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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

We derive an intuitionistic version of Gödel-Löb modal logic (GL) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, ℓIGL, by restricting a non-wellfounded labelled system for GL to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that GL’s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of GL are typically defined over only the box (and not the diamond), our presentation includes both modalities.
Our main result is that ℓIGL coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic IGL. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of IGL.

Anupam Das, Iris van der Giessen, and Sonia Marin. Intuitionistic Gödel-Löb Logic, à la Simpson: Labelled Systems and Birelational Semantics. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{das_et_al:LIPIcs.CSL.2024.22, author = {Das, Anupam and van der Giessen, Iris and Marin, Sonia}, title = {{Intuitionistic G\"{o}del-L\"{o}b Logic, \`{a} la Simpson: Labelled Systems and Birelational Semantics}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {22:1--22:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.22}, URN = {urn:nbn:de:0030-drops-196654}, doi = {10.4230/LIPIcs.CSL.2024.22}, annote = {Keywords: provability logic, proof theory, intuitionistic modal logic, cyclic proofs, non-wellfounded proofs, proof search, cut-elimination, labelled sequents} }

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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Extensions of Girard’s linear logic by least and greatest fixed point operators (μMALL) have been an active field of research for almost two decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we compare the relative expressivity, at the level of provability, of two complementary infinitary proof systems: finitely branching non-wellfounded proofs (μMALL^∞) vs. infinitely branching well-founded proofs (μMALL_{ω,∞}). Our main result is that μMALL^∞ is strictly contained in μMALL_{ω,∞}.
For inclusion, we devise a novel technique involving infinitary rewriting of non-wellfounded proofs that yields a wellfounded proof in the limit. For strictness of the inclusion, we improve previously known lower bounds on μMALL^∞ provability from Π⁰₁-hard to Σ¹₁-hard, by encoding a sort of Büchi condition for Minsky machines.

Anupam Das, Abhishek De, and Alexis Saurin. Comparing Infinitary Systems for Linear Logic with Fixed Points. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{das_et_al:LIPIcs.FSTTCS.2023.40, author = {Das, Anupam and De, Abhishek and Saurin, Alexis}, title = {{Comparing Infinitary Systems for Linear Logic with Fixed Points}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {40:1--40:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.40}, URN = {urn:nbn:de:0030-drops-194131}, doi = {10.4230/LIPIcs.FSTTCS.2023.40}, annote = {Keywords: linear logic, fixed points, non-wellfounded proofs, omega-branching proofs, analytical hierarchy} }

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**Published in:** LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain "impredicative" theories; moreover, our cyclic systems naturally subsume Simpson’s Cyclic Arithmetic.
Our main result is that cyclic and inductive systems for arithmetical inductive definitions are equally powerful. We conduct a metamathematical argument, formalising the soundness of cyclic proofs within second-order arithmetic by a form of induction on closure ordinals, thence appealing to conservativity results. This approach is inspired by those of Simpson and Das for Cyclic Arithmetic, however we must further address a difficulty: the closure ordinals of our inductive definitions (around Church-Kleene) far exceed the proof theoretic ordinal of the appropriate metatheory (around Bachmann-Howard), so explicit induction on their notations is not possible. For this reason, we rather rely on formalisation of the theory of (recursive) ordinals within second-order arithmetic.

Anupam Das and Lukas Melgaard. Cyclic Proofs for Arithmetical Inductive Definitions. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{das_et_al:LIPIcs.FSCD.2023.27, author = {Das, Anupam and Melgaard, Lukas}, title = {{Cyclic Proofs for Arithmetical Inductive Definitions}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {27:1--27:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-277-8}, ISSN = {1868-8969}, year = {2023}, volume = {260}, editor = {Gaboardi, Marco and van Raamsdonk, Femke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.27}, URN = {urn:nbn:de:0030-drops-180119}, doi = {10.4230/LIPIcs.FSCD.2023.27}, annote = {Keywords: cyclic proofs, inductive definitions, arithmetic, fixed points, proof theory} }

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**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Cyclic and non-wellfounded proofs are now increasingly employed to establish metalogical results in a variety of settings, in particular for type systems with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic proof can be seen as a typing derivation "with loops", closer to low-level machine models, and so comprise a highly expressive computational model that nonetheless enjoys excellent metalogical properties.
In recent work, we showed how the cyclic proof setting can be further employed to model computational complexity, yielding characterisations of the polynomial time and elementary computable functions. These characterisations are "implicit", inspired by Bellantoni and Cook’s famous algebra of safe recursion, but exhibit greater expressivity thanks to the looping capacity of cyclic proofs.
In this work we investigate the capacity for non-wellfounded proofs, where finite presentability is relaxed, to model non-uniformity in complexity theory. In particular, we present a characterisation of the class FP/poly of functions computed by polynomial-size circuits. While relating non-wellfoundedness to non-uniformity is a natural idea, the precise amount of irregularity, informally speaking, required to capture FP/poly is given by proof-level conditions novel to cyclic proof theory. Along the way, we formalise some (presumably) folklore techniques for characterising non-uniform classes in relativised function algebras with appropriate oracles.

