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1. Preprocessing of Propagation Redundant Clauses

   Reeves, Joseph E. ; Bryant, Randal E. ; Heule, Marijn J. ( August 2022 , Lecture notes in computer science)

   Blanchette, Jasmin ; Kovacs, Laura ; Pattinson, Dirk (Ed.)

   The propagation redundant (PR) proof system generalizes the resolution and resolution asymmetric tautology proof systems used by conflict-driven clause learning (CDCL) solvers. PR allows short proofs of unsatisfiability for some problems that are difficult for CDCL solvers. Previous attempts to automate PR clause learning used hand-crafted heuristics that work well on some highly-structured problems. For example, the solver SaDiCaL incorporates PR clause learning into the CDCL loop, but it cannot compete with modern CDCL solvers due to its fragile heuristics. We present prExtract, a preprocessing technique that learns short PR clauses. Adding these clauses to a formula reduces the search space that the solver must explore. By performing PR clause learning as a preprocessing stage, PR clauses can be found efficiently without sacrificing the robustness of modern CDCL solvers. On a large portion of SAT competition benchmarks we found that preprocessing with prExtract improves solver performance. In addition, there were several satisfiable and unsatisfiable formulas that could only be solved after preprocessing with prExtract. prExtract supports proof logging, giving a high level of confidence in the results. 

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2. Simulating Strong Practical Proof Systems with Extended Resolution

   https://doi.org/10.1007/s10817-020-09554-z  

   Kiesl, Benjamin ; Rebola-Pardo, Adrián ; Heule, Marijn J. ; Biere, Armin ( July 2020 , Journal of Automated Reasoning)

   Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that—from the viewpoint of proof complexity—these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems. 

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3. Strong Extension-Free Proof Systems

   https://doi.org/10.1007/s10817-019-09516-0  

   Heule, Marijn J. ; Kiesl, Benjamin ; Biere, Armin ( February 2019 , Journal of Automated Reasoning)

   We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider various efficiently decidable redundancy criteria which we obtain by first characterizing clause redundancy in terms of a semantic implication relationship and then restricting this relationship so that it becomes decidable in polynomial time. As the restricted implication relation is based on unit propagation---a core technique of SAT solvers---it allows efficient proof checking too. The resulting proof systems are surprisingly strong, even without the introduction of new variables---a key feature of short proofs presented in the proof-complexity literature. We demonstrate the strength of our proof systems on the famous pigeon hole formulas by providing short clausal proofs without new variables. 

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4. Leibniz International Proceedings in Informatics (LIPIcs):14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

   https://doi.org/10.4230/LIPIcs.ITCS.2023.101  

   Yolcu, Emre ; Heule, Marijn J. ( January 2023 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Tauman Kalai, Yael (Ed.)

   We study the complexity of proof systems augmenting resolution with inference rules that allow, given a formula Γ in conjunctive normal form, deriving clauses that are not necessarily logically implied by Γ but whose addition to Γ preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in Γ, the systems we consider become equivalent to extended resolution. We are concerned with the versions of these systems without new variables. They are called BC⁻, RAT⁻, SBC⁻, and GER⁻, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold --- without loss of generality --- which is commonly used informally to simplify or shorten proofs. Except for SBC⁻, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. By taking advantage of this fact, we construct formulas that separate RAT⁻ from GER⁻ and vice versa. With the same strategy, we also separate SBC⁻ from RAT⁻. Additionally, we give polynomial-size SBC⁻ proofs of the pigeonhole principle, which separates SBC⁻ from GER⁻ by a previously known lower bound. These results also separate the three systems from BC⁻ since they all simulate it. We thus give an almost complete picture of their relative strengths. 

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5. Hopf Monoids and Generalized Permutahedra

   https://doi.org/10.1090/memo/1437  

   Aguiar, Marcelo ; Ardila, Federico ( September 2023 , Memoirs of the American Mathematical Society)

   Generalized permutahedra are polytopes that arise in combinatorics, algebraic geometry, representation theory, topology, and optimization. They possess a rich combinatorial structure. Out of this structure we build a Hopf monoid in the category of species.

   Species provide a unifying framework for organizing families of combinatorial objects. Many species carry a Hopf monoid structure and are related to generalized permutahedra by means of morphisms of Hopf monoids. This includes the species of graphs, matroids, posets, set partitions, linear graphs, hypergraphs, simplicial complexes, and building sets, among others. We employ this algebraic structure to define and study polynomial invariants of the various combinatorial structures.

   We pay special attention to the antipode of each Hopf monoid. This map is central to the structure of a Hopf monoid, and it interacts well with its characters and polynomial invariants. It also carries information on the values of the invariants on negative integers. For our Hopf monoid of generalized permutahedra, we show that the antipode maps each polytope to the alternating sum of its faces. This fact has numerous combinatorial consequences.

   We highlight some main applications:

   We obtain uniform proofs of numerous old and new results about the Hopf algebraic and combinatorial structures of these families. In particular, we give optimal formulas for the antipode of graphs, posets, matroids, hypergraphs, and building sets. They are optimal in the sense that they provide explicit descriptions for the integers entering in the expansion of the antipode, after all coefficients have been collected and all cancellations have been taken into account.

   We show that reciprocity theorems of Stanley and Billera–Jia–Reiner (BJR) on chromatic polynomials of graphs, order polynomials of posets, and BJR-polynomials of matroids are instances of one such result for generalized permutahedra.

   We explain why the formulas for the multiplicative and compositional inverses of power series are governed by the face structure of permutahedra and associahedra, respectively, providing an answer to a question of Loday.

   We answer a question of Humpert and Martin on certain invariants of graphs and another of Rota on a certain class of submodular functions.

   We hope our work serves as a quick introduction to the theory of Hopf monoids in species, particularly to the reader interested in combinatorial applications. It may be supplemented with Marcelo Aguiar and Swapneel Mahajan’s 2010 and 2013 works, which provide longer accounts with a more algebraic focus.

    

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