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This repository implements Kleene algebra modulo theories (KMT), a framework for deriving concrete Kleene algebras with tests (KATs), an algebraic framework for While-like programs with decidable program equivalence.

More plainly: KMT is a framework for building simple programming languages with structured control (if, while, etc.) where we can algorithmically decide whether or not two programs are equivalent. You can use equivalence to verify programs. If a is a nice property to have after running your program, then if p;a == p, you know that p satisfies a. Kleene algebra with tests subsumes Hoare logic: if a;p;~b == 0 then all runs starting from a either diverge or end with b, i.e., that equation corresponds to the partial correctness specification {a} p {b}. While prior work on KAT often focuses on abstract properties, we write programs over theories that assign concrete meanings to primitive tests and actions.

In addition to providing an OCaml library for defining KMTs over your own theories, we offer a command-line tool for testing equivalence in a variety of pre-defined theories.

Getting Started Guide

How do I build it?

KMT is available on OPAM; if you have OCaml and OPAM installed, œyou should be able to simply run:

$ opam install kmt

But you can also build a Docker container from the root of the repo:

$ docker build -t kmt .    # build KMT, run tests and evaluation

If your docker build command exits with status 137, that indicates that the build ran out of memory (typically when building Z3). We find that 12GB of RAM is sufficient, but more may be necessary on your machine. You might have to reconfigure Docker to have sufficient memory.

Building the image will automatically run unit tests as well as the PLDI 2022 evaluation. When running the image, you can use the kmt executable to test equivalence of various terms directly:

$ docker run -it kmt       # enter a shell
opam@b3043b7dca44:~/kmt$ kmt --boolean 'x=T' 'x=T + x=F;set(x,F);x=T'
[x=T parsed as x=T]
nf time: 0.000004s
lunf time: 0.000022s
[x=T + x=F;set(x,F);x=T parsed as x=T + x=F;set(x,F)[1];x=T]
nf time: 0.000008s
lunf time: 0.000006s
[1 equivalence class]
1: { x=T + x=F;set(x,F);x=T, x=T }

The message 1 equivalence class indicates that all terms given as command-line arguments form a single equivalence class, i.e., the two terms are equivalent. Each equivalence class is printed after:

opam@b3043b7dca44:~/kmt$ kmt --boolean 'x=T' 'x=T + x=F;set(x,T)'
[x=T parsed as x=T]
nf time: 0.000003s
lunf time: 0.000016s
[x=T + x=F;set(x,T) parsed as x=T + x=F;set(x,T)[1]]
nf time: 0.000007s
lunf time: 0.000010s
[2 equivalence classes]
1: { x=T + x=F;set(x,T) }
2: { x=T }

Note that b3043b7dca44 will be replaced by some new hash each time you run docker run -it kmt.

Running run_eval inside the Docker container will reproduce the evaluation from our paper. You can run the regression tests by running test_word (for regular expression word equivalence, part of our decision procedure) and test_equivalence (for KMT term equivalence). All of these steps are performed automatically during docker build.

The source code for all of these is in the src directory; see src/dune for the build script.

How do I use the kmt executable?

The default way of using the kmt executable is to give it a theory (here --boolean) and 2 or more KMT programs in that theory. It will give you the equivalence classes of those terms. The -v flag is useful when many terms are given:

opam@3ce9eaca9fb1:~/kmt$ kmt -v --boolean 'x=T' 'x=F' 'x=T + x=F' 'x=T + x=F;x=T'
[x=T parsed as x=T]
kmt: [INFO] nf = {(x=T,true)}
nf time: 0.000004s
kmt: [INFO] lunf = {(x=T,true), (x=F,false)}
lunf time: 0.000015s
[x=F parsed as x=F]
kmt: [INFO] nf = {(x=F,true)}
nf time: 0.000003s
kmt: [INFO] lunf = {(x=T,false), (x=F,true)}
lunf time: 0.000008s
[x=T + x=F parsed as true]
kmt: [INFO] nf = {(true,true)}
nf time: 0.000003s
kmt: [INFO] lunf = {(true,true)}
lunf time: 0.000014s
[x=T + x=F;x=T parsed as x=T]
kmt: [INFO] nf = {(x=T,true)}
nf time: 0.000003s
kmt: [INFO] lunf = {(x=T,true), (x=F,false)}
lunf time: 0.000006s
[3 equivalence classes]
kmt: [INFO] 1: {(x=T,true), (x=F,false)}; {(x=T,true), (x=F,false)}
kmt: [INFO] 2: {(true,true)}
kmt: [INFO] 3: {(x=T,false), (x=F,true)}

The last three lines identify the three equivalence classes in terms of their normal forms.

