From NP-Completeness to DP-Completeness: A Membrane Computing Perspective

<div>Transport of objects from <svg height="15.3797pt" id="M476" style="vertical-align:-3.9436pt" version="1.1" viewbox="-0.0498162 -11.4361 12.4525 15.3797" width="12.4525pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.488,-5.741)"><path d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.527,-5.741)"><path d="M310 541L304 571C290 586 211 619 185 610L80 76L131 52L310 541Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,4.901,3.784)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"></path></g></svg> to the corresponding <span class="nowrap"><svg height="13.7577pt" id="M477" style="vertical-align:-5.47413pt" version="1.1" viewbox="-0.0498162 -8.28357 37.3253 13.7577" width="37.3253pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M135 536C164 536 186 560 186 587C186 617 164 639 136 639C109 639 85 617 85 587C85 560 109 536 135 536ZM252 0V26C188 32 181 38 181 106V451C138 433 90 420 39 412V388C99 379 102 374 102 312V106C102 38 95 32 32 26V0H252Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.484,0)"><path d="M524 0V26C466 32 460 36 460 104V297C460 393 411 449 331 449C302 449 276 437 248 419C223 402 201 387 181 372V451C137 432 90 420 42 411V388C96 378 102 374 102 310V104C102 38 97 33 29 26V0H246V26C187 32 181 36 181 104V339C211 365 250 390 290 390C357 390 381 345 381 276V109C381 40 374 32 315 26V0H524Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.635,3.265)"><path d="M713 650H37V617C122 611 130 604 130 523V127C130 48 121 39 41 33V0H305V33C226 39 217 47 217 127V577C217 605 221 609 251 609H500C528 609 534 607 534 579V127C534 49 526 39 439 33V0H711V33C629 39 621 47 621 127V523C621 602 629 611 713 617V650Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,17.469,.068)"><path d="M318 -137C221 -59 159 100 159 272S220 600 318 680L294 710C146 612 80 452 80 273V272C80 91 146 -72 294 -168L318 -137Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,19.874,.068)"><path d="M256 606C256 635 239 657 207 657C171 657 147 619 147 593C147 563 173 543 194 543C234 543 256 573 256 606ZM233 98L216 125C191 101 156 75 139 75C133 75 132 82 139 107L214 377C226 420 224 454 198 454C169 454 98 415 31 351L46 323C81 353 110 372 116 372C129 372 126 356 121 338L57 101C31 5 47 -12 72 -12C115 -12 197 52 233 98Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,21.739,.068)"><path d="M291 272C291 452 228 611 76 710L51 680C147 602 211 444 211 272S151 -55 51 -137L76 -168C227 -66 291 91 291 272Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,24.667,3.265)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,27.925,3.265)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,31.037,5.366)"><path d="M256 606C256 635 239 657 207 657C171 657 147 619 147 593C147 563 173 543 194 543C234 543 256 573 256 606ZM233 98L216 125C191 101 156 75 139 75C133 75 132 82 139 107L214 377C226 420 224 454 198 454C169 454 98 415 31 351L46 323C81 353 110 372 116 372C129 372 126 356 121 338L57 101C31 5 47 -12 72 -12C115 -12 197 52 233 98Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.408,3.265)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"></path></g></svg>.</span> For this purpose, rules from 1 and 2 are applied. Besides these rules, rules from 3 are applied to make the objects from <svg height="17.2508pt" id="M478" style="vertical-align:-4.684699pt" version="1.1" viewbox="-0.0498162 -12.5661 23.6836 17.2508" width="23.6836pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M1048 672C1036 675 1028 674 1020 674C822 646 666 318 554 134C521 81 495 39 468 39C457 39 455 46 455 57C455 79 469 109 472 118C520 227 653 461 794 660L779 672C768 675 766 674 762 674C654 674 544 586 344 250C241 78 187 15 119 15C104 15 91 21 88 36H91C96 33 101 32 106 32C133 32 149 53 149 76C149 103 128 128 97 128C65 128 38 102 38 63C38 20 71 -15 118 -15C235 -15 307 132 473 397C542 508 638 615 712 644L713 643C646 565 574 455 521 373C460 281 378 144 378 66C378 23 399 -4 434 -4C518 -4 559 76 647 231C792 488 891 603 977 635L978 633C779 366 670 179 670 70C670 18 708 -12 756 -12C819 -12 883 42 959 134L940 150C868 66 810 18 770 18C751 18 737 28 737 53C737 111 865 359 1056 659L1048 672Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.936,-6.049)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,17.194,-6.049)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,19.742,-6.049)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.936,4.525)"><path d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"></path></g></svg> be available in their corresponding membranes only when objects from <svg height="11.5564pt" id="M479" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 21.3821 11.5564" width="21.3821pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 168L541 175C510 88 481 71 415 71H135L378 358L167 614H348C422 614 447 608 463 588C478 571 488 546 500 500L529 504C524 559 519 633 518 650H47V628L295 324L33 23V0H531L570 168Z"></path></g><g transform="matrix(.013,0,0,-0.013,7.774,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.272,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.705,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> are in their corresponding input membranes.</div>

Complexity

fig1

Figure 1

Figure 1: From NP-Completeness to DP-Completeness: A Membrane Computing Perspective 