Summary:
Strengthen normalization performed by Term and Formula constructors to
eliminate literal 0 subterms and true or false subformulas, as well as
cases where subterms or subformulas are either equal or opposite.
This strengthens the ability of Context.implies to prove formulas that
involve embedding or projecting between terms and formulas, as the
added normalization sometimes reduces if-then-else formulas to
literals that are then directly provable.
Reviewed By: ngorogiannis
Differential Revision: D23459512
fbshipit-source-id: 6d4d90399
Summary:
The dune build info embedded into executables is broken with dune <
2.7 and an flambda compiler.
Reviewed By: ngorogiannis
Differential Revision: D23459510
fbshipit-source-id: 18b3e3e15
Summary:
`Sh.norm` relies on `Eq (v, v)` formulas for `v` a variable to be
normalized to `Tt`, which is how eliminated variables are actually
removed from the representation.
Reviewed By: jvillard
Differential Revision: D22571132
fbshipit-source-id: 6d5f3efd7
Summary:
Move normalization that is necessary for `embed_into_fml` to be left
inverse to `embed_into_cnd` into the general `Fml` constructors.
Reviewed By: jvillard
Differential Revision: D22571137
fbshipit-source-id: 575c6bc45
Summary:
In order to ensure that the normalizing constructors are not
circumvented.
Reviewed By: jvillard
Differential Revision: D22571139
fbshipit-source-id: 32032c6fa
Summary:
The default printer is `pp_classes`, while `pp` is for debugging the
internal representation manipulation, so it is renamed to `pp_raw`.
Reviewed By: jvillard
Differential Revision: D22571135
fbshipit-source-id: 2d624e279
Summary:
`Formula.is_true` and `is_false` are now trivial one-line wrappers,
remove them for clarity.
Reviewed By: jvillard
Differential Revision: D22571143
fbshipit-source-id: 3f058eab4
Summary:
It is more confusing than necessary to use logical formula terminology
for the Context interface, considering that Formula represents
formulas and Context represents a (solver state resulting from a) set
of assumptions.
Reviewed By: jvillard
Differential Revision: D22571136
fbshipit-source-id: 087c97a02
Summary:
The unary forms are primitive in ICS, and in uncoming changes which
involve considering the product of a term and an equality relation, it
is more efficient to have unary constructors since the product is then
linear instead of quadratic in the size of the equality relation.
Reviewed By: jvillard
Differential Revision: D22571138
fbshipit-source-id: e0b745cc8
Summary:
Add a Predsym module for uninterpreted (unary) predicate symbols, and
positive and negative literals applying them to a term. As with
uninterpreted functions, tuple terms are used to represent predicates
of other arities.
Using this support, change the Ord and Uno formulas to uninterpreted
literals.
Reviewed By: jvillard
Differential Revision: D22571140
fbshipit-source-id: 5022a91e2
Summary:
`Context.difference` is now just a convenience function that does not
need to be defined internally.
Reviewed By: jvillard
Differential Revision: D22571141
fbshipit-source-id: 58aea9488
Summary: Also, when printing in raw mode, do not print the context.
Reviewed By: jvillard
Differential Revision: D22571145
fbshipit-source-id: b3596d9cc
Summary:
Make the relationship between Sh.is_empty and Sh.pure_approx stronger
and more precise. In particular:
> If [is_empty q], then [pure_approx q] is equivalent to
> [pure (pure_approx q)].
This enables replacing Solver.excise_pure with a simpler pure_entails
function. In particular, the heavy reliance on normalization of pure
formulas to true or false literals is eliminated, and only pure
entailment is needed.
Reviewed By: jvillard
Differential Revision: D22571146
fbshipit-source-id: 2fca64a61
Summary:
Generalize Fol interface to allow checking if a context implies any
formula, rather than restricting to only equalities.
Reviewed By: jvillard
Differential Revision: D22571144
fbshipit-source-id: 726bd87fd
Summary:
There is nothing specific to the Ses representation in the
implementation, and no uses within Ses.
Reviewed By: jvillard
Differential Revision: D22571150
fbshipit-source-id: 8952f0301
Summary:
In Ses, the constant term of a polynomial is represented as a
redundant multiplication by 1. Fix Fol.of_ses to recognize and
normalize this.
