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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Good-for-Games (GFG) automata offer a compromise between deterministic and nondeterministic automata. They can resolve nondeterministic choices in a step-by-step fashion, without needing any information about the remaining suffix of the word. These automata can be used to solve games with omega-regular conditions, and in particular were introduced as a tool to solve Church's synthesis problem. We focus here on the problem of recognizing Büchi GFG automata, that we call Büchi GFGness problem: given a nondeterministic Büchi automaton, is it GFG? We show that this problem can be decided in P, and more precisely in O(n^4m^2|Sigma|^2), where n is the number of states, m the number of transitions and |Sigma| is the size of the alphabet. We conjecture that a very similar algorithm solves the problem in polynomial time for any fixed parity acceptance condition.

Marc Bagnol and Denis Kuperberg. Büchi Good-for-Games Automata Are Efficiently Recognizable. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bagnol_et_al:LIPIcs.FSTTCS.2018.16, author = {Bagnol, Marc and Kuperberg, Denis}, title = {{B\"{u}chi Good-for-Games Automata Are Efficiently Recognizable}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.16}, URN = {urn:nbn:de:0030-drops-99157}, doi = {10.4230/LIPIcs.FSTTCS.2018.16}, annote = {Keywords: B\"{u}chi, automata, games, polynomial time, nondeterminism} }

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**Published in:** LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is Pspace-complete, which rules out any low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds(under AC_0 reductions)to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently.
We show these two equivalent problems to be Logspace-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.

Marc Bagnol. MALL Proof Equivalence is Logspace-Complete, via Binary Decision Diagrams. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 60-75, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bagnol:LIPIcs.TLCA.2015.60, author = {Bagnol, Marc}, title = {{MALL Proof Equivalence is Logspace-Complete, via Binary Decision Diagrams}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {60--75}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-87-3}, ISSN = {1868-8969}, year = {2015}, volume = {38}, editor = {Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.60}, URN = {urn:nbn:de:0030-drops-51558}, doi = {10.4230/LIPIcs.TLCA.2015.60}, annote = {Keywords: linear logic, proof equivalence, additive connectives, proofnets, binary decision diagrams, logarithmic space, AC0 reductions} }

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