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We are still working on updating the documentation, and completion of the ICS decision procedures. Please let us know of any bugs or suggestions you have by sending them to pvs-bugs@csl.sri.com
You can download it here.
In addition to the usual bug fixes, there are quite a few changes to this release. Most of these changes are backward compatible, but the new multiple proofs feature makes it difficult to run PVS 3.0 in a given context and then revert back to an earlier version. For this reason we strongly suggest that you copy existing directories (especially the proof files) before running PVS 3.0 on existing specifications.
There are a number of new features in PVS 3.0.
PVS 3.0 has been ported to the case-sensitive version of Allegro version 6.0. This was done in order to be able to use the XML support provided by Allegro 6.0. We plan to both write and read XML abstract syntax for PVS, which should make it easier to interact with other systems.
Note: for the most part, you may continue to define pvs-strategies (and the files they load) as case insensitive, but in general this cannot always be done correctly, and it means that you cannot load such files directly at the lisp prompt. If you suspect that your strategies are not being handled properly, try changing it to all lower case (except in specific cases), and see if that helps. If not, send the strategies file to pvs-bugs and we'll fix it as quickly as we can. Because there is no way to handle it robustly, and since case-sensitivity can actually be useful, in the future we may no longer support mixed cases in strategy files.
Theory interpretations are described fully in Theory Interpretations in PVS
NOTES:
foo: THEORY = bar[int, 3]should be changed to the new form
IMPORTING bar[int, 3] AS fooNote that `
AS
' is a new keyword, and may cause parse errors
where none existed before.
push
, top
, and pop
should be
changed to
push := LAMBDA (x: t, A: E[cstack, ce]): equiv_class[cstack,ce](cpush(x)(rep(A))), top := LAMBDA (A: E[cstack, ce] | cnonempty?(rep(A))): ctop(rep(A)), pop := LAMBDA (A: E[cstack, ce] | cnonempty?(rep(A))): equiv_class[cstack,ce](cpop(rep(A)))Otherwise unprovable TCCs result (e.g., every stack is nonempty).
PVS now supports multiple proofs for a given formula. When a proof attempt is completed, either by quitting or successfully completing the proof, the proof is checked for changes. If any changes have occured, the user is queried about whether to save the proof, and whether to overwrite the current proof or to create a new proof. If a new proof is created, the user is prompted for a proof identifier and description.
In addition to a proof identifier, description, and proof script,
the proof objects contain the status, the date of creation, the date
last run, and the run time. Note that this information is kept in the
.prf
files, which therefore look different from those of
earlier PVS versions.
Every formula that has proofs has a default proof, which is used for most of the existing commands, such as prove, prove-theory, and status-proofchain. Whenever a proof is saved, it automatically becomes the default.
Three new Emacs commands allow for browsing and manipulating multiple
proofs: display-proofs-formula
,
display-proofs-theory
,
and display-proofs-pvs-file
. These commands all pop up buffers
with a table of proofs. The default proof is marked with a `+
'.
Within such buffers, the following keys have the following effects.
At the end of a proof a number of questions may be asked:
This may be annoying to some users, so the command M-x
pvs-set-proof-prompt-behavior
was added to control this. The
possible values are:
:ask
:overwrite
:add
Note that the id and description may be modified later using the commands described earlier in this section.
PVS now uses the PVS_LIBRARY_PATH
environment variable to look
for library pathnames, allowing libraries to be specified as simple
(subdirectory) names. This is an extension of the way, for example,
the finite_sets
library is found relative to the PVS
installation path - in fact it is implicitly appended to the end the
PVS_LIBRARY_PATH
.
The .pvscontext
file stores, amongst other things, library
dependencies. Any library found as a subdirectory of a path in the
PVS_LIBRARY_PATH
is stored as simply the subdirectory name.
Thus if the .pvscontext
file is included in a tar file, it may
be untarred on a different machine as long as the needed libraries
may be found in the PVS_LIBRARY_PATH
. This makes libraries
much more portable.
In addition, the load-prelude-library
command now
automatically loads the pvs-lib.el
file, if it exists, into
Emacs and the pvs-lib.lisp
file, if it exists, into lisp,
allowing the library to add new features, e.g., key-bindings. Note
that the pvs-lib.lisp
file is not needed for new strategies,
which should go into the pvs-strategies
file as usual. The
difference is that the pvs-strategies
file is only loaded
when a proof is started, and it may be desirable to have some lisp code
that is loaded when the library is, i.e., to support some new Emacs
key-bindings.
