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Jakub Szymanik - One of the best experts on this subject based on the ideXlab platform.
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Computing Simple Quantifiers
Studies in Linguistics and Philosophy, 2016Co-Authors: Jakub SzymanikAbstract:In this Chapter I introduce the idea of semantic automata—simple computational devices corresponding to basic Quantifiers in natural language. In line with a procedural approach to semantics, given a quantified sentence and a finite model, a semantic automaton computes the truth-value of this sentence in that model. In order to build the semantic automata theory, I first show how to encode finite models as strings of symbols, translating between generalized Quantifier theory and formal language theory. With the help of this encoding I show what kind of automata correspond to particular Quantifiers. This leads to a number of characterization results, for instance, a classic theorem of Van Benthem establishing equivalence between Quantifiers definable in first-order logic (e.g., ‘more than 5’) and Quantifiers recognizable by finite-automata. Quantifier ‘most’, which is not definable in first-order logic, will require a recognition device with some sort of unbounded working memory, e.g., a push-down automaton. The question arises: are these logical characterizations cognitively plausible? In the next chapter, I will argue that the answer is positive.
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Complexity of Collective Quantification
Studies in Linguistics and Philosophy, 2016Co-Authors: Jakub SzymanikAbstract:Generalized Quantifier theory tends to focus on distributive readings of natural language determiners. In contrast, this last chapter is devoted to collective readings of Quantifiers, e.g., “Most students played poker together”. I start by introducing the common strategy of formalizing collective quantification by using certain type-shifting operations. I show that all these lifts turn out to be definable in second-order logic. Next, I introduce an alternative approach to modeling collective quantification by means of second-order generalized Quantifiers and develop a definability theory for them. I study the collective reading of the proportional Quantifier ‘most’ and prove that it is not definable in second-order logic. Therefore, there is no second-order definable lift expressing the collective meaning of the Quantifier ‘most’. This is clearly a restriction of the type-shifting approach. I finish by discussing various methodological interpretation of this result, touching again upon the issues of complexity in natural language and the semantic borders of everyday language.
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Computational complexity of polyadic lifts of generalized Quantifiers in natural language
Linguistics and Philosophy, 2010Co-Authors: Jakub SzymanikAbstract:We study the computational complexity of polyadic Quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-Quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex Quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-Quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility of revising the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-Quantifier sentences. The paper not only contributes to the field of formal semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science.
Tru H. Cao - One of the best experts on this subject based on the ideXlab platform.
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Modelling with Words - Conceptual Graphs for Modelling and Computing with Generally Quantified Statements
Lecture Notes in Computer Science, 2003Co-Authors: Tru H. CaoAbstract:Conceptual graphs have been shown to be a logic that has a smooth mapping to and from natural language, in particular generally quantified statements, which is one of its advantages over predicate logic. However, classical semantics of conceptual graphs cannot deal with intrinsically vague generalized Quantifiers like few, many, or most, which represent imprecise quantities that go beyond the capability of classical arithmetic. In this paper, we apply the fuzzy set-theoretic semantics of generalized Quantifiers and formally define the semantics of generally quantified fuzzy conceptual graphs as probabilistic logic rules comprising only simple fuzzy conceptual graphs. Then we derive inference rules performed directly on fuzzy conceptual graphs with either relative or absolute Quantifiers.
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ICCS - Generalized Quantifiers and Conceptual Graphs
Conceptual Structures: Broadening the Base, 2001Co-Authors: Tru H. CaoAbstract:Conceptual graphs have been shown to be a logic that has a smooth mapping to and from natural language, in particular generally quantified statements, which is one of its advantages over predicate logic. However, classical semantics of conceptual graphs cannot deal with intrinsically vague generalized Quantifiers like few, many, or most, which represent imprecise quantities that go beyond the capability of classical arithmetic. In this paper, we apply the fuzzy set-theoretic semantics of generalized Quantifiers and formally define the semantics of generally quantified fuzzy conceptual graphs as probabilistic logic rules comprising only simple fuzzy conceptual graphs. Then we derive inference rules performed directly on fuzzy conceptual graphs with either relative or absolute Quantifiers.
Cesare Tinelli - One of the best experts on this subject based on the ideXlab platform.
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On solving quantified bit-vector constraints using invertibility conditions
Formal Methods in System Design, 2021Co-Authors: Aina Niemetz, Andrew Reynolds, Clark Barrett, Mathias Preiner, Cesare TinelliAbstract:We present a novel approach for solving quantified bit-vector constraints in Satisfiability Modulo Theories (SMT) based on computing symbolic inverses of bit-vector operators. We derive conditions that precisely characterize when bit-vector constraints are invertible for a representative set of bit-vector operators commonly supported by SMT solvers. We utilize syntax-guided synthesis techniques to aid in establishing these conditions and verify them independently by using several SMT solvers. We show that invertibility conditions can be embedded into Quantifier instantiations using Hilbert choice expressions and give experimental evidence that a counterexample-guided approach for Quantifier instantiation utilizing these techniques leads to performance improvements with respect to state-of-the-art solvers for quantified bit-vector constraints.
