Some remarks on ! 2 e 00 E ! 2 e 00 E w equivalence of - - PowerPoint PPT Presentation

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Some remarks on ! 2 e 00 E ! 2 e 00 E w equivalence of - - PowerPoint PPT Presentation

Jun 02, 2023 460 likes •695 views

E e E e 0 E Some remarks on ! 2 e 00 E ! 2 e 00 E w equivalence of graph-to-graph transducers inc( w, e 0 ) inc( w, e 00 ) Schmude, J. Boja czyk, M. University of Warsaw Highlights of Logic,

slide-1
SLIDE 1

Some remarks on equivalence of graph-to-graph transducers

Schmude, J.

University of Warsaw

Highlights of Logic, Games and Automata 2020, September 15-18

Bojańczyk, M.

∃E e ∈ E ∧ ∀e0 ∈ E inc(w, e0) ∧ inc(w, e00) ∃!2 e00 ∈ E ∧ ∃!2 e00 ∈ E ∃w

slide-2
SLIDE 2

transducers

baaabab . . .

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abbaa . . .

<latexit sha1_base64="5vep7nCbjRhZ5rEQN9s1cI/sZFE=">ACGHicbVC7SgNBFJ2NrxhfUubwSBYhV1RtAzaWEYwD9iEcHd2Nhkyu7PM3BVCyGfYWOiv2ImtnX9i6STZwiQeGDic+zpzglQKg67RTW1jc2t4rbpZ3dvf2D8uFR06hM95gSirdDsBwKRLeQIGSt1PNIQ4kbwXDu2m9cS1ESp5xFHKuzH0ExEJBmglH4IAoCNDhaZXrhVdwa6SrycVEiOeq/80wkVy2KeIJNgjO+5KXbHoFEwySelTmZ4CmwIfe5bmkDMTXc8szyhZ1YJaS0fQnSmfp3YgyxMaM4sJ0x4MAs16bifzU/w+imOxZJmiFP2PxQlEmKik7/T0OhOUM5sgSYFtYrZQPQwNCmtHDFYAx6pEO7Y+qzI2Vg24YcSzYqbzmYVdK8qHqX1auHy0rtNg+tSE7IKTknHrkmNXJP6qRBGFHkmbySN+fFeXc+nM95a8HJZ47JApyvXxioOY=</latexit>

bab . . . bbab . . . aabb . . .

<latexit sha1_base64="goaEZfstFv1SxfyiQGmTzJCr1Q=">ACL3icbZC7SgNBFIZn4y3GW9TSZjAIVmFXFC2DNpYRzAWSJZyZnSRDZi/MnBXCktansbHQVxEbsfUNLJ1cQJN4YODjP9f5WaKkQd9d3Irq2vrG/nNwtb2zu5ecf+gbuJUc1HjsYp1k4ERSkaihKVaCZaQMiUaLDBzTjfeBDayDi6x2Ei/B6kexKDmilTpEyYG0VxGgo+0UAxuiUO8WSW3YnQZfBm0GJzKLaKX63g5inoYiQKzCm5bkJ+hlolFyJUaGdGpEAH0BPtCxGEArjZ5OfjOiJVQLajbV9EdKJ+rcjg9CYchsZQjYN4u5sfhfrpVi98rPZJSkKCI+XdRNFcWYjm2hgdSCoxpaAK6lvZXyPmjgaM2b2IwBD3UgZ0xvrOtFLNlA4EFa5W3aMwy1M/K3n54u68VLmemZYnR+SYnBKPXJIKuSVUiOcPJIn8kJenWfnzflwPqelOWfWc0jmwvn6ARTAqZg=</latexit>

automata that transform combinatorial objects (’60s)

Motivation

. .. . . . . . . . . . . . . . . .

slide-3
SLIDE 3

transducers

equivalence given , T1

<latexit sha1_base64="nOhyOz1a9+n1yoTrOBntVLqdNzQ=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8eK/YI2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szMwLEikMu63U1hb39jcKm6Xdnb39g/Kh0ctE6ea8SaLZaw7ATVcCsWbKFDyTqI5jQLJ28H4bua3n7g2IlYNnCTcj+hQiVAwilZ6bPS9frniVt05yCrxclKBHPV+as3iFkacYVMUmO6npugn1GNgk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP6Y3viZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpHVR9S6rVw+XldptHkcRTuAUzsGDa6jBPdShCQyG8Ayv8OZI58V5dz4WrQUnzmGP3A+fwDXY2E</latexit>