Gianluca Curzi and Anupam Das. Non-Uniform Complexity via Non-Wellfounded Proofs. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{curzi_et_al:LIPIcs.CSL.2023.16, author = {Curzi, Gianluca and Das, Anupam}, title = {{Non-Uniform Complexity via Non-Wellfounded Proofs}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.16}, URN = {urn:nbn:de:0030-drops-174770}, doi = {10.4230/LIPIcs.CSL.2023.16}, annote = {Keywords: Cyclic proofs, non-wellfounded proof-theory, non-uniform complexity, polynomial time, safe recursion, implicit complexity} }

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**Published in:** LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)

Linear logic is an important logic for modelling resources and decomposing computational interpretations of proofs. Decision problems for fragments of linear logic exhibiting "infinitary" behaviour (such as exponentials) are notoriously complicated. In this work, we address the decision problems for variations of linear logic with fixed points (μMALL), in particular, recent systems based on "circular" and "non-wellfounded" reasoning. In this paper, we show that μMALL is undecidable.
More explicitly, we show that the general non-wellfounded system is Π⁰₁-hard via a reduction to the non-halting of Minsky machines, and thus is strictly stronger than its circular counterpart (which is in Σ⁰₁). Moreover, we show that the restriction of these systems to theorems with only the least fixed points is already Σ⁰₁-complete via a reduction to the reachability problem of alternating vector addition systems with states. This implies that both the circular system and the finitary system (with explicit (co)induction) are Σ⁰₁-complete.

Anupam Das, Abhishek De, and Alexis Saurin. Decision Problems for Linear Logic with Least and Greatest Fixed Points. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{das_et_al:LIPIcs.FSCD.2022.20, author = {Das, Anupam and De, Abhishek and Saurin, Alexis}, title = {{Decision Problems for Linear Logic with Least and Greatest Fixed Points}}, booktitle = {7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-233-4}, ISSN = {1868-8969}, year = {2022}, volume = {228}, editor = {Felty, Amy P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.20}, URN = {urn:nbn:de:0030-drops-163019}, doi = {10.4230/LIPIcs.FSCD.2022.20}, annote = {Keywords: Linear logic, fixed points, decidability, vector addition systems} }

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**Published in:** LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems.
Systems of linear logic and, later, deep inference are founded upon two particular linear inferences, switch : x ∧ (y ∨ z) → (x ∧ y) ∨ z, and medial : (w ∧ x) ∨ (y ∧ z) → (w ∨ y) ∧ (x ∨ z). It is well-known that these two are not enough to derive all linear inferences (even modulo all valid linear equations), but beyond this little more is known about the structure of linear inferences in general. In particular despite recurring attention in the literature, the smallest linear inference not derivable under switch and medial ("switch-medial-independent") was not previously known.
In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find two "minimal" 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. One of these new inferences derives some previously found independent linear inferences. The other exhibits structure seemingly beyond the scope of previous approaches we are aware of; in particular, its existence contradicts a conjecture of Das and Strassburger.

Anupam Das and Alex A. Rice. New Minimal Linear Inferences in Boolean Logic Independent of Switch and Medial. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{das_et_al:LIPIcs.FSCD.2021.14, author = {Das, Anupam and Rice, Alex A.}, title = {{New Minimal Linear Inferences in Boolean Logic Independent of Switch and Medial}}, booktitle = {6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)}, pages = {14:1--14:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-191-7}, ISSN = {1868-8969}, year = {2021}, volume = {195}, editor = {Kobayashi, Naoki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.14}, URN = {urn:nbn:de:0030-drops-142525}, doi = {10.4230/LIPIcs.FSCD.2021.14}, annote = {Keywords: rewriting, linear inference, proof theory, linear logic, implementation} }

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**Published in:** LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

