Learning failure-free PRISM programs

PRISM is a probabilistic logic programming formalism which allows defining a probability distribution over possible worlds. This paper investigates learning a class of generative PRISM programs known as failure-free. The aim is to learn recursive PRISM programs which can be used to model stochastic processes. These programs generalise dynamic Bayesian networks by defining a halting distribution over the generative process. Dynamic Bayesian networks model infinite stochastic processes. Sampling from infinite process can only be done by specifying the length of sequences that the process generates. In this case, only observations of a fixed length of sequences can be obtained. On the other hand, the recursive PRISM programs considered in this paper are self-terminating upon some halting conditions. Thus, they generate observations of different lengths of sequences. The direction taken by this paper is to combine ideas from inductive logic programming and learning Bayesian networks to learn PRISM programs. It builds upon the inductive logic programming approach of learning from entailment.     

Separation logic and logics with team semantics

Separation logic is a successful logical system for formal reasoning about programs that mutate their data structures. Team semantics, on the other side, is the basis of modern logics of dependence and independence. Separation logic and team semantics have been introduced with quite different motivations, and are investigated by research communities with rather different backgrounds and objectives. Nevertheless, there are obvious similarities between these formalisms. Both separation logic and logics with team semantics involve the manipulation of second-order objects, such as heaps and teams, by first-order syntax without reference to second-order variables. Moreover, these semantical objects are closely related; it is for instance obvious that a heap can be seen as a team, and the separating conjunction of separation logic is (essentially) the same as the team-semantical disjunction. Based on such similarities, the possible connections between separation logic and team semantics have been raised as a question at several occasions, and lead to informal discussions between these research communities. The objective of this paper is to make this connection precise, and to study its potential but also its obstacles and limitations.     

Kripke‐style semantics for many‐valued logics

This paper deals with Kripke‐style semantics for many‐valued logics. We introduce various types of Kripke semantics, and we connect them with algebraic semantics. As for modal logics, we relate the axioms of logics extending MTL to properties of the Kripke frames in which they are valid. We show that in the propositional case most logics are complete but not strongly complete with respect to the corresponding class of complete Kripke frames, whereas in the predicate case there are important many‐valued logics like BL, ? and Π, which are not even complete with respect to the class of all predicate Kripke frames in which they are valid. Thus although very natural, Kripke semantics seems to be slightly less powerful than algebraic semantics. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterated team semantics for a hierarchy of informational types

In this paper, we introduce and study a framework that is inspired by the team semantics for propositional dependence logic but deviates from it in several respects. Most importantly, instead of the two semantic layers used in dependence logic – possible worlds and teams – a whole hierarchy of contexts is introduced and different types of formulas are evaluated at different levels of this hierarchy. This leads to a rich stratification of informational types. In this framework, the dependence operator of dependence logic can be defined by the standard propositional connectives (negation, conjunction, disjunction and implication). We explore the formal aspects of this approach and apply it to a number of puzzling phenomena related to modalities and conditionals.     

On elementary logics for quantitative dependencies

We define and study logics in the framework of probabilistic team semantics and over metafinite structures. Our work is paralleled by the recent development of novel axiomatizable and tractable logics in team semantics that are closed under the Boolean negation. Our logics employ new probabilistic atoms that resemble so-called extended atoms from the team semantics literature. We also define counterparts of our logics over metafinite structures and show that all of our logics can be translated into functional fixed point logic implying a polynomial time upper bound for data complexity with respect to BSS-computations.     

New reformulations for probabilistically constrained quadratic programs

The mixed integer quadratic programming (MIQP) reformulation by Zheng, Sun, Li, and Cui (2012) for probabilistically constrained quadratic programs (PCQP) recently published in EJOR significantly dominates the standard MIQP formulation ( and ) which has been widely adopted in the literature. Stimulated by the dimensionality problem which Zheng et al. (2012) acknowledge themselves for their reformulations, we study further the characteristics of PCQP and develop new MIQP reformulations for PCQP with fewer variables and constraints. The results from numerical tests demonstrate that our reformulations clearly outperform the state-of-the-art MIQP in Zheng et al. (2012).     

