IA Scholar Query: Pebble Games with Algebraic Rules.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 24 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440The two-sided approximation and counting method
https://scholar.archive.org/work/yflg4zro5zd3riwqph5togcdf4
Below, we summarize the appearances and possible uses of the two-sided approach and the two-sided counting in the most diverse areas of (secondary) school mathematics.Attila Máder, Máté Szalaiwork_yflg4zro5zd3riwqph5togcdf4Sat, 24 Sep 2022 00:00:00 GMTThe Objective Function: Science and Society in the Age of Machine Intelligence
https://scholar.archive.org/work/2unyehnlczcpfnkcxvnjp6wxye
Machine intelligence, or the use of complex computational and statistical practices to make predictions and classifications based on data representations of phenomena, has been applied to domains as disparate as criminal justice, commerce, medicine, media and the arts, mechanical engineering, among others. How has machine intelligence become able to glide so freely across, and to make such waves for, these domains? In this dissertation, I take up that question by ethnographically engaging with how the authority of machine learning has been constructed such that it can influence so many domains, and I investigate what the consequences are of it being able to do so. By examining the workplace practices of the applied machine learning researchers who produce machine intelligence, those they work with, and the artifacts they produce. The dissertation begins by arguing that machine intelligence proceeds from a naive form of empiricism with ties to positivist intellectual traditions of the 17th and 18th centuries. This naive empiricism eschews other forms of knowledge and theory formation in order for applied machine learning researchers to enact data performances that bring objects of analysis into existence as entities capable of being subjected to machine intelligence. By data performances, I mean generative enactments which bring into existence that which machine intelligence purports to analyze or describe. The enactment of data performances is analyzed as an agential cut into a representational field that produces both stable claims about the world and the interpretive frame in which those claims can hold true. The dissertation also examines how machine intelligence depends upon a range of accommodations from other institutions and organizations, from data collection and processing to organizational commitments to support the work of applied machine learning researchers.Emanuel Mosswork_2unyehnlczcpfnkcxvnjp6wxyeWed, 21 Sep 2022 00:00:00 GMTDownward Self-Reducibility in TFNP
https://scholar.archive.org/work/a7ggfa7gc5g7xaeuy7yh2rr26m
A problem is downward self-reducible if it can be solved efficiently given an oracle that returns solutions for strictly smaller instances. In the decisional landscape, downward self-reducibility is well studied and it is known that all downward self-reducible problems are in PSPACE. In this paper, we initiate the study of downward self-reducible search problems which are guaranteed to have a solution – that is, the downward self-reducible problems in TFNP. We show that most natural -complete problems are downward self-reducible and any downward self-reducible problem in TFNP is contained in PLS. Furthermore, if the downward self-reducible problem is in UTFNP (i.e. it has a unique solution), then it is actually contained in CLS. This implies that if integer factoring is downward self-reducible then it is in fact in CLS, suggesting that no efficient factoring algorithm exists using the factorization of smaller numbers.Prahladh Harsha, Daniel Mitropolsky, Alon Rosenwork_a7ggfa7gc5g7xaeuy7yh2rr26mWed, 21 Sep 2022 00:00:00 GMTManipulative Materials in Teaching Mathematics among Junior High School Teachers: A Literature Review
https://scholar.archive.org/work/uz5crhwk3ve6jf5wxfz7zxyn2q
