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1. ZDD Boolean Synthesis

   Lin, Yi; Tabajara, Lucas M.; Vardi, Moshe Y. (April 2022, International Conference on Tools and Algorithms for the Construction and Analysis of Systems)

   Dana Fisman and Grigore Rosu (Ed.)

   Motivated by applications in boolean-circuit design, boolean synthesis is the process of synthesizing a boolean function with multiple outputs, given a relation between its inputs and outputs. Previous work has attempted to solve boolean functional synthesis by converting a specification formula into a Binary Decision Diagram (BDD) and quantifying existentially the output variables. We make use of the fact that the specification is usually given in the form of a Conjunctive Normal Form (CNF) formula, and we can perform resolution on a symbolic representation of a CNF formula in the form of a Zero-suppressed Binary Decision Diagram (ZDD). We adapt the realizability test to the context of CNF and ZDD, and show that the Cross operation defined in earlier work can be used for witness construction. Experiments show that our approach is complementary to BDD-based Boolean synthesis. 

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2. Fridge Compiler: Optimal Circuits from Molecular Inventories

   https://doi.org/10.1007/978-3-031-42697-1_16  

   Wathieu, Lancelot; Smith, Gus; Ceze, Luis; Thachuk, Chris (September 2023, Proceedings of Computational Methods in Systems Biology. CMSB 2023.)

   Pang, J. (Ed.)

   Rationally designed molecular circuits describable by well-mixed chemical reaction kinetics can realize arbitrary Boolean function computation yet differ significantly from their electronic counterparts. The design, preparation, and purification of new molecular components poses significant barriers. Consequently, it is desirable to synthesize circuits from an existing “fridge” inventory of distinguishable parts, while satisfying constraints such as component compatibility. Heuristic synthesis techniques intended for large electronic circuits often result in non-optimal molecular circuits, invalid circuits that violate domain-specific constraints, or circuits that cannot be built with the current inventory. Existing “exact” synthesis techniques are able to find minimal feedforward Boolean circuits with complex constraints, but do not map to distinguishable inventory components. We present the Fridge Compiler, an SMT-based approach to find optimal Boolean circuits within a given molecular inventory. Empirical results demonstrate the Fridge Compiler’s versatility in synthesizing arbitrary Boolean functions using three different molecular architectures, while satisfying user-specified constraints. We showcase the successful synthesis of all 256 three-bit and 65,536 four-bit predicate functions using a large custom inventory, with worst-case completion times of only seconds on a modern laptop. In addition, we introduce a unique class of cyclic molecular circuits that cover a larger number of Boolean functions than their conventional counterparts over a common inventory, often with significantly smaller implementations. Importantly, and absent in previous approaches specific to molecular circuits, the Fridge Compiler is logically sound, complete, and optimal for the user-specified cost function and component inventory. 

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3. Distributed Implementation of Boolean Functions by Transcriptional Synthetic Circuits

   https://doi.org/10.1021/acssynbio.0c00228  

   Al-Radhawi, M. Ali; Tran, Anh Phong; Ernst, Elizabeth A.; Chen, Tianchi; Voigt, Christopher A.; Sontag, Eduardo D. (June 2020, ACS Synthetic Biology)

   Starting in the early 2000s, sophisticated technologies have been developed for the rational construction of synthetic genetic networks that implement specified logical functionalities. Despite impressive progress, however, the scaling necessary in order to achieve greater computational power has been hampered by many constraints, including repressor toxicity and the lack of large sets of mutually orthogonal repressors. As a consequence, a typical circuit contains no more than roughly seven repressor-based gates per cell. A possible way around this scalability problem is to distribute the computation among multiple cell types, each of which implements a small subcircuit, which communicate among themselves using diffusible small molecules (DSMs). Examples of DSMs are those employed by quorum sensing systems in bacteria. This paper focuses on systematic ways to implement this distributed approach, in the context of the evaluation of arbitrary Boolean functions. The unique characteristics of genetic circuits and the properties of DSMs require the development of new Boolean synthesis methods, distinct from those classically used in electronic circuit design. In this work, we propose a fast algorithm to synthesize distributed realizations for any Boolean function, under constraints on the number of gates per cell and the number of orthogonal DSMs. The method is based on an exact synthesis algorithm to find the minimal circuit per cell, which in turn allows us to build an extensive database of Boolean functions up to a given number of inputs. For concreteness, we will specifically focus on circuits of up to 4 inputs, which might represent, for example, two chemical inducers and two light inputs at different frequencies. Our method shows that, with a constraint of no more than seven gates per cell, the use of a single DSM increases the total number of realizable circuits by at least 7.58-fold compared to centralized computation. Moreover, when allowing two DSM’s, one can realize 99.995% of all possible 4-input Boolean functions, still with at most 7 gates per cell. The methodology introduced here can be readily adapted to complement recent genetic circuit design automation software. A toolbox that uses the proposed algorithm was created and made available at https://github. com/sontaglab/DBC/. 

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4. From Shapley Value to Model Counting and Back

   https://doi.org/10.1145/3651142  

   Kara, Ahmet; Olteanu, Dan; Suciu, Dan (May 2024, Proceedings of the ACM on Management of Data)

   In this paper we investigate the problem of quantifying the contribution of each variable to the satisfying assignments of a Boolean function based on the Shapley value. Our main result is a polynomial-time equivalence between computing Shapley values and model counting for any class of Boolean functions that are closed under substitutions of variables with disjunctions of fresh variables. This result settles an open problem raised in prior work, which sought to connect the Shapley value computation to probabilistic query evaluation. We show two applications of our result. First, the Shapley values can be computed in polynomial time over deterministic and decomposable circuits, since they are closed under OR-substitutions. Second, there is a polynomial-time equivalence between computing the Shapley value for the tuples contributing to the answer of a Boolean conjunctive query and counting the models in the lineage of the query. This equivalence allows us to immediately recover the dichotomy for Shapley value computation in case of self-join-free Boolean conjunctive queries; in particular, the hardness for non-hierarchical queries can now be shown using a simple reduction from the \#P-hard problem of model counting for lineage in positive bipartite disjunctive normal form. 

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5. Counting and Sampling Labeled Chordal Graphs in Polynomial Time

   https://doi.org/10.4230/LIPIcs.ESA.2023.58  

   Hébert-Johnson, Úrsula; Lokshtanov, Daniel; Vigoda, Eric (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   We present the first polynomial-time algorithm to exactly compute the number of labeled chordal graphs on n vertices. Our algorithm solves a more general problem: given n and ω as input, it computes the number of ω-colorable labeled chordal graphs on n vertices, using O(n⁷) arithmetic operations. A standard sampling-to-counting reduction then yields a polynomial-time exact sampler that generates an ω-colorable labeled chordal graph on n vertices uniformly at random. Our counting algorithm improves upon the previous best result by Wormald (1985), which computes the number of labeled chordal graphs on n vertices in time exponential in n. An implementation of the polynomial-time counting algorithm gives the number of labeled chordal graphs on up to 30 vertices in less than three minutes on a standard desktop computer. Previously, the number of labeled chordal graphs was only known for graphs on up to 15 vertices. 

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