The PVS_LIBRARY_PATH
is a colon-separated list of paths, and
the lib
subdirectory of the PVS path is added implicitly at
the end. Note that the paths given in the PVS_LIBRARY_PATH
are expected to have subdirectories, e.g., if you have put Ben Di Vito's
Manip-package
in ~/pvs-libs/Manip-1.0
, then your PVS_LIBRARY_PATH
should only include ~/pvs-libs
, not
~/pvs-libs/Manip-1.0
.
If the pvs-libs.lisp
file needs to load other files in other
libraries, use libload
. For example, C'esar Mu@~noz's
Field Package
loads the Manip-package
using (libload "Manip-1.0/manip-strategies")
A new command, M-x list-prelude-libraries
, has been added that
shows the prelude library and supplemental files that have been
loaded in the current context.
PVS now supports cotuple types (also known as coproduct or sum types)
directly. The syntax is similar to that for tuple types, but with the
`,
' replaced with a `+
'. For example,
cT: TYPE = [int + bool + [int -> int]]
Associated with a cotuple type are injections IN_
i,
predicates IN?_
i, and extractions
OUT_
i (none of these is case-sensitive). For
example, in this case we have
IN_1: [int -> cT] IN?_1: [cT -> bool] OUT_1: [(IN?_1) -> int]
Thus IN_2(true)
creates a cT
element, and an
arbitrary cT
element c
is processed using
CASES
, e.g.,
CASES c OF IN_1(i): i + 1, IN_2(b): IF b THEN 1 ELSE 0 ENDIF, IN_3(f): f(0) ENDCASES
This is very similar to using the union
datatype defined in the
prelude, but allows for any number of arguments, and doesn't generate
a datatype theory.
Typechecking expressions such as IN_1(3)
requires that the
context be known. This is similar to the problem of a standalone
PROJ_1
, and both are now supported:
F: [cT -> bool] FF: FORMULA F(IN_1(3)) G: [[int -> [int, bool, [int -> int]]] -> bool] GG: FORMULA G(PROJ_1)
This means it is easy to write terms that are ambiguous:
HH: FORMULA IN_1(3) = IN_1(4) HH: FORMULA PROJ_1 = PROJ_1
This can be disambiguated by providing the type explicitly:
HH: FORMULA IN_1[cT](3) = IN_1(4) HH: FORMULA PROJ_1 = PROJ_1[[int, int]]
This uses the same syntax as for actual parameters, but doesn't mean
the same thing, as the projections, injections, etc., are builtin, and
not provided by any theories. Note that coercions don't work in this
case, as PROJ_1::[[int, int] -> int]
is the same as
(LAMBDA (x: [[int, int] -> int]): x)(PROJ_1)
and not
LAMBDA (x: [int, int]): PROJ_1(x)
The prover has been updated to handle extensionality and reduction rules as expected.
Coinductive definitions are now supported. They are like inductive
definitions, but introduced with the keyword `COINDUCTIVE
', and
generate the greatest fixed point.
Update expressions now work on datatypes, in much the same way they work
on records. For example, if lst: list[nat]
, then lst WITH
[`car := 0]
returns the list with first element 0, and the rest the
same as the cdr of lst
. In this case there is also a TCC of the
form cons?(lst)
, as it makes no sense to set the car of
null
.
Complex datatypes with overloaded accessors and dependencies are also handled. For example,
dt: DATATYPE BEGIN c0: c0? c1(a: (even?), b: int): c1? c2(a: nat, c: int): c2? END dt datatype_update: THEORY BEGIN IMPORTING dt x: dt y: int f: dt = x WITH [a := y] END datatype_update
This generates the TCC
f_TCC1: OBLIGATION (c1?(x) AND IF c1?(x) THEN even?(y) ELSE y >= 0 ENDIF) OR (c2?(x) AND IF c1?(x) THEN even?(y) ELSE y >= 0 ENDIF);
There are two additions to the theory generated from a datatype: a new ord function, and an every relation. Both of these can be seen by examining the generated theories.
The new ord function is given as a constant followed by an ordinal axiom. The reason for this is that the disjointness axiom is not generated, and providing interpretations for datatype theories without it is not sound. However, for large numbers of constructors, the disjointness axiom gets unwieldy, and can significantly slow down typechecking. The ord axiom simply maps each constructor to a natural number, thus using the builtin disjointness of the natural numbers. For lists, the new ord function and axiom are
list_ord: [list -> upto(1)] list_ord_defaxiom: AXIOM list_ord(null) = 0 AND (FORALL (car: T, cdr: list): list_ord(cons(car, cdr)) = 1);
This means that to fully interpret the list datatype, list_ord
must be given a mapping and shown to satisfy the axiom.
