306 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On the relative asymptotic expressivity of inference frameworks

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    Let σ\sigma be a first-order signature and let Wn\mathbf{W}_n be the set of all σ\sigma-structures with domain {1,,n}\{1, \ldots, n\}. By an inference framework we mean a class F\mathbf{F} of pairs (P,L)(\mathbb{P}, L), where P=(Pn:n=1,2,3,)\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots) and Pn\mathbb{P}_n is a probability distribution on Wn\mathbf{W}_n, and LL is a logic with truth values in the unit interval [0,1][0, 1]. An inference framework F\mathbf{F}' is asymptotically at least as expressive as another inference framework F\mathbf{F} if for every (P,L)F(\mathbb{P}, L) \in \mathbf{F} there is (P,L)F(\mathbb{P}', L') \in \mathbf{F}' such that P\mathbb{P} is asymptotically total-variation-equivalent to P\mathbb{P}' and for every φ(xˉ)L\varphi(\bar{x}) \in L there is φ(xˉ)L\varphi'(\bar{x}) \in L' such that φ(xˉ)\varphi'(\bar{x}) is asymptotically equivalent to φ(xˉ)\varphi(\bar{x}) with respect to P\mathbb{P}. This relation is a preorder and we describe a partial order on the equivalence classes of some inference frameworks that seem natural in the context of machine learning and artificial intelligence. Several previous results about asymptotic (or almost sure) equivalence of formulas or convergence in probability can be formulated in terms of relative asymptotic strength of inference frameworks. We incorporate these results in our classification of inference frameworks and prove two new results. Both concern sequences of probability distributions defined by directed graphical models that use ``continuous'' aggregation functions. The first considers queries expressed by a logic with truth values in [0,1][0, 1] which employs continuous aggregation functions. The second considers queries expressed by a two-valued conditional logic that can express statements about relative frequencies.Comment: 52 page

    Bounded Relativization

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    Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible

    Crisp bi-G\"{o}del modal logic and its paraconsistent expansion

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    In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G\"{o}del modal logic \KbiG. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-G\"{o}del algebra on [0,1][0,1]. We also consider a paraconsistent expansion of \KbiG with a De Morgan negation ¬\neg which we dub \KGsquare. We devise a Hilbert-style calculus for this logic and, as a~con\-se\-quence of a~conservative translation from \KbiG to \KGsquare, prove its completeness w.r.t.\ crisp Kripke models with two valuations over [0,1][0,1] connected via ¬\neg. For these two logics, we establish that their decidability and validity are PSPACE\mathsf{PSPACE}-complete. We also study the semantical properties of \KbiG and \KGsquare. In particular, we show that Glivenko theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in K\mathbf{K} (the classical modal logic) and the crisp G\"{o}del modal logic \KG^c. We show that, among others, all Sahlqvist formulas and all formulas ϕχ\phi\rightarrow\chi where ϕ\phi and χ\chi are monotone, define the same classes of frames in K\mathbf{K} and \KG^c

    Search versus Search for Collapsing Electoral Control Types

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    Electoral control types are ways of trying to change the outcome of elections by altering aspects of their composition and structure [BTT92]. We say two compatible (i.e., having the same input types) control types that are about the same election system E form a collapsing pair if for every possible input (which typically consists of a candidate set, a vote set, a focus candidate, and sometimes other parameters related to the nature of the attempted alteration), either both or neither of the attempted attacks can be successfully carried out [HHM20]. For each of the seven general (i.e., holding for all election systems) electoral control type collapsing pairs found by Hemaspaandra, Hemaspaandra, and Menton [HHM20] and for each of the additional electoral control type collapsing pairs of Carleton et al. [CCH+ 22] for veto and approval (and many other election systems in light of that paper's Theorems 3.6 and 3.9), both members of the collapsing pair have the same complexity since as sets they are the same set. However, having the same complexity (as sets) is not enough to guarantee that as search problems they have the same complexity. In this paper, we explore the relationships between the search versions of collapsing pairs. For each of the collapsing pairs of Hemaspaandra, Hemaspaandra, and Menton [HHM20] and Carleton et al. [CCH+ 22], we prove that the pair's members' search-version complexities are polynomially related (given access, for cases when the winner problem itself is not in polynomial time, to an oracle for the winner problem). Beyond that, we give efficient reductions that from a solution to one compute a solution to the other. For the concrete systems plurality, veto, and approval, we completely determine which of their (due to our results) polynomially-related collapsing search-problem pairs are polynomial-time computable and which are NP-hard.Comment: The metadata's abstract is abridged due to arXiv.org's abstract-length limit. The paper itself has the unabridged (i.e., full) abstrac

    Recent Advances in Research on Island Phenomena

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    In natural languages, filler-gap dependencies can straddle across an unbounded distance. Since the 1960s, the term “island” has been used to describe syntactic structures from which extraction is impossible or impeded. While examples from English are ubiquitous, attested counterexamples in the Mainland Scandinavian languages have continuously been dismissed as illusory and alternative accounts for the underlying structure of such cases have been proposed. However, since such extractions are pervasive in spoken Mainland Scandinavian, these languages may not have been given the attention that they deserve in the syntax literature. In addition, recent research suggests that extraction from certain types of island structures in English might not be as unacceptable as previously assumed either. These findings break new empirical ground, question perceived knowledge, and may indeed have substantial ramifications for syntactic theory. This volume provides an overview of state-of-the-art research on island phenomena primarily in English and the Scandinavian languages, focusing on how languages compare to English, with the aim to shed new light on the nature of island constraints from different theoretical perspectives

    Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata

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    Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where "polynomial-size" refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.Comment: (A4, 10pt, 21 pages) This paper corrects and extends a preliminary report published in the Proceedings of the 24th International Symposium on Fundamentals of Computation Theory (FCT 2023), Trier, Germany, September 18-24, 2023, Lecture Notes in Computer Science, vol. 14292, pp. 421-435, Springer Cham, 202

    Ideal presentations and numberings of some classes of effective quasi-Polish spaces

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    The well known ideal presentations of countably based domains were recently extended to (effective) quasi-Polish spaces. Continuing these investigations, we explore some classes of effective quasi-Polish spaces. In particular, we prove an effective version of the domain-characterization of quasi-Polish spaces, describe effective extensions of quasi-Polish topologies, discover natural numberings of classes of effective quasi-Polish spaces, estimate the complexity of the (effective) homeomorphism relation and of some classes of spaces w.r.t. these numberings, and investigate degree spectra of continuous domains

    Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP

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    We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over NP\mathrm{NP}. Our main results concern DP\mathrm{DP}, i.e., the second level of this hierarchy: If all sets in DP\mathrm{DP} have p-optimal proof systems, then all sets in coDP\mathrm{coDP} have p-optimal proof systems. The analogous implication for optimal proof systems fails relative to an oracle. As a consequence, we clarify such implications for all classes C\mathcal{C} and D\mathcal{D} in the Boolean hierarchy over NP\mathrm{NP}: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets SAT\mathrm{SAT} and TAUT\mathrm{TAUT} have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over NP\mathrm{NP} have p-optimal proof systems which is a new characterization of a conjecture studied by Pudl\'ak

    Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP

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