+simply denotes the set of vectors which is nonnegative in each coordinate.Definition 2.2 (Lagrangian of an optimization problem).Let 0 : R R, 1 , . . . , 1 , . . . be a standardform convex optimization problem, as in Definition 2.1.Then the Lagrangian of this problem is the function : R R + R R defined by (;, ) = 0 () + () +

Formal reasoning about inductively defined relations and structures is widely recognized not only for its mathematical interest but also for its importance in computer science, and has applications in verifying properties of programs and algorithms. Recently, several proof systems of inductively defined predicates based on sequent calculus including the cyclic proof system CLKID-omega and the infinite-descent proof system LKID-omega have attracted much attention. Although the relation among their provabilities has been clarified so far, the logical complexity of these systems has not been much studied. The infinite-descent proof system LKID-omega is an infinite proof system for inductive definitions and allows infinite paths in proof figures. It serves as a basis for the cyclic proof system. This paper shows that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete. To show this, first it is shown that the validity for inductive definitions in standard models is equivalent to the validity for inductive definitions in standard term models. Next, using this equivalence, this paper extends the truth predicate of omega-languages, as given in Girard's textbook, to inductive definitions by employing arithmetical coding of inductive definitions. This shows that the validity of inductive definitions in standard models is a (Pi-1-1) relation. Then, using the completeness of LKID-omega for standard models, it is shown that the logical complexity of the provability in LKID-omega is (Pi-1-1)-complete.

Constraint Handling Rules (CHR) is a rule-based programming language which is typically embedded into a general-purpose language. There exists a plethora of implementations for numerous host languages. However, the existing implementations often re-invent the way to embed CHR, which impedes maintenance and weakens assertions of correctness. To formalize and thereby standardize the embedding into arbitrary host languages, we introduced the framework FreeCHR and proved it to be a valid representation of classical ground CHR. Until now, this framework only includes a translation of the very abstract operational semantics which, due to its abstract nature, is not a sufficient base for practical implementations. In this paper we present a translation of the refined operational semantics for FreeCHR and prove it to be both a valid concretization of the very abstract semantics of FreeCHR, and an equivalent representation of the refined semantics of CHR. This will establish implementations of FreeCHR as equivalent in behavior and expressiveness to existing implementations of CHR.

We present a framework for synthesising formulas in first-order logic (FOL) from examples, which unifies and advances state-of-the-art approaches for inference of transition system invariants. To do so, we study and categorise the existing methodologies, encoding techniques in their formula synthesis via answer set programming (ASP). Based on the derived categorisation, we propose orthogonal slices, a new technique for formula enumeration that partitions the search space into manageable chunks, enabling two approaches for incremental candidate pruning. Using a combination of existing techniques for first-order (FO) invariant synthesis and the orthogonal slices implemented in our framework FORCE, we significantly accelerate a state-of-the-art algorithm for distributed system invariant inference. We also show that our approach facilitates composition of different invariant inference frameworks, allowing for novel optimisations.

Quantified Answer Set Programming (QASP) extends Answer Set Programming (ASP) by allowing quantification over propositional variables, similar to Quantified Boolean Formulas (QBF). In this paper, we interpret models of QASP formulas in terms of policies, which represent decision-making strategies that determine how existentially quantified variables should be assigned, given the conditions set by universally quantified variables. As a main contribution, we present an algorithm for policy extraction under QASP semantics, inspired by the Equilibrium Logic semantics for general ASP theories.

Answer set programming (ASP) aims to realize the AI vision: The user specifies the problem, and the computer solves it. Indeed, ASP has made this vision true in many application domains. However, will current ASP solving techniques scale up for large configuration problems? As a benchmark for such problems, we investigated the configuration of electronic systems, which may comprise more than 30,000 components. We show the potential and limits of current ASP technology, focusing on methods that address the so-called grounding bottleneck, i.e., the sharp increase of memory demands in the size of the problem instances. To push the limits, we investigated the incremental solving approach, which proved effective in practice. However, even in the incremental approach, memory demands impose significant limits. Based on an analysis of grounding, we developed the method constraint-aware guessing, which significantly reduced the memory need.

The paper introduces a basic logic of knowledge and abduction by extending Levesque logic of only-knowing with an abduction modal operator defined via the combination of basic epistemic concepts. The upshot is an alternative approach to abduction that employs a modal vocabulary and explores the relation between abductive reasoning and epistemic states of only knowing. Furthermore, by incorporating a preferential relation into modal frames, we provide a non-monotonic extension of our basic framework capable of expressing different selection methods for abductive explanations. Core metatheoretic properties of non-monotonic consequence relations are explored within this setting and shown to provide a well-behaved foundation for abductive reasoning.

When designing new web applications, developers must cope with different kinds of constraints relative to the resources they rely on: software, hardware, network, online micro-services, or any combination of the mentioned entities. Together, these entities form a complex system of communicating interdependent processes, physical or logical. It is very desirable that such system ensures its robustness to provide a good quality of service. In this paper we introduce Chips, a language that aims at facilitating the design of models made of various entwined components. It allows the description of applications in the form of functional blocks. Chips mixes notions from control theory and general purpose programming languages to generate robust component-based models. This paper presents how to use Chips to systematically design, model and analyse a complex system project, using a variation of the Adaptable TeaStore application as running example.