Logic programming with pseudo-Boolean constraints

Boolean constraints play an important role in various constraint logic programming languages. In this paper we consider pseudo-Boolean constraints, that is equations and inequalities between pseudo-Boolean functions. A pseudo-Boolean function is an integer-valued function of Boolean variables and thus a generalization of a Boolean function. Pseudo-Boolean functions occur in many application areas, in particular in problems from operations research. An interesting connection to logic is that inference problems in propositional logic can be translated into linear pseudo-Boolean optimization problems. More generally, pseudo-Boolean constraints can be seen as a particular way of combining two of the most important domains in constraint logic programming: arithmetic and Boolean algebra. In this paper we define a new constraint logic programming language {\em CLP(PB)} for logic progamming with pseudo-Boolean constraints. The language is an instance of the general constraint logic programming language scheme {\em CLP(X)} and inherits all the typical semantic properties. We show that any pseudo-Boolean constraint has a most general solution and give variable elimination algorithms for pseudo-Boolean unification and unconstrained pseudo-Boolean optimization. Both algorithms subsume the well-known Boolean unification algorithm of B\"uttner and Simonis

Cutting planes and the elementary closure in fixed dimension

The elementary closure $P'$ of a polyhedron $P$ is the intersection of $P$ with all its Gomory-Chvátal cutting planes. $P'$ is a rational polyhedron provided that $P$ is rational. The known bounds for the number of inequalities defining $P'$ are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. If $P$ is a simplicial cone, we construct a polytope $Q$, whose integral elements correspond to cutting planes of $P$. The vertices of the integer hull $Q_I$ include the facets of $P'$. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of $Q_I$

Dynamic optimization of metabolic networks coupled with gene expression

The regulation of metabolic activity by tuning enzyme expression levels is crucial to sustain cellular growth in changing environments. Metabolic networks are often studied at steady state using constraint-based models and optimization techniques. However, metabolic adaptations driven by changes in gene expression cannot be analyzed by steady state models, as these do not account for temporal changes in biomass composition. Here we present a dynamic optimization framework that integrates the metabolic network with the dynamics of biomass production and composition, explicitly taking into account enzyme production costs and enzymatic capacity. In contrast to the established dynamic flux balance analysis, our approach allows predicting dynamic changes in both the metabolic fluxes and the biomass composition during metabolic adaptations. We applied our algorithm in two case studies: a minimal nutrient uptake network, and an abstraction of core metabolic processes in bacteria. In the minimal model, we show that the optimized uptake rates reproduce the empirical Monod growth for bacterial cultures. For the network of core metabolic processes, the dynamic optimization algorithm predicted commonly observed metabolic adaptations, such as a diauxic switch with a preference ranking for different nutrients, re-utilization of waste products after depletion of the original substrate, and metabolic adaptation to an impending nutrient depletion. These examples illustrate how dynamic adaptations of enzyme expression can be predicted solely from an optimization principle

Narrowing strategies for arbitrary canonical rewrite systems

Narrowing is a universal unification procedure for equational theories defined by a canonical term rewriting system. In its original form it is extremely inefficient. Therefore, many optimizations have been proposed during the last years. In this paper, we present the narrowing strategies for arbitrary canonical systems in a uniform framework and introduce the new narrowing strategy LSE narrowing. LSE narrowing is complete and improves all other strategies which are complete for arbitrary canonical systems. It is optimal in the sense that two different LSE narrowing derivations cannot generate the same narrowing substitution. Moreover, LSE narrowing computes only normalized narrowing substitutions

Double and multiple knockout simulations for genome-scale metabolic network reconstructions

Constraint-based modeling of genome-scale metabolic network reconstructions has become a widely used approach in computational biology. Flux coupling analysis is a constraint-based method that analyses the impact of single reaction knockouts on other reactions in the network. We present an extension of flux coupling analysis for double and multiple gene or reaction knockouts, and develop corresponding algorithms for an in silico simulation. To evaluate our method, we perform a full single and double knockout analysis on a selection of genome-scale metabolic network reconstructions and compare the results