, decide if T2

<latexit sha1_base64="Qs7Vli4OFN5bTSdrEC+NEFr6Js=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyGiB6DXjxGzAuSJcxOepMhs7PLzKwQj7BiwdFvPpF3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJsFrHqBFSj4BKbhuBnUQhjQKB7WB8N/fbT6g0j2XDTBL0IzqUPOSMGis9NvqVfrHklt0FyDrxMlKCDPV+8as3iFkaoTRMUK27npsYf0qV4UzgrNBLNSaUjekQu5ZKGqH2p4tTZ+TCKgMSxsqWNGSh/p6Y0kjrSRTYzoiakV715uJ/Xjc14Y0/5TJDUq2XBSmgpiYzP8mA6QGTGxhDLF7a2EjaizNh0CjYEb/XldKqlL1q+eqhWqrdZnHk4QzO4RI8uIYa3EMdmsBgCM/wCm+OcF6cd+dj2ZpzsplT+APn8wfY5Y2F</latexit>

JT1K = JT2K

<latexit sha1_base64="RmSGkFbnLuSrGUxU6gAjYXqFA=">ACRXicbVDLSgMxFM34rPU16tJNsAiuykypj41QdOyQl/QliGTSdvQzIPkjlCG/olf48aF/oAf4U7EnWbaEfvwQuDc+4rx40EV2BZb8bK6tr6xmZuK7+9s7u3bx4cNlQYS8rqNBShbLlEMcEDVgcOgrUiyYjvCtZ0h7ep3nxgUvEwqMEoYl2f9APe45SAphzoiOEKwkdMsA1x8YdKX/TazynlWa0vGMWrKI1CbwM7AwUBZVx/zqeCGNfRYAFUSptm1F0E2IBE4FG+c7sWKRnk36rK1hQHymusnkf2N8qhkP90KpXwB4ws52JMRXauS7utInMFCLWkr+p7Vj6F1Ex5EMbCAThf1YoEhxKlZ2OSURAjDQiVXN+K6YBoD0BbOrdFgU/kSHp6Rnrn3OpVfaiMcugUSra5eL5fblQuclMy6FjdILOkI0uUQXdoSqI4oe0RN6Qa/Gs/FufBif09IVI+s5QnNhfP8AViGxhA=</latexit>

baaabab . . .

<latexit sha1_base64="uWCbEYn82afSBWJvipScetRC9VA=">ACGnicbVC7SgNBFJ2NrxhfUubxSBYhV2JaBm0sYxgHpAs4c7sJBky+3DmrCEfIeNhf6Kndja+CeWziZbmMQDA4dz7p17ODSWQqPjfFuFtfWNza3idmlnd2/oHx41NJRohvskhGqkNBcylC3kSBkndixSGgkrfp+Dbz209caRGFD5jG3AtgGIqBYIBG8igAUKA96Ueo+WKU3VmsFeJm5MKydHol396fsSgIfIJGjdZ0YvQkoFEzyamXaB4DG8OQdw0NIeDam8xCT+0zo/j2IFLmhWjP1L8bEwi0TgNqJgPAkV72MvE/r5vg4NqbiDBOkIdsfmiQSBsjO2vA9oXiDGVqCDAlTFabjUABQ9PTwhWNAahU+eaPLGdPSmrGxhxLpip3uZhV0rqourXq5X2tUr/JSyuSE3JKzolLrkid3JEGaRJGHskzeSVv1ov1bn1Yn/PRgpXvHJMFWF+/sC2hvQ=</latexit>

abbaa . . .

<latexit sha1_base64="5vep7nCbjRhZ5rEQN9s1cI/sZFE=">ACGHicbVC7SgNBFJ2NrxhfUubwSBYhV1RtAzaWEYwD9iEcHd2Nhkyu7PM3BVCyGfYWOiv2ImtnX9i6STZwiQeGDic+zpzglQKg67RTW1jc2t4rbpZ3dvf2D8uFR06hM95gSirdDsBwKRLeQIGSt1PNIQ4kbwXDu2m9cS1ESp5xFHKuzH0ExEJBmglH4IAoCNDhaZXrhVdwa6SrycVEiOeq/80wkVy2KeIJNgjO+5KXbHoFEwySelTmZ4CmwIfe5bmkDMTXc8szyhZ1YJaS0fQnSmfp3YgyxMaM4sJ0x4MAs16bifzU/w+imOxZJmiFP2PxQlEmKik7/T0OhOUM5sgSYFtYrZQPQwNCmtHDFYAx6pEO7Y+qzI2Vg24YcSzYqbzmYVdK8qHqX1auHy0rtNg+tSE7IKTknHrkmNXJP6qRBGFHkmbySN+fFeXc+nM95a8HJZ47JApyvXxioOY=</latexit>

bab . . . bbab . . . aabb . . .

