On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝ We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles

Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556

On the relationship between satisfiability and partially observable Markov decision processes

Stochastic satisfiability (SSAT), Quantified Boolean Satisfiability (QBF) and decision-theoretic planning in finite horizon partially observable Markov decision processes (POMDPs) are all PSPACE-Complete problems. Since they are all complete for the same complexity class, I show how to convert them into one another in polynomial time and space. I discuss various properties of each encoding and how they get translated into equivalent constructs in the other encodings. An important lesson of these reductions is that the states in SSAT and flat POMDPs do not match. Therefore, comparing the scalability of satisfiability and flat POMDP solvers based on the size of the state spaces they can tackle is misleading. A new SSAT solver called SSAT-Prime is proposed and implemented. It includes improvements to watch literals, component caching and detecting symmetries with upper and lower bounds under certain conditions. SSAT-Prime is compared against a state of the art solver for probabilistic inference and a native POMDP solver on challenging benchmarks

Computing a Probabilistic Extension of Answer Set Program Language Using ASP and Markov Logic Solvers

abstract: LPMLN is a recent probabilistic logic programming language which combines both Answer Set Programming (ASP) and Markov Logic. It is a proper extension of Answer Set programs which allows for reasoning about uncertainty using weighted rules under the stable model semantics with a weight scheme that is adopted from Markov Logic. LPMLN has been shown to be related to several formalisms from the knowledge representation (KR) side such as ASP and P-Log, and the statistical relational learning (SRL) side such as Markov Logic Networks (MLN), Problog and Pearl’s causal models (PCM). Formalisms like ASP, P-Log, Problog, MLN, PCM have all been shown to embeddable in LPMLN which demonstrates the expressivity of the language. Interestingly, LPMLN has also been shown to reducible to ASP and MLN which is not only theoretically interesting, but also practically important from a computational point of view in that the reductions yield ways to compute LPMLN programs utilizing ASP and MLN solvers. Additionally, the reductions also allow the users to compute other formalisms which can be reduced to LPMLN. This thesis realizes two implementations of LPMLN based on the reductions from LPMLN to ASP and LPMLN to MLN. This thesis first presents an implementation of LPMLN called LPMLN2ASP that uses standard ASP solvers for computing MAP inference using weak constraints, and marginal and conditional probabilities using stable models enumeration. Next, in this thesis, another implementation of LPMLN called LPMLN2MLN is presented that uses MLN solvers which apply completion to compute the tight fragment of LPMLN programs for MAP inference, marginal and conditional probabilities. The computation using ASP solvers yields exact inference as opposed to approximate inference using MLN solvers. Using these implementations, the usefulness of LPMLN for computing other formalisms is demonstrated by reducing them to LPMLN. The thesis also shows how the implementations are better than the native solvers of some of these formalisms on certain domains. The implementations make use of the current state of the art solving technologies in ASP and MLN, and therefore they benefit from any theoretical and practical advances in these technologies, thereby also benefiting the computation of other formalisms that can be reduced to LPMLN. Furthermore, the implementation also allows for certain SRL formalisms to be computed by ASP solvers, and certain KR formalisms to be computed by MLN solvers.Dissertation/ThesisMasters Thesis Computer Science 201