Reviewed By: jvillard
Differential Revision: D22571131
fbshipit-source-id: 3e1a12e5f
Summary:
The Ses constructors might simplify terms when called from
Fol.to_ses. Fix Fol.ses_map to account for this.
Reviewed By: jvillard
Differential Revision: D22571151
fbshipit-source-id: 1d573ac5f
Summary:
There is nothing specific to the Ses representation in the
implementation, and no uses within Ses.
Reviewed By: jvillard
Differential Revision: D22455725
fbshipit-source-id: 6f0059873
Summary:
In preparation for more smoothly interoperating with ICS's functional
array theory.
Reviewed By: jvillard
Differential Revision: D22401039
fbshipit-source-id: 4de39c38a
Summary:
It is not needed or very meaningful since the addition of type-based
disambiguation.
Reviewed By: jvillard
Differential Revision: D22401035
fbshipit-source-id: 31996f946
Summary:
The first-order context is induced by the pure part, so no need to
compare it.
Reviewed By: jvillard
Differential Revision: D22381645
fbshipit-source-id: 29fff13a3
Summary:
In order to allow implementations of the single Fol interface using
multiple backend first-order logic solvers, add explicit definitions
of terms and formulas in the Fol module, and implement Context in
terms of them.
The Fol interface supports freely mixing Terms and Formulas, in
particular there is `Term.ite : cnd:Formula.t -> thn:Term.t ->
els:Term.t -> Term.t` which allows Formulas to appear in Terms. The
Fol implementation performs enough normalization to enable using an
internal representation of terms that is strictly partitioned into
"theory terms" and "formulas", which are stratified below "conditional
terms" and then below "general terms". This partitioning and
stratification enables using backend solvers that do not support
mixing formulas in terms.
Reviewed By: jvillard
Differential Revision: D22170506
fbshipit-source-id: a014ee7d7
Summary:
The eventual aim is for the conversion of Llair to Fol to be external
to Fol. Fol should not need to depend on Llair, nor vice versa. This
is not yet possible, but a step forward is to move the conversion
functions into separate modules outside the core Fol modules.
Reviewed By: ngorogiannis
Differential Revision: D22170522
fbshipit-source-id: 4860b4c07
Summary:
The `Subst` module is not used by `Ses`, so move it out of the
Ses-internal APIs.
Reviewed By: ngorogiannis
Differential Revision: D22170513
fbshipit-source-id: a189a1440
Summary:
Change Term and Equality rename functions to accept a partial
application of Var.Subst.apply to the renaming substitution.
Reviewed By: ngorogiannis
Differential Revision: D22170520
fbshipit-source-id: 003d8b27e
Summary:
The convention is for modules that are intended to be `open`ed, that
define syntax and infix operations, are named `Import`. This diff
combines the `Option.Monad_infix` and `Option.Monad_syntax` modules
into `Option.Import` to follow this convention.
Reviewed By: ngorogiannis
Differential Revision: D22170507
fbshipit-source-id: 44378fd56
Summary:
Exponents in polynomials are represented by rationals for code reuse
purposes, but only integral exponents are supported. Strengthen the
invariant check to clarify this.
Reviewed By: ngorogiannis
Differential Revision: D22170517
fbshipit-source-id: 81ae38f4e
Summary:
Refactor `Equality.ppx_classes_diff` into `diff_classes` to construct
the difference between classes maps explicitly, and change
`ppx_classes` to accept such a difference instead of computing it
internally.
This more flexibly composable interface allows elimination of extra
calls to `Equality.entails` to check if the difference is empty.
Reviewed By: jvillard
Differential Revision: D22038488
fbshipit-source-id: c19c18fc8
Summary:
The list interpreted as a conjunction of individual terms in `Sh.pure`
is now redundant with `Term.And`. This patch removes the redundant
list.
Reviewed By: jvillard
Differential Revision: D22035852
fbshipit-source-id: 49c01a078
Summary:
With the current handling of fresh variable generation during symbolic
execution, it is now possible to delay generating fresh variables in
individual small axioms until the precondition is known. In
particular, the existential variables of the precondition formula can
be bound, and then the small axiom can be generated with variables
fresh with respect to them. Previously, the small axioms were
generated with fresh variables that could later clash with the
precondition's existentials, necessitating renaming. This
double-freshening is now eliminated.
