Unified Mathematical Framework for Slicing and Symmetry Reduction over Event Structures

Algorithm 1

Conflict-free partition: 𝑐𝑓𝑝.

Input: a prime event structure: ℰ;

Output: the set of 𝑚𝑐𝑓𝐸𝑆𝑠: ℬ^{𝑚𝑐𝑓𝐸𝑆}(ℰ);

BEGIN

(1)  𝐶=∅;𝑖𝑛𝑡𝑖=1;𝑖𝑛𝑡𝑛=0;ℜ=∅;

(2)  𝑆𝑄=𝐸𝑛𝑡𝑒𝑟𝑄𝑢𝑒𝑢𝑒(𝐸_ℰ,1);

  /* Initialize the Queue */

(3) if (𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡𝐹𝑟𝑒𝑒(𝐸_ℰ)){

(5)  𝐶=∪{𝐸};

(6)  Goto BUILDES;

(7) } /* end if; */

  /* Expand and merge each conflict pair and build maximal conflict patterns:

   {ℐ^𝑖_#ℰ∶𝐴_𝑖#^𝑥𝑠𝐵_𝑖∣𝑖∈,1≤𝑖≤𝑚}; */

(8) for (𝑖=1;𝑖≤𝑚;𝑖++) { /* do */

(9)  Select a partition pattern: ℐ^𝑖_#ℰ∶𝐴_𝑖#^𝑥𝑠𝐵_𝑖;

(10) /* Current level is 𝑖, apply ℐ^𝑖_#ℰ, otherwise, skip while loop to the next one: 𝑖++; */

(11)  while (!𝐸𝑚𝑝𝑡𝑦𝑄𝑢𝑒𝑢𝑒(𝑆𝑄)  &&

   𝑖==𝑉𝑎𝑟𝑄𝑢𝑒𝑢𝑒(𝑆𝑄,𝐋𝐄𝐕𝐄𝐋)) {

  /* Get the head of the Queue: event set */

(12)    𝐸=𝑂𝑢𝑡𝑄𝑢𝑒𝑢𝑒(𝑆𝑄,𝐄𝐕𝐄𝐍𝐓);

(13)    𝐴^′_𝑖=𝑚𝑎𝑥𝑖𝑛𝑐𝑙(𝐴_𝑖,𝐸);

(14)    𝐵^′_𝑖=𝑚𝑎𝑥𝑖𝑛𝑐𝑙(𝐵_𝑖,𝐸);

(15)    𝐸₁=(𝐸−𝐴^′_𝑖);

(16)    𝐸₂=(𝐸−𝐵^′_𝑖);

  /*𝐸₁ is a conflict-free subset; */

(17)  if (𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡𝐹𝑟𝑒𝑒(𝐸₁)) {

(18)    𝐶=𝐶∪{𝐸₁};

  /* Remove these elements included in others; */

(19)   𝑅𝑒𝑚𝑜𝑣𝑒𝐼𝑛𝑐𝑙𝑢𝑑𝑒𝑑𝐸𝑙𝑒𝑚𝑒𝑛𝑡(𝐶);

(20)  }else{

  /* Continue next partition by the conflict pattern: ℐ_#^𝑖+1_ℰ; */

(21)     𝑆𝑄=𝐸𝑛𝑡𝑒𝑟𝑄𝑢𝑒𝑢𝑒(𝐸₁,𝑖+1);

(22)    } /* end if */

(23)    if (𝐼𝑠𝐶𝑜𝑛𝑓𝑙𝑖𝑐𝑡𝐹𝑟𝑒𝑒(𝐸₂)) {

(24)   𝐶=𝐶∪{𝐸₂};

(25)   𝑅𝑒𝑚𝑜𝑣𝑒𝐼𝑛𝑐𝑙𝑢𝑑𝑒𝑑𝐸𝑙𝑒𝑚𝑒𝑛𝑡(𝐶);

(26)    }else{

(27)   𝑆𝑄=𝐸𝑛𝑡𝑒𝑟𝑄𝑢𝑒𝑢𝑒(𝐸₂,𝑖+1);

(28)    } /* end if */

(29)  } /* end while */

(30) } /* end for */

(31) BUILDES: /* Build the sub-structure: ℱ=(𝐸_ℱ,⪯_ℱ,#_ℱ,𝑙_ℱ)  */

(32) while (¬𝐼𝑠𝐸𝑚𝑝𝑡𝑦(𝐶)) {

(33)   Select a conflict-free event subset 𝐴_!# from 𝐶;

(34)   𝐸_ℱ=𝐴_!#;

(35)   #_ℱ=∅;

(36)   ⪯_ℱ=⪯_ℰ∩(𝐴_!#×𝐴_!#);

(37)   𝑙_ℱ=𝑙|_𝐴!#;

(38)   𝐶=𝐶−{𝐴_!#};

(39)   ℜ=ℜ∪ℱ; 𝑛++;

(40) } /* end while */

(41) 𝑚𝑐𝑓𝑠𝑒𝑡𝑁(ℰ)=𝑛;

(42) return ℜ;

END;