Perfect Half Space Games Thomas Colcombet, Marcin Jurdzi nski, - PowerPoint PPT Presentation

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Perfect Half Space Games Thomas Colcombet, Marcin Jurdzi´ nski, Ranko Lazi´ c, and Sylvain Schmitz LSV, ENS Paris-Saclay & CNRS & Inria LICS 2017, June 23rd, 2017 1/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games What to do this week-end? Uncontrolled events Maximal dry temperature Discrete resources as a parity objective Uncontrolled events as a two-players game Discrete resources 6 o M M´ yvatn as a multi-energy objective 0 0 10 o H Hr´ utafj¨ ordur 1 1 (− 1,0 ) (− 1,0 ) (− 2, − 1 ) (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 0 4 o Vatnaj¨ okull V 0 0 Reykjavik R 9 o 13 o L Landmannalaugar (− 1,0 ) 2 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 1 12 o Th´ orsm¨ ork T 2/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games M Player 1 wins a play if both 0 H 1 (− 1,0 ) ◮ energy objective: no (− 1,0 ) (− 2, − 1 ) component goes negative (− 1,0 ) (− 1,0 ) ( 0,0 ) 0 V ◮ parity objective: the maximal 0 R L (− 1,0 ) priority is odd 2 (− 4, − 5 ) ( 1,0 ) ( 0,0 ) (− 4, − 3 ) 1 T Example ( 1,0 ) ( 1,0 ) (− 1,0 ) ( 0,0 ) R ( 0,0 ) − − − → R ( 1,0 ) − − − → R ( 2,0 ) − − − − → H ( 1,0 ) − − − → R ( 1,0 ) → ··· 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Applications ◮ contractive ( ⊕ ,! ) -Horn linear logic (Kanovich, APAL ’95) ◮ (weak) simulation of finite-state systems by Petri nets (Abdulla et al., Concur ’13) ◮ model-checking Petri nets with a fragment of µ -calculus (Abdulla et al., Concur ’13) ◮ resource-bounded agent temporal logic RB ± ATL ∗ (Alechina et al., RP ’16 & AI ’17) 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Complexity lower bound upper bound w. initial credit ∃ initial credit 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Complexity lower bound upper bound EXPSPACE TOWER w. initial credit (Lasota, IPL ’09) (Br´ azdil et al., ICALP ’10) coNP coNP ∃ initial credit (Chatterjee et al., FSTTCS ’10) (Chatterjee et al., FSTTCS ’10) 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Complexity lower bound upper bound 2-EXP TOWER w. initial credit (Courtois and S., MFCS ’14) (Br´ azdil et al., ICALP ’10) coNP coNP ∃ initial credit (Chatterjee et al., FSTTCS ’10) (Chatterjee et al., FSTTCS ’10) 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Complexity lower bound upper bound 2-EXP 2-EXP w. initial credit (Courtois and S., MFCS ’14) (Jurdzi´ nski et al., ICALP ’15) coNP coNP ∃ initial credit (Chatterjee et al., FSTTCS ’10) (Chatterjee et al., FSTTCS ’10) 3/10

Energy Parity Games Extended Energy Games Bounding Games Perfect Half Space Games Multi-Dimensional Energy Parity Games Complexity lower bound upper bound 2-EXP w. initial credit (Courtois and S., MFCS ’14) coNP coNP ∃ initial credit (Chatterjee et al., Concur ’12) (Chatterjee et al., Concur ’12) 3/10