Reviewed By: jvillard
Differential Revision: D21974022
fbshipit-source-id: f217bfb9f
Summary:
When fresh variables are generated to name the overwritten value in an
assignment, they should be included in the ghost variables of the
resulting small axiom. This change should have been included in the
elimination of SSA.
Also strengthen assertion checking of small specs during symbolic
execution.
Reviewed By: jvillard
Differential Revision: D21974019
fbshipit-source-id: a66d8dac6
Summary:
Currently the symbolic execution code in `Exec` manually threads
universal and existential variable contexts through virtually every
function. It is easy to mistakenly pass on a context that is not the
latest-extended one, or to forget to add generated variables to the
contexts.
This patch adds a state monad, `Fresh`, to manage the generation of
fresh variables in `Exec`. This is a standard state monad where the
state is two sets of variables: those to which fresh variables must be
chosen fresh, and those which have been generated. This yields an
abstraction where an `'a Fresh.t` value represents a value of type
`'a` which may contain as-yet-unnamed variables, and `Fresh.gen ~wrt
a` generates names that are fresh with respect to `wrt` for all
unnamed variables in `a`, and yields the set of generated variables
together with `a` expressed in terms of those variables.
Reviewed By: jvillard
Differential Revision: D21974018
fbshipit-source-id: 1917e82c0
Summary: Minor code simplification and optimization of `extend_us` in no-op case.
Reviewed By: jvillard
Differential Revision: D21974021
fbshipit-source-id: a8b12b564
Summary:
Change the `Var.Subst` `freshen` and `restrict` constructors to return
the domain and range of the substitution explicitly. Clients generally
need to compute them immediately, and they are at least partially
constructed during the initial substitution construction anyhow. This
may be an incidental minor optimization.
This allows removing the `apply_set` operation, as it's use can be
handled directly from the domain and range sets.
This also allows `Sh.rename` to be split into a function that assumes
that the substitution is restricted to the vocabulary of the formula,
and a wrapper that does this restriction and calls through. This
allows `Sh.freshen_xs` to be simplified slightly, and avoids some
redundant restriction, domain, and range computations.
Reviewed By: jvillard
Differential Revision: D21974017
fbshipit-source-id: aa8b3db24
Summary:
`Domain_sh.from_call.subst` is a substitution that replaces shadowed
variables with fresh ones, which is constructed by `Domain_sh.call`
and used by `Domain_sh.retn`, after inverting it. This patch changes
the stored substitution to the inverted one, and renames it to
`unshadow` for clarity.
After this change, the stored substitutions have the property that
they map variables to program variables. This is desirable since it
avoids the question about the uninverted substitution of whether the
variables in the range of the uninverted substitution are "fresh".
Reviewed By: jvillard
Differential Revision: D21974020
fbshipit-source-id: d469c89f9
Summary:
Logically there is nothing specific to memory contents (as
byte-arrays) or aggregate (struct/array) values, the theory is for
sequences of non-fixed sized elements.
Reviewed By: jvillard
Differential Revision: D21721019
fbshipit-source-id: b2b730a50
Summary:
`Term.eq_concat` is not primitive and complicates the `Term`
interface. Move it to a couple clients as a convenience wrapper.
Reviewed By: jvillard
Differential Revision: D21721026
fbshipit-source-id: 0d74aa251
Summary:
Previously `null` and `zero` had different sorts/types, but now they
are equivalent.
Reviewed By: jvillard
Differential Revision: D21721023
fbshipit-source-id: 485219f6a
Summary:
Multiplication by a constant is primitive in the linear arithmetic
solver, while general multiplication is not, so for clarity and
predictability, use constants where possible.
Reviewed By: jvillard
Differential Revision: D21721020
fbshipit-source-id: 3497d06c9
Summary:
It is suboptimal for `Sh` to destruct terms with detailed knowledge of
their representation. So add `Term.const_of` to obtain the constant
summand of a polynomial term.
Reviewed By: jvillard
Differential Revision: D21721022
fbshipit-source-id: 4af858896
Summary:
It is suboptimal for `Sh` to destruct terms with detailed knowledge of
their representation. So add `Term.d_int` to destruct an integer term.
Reviewed By: jvillard
Differential Revision: D21721024
fbshipit-source-id: 5f13794b6
Summary:
It is suboptimal for `Sh` to destruct terms with detailed knowledge of
their representation. So add `Term.disjuncts` to obtain the toplevel
disjuncts of a term.
