# lpsconfcheck¶

Given an LPS:

$\begin{split}\begin{array}{lll} P(d:D) &=& \ldots + \sum_{e_i:E_i} c_i(d,e_i) \to a_i(f_i(d,e_i)) \cdot P(g_i(d,e_i)) +\\ && \ldots + \sum_{e_j:E_j} c_j(d,e_j) \to \tau \cdot P(g_j(d,e_j)) + \ldots \end{array}\end{split}$

$$tau$$-summand $$j$$ is confluent with summand $$i$$ if the following condition holds:

$\begin{split}\begin{array}{l} \forall d{:}D.\forall e_i{:}E_i. \forall e_j{:}E_j . (inv(d) \land c_i(d,e_i) \land c_j(d,e_j)) \Rightarrow\\ (c_i(g_j(d,e_j),e_i) \land c_j(g_i(d,e_i),e_j) \land\\ f_i(d,e_i) = f_i(g_j(d,e_j),e_i) \land g_i(g_j(d,e_j),e_i) = g_j(g_i(d,e_i),e_j)) \end{array}\end{split}$

where $$inv$$ is the invariant specified using the option –invariant. In case $$a_i$$ is also a $$tau$$-action, this formula can be weakened to the following:

$\begin{split}\begin{array}{l} \forall d{:}D. \forall e_i{:}E_i. \forall e_j{:}E_j . (inv(d) \land c_i(d,e_i) \land c_j(d,e_j)) \Rightarrow\\ (g_i(d,e_i) = g_j(d,e_j) \lor (c_i(g_j(d,e_j),e_i) \land c_j(g_i(d, e_i),e_j) \land\\ g_i(g_j(d,e_j),e_i) = g_j(g_i(d,e_i),e_j))) \end{array}\end{split}$

If the option :option–invariant is not used, the invariant is equal to true.

The tool will generate these confluence conditions for all $$tau$$-summands and tries to prove that they are tautologies using a BDD based prover for propositional formulas. In some cases lpsconfcheck indicates that a $$tau$$-summand is not confluent even though it is. The option –verbose gives insight into what the prover is doing and can be used to see if rewrite rules have to be added to the specification in order to enable the prover to determine that certain condition are indeed tautologies.

In some cases it may be useful to use an SMT solver to assist the prover. The SMT solver can further reduce BDDs by removing inconsistent paths. A specific SMT solver can be chosen using the option –smt-solver=SOLVER. Either the SMT solver Ario or CVC3 can be used. To use one of these solvers, the directory containing the corresponding executable must be in the path.

The tool can determine whether two summands are confluent in three ways and will indicate which of the methods was used while proving confluence. The three ways of determining confluence are as follows:

• If summand number 1 has been proven confluent with summand number 2, summand number 2 is also confluent with summand number 1. This method of checking confluence is called checking confluence by symmetry. If two summands are confluent by symmetry, lpsconfcheck indicates this by printing a dot (‘.’).

• Another way of checking the confluence of two summands is determining whether the two summands are syntactically disjoint. Two summands are syntactically disjoint if the following holds:

• The set of variables used by one summand is disjoint from the set of variables changed by the other summand and vice versa.

• The set of variables changed by one summand is disjoint from the set of variables changed by the other summand.

If two summands are confluent because of syntactic disjointness, lpsconfcheck indicates this by printing a colon (‘:’).

• The most time consuming way of checking the confluence of two summands is generating the confluence condition and then checking if this condition is a tautology using the prover. If two summands are proven confluent using the prover, lpsconfcheck indicates this by printing a plus sign (‘+’). If the option –generate-invariants is used, the tool will try to prove that the reduced confluence condition is an invariant of the LPS, in case the confluence condition is not a tautology. If the reduced confluence condition is indeed an invariant, the two summands are proven confluent lpsconfcheck indicates this by printing an ‘i’.

If there already is an action named ‘’ctau’’ present in the LPS as found in ‘’INFILE’’, an error will be reported.

## Manual page for lpsconfcheck¶

### Usage¶

lpsconfcheck   [OPTION]... [INFILE [OUTFILE]]


### Description¶

Checks which tau-summands of the mCRL2 LPS in INFILE are confluent, marks them by renaming them to ctau, and write the result to OUTFILE. If INFILE is not present stdin is used. If OUTFILE is not present, stdout is used.

### Command line options¶

-a , --check-all

check the confluence of tau-summands regarding all other summands, instead of continuing with the next tau-summand as soon as a summand is encountered that is not confluent with the current tau-summand

-x[CONDITIONS] , --conditions[CONDITIONS]

use the confluence conditions specified by individual characters (c/C: commutative confluence with/without disjointness check; d: commutative confluence with disjointness check only; T: triangular confluence; Z: trivial confluence)

-c , --counter-example

display a valuation for which the confluence condition does not hold, in case the encountered condition is neither a contradiction nor a tautolgy

-g , --generate-invariants

try to prove that the reduced confluence condition is an invariant of the LPS, in case the confluence condition is not a tautology

-o , --induction

apply induction on lists

-iINVFILE , --invariantINVFILE

use the boolean formula (an mCRL2 data expression of sort Bool) in INVFILE as invariant

-n , --no-check

do not check if the invariant holds before checking for for confluence

-m , --no-marking

do not mark the confluent tau-summands; since there are no changes made to the LPS, nothing is written to OUTFILE

-u , --no-sums

do not rewrite summands with a sum operator and prove their confluence formula

-pPREFIX , --print-dotPREFIX

save a .dot file of the resulting BDD in case two summands cannot be proven confluent; PREFIX will be used as prefix of the output files

-QNUM , --qlimitNUM

limit enumeration of quantifiers to NUM iterations. (Default NUM=1000, NUM=0 for unlimited).

-rNAME , --rewriterNAME

use rewrite strategy NAME:

jitty

jitty rewriting

jittyc

compiled jitty rewriting

-zSOLVER , --smt-solverSOLVER

use SOLVER to remove inconsistent paths from the internally used BDDs (by default, no path elimination is applied):

cvc

the SMT solver CVC3

z3

the SMT solver Z3

-sNUM , --summandNUM

eliminate or simplify the summand with number NUM only

-tLIMIT , --time-limitLIMIT

spend at most LIMIT seconds on proving a single formula

--timings[FILE]

append timing measurements to FILE. Measurements are written to standard error if no FILE is provided

#### Standard options¶

-q , --quiet

do not display warning messages

-v , --verbose

display short intermediate messages

-d , --debug

display detailed intermediate messages

--log-levelLEVEL

display intermediate messages up to and including level

-h , --help

display help information

--version`

display version information

Luc Engelen