This review article visualizes the use of manipulative materials in teaching mathematics among junior high school teachers. The review correlated to the study in the following areas: The concept of manipulative materials, types of manipulative materials teachers use in teaching mathematics, How teachers obtain manipulative materials for teaching mathematics, teaching methods teachers use for teaching mathematics with the use of manipulative materials, teachers' perceived benefits of using manipulative materials in teaching mathematics, and challenges of using manipulative materials in teaching mathematics among teachers. In order to develop every student's mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulative materials into classroom instruction at all grades. Manipulative materials not only allow students to construct their own cognitive models for abstract mathematical ideas and processes, but they also provide a common language with which to communicate these models to the teacher and other students.James Kwabena Odum, Catholic University of Ghana, Fiapre-Sunyani, Bono Region, Ghanawork_uz5crhwk3ve6jf5wxfz7zxyn2qThu, 08 Sep 2022 00:00:00 GMTOn the parallel complexity of Group Isomorphism via Weisfeiler-Leman
https://scholar.archive.org/work/mg7roztjx5fdfmx4ryg4r6zbme
In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter Schweitzer, LICS 2020) can be fruitfully used to improve parallel complexity upper bounds on isomorphism testing for several families of groups. In particular, we show: - Groups with an Abelian normal Hall subgroup whose complement is O(1)-generated are identified by constant-dimensional Weisfeiler-Leman using only a constant number of rounds. This places isomorphism testing for this family of groups into ; the previous upper bound for isomorphism testing was (Qiao, Sarma, Tang, STACS 2011). - We use the individualize-and-refine paradigm to obtain a ^1 isomorphism test for groups without Abelian normal subgroups, previously only known to be in (Babai, Codenotti, Qiao, ICALP 2012). - We extend a result of Brachter Schweitzer (arXiv, 2021) on direct products of groups to the parallel setting. Namely, we also show that Weisfeiler-Leman can identify direct products in parallel, provided it can identify each of the indecomposable direct factors in parallel. They previously showed the analogous result for . We finally consider the count-free Weisfeiler-Leman algorithm, where we show that count-free WL is unable to even distinguish Abelian groups in polynomial-time. Nonetheless, we use count-free WL in tandem with bounded non-determinism and limited counting to obtain a new upper bound of β_1^0() for isomorphism testing of Abelian groups. This improves upon the previous ^0() upper bound due to Chattopadhyay, Torán, Wagner (ACM Trans. Comput. Theory, 2013).Joshua A. Grochow, Michael Levetwork_mg7roztjx5fdfmx4ryg4r6zbmeTue, 23 Aug 2022 00:00:00 GMTPolynomial Time Algorithm for ARRIVAL on Tree-Like Multigraphs
https://scholar.archive.org/work/ybcsy4bc5rgglhvwl2a2xjtnyy
A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, has been reached. The ARRIVAL problem, as defined by Dohrau et al. [Dohrau et al., 2017], consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP ∩ co-NP without being known to be in P. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that in this class, ARRIVAL can be solved in almost linear time, while the number of steps of a rotor walk can still be exponential. Then, we give an application of this result to solve some deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games).David Auger, Pierre Coucheney, Loric Duhazé, Stefan Szeider, Robert Ganian, Alexandra Silvawork_ybcsy4bc5rgglhvwl2a2xjtnyyMon, 22 Aug 2022 00:00:00 GMTGraded Monads and Behavioural Equivalence Games
https://scholar.archive.org/work/5vvsxffo3nelfjcksilzdw2xha
The framework of graded semantics uses graded monads to capture behavioural equivalences of varying granularity, for example as found in the linear-time / branching-time spectrum, over general system types. We describe a generic Spoiler-Duplicator game for graded semantics that is extracted from the given graded monad, and may be seen as playing out an equational proof; instances include standard pebble games for simulation and bisimulation as well as games for trace-like equivalences and coalgebraic behavioural equivalence. Considerations on an infinite variant of such games lead to a novel notion of infinite-depth graded semantics. Under reasonable restrictions, the infinite-depth graded semantics associated to a given graded equivalence can be characterized in terms of a determinization construction for coalgebras under the equivalence at hand.Chase Ford, Stefan Milius, Lutz Schröder, Harsh Beohar, Barbara Königwork_5vvsxffo3nelfjcksilzdw2xhaTue, 02 Aug 2022 00:00:00 GMTBlockchain-Based Gas Auctioning Coupled with a Novel Economic Dispatch Formulation for Gas-Deficient Thermal Plants