If you don't specify a theory, the default will be the theory of booleans.

If you give just one term, kmt will normalize it for you.

Run kmt --help for command-line help in a manpage-like format.

What is the syntax?

A Kleene algebra with tests breaks syntax into two parts: tests (or prediates) and actions. Actions are in some sense the 'top level', as every test is an action.

We use the following syntax, where a and b are tests and p and q are actions. The following is the core KAT notation; individual theories introduce their own notations.

Tests Interpretation
false always fails
true always succeeds
not a negation
a + b or, disjunction
a ; b and, conjunction
Actions Interpretation
false failed trace
true noop trace
a filter traces by test
p + q parallel composition
p ; q sequential composition
p* Kleene star; iteration

Whitespace is ignored, and comments are written with /* ... */.

Theory-specific forms

On its own, the Kleene algebra with tests above doesn't let you express any interesting programs: we need a notion of concrete predicates and actions. KMT builds a concrete KAT around a theory, which defines a predicates and actions. Our implementation has several predefined, and the library itself lets you define new theories.

Theories add predicates and actions of the form NAME(ARGS,...) and ARG1 OP ARG2. Each theory specificies its own language: an ARG will be a variable or a theory-specific constant of some kind; NAME will be a conventional function symbol name, like set; OP takes a variety of forms, like < or =.

Booleans

You can use the booleans by specifying --boolean on the kmt command line. It is the default theory, so you can also leave it off. The theory of booleans adds two forms, where x and y are variables. We write T and F for the boolean values true and false, which should not be confused with the KAT terms true and false.

  • x=T and y=F are tests that are true when x is true and y is false, respectively
  • set(x,T) and set(y,F) are actions that set x to true and y to false, respectively

Monotonic naturals

You can use the monotonically increasing naturals by specifying --incnat on the command line. Monotonic naturals have several theory-specific forms, where variables x, y, and z range over natural numbers; we write n to mean a constant natural number.

  • x > n is a test that is true when the variable x's value is greater than n
  • inc(y) is an action that increments the variable y
  • set(z, n) is an action that sets the variable z to n

Other theories

We have several other theories built in:

  • --addition is a theory of naturals with both < and >, along with inc(x,n)
  • --network is a theory of tracing NetKAT over natural-valued fields src, dst, pt, and sw; use FIELD <- n for assignment
  • --product is a product theory of booleans and monotonic naturals
  • --product-addition is a product theory of booleans and the --addition theory of naturals

You can add new theories to the kmt tool by updating the modes in src/kmt.ml.

Step-by-Step

The paper makes three core claims about the implementation.

  1. It is extensible.
  2. We have implemented some optimizations.
  3. The benchmarks according to our evaluation in Section 5.

How can I tell that the implementation is extensible?

Look at src/kat.ml. It defines several modules.

  • The KAT_IMPL signature characterizes what a KAT has. Here A is for theory tests and P is for theory actions. (The Test and Term modules are for defining comparison and hashing operations on the hashconsed KMT terms.)
  • The THEORY signature characterizes what a client theory must define to generate a KMT.
  • The KAT module is a functor that takes a THEORY and produces a KAT_IMPL.

That is, we use OCaml functors to transform a THEORY into a KAT.

You can see this process in action in src/boolean.ml. After some base definitions (outside the module to simplify things), we define the module Boolean recursively as a THEORY... where we use K = KAT (Boolean) inside our definition. That is, Boolean.K is the KMT over booleans. You can see that there is very little boilerplate: parsing is just a few lines; we define push_back in just a few lines. The satisfiability checker is somewhat complicated by our use of a 'fast' path in the satisfiable function, where we discharge simple queries (with just conjunction and negation of theory predicates, but no disjunction---see can_use_fast_solver) without calling Z3 at all.