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Quantifier instantiation techniques for finite model finding in smt
Conference on Automated Deduction, 2013Co-Authors: Andrew Reynolds, Cesare Tinelli, Amit Goel, Sava Krstic, Morgan Deters, Clark BarrettAbstract:SMT-based applications increasingly rely on SMT solvers being able to deal with quantified formulas. Current work shows that for formulas with Quantifiers over uninterpreted sorts counter-models can be obtained by integrating a finite model finding capability into the architecture of a modern SMT solver. We examine various strategies for on-demand Quantifier instantiation in this setting. Here, completeness can be achieved by considering all ground instances over the finite domain of each Quantifier. However, exhaustive instantiation quickly becomes unfeasible with larger domain sizes. We propose instantiation strategies to identify and consider only a selection of ground instances that suffices to determine the satisfiability of the input formula. We also examine heuristic Quantifier instantiation techniques such as E-matching for the purpose of accelerating the search. We give experimental evidence that our approach is practical for use in industrial applications and is competitive with other approaches.
Hoon Hong - One of the best experts on this subject based on the ideXlab platform.
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Real Quantifier elimination for the synthesis of optimal numerical algorithms (Case study
Journal of Symbolic Computation, 2016Co-Authors: Mădălina Eraşcu, Hoon HongAbstract:We report on our on-going efforts to apply real Quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound e, find a real interval such that it contains x and its width is less than or equal to e.A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a "refinement map" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm.This problem can be formulated as a real Quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real Quantifier elimination software.We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the Quantifiers from the resulting ones using the state-of-the-art Quantifier elimination software.As the result, we were able to synthesize semi-automatically an optimal quadratically1 convergent map, which is better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized map is not contracting as one would naturally expect.
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synthesis of optimal numerical algorithms using real Quantifier elimination case study square root computation
International Symposium on Symbolic and Algebraic Computation, 2014Co-Authors: Mădălina Eraşcu, Hoon HongAbstract:We report on on-going efforts to apply real Quantifier elimination to the synthesis of optimal numerical algorithms. In particular, we describe a case study on the square root problem: given a real number x and an error bound e, find a real interval such that it contains [EQUATION] and its width is less than or equal to e. A typical numerical algorithm starts with an initial interval and repeatedly updates it by applying a "refinement map" on it until it becomes narrow enough. Thus the synthesis amounts to finding a refinement map that ensures the correctness and optimality of the resulting algorithm. This problem can be formulated as a real Quantifier elimination. Hence, in principle, the synthesis can be carried out automatically. However, the computational requirement is huge, making the automatic synthesis practically impossible with the current general real Quantifier elimination software. We overcame the difficulty by (1) carefully reducing a complicated quantified formula into several simpler ones and (2) automatically eliminating the Quantifiers from the resulting ones using the state of the art Quantifier elimination software. As the result, we were able to synthesize semi-automatically, under mild assumptions, a class of optimal maps, which are significantly better than the well known hand-crafted Secant-Newton map. Interestingly, the optimal synthesized maps are not contracting as one would naturally expect.
Clark Barrett - One of the best experts on this subject based on the ideXlab platform.
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On solving quantified bit-vector constraints using invertibility conditions
Formal Methods in System Design, 2021Co-Authors: Aina Niemetz, Andrew Reynolds, Clark Barrett, Mathias Preiner, Cesare TinelliAbstract:We present a novel approach for solving quantified bit-vector constraints in Satisfiability Modulo Theories (SMT) based on computing symbolic inverses of bit-vector operators. We derive conditions that precisely characterize when bit-vector constraints are invertible for a representative set of bit-vector operators commonly supported by SMT solvers. We utilize syntax-guided synthesis techniques to aid in establishing these conditions and verify them independently by using several SMT solvers. We show that invertibility conditions can be embedded into Quantifier instantiations using Hilbert choice expressions and give experimental evidence that a counterexample-guided approach for Quantifier instantiation utilizing these techniques leads to performance improvements with respect to state-of-the-art solvers for quantified bit-vector constraints.
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Quantifier instantiation techniques for finite model finding in smt
Conference on Automated Deduction, 2013Co-Authors: Andrew Reynolds, Cesare Tinelli, Amit Goel, Sava Krstic, Morgan Deters, Clark BarrettAbstract:SMT-based applications increasingly rely on SMT solvers being able to deal with quantified formulas. Current work shows that for formulas with Quantifiers over uninterpreted sorts counter-models can be obtained by integrating a finite model finding capability into the architecture of a modern SMT solver. We examine various strategies for on-demand Quantifier instantiation in this setting. Here, completeness can be achieved by considering all ground instances over the finite domain of each Quantifier. However, exhaustive instantiation quickly becomes unfeasible with larger domain sizes. We propose instantiation strategies to identify and consider only a selection of ground instances that suffices to determine the satisfiability of the input formula. We also examine heuristic Quantifier instantiation techniques such as E-matching for the purpose of accelerating the search. We give experimental evidence that our approach is practical for use in industrial applications and is competitive with other approaches.
