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Register a new userHybrid-Logical Reasoning in False-Belief Tasks
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Added by
Burkhard C. Schipper
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Authors
Torben Brauner
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The main aim of the present paper is to use a proof system for hybrid modal logic to formalize what are called falsebelief tasks in cognitive psychology, thereby investigating the interplay between cognition and logical reasoning about belief. We consider two differe
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Appel and McAllesters step-indexed logical relations have proven to be a simple and effective technique for reasoning about programs in languages with semantically interesting types, such as general recursive types and general reference types. However, proofs using step-indexed models typically involve tedious, error-prone, and proof-obscuring step-index arithmetic, so it is important to develop clean, high-level, equational proof principles that avoid mention of step indices. In this paper, we show how to reason about binary step-indexed logical relations in an abstract and elegant way. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadis logic for parametricity, but also supports recursively defined relations by means of the modal later operator from Appel, Melli`es, Richards, and Vouillons very modal model paper. We encode in LSLR a logical relation for reasoning relationally about programs in call-by-value System F extended with general recursive types. Using this logical relation, we derive a set of useful rules with which we can prove contextual equivalence and approximation results without counting steps.
Logical reasoning, which is closely related to human cognition, is of vital importance in humans understanding of texts. Recent years have witnessed increasing attentions on machines logical reasoning abilities. However, previous studies commonly apply ad-hoc methods to model pre-defined relation patterns, such as linking named entities, which only considers global knowledge components that are related to commonsense, without local perception of complete facts or events. Such methodology is obviously insufficient to deal with complicated logical structures. Therefore, we argue that the natural logic units would be the group of backbone constituents of the sentence such as the subject-verb-object formed facts, covering both global and local knowledge pieces that are necessary as the basis for logical reasoning. Beyond building the ad-hoc graphs, we propose a more general and convenient fact-driven approach to construct a supergraph on top of our newly defined fact units, and enhance the supergraph with further explicit guidance of local question and option interactions. Experiments on two challenging logical reasoning benchmark datasets, ReClor and LogiQA, show that our proposed model, textsc{Focal Reasoner}, outperforms the baseline models dramatically. It can also be smoothly applied to other downstream tasks such as MuTual, a dialogue reasoning dataset, achieving competitive results.
Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowskis double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowskis algorithm to Moore and weighted automata over commutative semirings. In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata.
While quantum computers are expected to yield considerable advantages over classical devices, the precise features of quantum theory enabling these advantages remain unclear. Contextuality--the denial of a notion of classical physical reality--has emerged as a promising hypothesis. Magic states are quantum resources critical for practically achieving universal quantum computation. They exhibit the standard form of contextuality that is known to enable probabilistic advantages in a variety of computational and communicational tasks. Strong contextuality is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour. Here, we consider special magic states that deterministically enable quantum computation. After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large classes of strongly contextual magic states that enable deterministic implementation of gates from the Clifford hierarchy. These surprising discoveries bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes.
This paper combines two studies: a topological semantics for epistemic notions and abstract argumentation theory. In our combined setting, we use a topological semantics to represent the structure of an agents collection of evidence, and we use argumentation theory to single out the relevant sets of evidence through which a notion of beliefs grounded on arguments is defined. We discuss the formal properties of this newly defined notion, providing also a formal language with a matching modality together with a sound and complete axiom system for it. Despite the fact that our agent can combine her evidence in a rational way (captured via the topological structure), argument-based beliefs are not closed under conjunction. This illustrates the difference between an agents reasoning abilities (i.e. the way she is able to combine her available evidence) and the closure properties of her beliefs. We use this point to argue for why the failure of closure under conjunction of belief should not bear the burden of the failure of rationality.
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