Probabilistic Algorithmic Knowledge

The framework of algorithmic knowledge assumes that agents use deterministic knowledge algorithms to compute the facts they explicitly know. We extend the framework to allow for randomized knowledge algorithms. We then characterize the information provided by a randomized knowledge algorithm when its answers have some probability of being incorrect. We formalize this information in terms of evidence; a randomized knowledge algorithm returning ``Yes'' to a query about a fact \phi provides evidence for \phi being true. Finally, we discuss the extent to which this evidence can be used as a basis for decisions.Comment: 26 pages. A preliminary version appeared in Proc. 9th Conference on Theoretical Aspects of Rationality and Knowledge (TARK'03

Probabilistic Consensus of the Blockchain Protocol

We introduce a temporal epistemic logic with probabilities as an extension of temporal epistemic logic. This extension enables us to reason about properties that characterize the uncertain nature of knowledge, like “agent a will with high probability know after time s same fact”. To define semantics for the logic we enrich temporal epistemic Kripke models with probability functions defined on sets of possible worlds. We use this framework to model and reason about probabilistic properties of the blockchain protocol, which is in essence probabilistic since ledgers are immutable with high probabilities. We prove the probabilistic convergence for reaching the consensus of the protocol

Randomisation and Derandomisation in Descriptive Complexity Theory

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures

Cell-specific effects of insulin receptor substrate-1 deficiency on normal and IGF-I-mediated colon growth

Insulin-like growth factor I (IGF-I) potently stimulates intestinal growth. Insulinreceptorsubstrate-1(IRS-1)mediates proliferative and antiapoptotic actions of IGF-I in cell lines, but its in vivo relevance in intestine is not defined. This study tested the hypothesis that there is cell type-specific dependence on IRS-1 as a mediator of IGF-I action. Length, mass, crypt cell proliferation, and apoptosis were measured in small intestine and colon of IRS-1-null mice and wild-type (WT) littermates and in colon of IRS-1-null or WT mice expressing IGF-I transgenes. Expression of IGF-I receptor and signaling intermediates was examined in intestine of WT and IRS-1-null mice, cultured intestinal epithelial cells, and myofibroblasts. Absolute IRS-1 deficiency reduced mucosal mass in jejunum and colon, but effects were more pronounced in colon. Muscularis mass was decreased in both segments. In IGF-I transgenics, IRS-1 deficiency significantly attenuated IGF-I-stimulated growth of colonic mucosa and abolished antiapoptotic but not mitogenic effects of IGF-I transgene on crypt cells. IGF-I-induced muscularis growth was unaffected by IRS-1 deficiency. In intestinal epithelial cells, IRS-1 was expressed at higher levels than IRS-2 and was preferentially activated by IGF-I. In contrast, IGF-I activated both IRS-1 and IRS-2 in intestinal myofibroblasts and IRS-2 activation was upregulated in IRS-1-null myofibroblasts. We conclude that the intestinal epithelium but not the muscularis requires IRS-1 for normal trophic actions of IGF-I and that IRS-1 is required for antiapoptotic but not mitogenic effects of IGF-I in the intestinal crypts in vivo

Search Engine Similarity Analysis: A Combined Content and Rankings Approach

How different are search engines? The search engine wars are a favorite topic of on-line analysts, as two of the biggest companies in the world, Google and Microsoft, battle for prevalence of the web search space. Differences in search engine popularity can be explained by their effectiveness or other factors, such as familiarity with the most popular first engine, peer imitation, or force of habit. In this work we present a thorough analysis of the affinity of the two major search engines, Google and Bing, along with DuckDuckGo, which goes to great lengths to emphasize its privacy-friendly credentials. To do so, we collected search results using a comprehensive set of 300 unique queries for two time periods in 2016 and 2019, and developed a new similarity metric that leverages both the content and the ranking of search responses. We evaluated the characteristics of the metric against other metrics and approaches that have been proposed in the literature, and used it to (1) investigate the similarities of search engine results, (2) the evolution of their affinity over time, (3) what aspects of the results influence similarity, and (4) how the metric differs over different kinds of search services. We found that Google stands apart, but Bing and DuckDuckGo are largely indistinguishable from each other.Comment: Shorter version of this paper was accepted in the 21st International Conference on Web Information Systems Engineering (WISE 2020). The final authenticated version is available online at https://doi.org/10.1007/978-3-030-62008-0_

Logic, Probability and Action: A Situation Calculus Perspective

The unification of logic and probability is a long-standing concern in AI, and more generally, in the philosophy of science. In essence, logic provides an easy way to specify properties that must hold in every possible world, and probability allows us to further quantify the weight and ratio of the worlds that must satisfy a property. To that end, numerous developments have been undertaken, culminating in proposals such as probabilistic relational models. While this progress has been notable, a general-purpose first-order knowledge representation language to reason about probabilities and dynamics, including in continuous settings, is still to emerge. In this paper, we survey recent results pertaining to the integration of logic, probability and actions in the situation calculus, which is arguably one of the oldest and most well-known formalisms. We then explore reduction theorems and programming interfaces for the language. These results are motivated in the context of cognitive robotics (as envisioned by Reiter and his colleagues) for the sake of concreteness. Overall, the advantage of proving results for such a general language is that it becomes possible to adapt them to any special-purpose fragment, including but not limited to popular probabilistic relational models

Does Treewidth Help in Modal Satisfiability?

Many tractable algorithms for solving the Constraint Satisfaction Problem (CSP) have been developed using the notion of the treewidth of some graph derived from the input CSP instance. In particular, the incidence graph of the CSP instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we show either that modal satisfiability is FPT, or that it is W[1]-hard. In particular, modal satisfiability in general models is FPT, while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1: improved section 5 to avoid exponential blow-up in formula siz

Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c times the cost of the optimum set of size k. We give improved incremental algorithms for the metric version: an 8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized algorithm. The algorithm is s-size-competitive if the cost of the kth set is at most the minimum cost of any set of size k, and has size at most s k. The optimal size-competitive ratios for this problem are 4 (deterministic) and e (randomized). We present the first poly-time O(log m)-size-approximation algorithm for the offline problem and first poly-time O(log m)-size-competitive algorithm for the incremental problem. Our proofs reduce incremental medians to the following online bidding problem: faced with an unknown threshold T, an algorithm submits "bids" until it submits a bid that is at least the threshold. It pays the sum of all its bids. We prove that folklore algorithms for online bidding are optimally competitive.Comment: conference version appeared in LATIN 2006 as "Oblivious Medians via Online Bidding

Thinking About Causation : A Causal Language with Epistemic Operators

In this paper we propose a formal framework for modeling the interaction of causal and (qualitative) epistemic reasoning. To this purpose, we extend the notion of a causal model [11, 16, 17, 26] with a representation of the epistemic state of an agent. On the side of the object language, we add operators to express knowledge and the act of observing new information. We provide a sound and complete axiomatization of the logic, and discuss the relation of this framework to causal team semantics.Peer reviewe