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09:00 | The Logic of Real Proofs ABSTRACT. TBA |

11:00 | SPEAKER: Jorge Fandinno ABSTRACT. In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions (QELF). Then, we proceed to incorporate a new kind of logical term, 'intensional set' (a construct commonly used to denote the set of objects characterised by a given formula), and to extend QELF semantics for this new type of expressions. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count, sum, maximum, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the QELF interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to define sum in terms of the integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the Alog language, when we make the restriction to that syntactic fragment. |

11:30 | SPEAKER: Carmine Dodaro ABSTRACT. Aggregates are among the most frequently used linguistic extensions of answer set programming. The result of an aggregation may introduce new constants during the instantiation of the input program, a feature known as value invention. When the aggregation involves literals whose truth value is undefined at instantiation time, modern grounders introduce several instances of the aggregate, one for each possible interpretation of the undefined literals. This paper introduces new data structures and techniques to handle such cases, and more in general aggregations on the same aggregate set identified in the ground program in input. The proposed solution reduces the memory footprint of the solver without sacrificing efficiency. On the contrary, the performance of the solver may improve thanks to the addition of some simple entailed clauses which are not easily discovered otherwise, and since redundant computation is avoided during propagation. Empirical evidence of the potential impact of the proposed solution is given. |

12:00 | SPEAKER: Panos Rondogiannis ABSTRACT. We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal a serious candidate for the title of the well-founded semantics for higher-order logic programs. |

14:00 | Proof-relevant Horn Clauses for Dependent Type Inference and Term Synthesis SPEAKER: Frantisek Farka ABSTRACT. First-order resolution has been used for type inference for many years, including in Hindley-Milner type inference, type-classes, and constrained data types, to name but a few. Dependent types are a new trend in functional languages. In this paper, we show that proof-relevant first-order resolution can play an important role in automating type inference and term synthesis for dependently typed languages. We propose a calculus that translates type inference and term synthesis problems in a dependently typed language to a logic program and a goal in the proof-relevant first-order Horn clause logic. The computed answer substitution and proof term then provide a solution to the given type inference and term synthesis problem. We prove the decidability and soundness of our method. |

14:30 | SPEAKER: Aleksy Schubert ABSTRACT. We propose an interpretation of the first-order answer set programming (FOASP) in terms of intuitionistic proof theory. It is obtained by two polynomial translations between FOASP and the bounded-arity fragment of the \Sigma_1 level of the Mints hierarchy in first-order intuitionistic logic. It follows that \Sigma_1 formulas using predicates of fixed arity (in particular unary) are of the same strength as FOASP. Our construction reveals a close similarity between constructive provability and stable entailment, or equivalently, between the construction of an answer set and an intuitionistic refutation. |

15:00 | SPEAKER: Maurizio Proietti ABSTRACT. We address the problem of verifying the satisfiability of Constrained Horn Clauses (CHCs) based on theories of inductively defined data structures, such as lists and trees. We propose a transformation technique whose objective is the removal of these data structures from CHCs, hence reducing their satisfiability to a satisfiability problem for CHCs on integers and booleans. We propose a transformation algorithm and identify a class of clauses where it always succeeds. We also consider an extension of that |

17:00 | SPEAKER: Joaquin Arias ABSTRACT. Extending ASP with constraints (CASP) enhances its expressiveness and performance. This extension is not straightforward as the grounding phase, present in most ASP systems, removes variables and the links among them, and also causes a combinatorial explosion in the size of the program. This has led CASP systems to devise several methods to overcome this issue: restricting the constraint domains (e.g., discrete instead of dense), where constraints can appear, or the type (or number) of models that can be returned. In this paper we propose to incorporate constraints into s(ASP), a goal-directed, top-down execution model which implements ASP while retaining logical variables both during execution and in the answer sets. The resulting model, s(CASP), can constrain variables that (as in CLP) are kept during the execution and in the answer sets. s(CASP) inherits and generalizes the execution model of s(ASP) while parameterizing the constraint solver. We describe this novel execution model and show, through several examples, the enhanced expressiveness of s(CASP) w.r.t. ASP, CLP, and other ASP systems with constraints. We also report improved performance w.r.t. other very mature, highly optimized ASP systems in some benchmarks. |

17:30 | SPEAKER: Zhun Yang ABSTRACT. Logic Programs with Ordered Disjunction (LPOD) is an extension of standard answer set programs to handle preference using the concept of ordered disjunction, and CR-Prolog2 is an extension of standard answer set programs with consistency restoring rules and LPOD-like ordered disjunction. We present reductions of each of these languages into the standard ASP language, which gives us an alternative insight into the semantics of the extensions in terms of the standard ASP language. |

FLoC reception at Oxford Town Hall. Drinks and canapés available from 7pm (pre-booking via FLoC registration system required; guests welcome).