Weakly complete axiomatization of exogenous quantum propositional logic

A weakly complete finitary axiomatization for EQPL (exogenous quantum propositional logic) is presented. The proof is carried out using a non trivial extension of the Fagin-Halpern-Megiddo technique together with three Henkin style completions.Comment: 28 page

On Counting Propositional Logic and Wagner's Hierarchy

We introduce and study counting propositional logic, an extension of propositional logic with counting quantifiers. This new kind of quantification makes it possible to express that the argument formula is true in a certain portion of all possible interpretations. We show that this logic, beyond admitting a satisfactory proof-theoretical treatment, can be related to computational complexity: the complexity of the underlying decision problem perfectly matches the appropriate level of Wagner's counting hierarchy

Design of quantum optical experiments with logic artificial intelligence

Logic artificial intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and be solved by checking their satisfiability (SAT). Recently, SAT solvers have become a sophisticated and powerful computational tool capable, among other things, of solving long-standing mathematical conjectures. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI improves significantly the resolution of this problem, paving the path to develop more formal-based approaches in the context of quantum physics experiments

Merging the local and global approaches to probabilistic satisfiability

AbstractThe probabilistic satisfiability problem is to verify the consistency of a set of probability values or intervals for logical propositions. The (tight) probabilistic entailment problem is to find best bounds on the probability of an additional proposition. The local approach to these problems applies rules on small sets of logical sentences and probabilities to tighten given probability intervals. The global approach uses linear programming to find best bounds. We show that merging these approaches is profitable to both: local solutions can be used to find global solutions more quickly through stabilized column generation, and global solutions can be used to confirm or refute the optimality of the local solutions found. As a result, best bounds are found, together with their step-by-step justification

Design of quantum optical experiments with logic artificial intelligence

Logic Artificial Intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and solved by checking their satisfiability (SAT). In contrast to machine learning approaches where the results can be approximations or local minima, Logic AI delivers formal and mathematically exact solutions to those problems. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI significantly improves the resolution of this problem, paving the path to developing more formal-based approaches in the context of quantum physics experiments

Байесовские сети доверия: дерево сочленений и его вероятностная семантика

We consider Bayesian belief networks (BBN) with binary variables in nodes. We assume the condition independence and compare semantics of global joint probability and a set of local joint probabilities corresponded to junction tree nodesРассматриваются байесовские сети доверия (БСД) с бинарными переменными в вершинах. В предположении условной независимости сравниваются семантики глобального распределения вероятностей и локальных распределений, соответствующих узлам дерева сочленений

On Counting Propositional Logic and Wagner's Hierarchy

We introduce an extension of classical propositional logic with counting quantifiers. These forms of quantification make it possible to express that a formula is true in a certain portion of the set of all its interpretations. Beyond providing a sound and complete proof system for this logic, we show that validity problems for counting propositional logic can be used to capture counting complexity classes. More precisely, we show that the complexity of the decision problems for validity of prenex formulas of this logic perfectly match the appropriate levels of Wagner's counting hierarchy

Вероятностная семантика байесовских сетей в случае линейной цепочки фрагментов знаний

The article considers in detail Bayesian networks that may be represented as linear chains of knowledge patterns. We study both Bayesian belief networks, based on the conditional probabilities, and algebraic Bayesian networks, based on the joint marginal probabilities. We show relations between these two objects and explicitly describe the families of distributions corresponding to linear chains of knowledge patters in both cases.В статье подробно рассмотрены байесовские сети, представляемые в виде линейной цепочки фрагментов знании. Рассмотрены как байесовские сети доверия, основанные на условных вероятностях, так и алгебраические байесовские сети, основанные на маргинальных совместных вероятностях. Показана взаимосвязь между этими объектами. Явно выписаны семейства вероятностей, отвечающих линейным цепочкам фрагментов знаний в обоих случаях