Corrigendum to “Kripke‐style semantics for many‐valued logics”

This note contains a correct proof of the fact that the set of all first‐order formulas which are valid in all predicate Kripke frames for Hájek's many‐valued logic BL is not arithmetical. The result was claimed in [5], but the proof given there was incorrect. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semantics for possibilistic answer set programs: Uncertain rules versus rules with uncertain conclusions

Although Answer Set Programming (ASP) is a powerful framework for declarative problem solving, it cannot in an intuitive way handle situations in which some rules are uncertain, or in which it is more important to satisfy some constraints than others. Possibilistic ASP (PASP) is a natural extension of ASP in which certainty weights are associated with each rule. In this paper we contrast two different views on interpreting the weights attached to rules. Under the first view, weights reflect the certainty with which we can conclude the head of a rule when its body is satisfied. Under the second view, weights reflect the certainty that a given rule restricts the considered epistemic states of an agent in a valid way, i.e. it is the certainty that the rule itself is correct. The first view gives rise to a set of weighted answer sets, whereas the second view gives rise to a weighted set of classical answer sets.     

An OpenCL implementation of a forward sampling algorithm for CP-logic

We present an approximate query answering algorithm for the Probabilistic Logic Programming language CP-logic. It complements existing sampling algorithms by using the rules from body to head instead of in the other direction. We present an implementation in OpenCL, which is able to exploit the multicore architecture of modern GPUs to compute a large number of samples in parallel, and demonstrate that this is competitive with existing inference algorithms.     

On the optimal value function for certain linear programs with unbounded optimal solution sets

Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.The author would like to thank two anonymous reviewers for their most useful comments on earlier versions of this paper.     

Lagrangian decomposition and mixed-integer quadratic programming reformulations for probabilistically constrained quadratic programs

Probabilistically constrained quadratic programming (PCQP) problems arise naturally from many real-world applications and have posed a great challenge in front of the optimization society for years due to the nonconvex and discrete nature of its feasible set. We consider in this paper a special case of PCQP where the random vector has a finite discrete distribution. We first derive second-order cone programming (SOCP) relaxation and semidefinite programming (SDP) relaxation for the problem via a new Lagrangian decomposition scheme. We then give a mixed integer quadratic programming (MIQP) reformulation of the PCQP and show that the continuous relaxation of the MIQP is exactly the SOCP relaxation. This new MIQP reformulation is more efficient than the standard MIQP reformulation in the sense that its continuous relaxation is tighter than or at least as tight as that of the standard MIQP. We report preliminary computational results to demonstrate the tightness of the new convex relaxations and the effectiveness of the new MIQP reformulation.     

The impact of distribution system characteristics on computational tractability

Mathematical programming models that seek optimal design and operational plans for distribution systems can be computationally intractable. This paper examines the extent to which distribution configuration and demand characteristics affect the ease of obtaining an optimal solution. Problem characteristics, which are reflected in a model by the parameter values, can render the model for one distribution scenario to be computationally intractable and that for another to yield an optimal solution easily. We introduce echelon-flow-based valid inequalities and use them to explicate the extent to which problem characteristics impact computational tractability.     

A penalty function approach for solving bi-level linear programs

The paper presents an approach to bi-level programming using a duality gap—penalty function format. A new exact penalty function exists for obtaining a global optimal solution for the linear case, and an algorithm is given for doing this, making use of some new theoretical properties. For each penalty parameter value, the central optimisation problem is one of maximising a convex function over a polytope, for which a modification of an algorithm of Tuy (1964) is used. Some numerical results are given. The approach has other features which assist the actual decisionmaking process, which make use of the natural roles of duality gaps and penalty parameters. The approach also allows a natural generalization to nonlinear problems.     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

An Omitting Types Theorem for first order logic with infinitary relation symbols

In this paper, an extension of first order logic is introduced. In such logics atomic formulas may have infinite lengths. An Omitting Types Theorem is proved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Easy distributions for combinatorial optimization problems with probabilistic constraints

We show how we can linearize individual probabilistic linear constraints with binary variables when all coefficients are independently distributed according to either N(μ_i,λμ_i), for some λ>0 and μ_i>0, or Γ(k_i,θ) for some θ>0 and k_i>0. The constraint can also be linearized when the coefficients are independent and identically distributed and either positive or strictly stable random variables.