<latexit sha1_base64="goaEZfstFv1SxfyiQGmTzJCr1Q=">ACL3icbZC7SgNBFIZn4y3GW9TSZjAIVmFXFC2DNpYRzAWSJZyZnSRDZi/MnBXCktansbHQVxEbsfUNLJ1cQJN4YODjP9f5WaKkQd9d3Irq2vrG/nNwtb2zu5ecf+gbuJUc1HjsYp1k4ERSkaihKVaCZaQMiUaLDBzTjfeBDayDi6x2Ei/B6kexKDmilTpEyYG0VxGgo+0UAxuiUO8WSW3YnQZfBm0GJzKLaKX63g5inoYiQKzCm5bkJ+hlolFyJUaGdGpEAH0BPtCxGEArjZ5OfjOiJVQLajbV9EdKJ+rcjg9CYchsZQjYN4u5sfhfrpVi98rPZJSkKCI+XdRNFcWYjm2hgdSCoxpaAK6lvZXyPmjgaM2b2IwBD3UgZ0xvrOtFLNlA4EFa5W3aMwy1M/K3n54u68VLmemZYnR+SYnBKPXJIKuSVUiOcPJIn8kJenWfnzflwPqelOWfWc0jmwvn6ARTAqZg=</latexit>

automata that transform combinatorial objects (’60s)

Motivation

. .. . . . . . . . . . . . . . . .

slide-4
SLIDE 4

transducers

equivalence given , T1

<latexit sha1_base64="nOhyOz1a9+n1yoTrOBntVLqdNzQ=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8eK/YI2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szMwLEikMu63U1hb39jcKm6Xdnb39g/Kh0ctE6ea8SaLZaw7ATVcCsWbKFDyTqI5jQLJ28H4bua3n7g2IlYNnCTcj+hQiVAwilZ6bPS9frniVt05yCrxclKBHPV+as3iFkacYVMUmO6npugn1GNgk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP6Y3viZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpHVR9S6rVw+XldptHkcRTuAUzsGDa6jBPdShCQyG8Ayv8OZI58V5dz4WrQUnzmGP3A+fwDXY2E</latexit>

, decide if T2

<latexit sha1_base64="Qs7Vli4OFN5bTSdrEC+NEFr6Js=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyGiB6DXjxGzAuSJcxOepMhs7PLzKwQj7BiwdFvPpF3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJsFrHqBFSj4BKbhuBnUQhjQKB7WB8N/fbT6g0j2XDTBL0IzqUPOSMGis9NvqVfrHklt0FyDrxMlKCDPV+8as3iFkaoTRMUK27npsYf0qV4UzgrNBLNSaUjekQu5ZKGqH2p4tTZ+TCKgMSxsqWNGSh/p6Y0kjrSRTYzoiakV715uJ/Xjc14Y0/5TJDUq2XBSmgpiYzP8mA6QGTGxhDLF7a2EjaizNh0CjYEb/XldKqlL1q+eqhWqrdZnHk4QzO4RI8uIYa3EMdmsBgCM/wCm+OcF6cd+dj2ZpzsplT+APn8wfY5Y2F</latexit>

JT1K = JT2K

<latexit sha1_base64="RmSGkFbnLuSrGUxU6gAjYXqFA=">ACRXicbVDLSgMxFM34rPU16tJNsAiuykypj41QdOyQl/QliGTSdvQzIPkjlCG/olf48aF/oAf4U7EnWbaEfvwQuDc+4rx40EV2BZb8bK6tr6xmZuK7+9s7u3bx4cNlQYS8rqNBShbLlEMcEDVgcOgrUiyYjvCtZ0h7ep3nxgUvEwqMEoYl2f9APe45SAphzoiOEKwkdMsA1x8YdKX/TazynlWa0vGMWrKI1CbwM7AwUBZVx/zqeCGNfRYAFUSptm1F0E2IBE4FG+c7sWKRnk36rK1hQHymusnkf2N8qhkP90KpXwB4ws52JMRXauS7utInMFCLWkr+p7Vj6F1Ex5EMbCAThf1YoEhxKlZ2OSURAjDQiVXN+K6YBoD0BbOrdFgU/kSHp6Rnrn3OpVfaiMcugUSra5eL5fblQuclMy6FjdILOkI0uUQXdoSqI4oe0RN6Qa/Gs/FufBif09IVI+s5QnNhfP8AViGxhA=</latexit>

baaabab . . .