https://scholar.archive.org/work/kbr2upndfvg5bg3fqbhjvg2wzu
Inadequate gas supply is partly responsible for the energy shortfall experienced in some energy-poor nations. Favorable market conditions would boost investment in the gas supply sector; hence, we propose a blockchain-based fair, transparent, and secure gas trading scheme that facilitates peer-to-peer trading of gas. The scheme is developed using an Ethereum-based smart contract that receives offers from gas suppliers and bid(s) from the thermal plant operator. Giving priority to the cheapest offers, the smart contract determines the winning suppliers. This paper also proposes an economic dispatch model for gas-deficient plants. Conventional economic dispatch seeks to satisfy electric load demand whilst minimizing the total gas cost of generating units. Implicit in its formulation is the assumption that gas supply to generating units is sufficient to satisfy available demand. In energy poor nations, this is hardly the case as there is often inadequate gas supply and conventional economic dispatch is of little practical value. The proposed economic dispatch model's objective function maximizes the quantity of available gas and determines the optimal power output of each generating unit. The mathematical formulation is verified using data from the Egbin thermal station which is the largest thermal station in Nigeria and is solved using the General Algebraic Modeling System (GAMS). Obtained results indicate the viability of the novel approach as it results in a net power gain of 35 MW. On the other hand, the smart contract proved effective in accurately selecting winning suppliers and making payment.Uyikumhe Damisa, Peter Olabisi Oluseyi, Nnamdi Ikechi Nwuluwork_kbr2upndfvg5bg3fqbhjvg2wzuFri, 15 Jul 2022 00:00:00 GMTI/O-Optimal Algorithms for Symmetric Linear Algebra Kernels
https://scholar.archive.org/work/gcrwzywkizd6jcxnzznse6xpy4
In this paper, we consider two fundamental symmetric kernels in linear algebra: the Cholesky factorization and the symmetric rank-𝑘 update (SYRK), with the classical three nested loops algorithms for these kernels. In addition, we consider a machine model with a fast memory of size 𝑆 and an unbounded slow memory. In this model, all computations must be performed on operands in fast memory, and the goal is to minimize the amount of communication between slow and fast memories. As the set of computations is fixed by the choice of the algorithm, only the ordering of the computations (the schedule) directly influences the volume of communications. We prove lower bounds of 1 3 CCS CONCEPTS • Theory of computation → Shared memory algorithms; Communication complexity; • Mathematics of computing → Solvers.Olivier Beaumont, Lionel Eyraud-Dubois, Julien Langou, Mathieu Véritéwork_gcrwzywkizd6jcxnzznse6xpy4Mon, 11 Jul 2022 00:00:00 GMTGraded Monads and Behavioural Equivalence Games
https://scholar.archive.org/work/xynqua6runa7phewtazidl2nia
The framework of graded semantics uses graded monads to capture behavioural equivalences of varying granularity, for example as found on the linear-time/branching-time spectrum, over general system types. We describe a generic Spoiler-Duplicator game for graded semantics that is extracted from the given graded monad, and may be seen as playing out an equational proof; instances include standard pebble games for simulation and bisimulation as well as games for trace-like equivalences and coalgebraic behavioural equivalence. Considerations on an infinite variant of such games lead to a novel notion of infinite-depth graded semantics. Under reasonable restrictions, the infinite-depth graded semantics associated to a given graded equivalence can be characterized in terms of a determinization construction for coalgebras under the equivalence at hand.Harsh Beohar, Chase Ford, Barbara König, Stefan Milius, Lutz Schröderwork_xynqua6runa7phewtazidl2niaWed, 29 Jun 2022 00:00:00 GMTSpace Characterizations of Complexity Measures and Size-Space Trade-Offs in Propositional Proof Systems
https://scholar.archive.org/work/4bnhkpv5uvf25mzct7jbiggyae
