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  • 1. Alwen, Joël
    et al.
    de Rezende, Susanna F.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Cumulative Space in Black-White Pebbling and Resolution2017In: Leibniz International Proceedings in Informatics, LIPIcs, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2017Conference paper (Refereed)
    Abstract [en]

    We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko 2015] as a tool for obtaining results in cryptography. We consider instead the nondeterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10-15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström 2008, 2011], we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure.

  • 2. Chan, S. M.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordstrom, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Hardness of Approximation in PSPACE and Separation Results for Pebble Games2015In: Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, Institute of Electrical and Electronics Engineers (IEEE), 2015, p. 466-485Conference paper (Refereed)
    Abstract [en]

    We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond - Tompa and Raz - McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time. © 2015 IEEE.

  • 3.
    de Rezende, Susanna F.
    et al.
    KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.
    Meir, Or
    Nordström, Jakob
    KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.
    Toniann, Pitassi
    Robere, Robert
    Vinyals, Marc
    Lifting with Simple Gadgets and Applications to Circuit and Proof ComplexityManuscript (preprint) (Other academic)
    Abstract [en]

    We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems:

    • We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomialline space if coefficients are restricted to be of polynomial magnitude.
    • We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known.

    An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.

  • 4.
    de Rezende, Susanna F.
    et al.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)2016In: 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2016, Vol. 2016, p. 295-304Conference paper (Refereed)
    Abstract [en]

    We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constant-size coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large coefficients. These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers. We prove our results by a reduction to communication lower bounds in a round-efficient version of the real communication model of [Krajicek ' 98], drawing on and extending techniques in [Raz and McKenzie ' 99] and [Goos et al. '15]. The communication lower bounds are in turn established by a reduction to trade-offs between cost and number of rounds in the game of [Dymond and Tompa '85] played on directed acyclic graphs. As a by-product of the techniques developed to show these proof complexity trade-off results, we also obtain an exponential separation between monotone-AC(i-1) and monotone-AC(i), improving exponentially over the superpolynomial separation in [Raz and McKenzie ' 99]. That is, we give an explicit Boolean function that can be computed by monotone Boolean circuits of depth log(i) n and polynomial size, but for which circuits of depth O (log(i-1) n) require exponential size.

  • 5.
    Elffers, Jan
    et al.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Johannsen, J.
    Lauria, M.
    Magnard, T.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Trade-offs between time and memory in a tighter model of CDCL SAT solvers2016In: 19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016, Springer, 2016, p. 160-176Conference paper (Refereed)
    Abstract [en]

    A long line of research has studied the power of conflict- driven clause learning (CDCL) and how it compares to the resolution proof system in which it searches for proofs. It has been shown that CDCL can polynomially simulate resolution even with an adversarially chosen learning scheme as long as it is asserting. However, the simulation only works under the assumption that no learned clauses are ever forgot- ten, and the polynomial blow-up is significant. Moreover, the simulation requires very frequent restarts, whereas the power of CDCL with less frequent or entirely without restarts remains poorly understood. With a view towards obtaining results with tighter relations between CDCL and resolution, we introduce a more fine-grained model of CDCL that cap- tures not only time but also memory usage and number of restarts. We show how previously established strong size-space trade-offs for resolution can be transformed into equally strong trade-offs between time and memory usage for CDCL, where the upper bounds hold for CDCL with- out any restarts using the standard 1UIP clause learning scheme, and the (in some cases tightly matching) lower bounds hold for arbitrarily frequent restarts and arbitrary clause learning schemes.

  • 6. Filmus, Y.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Mikša, Mladen
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    From small space to small width in resolution2014In: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2014, Vol. 25, p. 300-311Conference paper (Refereed)
    Abstract [en]

    In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexity measure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

  • 7. Filmus, Yuval
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Mikša, Mladen
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Towards an understanding of polynomial calculus: New separations and lower bounds (extended abstract)2013In: Automata, Languages, and Programming: 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, Springer, 2013, no PART 1, p. 437-448Conference paper (Refereed)
    Abstract [en]

    During the last decade, an active line of research in proof complexity has been into the space complexity of proofs and how space is related to other measures. By now these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard "benchmark formulas" is still open, as well as the relation of space to size and degree in PC/PCR. We prove that if a formula requires large resolution width, then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least Ω(√n). Our proofs use techniques recently introduced in [Bonacina-Galesi '13]. Our final contribution, however, is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework and suggesting that we are still far from characterizing PC/PCR space.

  • 8.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Space in Proof Complexity2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    ropositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter.

    Different approaches to reasoning are captured by corresponding proof systems. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution.

    In resolution we can prove a space lower bound just by showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that lets us derive space lower bounds, and we use it to separate degree and space.

    For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time.

    We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is exploratory work, but we can also prove new results for the usual space measure.

    We define a new proof system that aims to capture the power of current SAT solvers, and we show a landscape of length-space trade-offs comparable to those in resolution.

    To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, very simple computational models that are useful for modelling space. In addition to results with applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate.

    Another area is communication complexity, the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function.

1 - 8 of 8
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