Reviewed By: jvillard
Differential Revision: D21721016
fbshipit-source-id: 809da9b1b
Summary:
It has no dependencies on the rest of the sledge codebase and might be
more generally useful.
Reviewed By: jvillard
Differential Revision: D21720980
fbshipit-source-id: b4f061e73
Summary:
Now that the frontend translates LLVM's undef to nondet instructions,
not expressions, Nondet in Exp and Term are not needed.
Reviewed By: jvillard
Differential Revision: D21720969
fbshipit-source-id: e8acaf432
Summary:
Instead of relying on Exp.nondet to encode the semantics of LLVM's
undef, translate each to a per-function unique register, with a nondet
assignment to it prior to each use. This avoids the need for
Exp.nondet, which is ill-formed in the sense that expressions denote
values, not sets of values (with particular constraints on what ways
in which the choice must be angelic vs demonic). This change
essentially allows the backend to be sane, and makes it the frontend's
problem to deal with LLVM's undef.
This treatment, like the treatment based on Exp.nondet, is expected to
result in LLAIR code with different semantics of undef compared to the
semantics of LLVM described in the
[LangRef](https://llvm.org/docs/LangRef.html#undefined-values). In
particular, the LLVM LangRef states
> An ‘undef’ “variable” can arbitrarily change its value over its
> “live range”. This is true because the variable doesn’t actually
> have a live range. Instead, the value is logically read from
> arbitrary registers that happen to be around when needed, so the
> value is not necessarily consistent over time.
To model this ability of undef to arbitrarily change its value over
its live range, it is likely that additional nondet assignments would
need to be added. Exactly where it not currently known.
Reviewed By: jvillard
Differential Revision: D21720976
fbshipit-source-id: 90c2a0d26
Summary:
Refactor frontend translation of LLVM values, opcodes, etc. to support
emitting not only a LLAIR expression, but also a sequence of
instructions to be prefixed onto the uses of the resulting expression.
This is currently unused, as all prefixes are empty. Later, it will be
used to translate e.g. `undef` to `r := nondet(); r`.
Reviewed By: jvillard
Differential Revision: D21720972
fbshipit-source-id: b89bb57de
Summary:
Refer to Llair modules using `Llair.` qualifier, except for in
`Frontend`, which makes so much use of `Llair` that it is now opened
(`Llair` only contains types and modules, so `open` is safe).
Reviewed By: jvillard
Differential Revision: D21720979
fbshipit-source-id: dd42075d9
Summary:
The term representing an exp should not rely on more info than is
carried by the exp.
Reviewed By: jvillard
Differential Revision: D21720989
fbshipit-source-id: b65bf3678
Summary:
It is now possible to not spew dune files all over the repo, and opam
files aren't needed either.
Reviewed By: jvillard
Differential Revision: D21720978
fbshipit-source-id: 553e1d154
Summary:
```
val ( let@ ) : ('a -> 'b) -> 'a -> 'b
(** [let@ x = e in b] is equivalent to [e @@ fun x -> b], that is,
[e (fun x -> b)] *)
```
Reviewed By: jvillard
Differential Revision: D21721025
fbshipit-source-id: d8efdebbe
Summary:
Rather than compute the size of the llair type of the llair
initializer expression, compute the size of the llvm initializer
directly.
Reviewed By: ngorogiannis
Differential Revision: D21720982
fbshipit-source-id: 4364baf38
Summary:
Rather than translate an llvm type to a llair one and then compute the
size of the llair type, obtain the size of the llvm type directly.
Reviewed By: ngorogiannis
Differential Revision: D21720968
fbshipit-source-id: ad98112a7
Summary:
In LLVM it is possible for struct constant values to be directly
recursive, with no pointer dereference to close the cycle. These
appear for example as the values of vtables from C++ code.
Currently such recursive records in the Exp and Term languages are
represented as genuinely cyclic values. Compared to a standard term
representation, the presence of cyclic values is a significant
complication everywhere. Since the backend solver does not do anything
such as induction over these, they have to be treated as essentially
atomic.
This patch changes the representation to a standard non-recursive tree
term structure. Instead of cyclic references, an explicit constructor
is added for the "non-tree edges", which simply indicates which
ancestor record value to which the recursive reference points.
