a fee function before it can be used
+// elsewhere in the model.
+sig ActiveFeeFunctions {
+   // activeFuncs is the set of all active functions.
+   var activeFuncs: set LinearFeeFunction
+}
+
+// feeFuncInitZeroValues is a basic fact that capture the semantics of fee
+// functions and initialization.
+fact feeFuncInitZeroValues {
+   // If a function isn't in the set of active functions, then its position
+   // must be zero.
+   all f: LinearFeeFunction | 
+       f not in ActiveFeeFunctions.activeFuncs => f.position = 0
+}
+
+// feeFuncSetStartsEmpty captures a basic fact that at the very start of the
+// trace, there are no active fee functions. There're no temporal operators in
+// this fact, so it only applies to the very start of the trace.
+fact feeFuncSetStartsEmpty {
+  no activeFuncs
+}
+
+// init is a predicate to initialize the params of the fee function.
+pred init[f: LinearFeeFunction, maxFeeRate, startFeeRate: Int, confTarget: Int] {
+  // We only want to initiate once, so we'll have a guard that only proceeds if
+  // f isn't in the set of active functions.
+  f not in ActiveFeeFunctions.activeFuncs
+
+  // The starting rate and the ending rate shouldn't be equal to each other. 
+  maxFeeRate != startFeeRate
+  maxFeeRate > startFeeRate
+
+  // If the conf target is zero, then we'll just jump straight to max params.
+  confTarget = 0 implies {
+    f.startingFeeRate' = maxFeeRate
+    f.endingFeeRate' = maxFeeRate
+    f.currentFeeRate' = maxFeeRate
+    f.position' = 0
+    f.width' = confTarget
+    f.deltaFeeRate' = 1
+    ActiveFeeFunctions.activeFuncs' = ActiveFeeFunctions.activeFuncs + f
+  } else {
+    // Otherwise, we'll be starting from a position that we can use to drive
+    // forward the system. 
+    confTarget != 0 
+
+    f.startingFeeRate' = startFeeRate
+    f.currentFeeRate' = startFeeRate
+    f.endingFeeRate' = maxFeeRate
+    f.width' = confTarget
+    f.position' = 0
+
+    // Our delta fee rate is just always 1, we'll take a single step towards
+    // the final solution at a time. 
+    f.deltaFeeRate' = 1
+    ActiveFeeFunctions.activeFuncs' = ActiveFeeFunctions.activeFuncs + f
+  }
+}
+
+// increment is a predicate that implements our fee bumping routine.
+pred increment[f: LinearFeeFunction] {
+  // Update our fee rate to take into account our new position.  Increase our
+  // position by one, as a single block has past.
+  increase_fee_rate[f, f.position.add[1]]
+}
+
+// increase_fee_rate takes a new position, and our fee function, then updates
+// based on this relationship:
+//  feeRate = startingFeeRate + position * delta.
+//	- width: deadlineBlockHeight - startingBlockHeight
+//	- delta: (endingFeeRate - startingFeeRate) / width
+//	 - position: currentBlockHeight - startingBlockHeight
+pred increase_fee_rate[f : LinearFeeFunction, newPosition: Int] {
+  // Update the position of the model in the next timestep.
+  f.position' = newPosition
+
+  // Ensure that our position hasn't passed our width yet. This is an error
+  // scenario in the original Go code.
+  f.position' <= f.width
+
+  // Our new position is the distance between the 
+  f.currentFeeRate' = fee_rate_at_position[f, newPosition]
+ 
+  f.startingFeeRate' = f.startingFeeRate
+  f.endingFeeRate' = f.endingFeeRate
+  f.width' = f.width
+  f.deltaFeeRate' = f.deltaFeeRate
+  activeFuncs' = activeFuncs
+}
+
+// fee_rate_at_position computes the new fee rate based on an updated position.
+fun fee_rate_at_position[f: LinearFeeFunction, p: Int]: Int {
+  // If the position is equal to the width, then we'll go straight to the max
+  // fee rate.
+  p >= f.width.sub[1] => f.endingFeeRate
+  //p >= f.width => f.endingFeeRate -- NOTE: Uncomment this to re-introduce the original bug.
+  else 
+    // Otherwise, we'll do the fee rate bump.
+    let deltaRate = f.deltaFeeRate,
+        newFeeRate = f.currentFeeRate.add[deltaRate] |
+
+        // If the new fee rate would exceed the ending rate, then we clamp it
+        // to our max fee rate.  Otherwise we return our normal fee rate.
+        newFeeRate > f.endingFeeRate => f.endingFeeRate else newFeeRate
+}
+
+// fee_bump is a predicate that executes a fee bump step.
+pred fee_bump[f: LinearFeeFunction] {
+  // We use a guard to ensure that fee_bump can only be called after init
+  // (which will add the func to the set of active funcs).
+  f in ActiveFeeFunctions.activeFuncs
+
+  increment[f]
+}
+
+// stutter does nothing. This just assigns all variables to themselves, this is
+// useful in a trace to allow it to noop.
+pred stutter[f: LinearFeeFunction] {
+  // No change in any of our variables.