<latexit sha1_base64="uWCbEYn82afSBWJvipScetRC9VA=">ACGnicbVC7SgNBFJ2NrxhfUubxSBYhV2JaBm0sYxgHpAs4c7sJBky+3DmrCEfIeNhf6Kndja+CeWziZbmMQDA4dz7p17ODSWQqPjfFuFtfWNza3idmlnd2/oHx41NJRohvskhGqkNBcylC3kSBkndixSGgkrfp+Dbz209caRGFD5jG3AtgGIqBYIBG8igAUKA96Ueo+WKU3VmsFeJm5MKydHol396fsSgIfIJGjdZ0YvQkoFEzyamXaB4DG8OQdw0NIeDam8xCT+0zo/j2IFLmhWjP1L8bEwi0TgNqJgPAkV72MvE/r5vg4NqbiDBOkIdsfmiQSBsjO2vA9oXiDGVqCDAlTFabjUABQ9PTwhWNAahU+eaPLGdPSmrGxhxLpip3uZhV0rqourXq5X2tUr/JSyuSE3JKzolLrkid3JEGaRJGHskzeSVv1ov1bn1Yn/PRgpXvHJMFWF+/sC2hvQ=</latexit>

abbaa . . .

<latexit sha1_base64="5vep7nCbjRhZ5rEQN9s1cI/sZFE=">ACGHicbVC7SgNBFJ2NrxhfUubwSBYhV1RtAzaWEYwD9iEcHd2Nhkyu7PM3BVCyGfYWOiv2ImtnX9i6STZwiQeGDic+zpzglQKg67RTW1jc2t4rbpZ3dvf2D8uFR06hM95gSirdDsBwKRLeQIGSt1PNIQ4kbwXDu2m9cS1ESp5xFHKuzH0ExEJBmglH4IAoCNDhaZXrhVdwa6SrycVEiOeq/80wkVy2KeIJNgjO+5KXbHoFEwySelTmZ4CmwIfe5bmkDMTXc8szyhZ1YJaS0fQnSmfp3YgyxMaM4sJ0x4MAs16bifzU/w+imOxZJmiFP2PxQlEmKik7/T0OhOUM5sgSYFtYrZQPQwNCmtHDFYAx6pEO7Y+qzI2Vg24YcSzYqbzmYVdK8qHqX1auHy0rtNg+tSE7IKTknHrkmNXJP6qRBGFHkmbySN+fFeXc+nM95a8HJZ47JApyvXxioOY=</latexit>

bab . . . bbab . . . aabb . . .

<latexit sha1_base64="goaEZfstFv1SxfyiQGmTzJCr1Q=">ACL3icbZC7SgNBFIZn4y3GW9TSZjAIVmFXFC2DNpYRzAWSJZyZnSRDZi/MnBXCktansbHQVxEbsfUNLJ1cQJN4YODjP9f5WaKkQd9d3Irq2vrG/nNwtb2zu5ecf+gbuJUc1HjsYp1k4ERSkaihKVaCZaQMiUaLDBzTjfeBDayDi6x2Ei/B6kexKDmilTpEyYG0VxGgo+0UAxuiUO8WSW3YnQZfBm0GJzKLaKX63g5inoYiQKzCm5bkJ+hlolFyJUaGdGpEAH0BPtCxGEArjZ5OfjOiJVQLajbV9EdKJ+rcjg9CYchsZQjYN4u5sfhfrpVi98rPZJSkKCI+XdRNFcWYjm2hgdSCoxpaAK6lvZXyPmjgaM2b2IwBD3UgZ0xvrOtFLNlA4EFa5W3aMwy1M/K3n54u68VLmemZYnR+SYnBKPXJIKuSVUiOcPJIn8kJenWfnzflwPqelOWfWc0jmwvn6ARTAqZg=</latexit>

automata that transform combinatorial objects (’60s)

Motivation

. .. . . . . . . . . . . . . . . .