We identify two new big clusters of proof complexity measures equivalent up to polynomial and log n factors. The first cluster contains, among others, the logarithm of tree-like resolution size, regularized (that is, multiplied by the logarithm of proof length) clause and monomial space, and clause space, both ordinary and regularized, in regular and tree-like resolution. As a consequence, separating clause or monomial space from the (logarithm of) tree-like resolution size is the same as showing a strong trade-off between clause or monomial space and proof length, and is the same as showing a super-critical trade-off between clause space and depth. The second cluster contains width, Σ₂ space (a generalization of clause space to depth 2 Frege systems), both ordinary and regularized, as well as the logarithm of tree-like size in the system R(log). As an application of some of these simulations, we improve a known size-space trade-off for polynomial calculus with resolution. In terms of lower bounds, we show a quadratic lower bound on tree-like resolution size for formulas refutable in clause space 4. We introduce on our way yet another proof complexity measure intermediate between depth and the logarithm of tree-like size that might be of independent interest.Theodoros Papamakarios, Alexander Razborov, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruffwork_4bnhkpv5uvf25mzct7jbiggyaeTue, 28 Jun 2022 00:00:00 GMTFrameworks with coordinated edge motions
https://scholar.archive.org/work/gio7db4sejgtfi7micvaa3aadu
We develop a rigidity theory for bar-joint frameworks in Euclidean d-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class. Rigidity for these coordinated frameworks is a generic property, and we characterize the rigid graphs in terms of redundant rigidity in the standard d-dimensional rigidity matroid. We also interpret our main results in terms of matroid unions.Bernd Schulze, Hattie Serocold, Louis Theranwork_gio7db4sejgtfi7micvaa3aaduSat, 11 Jun 2022 00:00:00 GMTFinite Model Theory and Proof Complexity revisited: Distinguishing graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus
https://scholar.archive.org/work/efqzflkcrjazxgya5tyqiugt5a
This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a PTIME-instance of the graph isomorphism problem would separate CPT from PTIME and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus.Benedikt Pagowork_efqzflkcrjazxgya5tyqiugt5aFri, 10 Jun 2022 00:00:00 GMTSeparations in Proof Complexity and TFNP
https://scholar.archive.org/work/galb7hig65b2rbgnobzuolxxdi
It is well-known that Resolution proofs can be efficiently simulated by Sherali-Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total NP search problems. First, we characterise the classes PPADS, PPAD, SOPL by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, PLS ⊈ PPP, SOPL ⊈ PPA, and EOPL ⊈ UEOPL. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical TFNP classes introduced in the 1990s.Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, Ran Taowork_galb7hig65b2rbgnobzuolxxdiFri, 10 Jun 2022 00:00:00 GMTOn the Descriptive Complexity of Temporal Constraint Satisfaction Problems
https://scholar.archive.org/work/77vhf5uz5rdepjc2wiidbmp3ji
Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs, the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterizes Datalog already for the CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be expressed in one of the following logical formalisms: Datalog, fixed-point logic (with or without counting), or fixed-point logic with the Boolean rank operator. The classification shows that many of the equivalent conditions in the finite fail to capture expressibility in Datalog or fixed-point logic already for temporal CSPs.Manuel Bodirsky, Jakub Rydvalwork_77vhf5uz5rdepjc2wiidbmp3jiTue, 07 Jun 2022 00:00:00 GMTThe EPFL Logic Synthesis Libraries
https://scholar.archive.org/work/qyuakvlxmvf3fhitnl7p6awiwu
We present a collection of modular open source C++ libraries for the development of logic synthesis applications. These libraries can be used to develop applications for the design of classical and emerging technologies, as well as for the implementation of quantum compilers. All libraries are well documented and well tested. Furthermore, being header-only, the libraries can be readily used as core components in complex logic synthesis systems.Mathias Soeken, Heinz Riener, Winston Haaswijk, Eleonora Testa, Bruno Schmitt, Giulia Meuli, Fereshte Mozafari, Siang-Yun Lee, Alessandro Tempia Calvino, Dewmini Sudara Marakkalage, Giovanni De Micheliwork_qyuakvlxmvf3fhitnl7p6awiwuFri, 03 Jun 2022 00:00:00 GMTWhen Can We Answer Queries Using Result-Bounded Data Interfaces?