In this work we address the logical strength of confluence and normalisation for non-wellfounded typing derivations, in the tradition of "cyclic proof theory". We present a circular version CT of Gödel's system T, with the aim of comparing the relative expressivity of the theories CT and T. We approach this problem by formalising rewriting-theoretic results such as confluence and normalisation for the underlying "coterm" rewriting system of CT within fragments of second-order arithmetic.
We establish confluence of CT within the theory RCA₀, a conservative extension of primitive recursive arithmetic and IΣ1. This allows us to recast type structures of hereditarily recursive operations as "coterm" models of T. We show that these also form models of CT, by formalising a totality argument for circular typing derivations within suitable fragments of second-order arithmetic. Relying on well-known proof mining results, we thus obtain an interpretation of CT into T; in fact, more precisely, we interpret level-n-CT into level-(n+1)-T, an optimum result in terms of abstraction complexity.
A direct consequence of these model-theoretic results is weak normalisation for CT. As further results, we also show strong normalisation for CT and continuity of functionals computed by its type 2 coterms.

Anupam Das. On the Logical Strength of Confluence and Normalisation for Cyclic Proofs. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 29:1-29:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{das:LIPIcs.FSCD.2021.29, author = {Das, Anupam}, title = {{On the Logical Strength of Confluence and Normalisation for Cyclic Proofs}}, booktitle = {6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)}, pages = {29:1--29:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-191-7}, ISSN = {1868-8969}, year = {2021}, volume = {195}, editor = {Kobayashi, Naoki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.29}, URN = {urn:nbn:de:0030-drops-142678}, doi = {10.4230/LIPIcs.FSCD.2021.29}, annote = {Keywords: confluence, normalisation, system T, circular proofs, reverse mathematics, type structures} }

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**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs, deterministic or non-deterministic. Decision trees (DTs) are represented by a natural term syntax, inducing the system LDT, and non-determinism is modelled by including disjunction, ∨, as primitive (system LNDT). Branching programs generalise DTs to dag-like structures and are duly handled by extension variables in our setting, as is common in proof complexity (systems eLDT and eLNDT).
Deterministic and non-deterministic branching programs are natural nonuniform analogues of log-space (L) and nondeterministic log-space (NL), respectively. Thus eLDT and eLNDT serve as natural systems of reasoning corresponding to L and NL, respectively.
The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in LDT, LNDT, eLDT and eLNDT. We also compare them with Frege systems, constant-depth Frege systems and extended Frege systems.

Sam Buss, Anupam Das, and Alexander Knop. Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buss_et_al:LIPIcs.CSL.2020.12, author = {Buss, Sam and Das, Anupam and Knop, Alexander}, title = {{Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.12}, URN = {urn:nbn:de:0030-drops-116553}, doi = {10.4230/LIPIcs.CSL.2020.12}, annote = {Keywords: proof complexity, decision trees, branching programs, logspace, sequent calculus, non-determinism, low-depth complexity} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We extend work of Lautemann, Schwentick and Stewart [Clemens Lautemann et al., 1996] on characterisations of the "positive" polynomial-time predicates (posP, also called mP by Grigni and Sipser [Grigni and Sipser, 1992]) to function classes. Our main result is the obtention of a function algebra for the positive polynomial-time functions (posFP) by imposing a simple uniformity constraint on the bounded recursion operator in Cobham's characterisation of FP. We show that a similar constraint on a function algebra based on safe recursion, in the style of Bellantoni and Cook [Stephen Bellantoni and Stephen A. Cook, 1992], yields an "implicit" characterisation of posFP, mentioning neither explicit bounds nor explicit monotonicity constraints.

Anupam Das and Isabel Oitavem. A Recursion-Theoretic Characterisation of the Positive Polynomial-Time Functions. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.CSL.2018.18, author = {Das, Anupam and Oitavem, Isabel}, title = {{A Recursion-Theoretic Characterisation of the Positive Polynomial-Time Functions}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.18}, URN = {urn:nbn:de:0030-drops-96851}, doi = {10.4230/LIPIcs.CSL.2018.18}, annote = {Keywords: Monotone complexity, Positive complexity, Function classes, Function algebras, Recursion-theoretic characterisations, Implicit complexity, Logic} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We prove cut-elimination for a sequent-style proof system which is sound and complete for the equational theory of Kleene algebra, and where proofs are (potentially) non-wellfounded infinite trees. We extend these results to systems with meets and residuals, capturing `star-continuous' action lattices in a similar way. We recover the equational theory of all action lattices by restricting to regular proofs (with cut) - those proofs that are unfoldings of finite graphs.