Important classes of transductions can be described using both logic and automata.

slide-5
SLIDE 5

transducers

equivalence given , T1

<latexit sha1_base64="nOhyOz1a9+n1yoTrOBntVLqdNzQ=">AB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8eK/YI2lM120y7dbMLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szMwLEikMu63U1hb39jcKm6Xdnb39g/Kh0ctE6ea8SaLZaw7ATVcCsWbKFDyTqI5jQLJ28H4bua3n7g2IlYNnCTcj+hQiVAwilZ6bPS9frniVt05yCrxclKBHPV+as3iFkacYVMUmO6npugn1GNgk+LfVSwxPKxnTIu5YqGnHjZ/NTp+TMKgMSxtqWQjJXf09kNDJmEgW2M6I4MsveTPzP6Y3viZUEmKXLHFojCVBGMy+5sMhOYM5cQSyrSwtxI2opoytOmUbAje8surpHVR9S6rVw+XldptHkcRTuAUzsGDa6jBPdShCQyG8Ayv8OZI58V5dz4WrQUnzmGP3A+fwDXY2E</latexit>

, decide if T2

<latexit sha1_base64="Qs7Vli4OFN5bTSdrEC+NEFr6Js=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKeyGiB6DXjxGzAuSJcxOepMhs7PLzKwQj7BiwdFvPpF3vwbJ8keNLGgoajqprsrSATXxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ThVDJsFrHqBFSj4BKbhuBnUQhjQKB7WB8N/fbT6g0j2XDTBL0IzqUPOSMGis9NvqVfrHklt0FyDrxMlKCDPV+8as3iFkaoTRMUK27npsYf0qV4UzgrNBLNSaUjekQu5ZKGqH2p4tTZ+TCKgMSxsqWNGSh/p6Y0kjrSRTYzoiakV715uJ/Xjc14Y0/5TJDUq2XBSmgpiYzP8mA6QGTGxhDLF7a2EjaizNh0CjYEb/XldKqlL1q+eqhWqrdZnHk4QzO4RI8uIYa3EMdmsBgCM/wCm+OcF6cd+dj2ZpzsplT+APn8wfY5Y2F</latexit>

JT1K = JT2K

<latexit sha1_base64="RmSGkFbnLuSrGUxU6gAjYXqFA=">ACRXicbVDLSgMxFM34rPU16tJNsAiuykypj41QdOyQl/QliGTSdvQzIPkjlCG/olf48aF/oAf4U7EnWbaEfvwQuDc+4rx40EV2BZb8bK6tr6xmZuK7+9s7u3bx4cNlQYS8rqNBShbLlEMcEDVgcOgrUiyYjvCtZ0h7ep3nxgUvEwqMEoYl2f9APe45SAphzoiOEKwkdMsA1x8YdKX/TazynlWa0vGMWrKI1CbwM7AwUBZVx/zqeCGNfRYAFUSptm1F0E2IBE4FG+c7sWKRnk36rK1hQHymusnkf2N8qhkP90KpXwB4ws52JMRXauS7utInMFCLWkr+p7Vj6F1Ex5EMbCAThf1YoEhxKlZ2OSURAjDQiVXN+K6YBoD0BbOrdFgU/kSHp6Rnrn3OpVfaiMcugUSra5eL5fblQuclMy6FjdILOkI0uUQXdoSqI4oe0RN6Qa/Gs/FufBif09IVI+s5QnNhfP8AViGxhA=</latexit>

this talk graph-to-graph transformations (c. ’90)

baaabab . . .

<latexit sha1_base64="uWCbEYn82afSBWJvipScetRC9VA=">ACGnicbVC7SgNBFJ2NrxhfUubxSBYhV2JaBm0sYxgHpAs4c7sJBky+3DmrCEfIeNhf6Kndja+CeWziZbmMQDA4dz7p17ODSWQqPjfFuFtfWNza3idmlnd2/oHx41NJRohvskhGqkNBcylC3kSBkndixSGgkrfp+Dbz209caRGFD5jG3AtgGIqBYIBG8igAUKA96Ueo+WKU3VmsFeJm5MKydHol396fsSgIfIJGjdZ0YvQkoFEzyamXaB4DG8OQdw0NIeDam8xCT+0zo/j2IFLmhWjP1L8bEwi0TgNqJgPAkV72MvE/r5vg4NqbiDBOkIdsfmiQSBsjO2vA9oXiDGVqCDAlTFabjUABQ9PTwhWNAahU+eaPLGdPSmrGxhxLpip3uZhV0rqourXq5X2tUr/JSyuSE3JKzolLrkid3JEGaRJGHskzeSVv1ov1bn1Yn/PRgpXvHJMFWF+/sC2hvQ=</latexit>

abbaa . . .

