Abhisekh Sankaran

Selected Publications

• Feferman-Vaught decompositions for prefix classes of first order logic   [abstract] [paper]
  

  Journal of Logic, Language and Information (JLLI), 32(1): 147-174, 2023

  The Feferman-Vaught theorem provides a way of evaluating a first order sentence φ on a disjoint union of structures by producing a decomposition of φ into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than φ. We show that for first order sentences in prenex normal form with a fixed number of quantifier alternations, such a decomposition, further with the same number of quantifier alternations, can be obtained in time elementary in the size of φ. We obtain this result as a consequence of a more general decomposition theorem that we prove for a family of infinitary logics we define. We extend these results by considering binary operations other than disjoint union, in particular sum-like operations such as ordered sum and NLC-sum, that are definable using quantifier-free interpretations.
• Clique-width of point configurations (with Onur Çağırıcı, Petr Hliněný, and Filip Pokrývka) [abstract] [ paper]
  

  Journal of Combinatorial Theory, Series B, Vol. 158 (Part 1): 43--73, 2023

  While structural width parameters (of the input) belong to the standard toolbox of graph algorithms, it is not the usual case in computational geometry. As a case study we propose a natural extension of the structural graph parameter of clique-width to geometric point configuration represented by their order type. We study basic properties of this clique-width notion, and show that it is aligned with the general concept of clique-width of relational structures by Blumensath and Courcelle (2006). We also relate the new notion to monadic second-order logic of point configurations. As an application, we provide several linear FPT time algorithms for geometric point problems which are NP-hard in general, in the special case that the input point set is of bounded clique-width and the clique-width expression is also given.
• MSO undecidability for some hereditary classes of unbounded clique-width (with Anuj Dawar) [abstract] [paper] [pdf] [slides] [arxiv]
  

  Proc. of the 30^th Computer Science Logic (CSL), Göttingen, Germany, pp. 17:1-17:17, February 2022

  Seese's conjecture for finite graphs states that monadic second-order logic (MSO) is undecidable on all graph classes of unbounded clique-width. We show that to establish this it would suffice to show that grids of unbounded size can be interpreted in two families of graph classes: minimal hereditary classes of unbounded clique-width; and antichains of unbounded clique-width under the induced subgraph relation. We explore all the currently known classes of the former category and establish that grids of unbounded size can indeed be interpreted in them.
• Pseudo-finiteness of arbitrary graphs of bounded shrub-depth [abstract] [arxiv]
  

  ArXiv preprint, February 2022

  We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the classes TM_r(d) of arbitrary graphs that have tree models of height d and r labels. We show that the graphs of TM_r(d) are MSO-pseudo-finite relative to the class TM^f_r(d) of finite graphs of TM_r(d); that is, that every MSO sentence true in a graph of TM_r(d) is also true in a graph of TM^f_r(d). We also show that TM_r(d) is closed under ultraproducts and ultraroots. These results have two consequences. The first is that the index of the MSO[m]-equivalence relation on graphs of TM_r(d) is bounded by a (d+1)-fold exponential in m. The second is that TM_r(d) is exactly the class of all graphs that are MSO-pseudo-finite relative to TM^f_r(d).
• Feferman-Vaught decompositions for prefix classes of first order logic   [abstract] [paper] [pdf] [slides] [arxiv]
  

  Proc. of the 9^th Indian Conference on Logic and Applications (ICLA), Chennai, India, March 2021, pp. 111-116

  The Feferman-Vaught theorem provides a way of evaluating a first order sentence φ on a disjoint union of structures by producing a decomposition of φ into sentences which can be evaluated on the individual structures and the results of these evaluations combined using a propositional formula. This decomposition can in general be non-elementarily larger than φ. We show that for FO sentences in prenex normal form with a fixed number of quantifier alternations, such a decomposition can be obtained in time elementary in the size of φ.
• Extension preservation in the finite and prefix classes of first order logic (with Anuj Dawar)  [abstract] [paper] [pdf] [slides] [video]
  

  Proc. of the 29^th Computer Science Logic (CSL), Ljbuljana, Slovenia, January 2021, pp. 18:1-18:13, Schloss Dagstuhl - LZI

  It is well known that the classic Łoś-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential fragment of first-order logic. We strengthen this by constructing for every n, first-order definable classes of finite structures closed under extensions which are not definable with n quantifier alternations. The classes we construct are definable in the extension of Datalog with negation and indeed in the existential fragment of transitive-closure logic. This answers negatively an open question posed by Rosen and Weinstein.
• Some classical model theoretic aspects of bounded shrub-depth classes  [abstract] []paper [pdf]
  