+  f.startingFeeRate' = f.startingFeeRate
+  f.endingFeeRate' = f.endingFeeRate
+  f.currentFeeRate' = f.currentFeeRate
+  f.position' = f.position 
+  f.width' = f.width
+  f.deltaFeeRate' = f.deltaFeeRate
+  activeFuncs' = activeFuncs
+}
+
+// Event is used to generate easier to follow traces. These these enums are
+// events that correspond to the main state transition predicates above.
+//
+// See this portion of documentation for more information on this idiom:
+// https://haslab.github.io/formal-software-design/modelling-tips/index.html#an-idiom-to-depict-events.
+enum Event { Stutter, Init, FeeBump }
+
+// stutter is a derived relation that's populated whenever a stutter event
+// happens.
+fun stutter : set Event -> LinearFeeFunction {
+  { e: Stutter, f: LinearFeeFunction | stutter[f] }
+}
+
+// init is a derived relation that's populated whenever an init event happens.
+fun init : Event -> LinearFeeFunction -> Int -> Int -> Int {
+  { e: Init, f: LinearFeeFunction, m: Int, s: Int, c: Int | init[f, m, s, c] }
+}
+
+// fee_bump is a derived relation that's populated whenever a fee_bump event
+// happens.
+fun fee_bump : Event -> LinearFeeFunction {
+  { e: FeeBump, f: LinearFeeFunction | fee_bump[f] }
+}
+
+// events is our final derived relation that stores all the possible event
+// types.
+fun events : set Event {
+  stutter.LinearFeeFunction + init.Int.Int.Int.LinearFeeFunction + fee_bump.LinearFeeFunction
+}
+
+// traces is a fact to ensure that we always get one of the above events in
+// traces we generate.
+fact traces {
+  always some events
+}
+
+// init_traces builds on the fact traces above, and also asserts that there's
+// always at least one init event (to ensure proper initialization).
+fact init_traces {
+  eventually (one init)
+}
+
+// fee_bump_happens ensures that within our trace, there's always at least a
+// single fee bump event.
+fact fee_bump_happens {
+  eventually (some fee_bump)
+}
+
+
+// valid_fee_rates is a fact that ensures that the ending fee rate is always
+// greater than the starting fee rate.
+fact valid_fee_rates {
+  all f: LinearFeeFunction | f.endingFeeRate > f.startingFeeRate
+}
+
+// init_correctness is an assertion that makes sure once an init event happens,
+// then we have the function in our set of active functions.
+assert init_correctness {
+  all f: LinearFeeFunction | {
+    eventually (some init) implies 
+      eventually f in ActiveFeeFunctions.activeFuncs
+  }
+}
+
+// check init_correctness
+
+// init_always_happens is a trivial assertion that ensures that an init
+// eventually happens in a trace.
+assert init_always_happens {
+   eventually (some init)
+}
+
+// check init_always_happens
+
+// init_then_fee_bump asserts that within our trace, we'll always have at least
+// a single fee bump event after an init.
+assert init_then_fee_bump {
+  eventually (some init => some fee_bump)
+}
+
+// check init_then_fee_bump
+
+// bump_to_completion is a predicate that can be used to force fee_bump events
+// to be emitted until we're right at our confirmation deadline.
+pred bump_to_completion {
+  always (all f: LinearFeeFunction | f.position < f.width => eventually (some fee_bump))
+}
+
+// bump_to_final_block is a predicate that can be used to force fee_bump events
+// to be emitted until we're one block before our confirmation deadline.
+pred bump_to_final_block {
+  always (all f: LinearFeeFunction | f.position < f.width.sub[1] => eventually (some fee_bump))
+}
+
+// req_num_blocks_to_conf is a predicate that can be used to restrict the
+// functions in a fact to a given confirmation target.
+pred req_num_blocks_to_conf[n: Int] {
+  all f: LinearFeeFunction | f.width = n
+}
+
+// req_starting_fee_rate is a predicate that can be used to restrict functions
+// in a fact to a given starting fee rate.
+pred req_starting_fee_rate[n: Int] {
+  all f: LinearFeeFunction | f.startingFeeRate = n
+}
+
+// max_fee_rate_before_deadline is the main assertion in this model. This
+// captures a model violation for our fee function, but only if the line in
+// fee_rate_at_position is uncommented.
+//
+// In this assertion, we declare that if we have a fee function that has a conf
+// target of 4 (we want a few fee bumps), and we bump to the final block, then
+// at that point our current fee rate is the ending fee rate. In the original
+// code, assertion isn't upheld, due to an off by one error.
+assert max_fee_rate_before_deadline {
+  always req_num_blocks_to_conf[4] => bump_to_final_block => eventually (
+    all f: LinearFeeFunction | f.position = f.width.sub[1] => f.currentFeeRate = f.endingFeeRate
+  )
+}
+
+check max_fee_rate_before_deadline
+
+run {
+  // Our default command just shows a trace that has at least 4 fee bump
+  // events.
+  always req_num_blocks_to_conf[4] 
+  always bump_to_completion
+}