If a top level datatype generates a map theory, the theory also contains
an every
relation. For lists, for example, it is defined as
every(R: [[T, T1] -> boolean])(x: list[T], y: list[T1]): boolean = null?(x) AND null?(y) OR cons?(x) AND cons?(y) AND R(car(x), car(y)) AND every(R)(cdr(x), cdr(y));
Thus, every(<)(x, y: list[nat])
returns true if the
lists x and y are of the same length, and each element of x
is
less than the corresponding element of y
.
Conversions are now applied to the components of tuple, record, and
function types. For example, if c1
is a conversion from
nat
to bool
, and c2
from
nat
to list[bool]
, the tuple (1, 2,
3)
will be converted to (c1(1), 2, c2(3))
if the
expected type is [bool, nat, list[bool]]
. Records are
treated the same way, but functions are contravariant in the domain; if
f
is a function of type [bool ->
list[bool]]
, and the expected type is [nat ->
bool]
, then the conversion applied is LAMBDA (x: nat):
c2(f(c1(x)))
.
Conversions now apply pointwise where possible. In the past, if
x
and y
were state variables, and
K_conversions
enabled, then x < y
would be
converted to LAMBDA (s: state): x(s) < y(s)
, but x
= y
would be converted to LAMBDA (s: state): x = y
,
since the equality typechecks without applying the conversion pointwise.
Of course, this is rarely what is intended; it says that the two state
variables are the same, i.e., aliases. The conversion mechanism has been
modified to deal with this properly.
Messages related to conversions have been separated out, so that if any are generated a message is produced such as
po_lems typechecked in 9.56s: 10 TCCs, 0 proved, 3 subsumed, 7 unproved; 4 conversions; 2 warnings; 3 msgs
In addition, the commands M-x show-theory-conversions
and
M-x show-pvs-file-conversions
have been added to view the
conversions.
Trivial TCCs of the form x /= 0 IMPLIES x /= 0
and 45 <
256
used to quietly be suppressed. Now they are added to the
messages associated with a theory, along with subsumed TCCs. In addition,
both trivial and subsumed TCCs are now displayed in commented form in the
show-tccs buffer.
The command M-x show-declaration-tccs
has been added. It shows
the TCCs associated with the declaration at the cursor, including the
trivial and subsumed TCCs as described above.
Numbers may now be declared as constants, e.g.,
42: [int -> int] = LAMBDA (x: int): 42
This is most useful in defining algebraic structures (groups, rings, etc.), where overloading 0 and 1 is common mathematical practice. It's usually a bad idea to declare a constant to be of a number type, e.g.,
42: int = 57
Even if the typechecker doesn't get confused, most users would.
When the parser encounters an importing for a theory foo
that
has not yet been typechecked, it looks first in the .pvscontext
file, then looks for foo.pvs
. In previous versions, if the theory
wasn't found at this point an error would result. The problem is that
file names often don't match the theory names, either because a given file
may have multiple theories, or a naming convention (e.g., the file is
lower case, but theories are capitalized)
Now the system will parse every .pvs
file in the current
context, and if there is only one file with that theory id in it, it will
be used. If multiple files are found, a message is produced indicating
which files contain a theory of that name, so that one of those may be
selected and typechecked.
NOTES:
.pvscontext
is
updated accordingly, and this check is no longer needed.
.pvs
files that contain parse errors will be ignored.
The existing (named Shostak, for the original author) decision procedures have been made more complete. Note that this sometimes breaks existing proofs, though they are generally easy to repair, especially if the proof is rerun in parallel with the older PVS version. If you have difficulties repairing your proofs, please let us know.
PVS 3.0 now has an alpha test integration of the
ICS decision procedure. Use M-x
set-decision-procedure ics
to try it out. Note that this is subject
to change, so don't count on proofs created using ICS to work in
future releases. Please let us know of any bugs encountered.
The BETA
and SIMPLIFY
rules, and the
ASSERT
, BASH
, REDUCE
,
SMASH
, GRIND
, GROUND
,
USE
, and LAZY-GRIND
strategies now all take an
optional LET-REDUCE?
flag. It defaults to t
,
and if set to nil
keeps LET
expressions from
being reduced.
restrict_props, extend_props
restrict
and extend
are identities when the subtype equals the supertype.
indexed_sets
number_fields
real
theory was split into two, with
number_fields
providing the field axioms and the subtype
reals
providing the ordering axioms. This allows for
theories such as complex numbers to be inserted in between, thus
allowing reals to be a subtype of complex numbers without having to
encode them.
nat_fun_props
finite_sets
finite_sets_def
(which was in the 2.4
prelude), card_def
, and finite_sets
(from
the finite_sets library)
bitvectors:
bit
, bv
, exp2
,
bv_cnv
, bv_concat_def
,
bv_bitwise
, bv_nat
, empty_bv
,
and bv_caret
.
finite_sets_of_sets
equivalence classes
EquivalenceClosure
,
QuotientDefinition
, KernelDefinition
,
QuotientKernelProperties
,
QuotientSubDefinition
,
QuotientExtensionProperties
,
QuotientDistributive
, and
QuotientIteration
.