Time-Optimal Adaptation in Metabolic Network Models

Analysis of metabolic models using constraint-based optimization has emerged as an important computational technique to elucidate and eventually predict cellular metabolism and growth. In this work, we introduce time-optimal adaptation (TOA), a new constraint-based modeling approach that allows us to evaluate the fastest possible adaptation to a pre-defined cellular state while fulfilling a given set of dynamic and static constraints. TOA falls into the mathematical problem class of time-optimal control problems, and, in its general form, can be broadly applied and thereby extends most existing constraint-based modeling frameworks. Specifically, we introduce a general mathematical framework that captures many existing constraint-based methods and define TOA within this framework. We then exemplify TOA using a coarse-grained self-replicator model and demonstrate that TOA allows us to explain several well-known experimental phenomena that are difficult to explore using existing constraint-based analysis methods. We show that TOA predicts accumulation of storage compounds in constant environments, as well as overshoot uptake metabolism after periods of nutrient scarcity. TOA shows that organisms with internal temporal degrees of freedom, such as storage, can in most environments outperform organisms with a static intracellular composition. Furthermore, TOA reveals that organisms adapted to better growth conditions than present in the environment (“optimists”) typically outperform organisms adapted to poorer growth conditions (“pessimists”).Peer Reviewe

Time-Optimal Adaptation in Metabolic Network Models

Analysis of metabolic models using constraint-based optimization has emerged as an important computational technique to elucidate and eventually predict cellular metabolism and growth. In this work, we introduce time-optimal adaptation (TOA), a new constraint-based modeling approach that allows us to evaluate the fastest possible adaptation to a pre-defined cellular state while fulfilling a given set of dynamic and static constraints. TOA falls into the mathematical problem class of time-optimal control problems, and, in its general form, can be broadly applied and thereby extends most existing constraint-based modeling frameworks. Specifically, we introduce a general mathematical framework that captures many existing constraint-based methods and define TOA within this framework. We then exemplify TOA using a coarse-grained self-replicator model and demonstrate that TOA allows us to explain several well-known experimental phenomena that are difficult to explore using existing constraint-based analysis methods. We show that TOA predicts accumulation of storage compounds in constant environments, as well as overshoot uptake metabolism after periods of nutrient scarcity. TOA shows that organisms with internal temporal degrees of freedom, such as storage, can in most environments outperform organisms with a static intracellular composition. Furthermore, TOA reveals that organisms adapted to better growth conditions than present in the environment (“optimists”) typically outperform organisms adapted to poorer growth conditions (“pessimists”)

F2C2: a fast tool for the computation of flux coupling in genome-scale metabolic networks

Background: Flux coupling analysis (FCA) has become a useful tool in the constraint-based analysis of genome-scale metabolic networks. FCA allows detecting dependencies between reaction fluxes of metabolic networks at steady-state. On the one hand, this can help in the curation of reconstructed metabolic networks by verifying whether the coupling between reactions is in agreement with the experimental findings. On the other hand, FCA can aid in defining intervention strategies to knock out target reactions. Results: We present a new method F2C2 for FCA, which is orders of magnitude faster than previous approaches. As a consequence, FCA of genome-scale metabolic networks can now be performed in a routine manner. Conclusions: We propose F2C2 as a fast tool for the computation of flux coupling in genome-scale metabolic networks. F2C2 is freely available for non-commercial use at https://sourceforge.net/projects/f2c2/files/

Reverse Chv\'atal-Gomory rank

We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all rational polyhedra whose integer hull is P. A well-known example in dimension two shows that there exist integral polytopes P with r*(P) equal to infinity. We provide a geometric characterization of polyhedra with this property in general dimension, and investigate upper bounds on r*(P) when this value is finite.Comment: 21 pages, 4 figure

Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APP's with at most demonic non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201