https://scholar.archive.org/work/2swct5duffb75avfokkv4q7hle
We consider answering queries on data available through access methods, that provide lookup access to the tuples matching a given binding. Such interfaces are common on the Web; further, they often have bounds on how many results they can return, e.g., because of pagination or rate limits. We thus study result-bounded methods, which may return only a limited number of tuples. We study how to decide if a query is answerable using result-bounded methods, i.e., how to compute a plan that returns all answers to the query using the methods, assuming that the underlying data satisfies some integrity constraints. We first show how to reduce answerability to a query containment problem with constraints. Second, we show "schema simplification" theorems describing when and how result-bounded services can be used. Finally, we use these theorems to give decidability and complexity results about answerability for common constraint classes.Antoine Amarilli, Michael Benediktwork_2swct5duffb75avfokkv4q7hleWed, 01 Jun 2022 00:00:00 GMTQuantum-Inspired Combinatorial Games: Algorithms and Complexity
https://scholar.archive.org/work/rzmrcpyvafhe7pim3zuxxb4cym
Recently, quantum concepts inspired a new framework in combinatorial game theory. This transformation uses discrete superpositions to yield beautiful new rulesets with succinct representations that require sophisticated strategies. In this paper, we address the following fundamental questions: - Complexity Leap: Can this framework transform polynomial-time solvable games into intractable games? - Complexity Collapse: Can this framework transform PSPACE-complete games into ones with complexity in the lower levels of the polynomial-time hierarchy? We focus our study on how it affects two extensively studied polynomial-time-solvable games: Nim and Undirected Geography. We prove that both Nim and Undirected Geography make a complexity leap over NP, when starting with superpositions: The former becomes Σ₂^p-hard and the latter becomes PSPACE-complete. We further give an algorithm to prove that from any classical starting position, quantumized Undirected Geography remains polynomial-time solvable. Together they provide a nearly-complete characterization for Undirected Geography. Both our algorithm and its correctness proof require strategic moves and graph contraction to extend the matching-based theory for classical Undirected Geography. Our constructive proofs for both games highlight the intricacy of this framework. The polynomial time robustness of Undirected Geography in this quantum-inspired setting provides a striking contrast to the recent result that the disjunctive sum of two Undirected Geography games is PSPACE-complete. We give a Σ₂^p-hardness analysis of quantumized Nim, even if there are no pile sizes of more than 1.Kyle W. Burke, Matthew Ferland, Shang-Hua Teng, Pierre Fraigniaud, Yushi Unowork_rzmrcpyvafhe7pim3zuxxb4cymMon, 23 May 2022 00:00:00 GMTPolynomial Time Algorithm for ARRIVAL on Tree-like Multigraphs
https://scholar.archive.org/work/g6cq2sjn25ffpifyf65luwimue
A rotor walk in a directed graph can be thought of as a deterministic version of a Markov Chain, where a pebble moves from vertex to vertex following a simple rule until a terminal vertex, or sink, is reached. The ARRIVAL problem, as defined by Dohrau and al., consists in determining which sink will be reached. While the walk itself can take an exponential number of steps, this problem belongs to the complexity class NP∩co-NP without being known to be in P. Several variants have been studied where we add one or two players to the model, defining deterministic analogs of stochastic models (e.g., Markovian decision processes, Stochastic Games) with rotor-routing rules instead of random transitions. The corresponding decision problem address the existence of strategies for players that ensure some condition on the reached sink. These problems are known to be NP-complete for one player and PSPACE-complete for two players. In this work, we define a class of directed graphs, namely tree-like multigraphs, which are multigraphs having the global shape of an undirected tree. We prove that the different variants of the reachability problem with zero, one, or two players can be solved in linear time, while the number of steps of rotor walks can still be exponential. To achieve this, we define a notion of return flow, which counts the number of times the pebble will bounce back in subtrees of the graph.David Auger, Pierre Coucheney, Loric Duhazework_g6cq2sjn25ffpifyf65luwimueSat, 30 Apr 2022 00:00:00 GMTConvergence of Datalog over (Pre-) Semirings
https://scholar.archive.org/work/3v2royvyzvdwfjhijitapae5y4
Recursive queries have been traditionally studied in the framework of datalog, a language that restricts recursion to monotone queries over sets, which is guaranteed to converge in polynomial time in the size of the input. But modern big data systems require recursive computations beyond the Boolean space. In this paper we study the convergence of datalog when it is interpreted over an arbitrary semiring. We consider an ordered semiring, define the semantics of a datalog program as a least fixpoint in this semiring, and study the number of steps required to reach that fixpoint, if ever. We identify algebraic properties of the semiring that correspond to certain convergence properties of datalog programs. Finally, we describe a class of ordered semirings on which one can use the semi-naive evaluation algorithm on any datalog program.Mahmoud Abo Khamis, Hung Q. Ngo, Reinhard Pichler, Dan Suciu, Yisu Remy Wangwork_3v2royvyzvdwfjhijitapae5y4Tue, 26 Apr 2022 00:00:00 GMT