<latexit sha1_base64="5vep7nCbjRhZ5rEQN9s1cI/sZFE=">ACGHicbVC7SgNBFJ2NrxhfUubwSBYhV1RtAzaWEYwD9iEcHd2Nhkyu7PM3BVCyGfYWOiv2ImtnX9i6STZwiQeGDic+zpzglQKg67RTW1jc2t4rbpZ3dvf2D8uFR06hM95gSirdDsBwKRLeQIGSt1PNIQ4kbwXDu2m9cS1ESp5xFHKuzH0ExEJBmglH4IAoCNDhaZXrhVdwa6SrycVEiOeq/80wkVy2KeIJNgjO+5KXbHoFEwySelTmZ4CmwIfe5bmkDMTXc8szyhZ1YJaS0fQnSmfp3YgyxMaM4sJ0x4MAs16bifzU/w+imOxZJmiFP2PxQlEmKik7/T0OhOUM5sgSYFtYrZQPQwNCmtHDFYAx6pEO7Y+qzI2Vg24YcSzYqbzmYVdK8qHqX1auHy0rtNg+tSE7IKTknHrkmNXJP6qRBGFHkmbySN+fFeXc+nM95a8HJZ47JApyvXxioOY=</latexit>

bab . . . bbab . . . aabb . . .

<latexit sha1_base64="goaEZfstFv1SxfyiQGmTzJCr1Q=">ACL3icbZC7SgNBFIZn4y3GW9TSZjAIVmFXFC2DNpYRzAWSJZyZnSRDZi/MnBXCktansbHQVxEbsfUNLJ1cQJN4YODjP9f5WaKkQd9d3Irq2vrG/nNwtb2zu5ecf+gbuJUc1HjsYp1k4ERSkaihKVaCZaQMiUaLDBzTjfeBDayDi6x2Ei/B6kexKDmilTpEyYG0VxGgo+0UAxuiUO8WSW3YnQZfBm0GJzKLaKX63g5inoYiQKzCm5bkJ+hlolFyJUaGdGpEAH0BPtCxGEArjZ5OfjOiJVQLajbV9EdKJ+rcjg9CYchsZQjYN4u5sfhfrpVi98rPZJSkKCI+XdRNFcWYjm2hgdSCoxpaAK6lvZXyPmjgaM2b2IwBD3UgZ0xvrOtFLNlA4EFa5W3aMwy1M/K3n54u68VLmemZYnR+SYnBKPXJIKuSVUiOcPJIn8kJenWfnzflwPqelOWfWc0jmwvn6ARTAqZg=</latexit>

automata that transform combinatorial objects (’60s)

Motivation

. .. . . . . . . . . . . . . . . .

Important classes of transductions can be described using both logic and automata.

slide-6
SLIDE 6

Graph-to-graph MSO transductions

subdivide every edge

Example:

slide-7
SLIDE 7

Graph-to-graph MSO transductions

subdivide every edge

Example:

disconnect a cycle

slide-8
SLIDE 8

Main results

slide-9
SLIDE 9

(1) An automaton model for graph-to-graph MSO transductions of graphs of bounded treewidth

Main results

slide-10
SLIDE 10

(1) An automaton model for graph-to-graph MSO transductions of graphs of bounded treewidth

Main results

undecidable otherwise

slide-11
SLIDE 11

(1) An automaton model for graph-to-graph MSO transductions of graphs of bounded treewidth (2) Decidability of equivalence of such transductions up to same number of walks in output graphs

Main results

≡

<latexit sha1_base64="M0nREVpT57DIrZNWbhpucYGmKs=">ACE3icbVDLTsJAFJ3iC/GFunTSExckdZgdEl04xITeS0IdPpLYxMH87ckjSEf3DjQn/FnXHrB/gnLp0CwFPMsnJua8zx0sEV2hZ30ZhbX1jc6u4XdrZ3ds/KB8etVScSgZNFotYdjyqQPAImshRQCeRQENPQNsb3ub19gik4nH0gFkCbkj7EQ84o6ilgNPKR/1yhWrak1hrhJ7Tipkjkav/OP4MUtDiJAJqlTXthJ0x1QiZwImJSdVkFA2pH3oahrREJQ7nrqdmGda8c0glvpFaE7VvxNjGiqVhZ7uDCkO1HItF/+rdVMrt0xj5IUIWKzQ0EqTIzN/OumzyUwFJkmlEmuvZpsQCVlqANauKIwpDKTvt6R+3SE8HTbELCko7KXg1klrYuqXate3tcq9Zt5aEVyQk7JObHJFamTO9IgTcLI3kmr+TNeDHejQ/jc9ZaMOYzx2QBxtcvLAOe0Q=</latexit>

7 walks of length 0, 12 walks of length 1, 24 walks of length 2, …

undecidable otherwise

slide-12
SLIDE 12

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

abbaa , − → (19, 32)