  Arxiv preprint, October 2020

  We consider classes of arbitrary (finite or infinite) graphs of bounded shrub-depth, specifically the class TM_{r, p}(d) of p-labeled arbitrary graphs whose underlying unlabeled graphs have tree models of height d and r labels. We show that this class satisfies an extension of the classical Löwenheim-Skolem property into the finite and for MSO. This extension being a generalization of the small model property, we obtain that the graphs of TM_{r, p}(d) are pseudo-finite. In addition, we obtain as consequences entirely new proofs of a number of known results concerning bounded shrub-depth classes (of finite graphs) and TM_{r, p}(d). These include the small model property for MSO with elementary bounds, the classical compactness theorem from model theory over TM_{r, p}(d), and the equivalence of MSO and FO over TM_{r, p}(d) and hence over bounded shrub-depth classes. The proof for the last of these is via an adaptation of the proof of the classical Lindström's theorem characterizing FO over arbitrary structures.
• Clique-width of point configurations (with Onur Çağırıcı, Petr Hliněný, and Filip Pokrývka)  [abstract] [paper] [pdf]
  

  Proc. of the 46^th International Workshop on Graph Theoretic Concepts in Computer Science, Leeds, UK, June 2020, pp. 54-66, Springer

  While structural width parameters (of the input) belong to the standard toolbox of graph algorithms, it is not the usual case in computational geometry. As a case study we propose a natural extension of the structural graph parameter of clique-width to geometric point configuration represented by their order type. We study basic properties of this clique-width notion, and show that it is aligned with the general concept of clique-width of relational structures by Blumensath and Courcelle (2006). We also relate the new notion to monadic second-order logic of point configurations. As an application, we provide several linear FPT time algorithms for geometric point problems which are NP-hard in general, in the special case that the input point set is of bounded clique-width and the clique-width expression is also given.
• Exact crossing number parameterized by vertex cover (with Petr Hliněný)  [abstract] [paper] [pdf] [slides]
  

  Proc. of the 27^th International Symposium on Graph Drawing and Network Visualization (GD), Prague, Czech Republic, September 2019, pp.307-319, Springer

  We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of graph classes with unbounded and yet efficiently computable crossing number. Our result can be viewed as a strengthening of a previous result of Lokshtanov [arXiv, 2015] that Optimal Linear Arrangement is in FPT when parameterized by the vertex cover size, and we use a similar approach of reducing the problem to a tractable instance of Integer Quadratic Programming as in Lokshtanov’s paper.
• Revisiting the generalized Łoś-Tarski theorem   [abstract] [paper] [pdf] [slides]
  

  Proc. of the 8^th Indian Conference on Logic and its Applications (ICLA), Delhi, India, March 2019, pp. 76-88, Springer

  We present a new proof of the generalized Łoś-Tarski theorem (GLT(k)) from [1], over arbitrary structures. Instead of using λ-saturation as in [1], we construct just the "required saturation" directly using ascending chains of structures. We also strengthen the failure of GLT(k) in the finite shown in [2], by strengthening the failure of the Łoś-Tarski theorem in this context. In particular, we prove that not just universal sentences, but for each fixed k, even Σ⁰₂ sentences containing k existential quantifiers fail to capture hereditariness in the finite. We conclude with two problems as future directions, concerning the Łoś-Tarski theorem and GLT(k), both in the context of all finite structures.
  
  [1] Sankaran, A., Adsul, B., Chakraborty, S.: A generalization of the Łoś-Tarski preservation theorem. Annals of Pure and Applied Logic, 167(3), 189-210 (2016)
  [2] Sankaran, A., Adsul, B., Madan, V., Kamath, P., Chakraborty, S.: Preservation under substructures modulo bounded cores. Proceedings of WoLLIC 2012, Buenos Aires, Argentina, 3-6 September, 2012, pp. 291–305 (2012)
• A finitary analogue of the downward Löwenheim-Skolem property  [abstract] [paper] [pdf] [slides] [Talk venues]
  

  Proc. of the 26^th Computer Science Logic (CSL), Stockholm, Sweden, August 2017, pp. 37:1-37:21, Schloss Dagstuhl - LZI