What optimizations are implemented?

All KAT terms are hashconsed. The library for that is in src/hashcons.ml; KAT terms are hashconsed using 'a pred/'a pred_hons and ('a, 'p) kat and ('a, 'p) kat_hons in src/kat.ml. We use smart constructors extensively in the KAT module (see not, ppar, pseq, etc.).

When we check word equivalence of actions in src/decide.ml (see same_actions), we use the equivalent_words function in src/word.ml. That method uses the Brzozowski derivative to generate word automata lazily during checking (see derivative and accepting in that src/word.ml).

Finally, several theories implement custom satisfiability checkers that don't merely defer to Z3: boolean.ml, incnat.ml, and addition.ml.

How do I reproduce the paper's evaluation?

By default, the Docker build will run the evaluation from Section 5, using a 30s timeout. Here is sample output (your hash and exact times will differ):

Step 14/18 : RUN opam exec -- dune exec -- src/kmt_eval
 ---> Running in f609aca22e92
test                      time (seconds)
                             30s timeout
----------------------------------------
a* != a (10 random `a`s)          0.0399
count twice                       0.0006
count order                       0.0008
parity loop                       0.0003
boolean tree                      0.0004
population count                  0.3677
toggle three bits                timeout

These numbers are slightly higher than those in the paper, which reports numbers from a local installation. Times will of course vary: machines differ (the original eval is on a 2014 MacBook Pro with 16GB of RAM); Docker on macOS is really a VM, and will be substantially slower than Docker on Linux; Docker will always be slower than a local installation. It should, however, be the case that these benchmarks will have the same relative performance.

You can change the evaluation timeout by passing -t SECONDS or --timeout SECONDS to kmt_eval. In Docker on macOS 10.13 on the 2014 MacBook Pro, we find a high timeout is necessary to get the last benchmark to terminate:

opam@6792c093ed91:~/kmt$ kmt_eval -t 3600
test                      time (seconds)
                           3600s timeout
----------------------------------------
a* != a (10 random `a`s)          0.0682
count twice                       0.0006
count order                       0.0008
parity loop                       0.0005
boolean tree                      0.0008
population count                  0.4311
toggle three bits                1175.1909

What isn't evaluated?

Not every theory described in the paper is completely implemented in KMT. Namely:

  • The implementation of the tracing NetKAT theory uses restricted fields and natural numbers as values, rather than the richer domain NetKAT enjoys.
  • LTLf is not implemented, and neither is Temporal NetKAT. (But the PLDI 2016 implementation is available on GitHub.)
  • Sets and maps are not implemented.

Building locally

The simplest way to play with KMT right away is to use Docker. If for some reason you would prefer to run KMT on your own Linux machine, run the following commands from a clone of the repo:

$ sudo apt-get install -y libgmp-dev python3
$ opam install ocamlfind ppx_deriving batteries ANSIterminal fmt alcotest cmdliner logs zarith z3 dune
$ eval $(opam env)
$ dune build -- src/kmt      # build the CLI
$ dune test                  # unit tests on regex word equivalence and KMT equivalence
$ dune exec -- src/kmt_eval  # PLDI2022 eval

On macOS, brew install gmp python3 ; sudo mkdir -p /opt/local/lib should replace the call to apt-get.

If the above fails, the CI automation is a good guide for manual installation: see the Dockerfile and .github/workflows/build.yml.

What do I have to provide to write my own theory?

The source code in src/incnat.ml is a nice example. You have to provide:

  • sub-modules P and A for the primitive parts of your language
  • a parse function to indicate how to parse the syntax of your primitives; return Left for tests and Right for actions
  • a push_back operation that calculates weakest preconditions on a pair of a primitive and a predicate
  • a satisfiable function to test whether a predicate is satisfiable

To use the Z3 backend, your theory can describe how it extracts to Z3 using functions variable, variable_test, create_z3_var, and theory_to_z3_expr.