Partial Functions
PartialFunctionDefinitions
and
PartialFunctionComposition
.
The following declarations have been added to the prelude:
relations.equivalence type
, sets.setofsets
,
sets.powerset
, sets.Union
,
sets.Intersection
, sets_lemmas.subset_powerset
,
sets_lemmas.empty_powerset
,
sets_lemmas.nonempty_powerset
,
real_props.div_cancel4
, and
rational_props.rational_pred_ax2
.
The following declarations have been modified.
finite_sets.is_finite_surj
was turned into an IFF and extended
from posnat to nat.
The fixpoint declarations of the mucalculus
theory have been
restricted to monotonic predicates. This affects the declarations
fixpoint?
, lfp
, mu
,
lfp?
, gfp
, nu
, and gfp?
.
Conversions may now be any function valued expression, for example,
CONVERSION+ EquivClass(ce), lift(ce), rep(ce)
This introduces a possible incompatibility if the following
declaration is for an infix operator. In that case the conversion
must be followed with a semi-colon ';
'.
Judgement TCCs may now be proved directly, without having to show the
TCCs using M-x show-tccs
or M-x prettyprint-expanded
.
Simple place the cursor on the judgement, and run one of the proof
commands. Note that there may be several TCCs associated with the
judgement, but only one of them is the judgement TCC. To prove the
others you still need to show the TCCs first.
On startup, PVS no longer asks whether to create a context file if
none exists, and if you simply change to another directory no
.pvscontext
file is created. This fixes a subtle bug in which
typing input before the question is asked caused PVS to get into a bad
state.
The M-x dump-pvs-files
command now includes PVS version
information, Allegro build information, and prelude library
dependencies. Note that since the proof files have changed, the dumps
may look quite different. See the Multiple Proofs section for details.
Bart Jacobs kindly provided some additional theories for the bitvector
library. These were used as an aid to Java code verification, but are
generally useful. The new files are BitvectorUtil
,
BitvectorMultiplication
,
BitvectorMultiplicationWidenNarrow
, DivisionUtil
,
BitvectorOneComplementDivision
,
BitvectorTwoComplementDivision
, and
BitvectorTwoComplementDivisionWidenNarrow
, and are included in
the libraries tar file.
Although there are still a number of bugs still outstanding, a large number of bugs have been fixed in this release. All those in the pvs-bugs list that are marked as analyzed have been fixed, at least for the specific specs that caused the bugs.
Most of these are covered elsewhere, they are collected here for easy reference.
The decision procedures are more complete. Though this is usually a good thing, some existing proofs may fail. For example, a given auto-rewrite may have worked in the past, but now the key term has been simplified and the rewrite no longer matches.
These are given in See section Prelude Changes. Theory identifiers used in the prelude may not be used for library or user theories, some existing theories may need to be adjusted.
The theories finite_sets
, finite_sets_def
, and
card_def
were once a part of the finite_sets
library,
but have been merged into a single finite_sets
theory and moved
to the prelude. This means that the library references such as
IMPORTING finite_sets@finite_sets IMPORTING fsets@card_def
must be changed. In the first case just drop the prefix, drop the
prefix and change card_def
to finite_sets
in the
second.
The reals
theory was split in two, separating out the field
axioms into the number_fields
theory. There is the possibility
that proofs could fail because of adjustments related to this, though
this did not show up in our validations.
Theory abbreviations such as
foo: THEORY = bar[int, 3]
should be changed to the new form
IMPORTING bar[int, 3] AS foo
Note that `AS
' is a new keyword, and may cause parse errors
where none existed before.
Since conversions may now be arbitrary function-valued expressions, if
the declaration following is an infix operator it leads to ambiguity.
In that case the conversion must be followed with a semi-colon
';
'.
expand
proof commandDefined infix operators were difficult to expand in the past, as the left to right count was not generally correct; the arguments were looked at before the operator, which meant that the parser tree had to be envisioned in order to get the occurrence number correct. This bug has been fixed, but it does mean that proofs may need to be adjusted. This is another case where it helps to run an earlier PVS version in parallel to find out which occurrence is actually intended.
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Last modified: Thu 30 Nov 2006 12:15 UTC
Maintainer: Sam Owre