  We present a model-theoretic property of finite structures, that can be seen to be a finitary analogue of the well-studied downward Löwenheim-Skolem property from classical model theory. We call this property the L-equivalent bounded substructure property, denoted L-EBSP, where L is either FO or MSO. Intuitively, L-EBSP states that a large finite structure contains a small "logically similar" substructure, where logical similarity means indistinguishability with respect to sentences of L having a given quantifier nesting depth. It turns out that this simply stated property is enjoyed by a variety of classes of interest in computer science: examples include regular languages of words, trees (unordered, ordered or ranked) and nested words, and various classes of graphs, such as cographs, graph classes of bounded tree-depth, those of bounded shrub-depth and n-partite cographs.
  Further, L-EBSP remains preserved in the classes generated from the above by operations that are implementable using quantifier-free translation schemes. All of the aforementioned classes admit natural tree representations for their structures. We use this fact to show that the small and logically similar substructure of a large structure in any of these classes, can be computed in time linear in the size of the tree representation of the structure, giving linear time fixed parameter tractable (f.p.t.) algorithms for checking L-definable properties of the large structure. We conclude by presenting a strengthening of L-EBSP, that asserts "logical self-similarity at all scales" for a suitable notion of scale. We call this the logical fractal property and show that most of the classes mentioned above are indeed, logical fractals.
  • Oct '17: (Invited talk) The annual meet of the Calcutta Logic Circle, Kolkata, India
  • Sep '17: Department of Informatics, Rheinisch-Westfälische Technische Hochschule (RWTH), Aachen, Germany
  • Aug '17: Institute of Informatics, University of Warsaw, Poland
  • Aug '17: (*) The 26^th International Conference on Computer Science Logic (CSL) 2017, Stockholm, Sweden
  • Dec '16: (Invited talk) International Conference of the Indian Mathematics Consortium (TIMC) in co-operation with the American Mathematical Society (AMS), Varanasi, India
  • Nov '16: Department of Mathematics, University of California at Berkeley, USA
  • Nov '16: The Graduate Center, City University of New York (CUNY), USA
  • Oct '16: Department of Computer Science, IIT Patna, India
• A generalization of the Łoś-Tarski preservation theorem (with Bharat Adsul and Supratik Chakraborty) [abstract] [paper] [pdf]
  

  Annals of Pure and Applied Logic, Vol. 167(3), 189-210, 2016

  We present new parameterized preservation properties that provide for each natural number k, semantic characterizations of the ∃^k∀^* and ∀^k∃^* prefix classes of first order logic sentences, over the class of all structures and for arbitrary finite vocabularies. These properties, that we call preservation under substructures modulo k-cruxes and preservation under k-ary covered extensions respectively, correspond exactly to the classical properties of preservation under substructures and preservation under extensions, when k equals 0. As a consequence, we get a parameterized generalization of the Łoś–Tarski preservation theorem for sentences, in both its substructural and extensional forms. We call our characterizations collectively the generalized Łoś–Tarski theorem for sentences. We generalize this theorem to theories, by showing that theories that are preserved under k-ary covered extensions are characterized by theories of ∀^k∃^* sentences, and theories that are preserved under substructures modulo k-cruxes, are equivalent, under a well-motivated model-theoretic hypothesis, to theories of ∃^k∀^* sentences. In contrast to existing preservation properties in the literature that characterize the Σ⁰₂ and Π⁰₂ prefix classes of FO sentences, our preservation properties are combinatorial and finitary in nature, and stay non-trivial over finite structures as well.
• A generalization of the Łoś-Tarski preservation theorem over classes of finite structures (with Bharat Adsul and Supratik Chakraborty) [abstract] [paper] [pdf] [slides] [Talk venues]
  

  Proc. of the 39^th Mathematical Foundations of Computer Science (MFCS), Budapest, Hungary, August 2014, pp. 474-485, Springer

  We present a logic-based combinatorial property of classes of finite structures that allows an effective generalization of the Łoś-Tarski preservation theorem to hold over classes satisfying the property. The well-studied classes of words and trees, and structures of bounded tree-depth are shown to satisfy the property. We also show that starting with classes satisfying this property, the classes generated by applying composition operations like disjoint union, cartesian and tensor products, inherit the property. We finally show that all classes of structures that are well-quasi-ordered under the embedding relation satisfy a natural generalization of our property.
  • Sep '15: Department of Mathematics, Rheinisch-Westfälische Technische Hochschule (RWTH), Aachen, Germany
  • Aug '15: (Poster presentation) Heidelberg Laureate Forum 2015, Heidelberg, Germany
  • Aug ’14: (*) The 39^th International Symposium on the Mathematical Foundations of Computer Science (MFCS), Budapest, Hungary
• Preservation under substructures modulo bounded cores (with Bharat Adsul, Vivek Madan, Pritish Kamath and Supratik Chakraborty) [abstract] [paper] [pdf] [slides] [Talk venues]
  

  Proc. of the 19^th Workshop on Logic, Language, Information and Computation (WoLLIC), Buenos Aires, Argentina, September 2012, Vol. 7456, pp. 291-305, Springer

  We investigate a model-theoretic property that generalizes the classical notion of preservation under substructures. We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via Σ⁰₂ sentences for properties of arbitrary structures definable by FO sentences. Towards a sharper characterization, we conjecture that the count of existential quantifiers in the and Σ⁰₂ sentence equals the size of the smallest bounded core. We show that this conjecture holds for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the conjecture fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.
  • Oct '12: Model Theory Seminar, Graduate Center, City University of New York (CUNY), USA
  • Oct '12: Computational Logic Seminar, Graduate Center, City University of New York (CUNY), USA
  • Sep ’12: (*) The 19^th Workshop for Logic, Language, Information and Computation (WoLLIC), Buenos Aires, Argentina

Thesis

• A generalization of the Łoś-Tarski preservation theorem [summary] [synopsis] [dissertation] [arxiv] [defence slides]
  Doctoral dissertation submitted to the Department of Computer Science and Technology, IIT Bombay, India, August 2016.