<latexit sha1_base64="LcPhZuA21oHi0ofsa5rSnj5H19U=">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</latexit>

compatible with composition

  • f output objects

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

x((x + 2)(x(x + 2)2 + 2)) + 2

<latexit sha1_base64="vGhXB23nKhxIqPGFGpnWcCMctMs=">ACJXicbVDLSgMxFM3UV62vUcGNm8EiTCmUmVLRZdGNywr2Ae1YMmnahmYeJHekw9ifceNCf8WdCK78DZem0y5s64Ukh3PzT0cN+RMgmV9aZm19Y3Nrex2bmd3b/9APzxqyCAShNZJwAPRcrGknPm0Dgw4bYWCYs/ltOmObqb95iMVkgX+PcQhdTw8FmfEQyK6uonY9McF8sFc5w+D2V1FYrlrp63SlZaxiqw5yCP5lXr6j+dXkAij/pAOJaybVshOAkWwAink1wnkjTEZIQHtK2gjz0qnST1PzHOFdMz+oFQxwcjZf9OJNiTMvZcpfQwDOVyb0r+12tH0L9yEuaHEVCfzBb1I25AYEzDMHpMUAI8VgATwZRXgwyxwARUZAtbJHhYxKn/pj67HDuKtmIQk5FZS8Hswoa5ZJdKV3cVfLV63loWXSKzpCJbHSJqugW1VAdEfSEntEretNetHftQ/ucSTPafOYLZT2/QuScaJE</latexit>

=

<latexit sha1_base64="L7u0qcqEKS2TcZHTXvGLoFmoZ48=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4KokouhGKbly2YB/QhjKZ3LRDJw9mboQ+gVuXOivuBO3/oJ/4tJm4VtPTBwOPd15rix4Aot69sora1vbG6Vtys7u3v7B9XDo46KEsmgzSIRyZ5LFQgeQhs5CujFEmjgCui6k/u83n0CqXgUPmIagxPQUch9zihqXU7rNasujWDuUrsgtRIgeaw+jPwIpYECITVKm+bcXoZFQiZwKmlUGiIKZsQkfQ1zSkASgnmxmdmda8Uw/kvqFaM7UvxMZDZRKA1d3BhTHarmWi/V+gn6N07GwzhBCNn8kJ8IEyMz/7XpcQkMRaoJZJryYbU0kZ6mwWrigMqEylp3fkPgdCuLptAljRUdnLwaySzkXdvqxftS5rjbsitDI5IafknNjkmjTIA2mSNmEyDN5JW/Gi/FufBif89aSUcwckwUYX7+e8pxW</latexit>

’18, Boiret et al., MSO ’99 Senizergues, DPDT ’00, Honkala, HDT0L sequence

?

slide-13
SLIDE 13

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

?

slide-14
SLIDE 14

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence:

slide-15
SLIDE 15

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: walk power series

slide-16
SLIDE 16

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>
slide-17
SLIDE 17

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: isomorphism: ? walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>
slide-18
SLIDE 18

Further research:

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: isomorphism: ? walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>
slide-19
SLIDE 19

Further research:

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: isomorphism: ? walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>

find a better power series

slide-20
SLIDE 20

Further research:

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: isomorphism: ? walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>

Known graph polynomials ? find a better power series

slide-21
SLIDE 21

Further research:

, − →

<latexit sha1_base64="v9ObQXrgJD4Jv/s14PzHYiDsck=">ACLHicbVC7SgNBFJ2NrxhfUsRFoNgFXYlomXQxjKCeUA2hNnJTJmdmaZuauEkMqvsbHQX7ERsfUbLJ1NUpjogYHDuec+5oSx4AY9793JLC2vrK5l13Mbm1vbO/ndvZpRiWZQZUo3QipAcElVJGjgEasgUahgHo4uEr9XvQhit5i8MYWhHtSd7ljKV2vnDQPSVGgR3iksNIhBK9jTv9ZFqrR7a+YJX9CZw/xJ/Rgpkhko7/x10FEsikMgENabpezG2RlQjZwLGuSAxEFM2oD1oWipBKY1mnxj7B5bpeN2lbZPojtRf3eMaGTMAqtM6LYN4u1VPyv1kywe9EacRknCJNF3UT4aJy0zcDtfAUAwtoUxze6vL+lRThja5uS0GI6qHumNnpHcGQoTWNgDM2aj8xWD+ktp0S8Vz25KhfLlLQsOSBH5IT45JyUyTWpkCph5JE8kRfy6jw7b86H8zm1ZpxZz6Zg/P1A5iNqX8=</latexit>

Idea that we explore

combinatorial object algebraic object

abbaa

(15, 16) x2 + 2x + 2

∞

X

n=1

xn

compatible with composition

  • f output objects

walk-equivalence: isomorphism: ? walk power series

7 + 12x + 24x2 + 48x3 + . . .