There is a potential issue with this representation, since for
mutually recursive records, the representation is not canonical: it
chooses one of the records in the cycle to start from and expresses
the cycles relative to that. Currently the choice of representation is
dictated by the frontend. For the case of vtables, the frontend
translates globals in the same order they appear in the LLVM IR, so
the representation choice is fixed.
It may turn out that other potential uses require more reasoning
support in the backend solver, which would involve a theory of
equality of record values induced by equating the representations
resulting from different rotations of the cycle of records.
Reviewed By: jvillard
Differential Revision: D21441533
fbshipit-source-id: 0c5a11378
Summary:
Having `val size_of : Typ.t -> t` in the signature of `Term` and `val
size_of : t -> t` in the signature of `Exp` gives the impression that
`Term` and `Exp` know something about `Typ`. But they don't, those
functions are only trivial convenience wrappers, and only have a few
uses, so just inline them to clarify that it is `Typ` that knows about
the sizes of types.
Reviewed By: jvillard
Differential Revision: D21441535
fbshipit-source-id: 09b135a8c
Summary:
Should have been included in:
22cbec493 [sledge] Rename contexts/profiles to be more conventional
Reviewed By: jvillard
Differential Revision: D21441538
fbshipit-source-id: 92b591171
Summary:
Dune does not track the ocamlformat binary as a dependency of the
`fmt` rules, so when testing or upgrading ocamlformat, the existing
`fmt` target is not reliable.
Reviewed By: jvillard
Differential Revision: D21441539
fbshipit-source-id: 40bea4447
Summary:
Add a path condition to each symbolic state, represented in sledge's arithmetic domain. This gives a precise account of arithmetic constraints. In particular, it is relation and thus is more robust in the face of inter-procedural analysis.
This is gated behind a flag for now as there are performance issues with the new arithmetic.
Reviewed By: jberdine
Differential Revision: D20393947
fbshipit-source-id: b780de22a
Summary:
Show that the SSA restrictions imply that the blocks can be
topologically sorted so that any use of a variable follows its
definition in the list of blocks. This showed a slight flaw in my
definition of SSA form. It used to treat program counters up to
equality. However, PCs that point to a block header contain the location
that control came from (so that the Phi instructions can be executed),
but the SSA restrictions shouldn't pay attention to that, so now the
definition of SSA introduces a equivalence on PCs that ignores that
additional information.
Reviewed By: jberdine
Differential Revision: D21066873
fbshipit-source-id: 735575a1f
Summary:
Introduce an explicit assumption that all of the instructions are ones
that are implemented in the translation so far, rather than just
cheating proof cases.
Reviewed By: jberdine
Differential Revision: D20891352
fbshipit-source-id: 37598bd8f
Summary:
Separate into separate files the theorems that are just about the
translation (mostly about the structure of the variable->expression
mapping that the translation builds) from theorems about the translation
and the semantics.
Also move the stuff about dominator_ordered into the SSA Theory, since
it only makes sense for SSA programs, but doesn't have anything to do
with the translation.
Reviewed By: jberdine
Differential Revision: D20673124
fbshipit-source-id: 9d8b08164
Summary:
The LLVM->LLAIR translation keeps a mapping of variables to expressions.
Previously, the invariants related to that mapping were kept in the
state relation, and so the proof needed show that they were preserved
along execution traces. This wasn't obvious as the state changes in
non-SSA ways during evaluation, but the correctness of the mappings is
heavily based on the program being in SSA form. This change separates
out the invariants, and the proof uses the final mapping that the
compiler builds, which contains all of the relevant bindings that might
be needed during execution.
Reviewed By: jberdine
Differential Revision: D20625109
fbshipit-source-id: d4c2dfe19
Summary:
Treat the remainder of dividing a rational by an integer as if the
rational was an integer division.
Reviewed By: jvillard
Differential Revision: D21042515
fbshipit-source-id: b5d42ddec
Summary:
The partial treatment of Mul and Div terms can simplify some cases,
but since it is only a partial treatment that is not producing a
normal form, in other cases the "simplification" results in large and
non-canonical terms. It is safer to leave them uninterpreted.
Reviewed By: jvillard
Differential Revision: D21042521
fbshipit-source-id: 04fc37f1a
Summary:
The heights of And and Or terms can grow high. This interacts poorly
with some unoptimized Equality operations such as normalization that
do some processing at every subterm.