Note that incnat.ml's theory solver in satisfiable has two cases: a fast path that need not use Z3, and a more general decision procedure in Z3.

Which example theories should I look at first?

The code in src/boolean.ml is for a simple language with boolean-valued variables.

Check out src/incnat.ml for a simple language with increment and assignment operations. It defines types a and p for the primitive parts of the language (one predicate, which tests whether a variable is greater than a number, and two actions, which increment and set variables).

The code in src/product.ml is for a higher-order theory, combining two theories into one. You can see it in action using the --product and --product-addition flags for KMT.

How is equivalence decided?

We decide equivalence via normalization. We convert KMT terms to a normal form using the novel push_back operation; to compare two such normal forms, we disambiguate the tests and compare the terms pointwise. When this procedure is fast, it's quite fast... but deeply nested loops or loops with lots of conditionals slow it down severely.

In more detail, see src/decide.ml. The top-level function is:

let equivalent (p: K.Term.t) (q: K.Term.t) : bool =
  let nx = normalize_term 0 p in
  let ny = normalize_term 0 q in
  equivalent_nf nx ny

That is, we normalize and then compare normal forms.

let equivalent_nf (nx: nf) (ny: nf) : bool =
  (* optimization: just if syntactically equal first *)
  if PSet.equal nx ny
  then
    begin
      Log.debug (fun m -> m "syntactic equality on %s" (show_nf nx));
      true
    end
  else begin
      Log.debug (fun m -> m
                         "running cross product on %s and %s"
                         (show_nf nx) (show_nf ny));
      let xhat = locally_unambiguous_form nx in
      Log.debug (fun m -> m "%s is locally unambiguous as %s" (show_nf nx) (show_nf xhat));
      let yhat = locally_unambiguous_form ny in
      Log.debug (fun m -> m "%s is locally unambiguous as %s" (show_nf ny) (show_nf yhat));
      equivalent_lunf xhat yhat
  end

It may be easier to understand without the logging/optimization:

let equivalent_nf (nx: nf) (ny: nf) : bool =
  let xhat = locally_unambiguous_form nx in
  let yhat = locally_unambiguous_form ny in
  equivalent_lunf xhat yhat

Given normal forms nx and ny, we first compute locally unambiguous forms xhat and yhat; we then check those for equivalence.

To generate locally unambiguous forms, suppose the normal form nx is equal to a1;m1 + a2;m2 + ... + an;mj. We generate xhat by considering every possibly combination of the tests ai, which engender every possibly combination of the actions mi. That is:

xhat =     a1 ;     a2 ; ... ;     aj ; (m1 + m2 + ... + mj)
     + not a1 ;     a2 ; ... ;     aj ; (     m2 + ... + mj)
     +     a1 ; not a2 ; ... ;     aj ; (m1 +      ... + mj)
     + ...                         
     + not a1 ; not a2 ; ... ;     aj ; (                mj)
     + not a1 ; not a2 ; ... ; not aj ; false

We build yhat from y = b1;n1 + ... + bk;nk similarly:

yhat =     b1 ;     b2 ; ... ;     bk ; (n1 + n2 + ... + nk)
     + not b1 ;     b2 ; ... ;     bk ; (     n2 + ... + nk)
     +     b1 ; not b2 ; ... ;     bk ; (n1 +      ... + nk)
     + ...                         
     + not b1 ; not b2 ; ... ;     bk ; (                nk)
     + not b1 ; not b2 ; ... ; not bk ; false

We call these hatted forms "locally unambiguous" because each possible test in xhat is syntactically unambiguous.

Now we can compare xhat and yhat (in equivalent_lunf): consider every pair of a predicates from xhat and yhat. If the combination of the predicates is unsatisfiable, then we can ignore that case. If it's satisfiable, then for xhat and yhat to be equivalent, the actions on both sides must be equivalent. We can decide that equivalence using the Hopcroft-Karp algorithm (see equivalent_words in src/word.ml).

Congratulations, you read the whole thing! 😁

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Kleene algebra modulo theories

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PLDI 2022 AEC Latest
Mar 4, 2022

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