<latexit sha1_base64="Ob8Je95NgD0NVb+5C7A9V+9Oqk=">ACMHicbVDLTgIxFO3gC/GFunRTJSYmJmRmRGFJ4sYlJvJIYCdTgcaOo+0dwyEsPZr3LjQX9GVcesXuLQ8FgLepL0n57be3vcWHAFpvlhpNbWNza30tuZnd29/YPs4VFNRYmkrEojEcmGSxQTPGRV4CBYI5aMBK5gdbd/O6nXH5lUPAofYBgzJyDdkPucEtBUJ3taxJfYsgf6tguDtq1zoYQH7SsNWsKLQHWyOTNvTgOvAmsOcmgelU72p+VFNAlYCFQpZqWGYMzIhI4FWycaSWKxYT2SZc1NQxJwJQzmn5ljM8142E/kvqEgKfs34RCZQaBq5WBgR6ark2If+rNRPwS86Ih3ECLKSzQX4iMER4gv2uGQUxFADQiXu2LaI5JQ0O4tTFEQEDmUn5jsmdLCFfL+gwy2ipr2ZhVULPzViF/fV/IlW/mpqXRCTpDF8hCRVRGd6iCqoiJ/SMXtGb8WK8G5/G10yaMuY9x2ghjO9fjCSlwQ=</latexit>

= 7 − 2x 1 − 2x

<latexit sha1_base64="+9FzPs10S1IZHiPQ+SKQlbFIQY=">ACInicbVDLSsNAFJ3UV62vaJduBovgxpKUat0IBTcuK9haEKZTCft0MmDmRsxhH6LGxf6K+7EleCHuHTSdmFbL8xwOc+DseLBVdgWV9GYW19Y3OruF3a2d3bPzAPjzoqSiRlbRqJSHY9opjgIWsDB8G6sWQk8AR78MY3uf7wyKTiUXgPaczcgAxD7nNKQFN9s3yNHV8SmjXOa0+TzM7/vlmxqta08Cqw56C5tXqmz/OIKJwEKgijVs60Y3IxI4FSwSclJFIsJHZMh62kYkoApN5uan+BTzQywH0n9QsBT9u9ERgKl0sDTnQGBkVrWcvI/rZeAf+VmPIwTYCGdHfITgSHCeRJ4wCWjIFINCJVce8V0RHQWoPNauKIgIDKVA70j9+kI4em2MYOSjspeDmYVdGpVu169uKtXmpfz0IroGJ2gM2SjBmqiW9RCbURip7RK3ozXox348P4nLUWjPlMGS2U8f0LGg6jTQ=</latexit>

=

<latexit sha1_base64="+QDocTureoaieRs+d4eQu3hJ8K4=">ACDnicbVDLSsNAFJ3UV62vqks3wSK4Kon42gFNy5bsA9oQ5lMbtqhkwczN0I/QI3LvRX3Ilbf8E/cemkzcK2Hhg4nPs6c9xYcIW9W2U1tY3NrfK25Wd3b39g+rhUdFiWTQZpGIZM+lCgQPoY0cBfRiCTRwBXTdyX1e7z6BVDwKHzGNwQnoKOQ+ZxS1LobVmtW3ZrBXCV2QWqkQHNY/Rl4EUsCJEJqlTftmJ0MiqRMwHTyiBREFM2oSPoaxrSAJSTzYxOzTOteKYfSf1CNGfq34mMBkqlgas7A4pjtVzLxf9q/QT9WyfjYZwghGx+yE+EiZGZ/9r0uASGItWEMsm1V5ONqaQMdTYLVxQGVKbS0ztynwMhXN02AazoqOzlYFZJ56JuX9avWpe1xnURWpmckFNyTmxyQxrkgTRJmzAC5Jm8kjfjxXg3PozPeWvJKGaOyQKMr1+bVpxK</latexit>

different purposes Known graph polynomials find a better power series

slide-22
SLIDE 22

Thank you for your attention!

∃E e ∈ E ∧ ∀e0 ∈ E inc(w, e0) ∧ inc(w, e00) ∃!2 e00 ∈ E ∧ ∃!2 e00 ∈ E ∃w