Reviewed By: jvillard
Differential Revision: D21042518
fbshipit-source-id: 55e6acbb1
Summary:
These are map and folding map that perform a cycle-preserving
pre-order transformation.
Reviewed By: jvillard
Differential Revision: D20877974
fbshipit-source-id: 251288228
Summary:
The current notion of "simplified" function symbols, which are treated
as a hybrid between interpreted and uninterpreted, has no logical
basis. Normalization is now strong enough, due to stronger handling of
the changing carrier set, that the "simplified" classification can be
removed.
Reviewed By: jvillard
Differential Revision: D20726961
fbshipit-source-id: 9962ea323
Summary:
It is possible for the canonical form of a term in (the carrier set
of) a relation to be a term that is not in the relation. It is also
possible for this term to be equal to a term in the relation. It is
the job of the lookup subroutine of normalization to find these
equations.
The current implementation of entails_eq is incomplete in this case,
in particular, when an uninterpreted term has an interpreted subterm,
which itself has a subterm that is not a representative.
This diff strengthens normalization (and hence it's callers such as
entails_eq) to handle such cases. This makes lookup work slightly
harder. An alternative would be to extend the carrier with new terms
from normalization to the carrier and closing the relation, and then
re-normalizing. That would lead to much inflated representation sizes,
and is inefficient.
Reviewed By: jvillard
Differential Revision: D20612565
fbshipit-source-id: 3b7534a62
Summary:
This diff adds enough interpretation of Mul and Div terms to be able
to exclude them from the domain of solution substitutions. While
non-linear arithmetic is still treated very incompletely, this change
increases the propagation power of the equality constraints that are
deduced. Mainly, this appears to be enough to avoid operations that
are semantically equivalence-preserving such as solve_for_vars from
producing equalities that are unprovable from their inputs.
Reviewed By: jvillard
Differential Revision: D20863528
fbshipit-source-id: fca74cba3
Summary:
For non-linear polynomial equations, the solver can currently choose
to solve for a variable that occurs in a non-linear subterm, leading
to a non-idepotent solution substitution. For example, `x - y + (x ×
z) = 0` could be solved for `x`, leading to `x ↦ y - (x × z)` where
`x` is solved in terms of itself. This diff refines Term.solve_zero_eq
to avoid such cyclic solutions.
Reviewed By: jvillard
Differential Revision: D20637482
fbshipit-source-id: 6d7df85c3
Summary:
Strengthen the canonizer for division and rational constant terms. It
is important to avoid constructing `Q.t` values with 0 denominators,
as they do not represent real numbers, and the algebraic manipulation
done by the rest of the solver is incorrect in that case.
Additionally strengthen the Term representation invariants to check
that all coefficients and exponents are valid real rational numbers.
Also normalize division by an integer or rational to multiplication by
a rational.
Reviewed By: jvillard
Differential Revision: D20663964
fbshipit-source-id: 210962fe0
Summary:
Equations of the form `a = b` where `a` is a proper subterm of `b` are
possible when uninterpreted functions are involved. Internally,
Equality does not eagerly substitute `b` for `a`, but external clients
can repeatedly `Equality.normalize` terms and thereby incrementally
blow up the sizes of terms.
This diff uses the heights of uninterpreted terms to choose equality
class representatives to avoid such blow-ups, by orienting equations
so that tall terms are represented by short terms, so that repeated
normalization cannot increase term height indefinitely.
Reviewed By: jvillard
Differential Revision: D20785632
fbshipit-source-id: ff4c5bacd
Summary:
The types are constants and need not be re-checked for equality during
normalization, etc.
Reviewed By: jvillard
Differential Revision: D20863526
fbshipit-source-id: 1adde5ee0
Summary:
Ensure all entry points check the representation invariant before
returning, and strengthen it to check the constraints on preference
between representative terms, and to check that the relation is
closed.
Reviewed By: jvillard
Differential Revision: D20612566
fbshipit-source-id: b345397c4
Summary:
Change the implementation of `Equality.Subst.compose` to preserve
physical equality if the result is `Subst.equal` to the first
argument.
Reviewed By: jvillard
Differential Revision: D20752671
fbshipit-source-id: 4641a298a
Josh Berdine
Scott Owens
Jules Villard