Anupam Das and Damien Pous. Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices). In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{das_et_al:LIPIcs.CSL.2018.19, author = {Das, Anupam and Pous, Damien}, title = {{Non-Wellfounded Proof Theory For (Kleene+Action)(Algebras+Lattices)}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {19:1--19:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.19}, URN = {urn:nbn:de:0030-drops-96869}, doi = {10.4230/LIPIcs.CSL.2018.19}, annote = {Keywords: Kleene algebra, proof theory, sequent system, non-wellfounded proofs} }

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**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

We prove a general form of 'free-cut elimination' for first-order theories in linear logic, yielding normal forms of proofs where cuts are anchored to nonlogical steps. To demonstrate the usefulness of this result, we consider a version of arithmetic in linear logic, based on a previous axiomatisation by Bellantoni and Hofmann. We prove a witnessing theorem for a fragment of this arithmetic via the `witness function method', showing that the provably convergent functions are precisely the polynomial-time functions. The programs extracted are implemented in the framework of 'safe' recursive functions, due to Bellantoni and Cook, where the ! modality of linear logic corresponds to normal inputs of a safe recursive program.

Patrick Baillot and Anupam Das. Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 40:1-40:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baillot_et_al:LIPIcs.CSL.2016.40, author = {Baillot, Patrick and Das, Anupam}, title = {{Free-Cut Elimination in Linear Logic and an Application to a Feasible Arithmetic}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {40:1--40:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.40}, URN = {urn:nbn:de:0030-drops-65807}, doi = {10.4230/LIPIcs.CSL.2016.40}, annote = {Keywords: proof theory, linear logic, bounded arithmetic, polynomial time computation, implicit computational complexity} }

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**Published in:** LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)

Recently it has been observed that the set of all sound linear inference rules in propositional logic is already coNP-complete, i.e. that every Boolean tautology can be written as a (left- and right-) linear rewrite rule. This raises the question of whether there is a rewriting system on linear terms of propositional logic that is sound and complete for the set of all such rewrite rules. We show in this paper that, as long as reduction steps are polynomial-time decidable, such a rewriting system does not exist unless coNP=NP.
We draw tools and concepts from term rewriting, Boolean function theory and graph theory in order to access the required intermediate results. At the same time we make several connections between these areas that, to our knowledge, have not yet been presented and constitute a rich theoretical framework for reasoning about linear TRSs for propositional logic.

Anupam Das and Lutz Straßburger. No complete linear term rewriting system for propositional logic. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 127-142, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{das_et_al:LIPIcs.RTA.2015.127, author = {Das, Anupam and Stra{\ss}burger, Lutz}, title = {{No complete linear term rewriting system for propositional logic}}, booktitle = {26th International Conference on Rewriting Techniques and Applications (RTA 2015)}, pages = {127--142}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-85-9}, ISSN = {1868-8969}, year = {2015}, volume = {36}, editor = {Fern\'{a}ndez, Maribel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.127}, URN = {urn:nbn:de:0030-drops-51935}, doi = {10.4230/LIPIcs.RTA.2015.127}, annote = {Keywords: Linear rules, Term rewriting, Propositional logic, Proof theory, Deep inference} }

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**Published in:** LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)

Linear inferences are sound implications of propositional logic where each variable appears exactly once in the premiss and conclusion. We consider a specific set of these inferences, MS, first studied by Straßburger, corresponding to the logical rules in deep inference proof theory. Despite previous results characterising the individual rules of MS, we show that there is no polynomial-time characterisation of MS, assuming that integers cannot be factorised in polynomial time. We also examine the length of rewrite paths in an extended system MSU that also has unit equations, utilising a notion dubbed trivialisation to reduce the case with units to the case without, amongst other observations on MS-rewriting and the set of linear inferences in general.

Anupam Das. Rewriting with Linear Inferences in Propositional Logic. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 158-173, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{das:LIPIcs.RTA.2013.158, author = {Das, Anupam}, title = {{Rewriting with Linear Inferences in Propositional Logic}}, booktitle = {24th International Conference on Rewriting Techniques and Applications (RTA 2013)}, pages = {158--173}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-53-8}, ISSN = {1868-8969}, year = {2013}, volume = {21}, editor = {van Raamsdonk, Femke}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.158}, URN = {urn:nbn:de:0030-drops-40609}, doi = {10.4230/LIPIcs.RTA.2013.158}, annote = {Keywords: proof theory, propositional logic, complexity of proofs, deep inference} }

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