Answer Set Programming (ASP) is a prominent rule-based language for knowledge representation and reasoning with roots in logic programming and non-monotonic reasoning. The aim to capture the essence of removing (ir)relevant details in ASP programs led to the investigation of different notions, from strong persistence (SP) forgetting, to faithful abstractions, and, recently, strong simplifications, where the latter two can be seen as relaxed and strengthened notions of forgetting, respectively. Although it was observed that these notions are related, especially given that they have characterizations through the semantics for strong equivalence, it remained unclear whether they can be brought together. In this work, we bridge this gap by introducing a novel relativized equivalence notion, which is a relaxation of the recent simplification notion, that is able to capture all related notions from the literature. We provide the necessary and sufficient conditions for relativized simplifiability, which shows that the challenging part is for when the context programs do not contain all the atoms to remove. We then introduce an operator that combines projection and a relaxation of SP-forgetting to obtain the relativized simplifications. We furthermore provide complexity results that complete the overall picture.
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Reactive synthesis is the process of generate correct con- trollers from temporal logic specifications. Typically, synthesis is restricted to Boolean specifications in LTL. Recently, a Boolean abstraction technique allows to translate LTLT specifications that contain literals in theories into equi-realizable LTL specifications, but no full synthesis procedure exists yet. In synthesis modulo theories, the system receives valuations of environment variables (from a first-order theory T ) and outputs valuations of system variables from T . In this paper, we address how to syntheize a full controller using a combination of the static Boolean controller obtained from the Booleanized LTL specification together with on-the-fly queries to a solver that produces models of satisfiable existential T formulae. This is the first synthesis method for LTL modulo theories. Additionally, our method can produce adaptive responses which increases explainability and can improve runtime properties like performance. Our approach is applicable to both LTL modulo theories and LTLf modulo theories.
Program synthesis is the task of automatically generating code based on a specification. In Syntax-Guided Synthesis (SyGuS) this specification is a combination of a syntactic template and a logical formula, and the result is guaranteed to satisfy both. We present a reinforcement-learning guided algorithm for SyGuS which uses Monte-Carlo Tree Search (MCTS) to search the space of candidate solutions. Our algorithm learns policy and value functions which, combined with the upper confidence bound for trees, allow it to balance exploration and exploitation. A common challenge in applying machine learning approaches to syntax-guided synthesis is the scarcity of training data. To address this, we present a method for automatically generating training data for SyGuS based on anti-unification of existing first-order satisfiability problems, which we use to train our MCTS policy. We implement and evaluate this setup and demonstrate that learned policy and value improve the synthesis performance over a baseline by over 26 percentage points in the training and testing sets. Our tool outperforms state-of-the-art tool cvc5 on the training set and performs comparably in terms of the total number of problems solved on the testing set (solving 23% of the benchmarks on which cvc5 fails). We make our data set publicly available, to enable further application of machine learning methods to the SyGuS problem.
We develop a general algorithmic framework that allows us to obtain fixed-parameter tractability for computing smallest symbolic models that represent given data. Our framework applies to all ML model types that admit a certain extension property. By showing this extension property for decision trees, decision sets, decision lists, and binary decision diagrams, we obtain that minimizing these fundamental model types is fixed-parameter tractable. Our framework even applies to ensembles, which combine individual models by majority decision.
The problem of model counting, i.e., counting satisfying assignments of a Boolean formula, is a fundamental problem in computer science, with diverse applications. Given #P-hardness of the problem, many algorithms have been developed over the years to provide an approximate model count. Recently, building on the practical success of SAT-solvers used as NP oracles, the focus has shifted from theory to practical implementations of such algorithms. This has brought to focus new challenges. In this paper, we consider one such challenge – that of auditable deterministic approximate model counters wherein a counter should also generate a certificate, which allows a user (often with limited computational power) to independently audit whether the count returned by an invocation of the algorithm is indeed within the promised bounds. We start by examining a celebrated approximate model counting algorithm due to Stockmeyer that uses polynomially many calls to a \Sigma^2_P oracle, and show that it can be audited via a \Pi^2_P formula on (n^2 log^2 n) variables, where n is the number of variables in the original formula. Since n is often large (10’s to 100’s of thousands) for typical instances, we ask if the count of variables in the certificate formula can be reduced – a critical question towards potential implementation. We show that this improvement in certification can be achieved with a tradeoff in the counting algorithm’s complexity. Specifically, we develop new deterministic approximate model counting algorithms that invoke a \Sigma^3_P oracle, but can be certified using a \Pi^2_P formula on fewer variables: our final algorithm uses just (n log n) variables. Our study demonstrates that one can simplify certificate checking significantly if we allow the counting algorithm to access a slightly more powerful oracle. We believe this shows for the first time how the audit complexity can be traded for the complexity of approximate counting.
MINES: Message Intercommunication for Inductive Relation Reasoning over Neighbor-Enhanced Subgraphs
PDF ↗GraIL and its variants have shown their promising capacities for inductive relation reasoning on knowledge graphs. However, the uni-directional message-passing mechanism hinders such models from exploiting hidden mutual relations between entities in directed graphs. Besides, the enclosing subgraph extraction in most GraIL-based models restricts the model from extracting enough discriminative information for reasoning. Consequently, the expressive ability of these models is limited. To address the problems, we propose a novel GraIL-based framework, termed MINES, by introducing a Message Intercommunication mechanism on the Neighbor-Enhanced Subgraph. Concretely, the message intercommunication mechanism is designed to capture the omitted hidden mutual information. It introduces bi-directed information interactions between connected entities by inserting an undirected/bi-directed GCN layer between uni-directed RGCN layers. Moreover, inspired by the success of involving more neighbors in other graph-based tasks, we extend the neighborhood area beyond the enclosing subgraph to enhance the information collection for inductive relation reasoning. Extensive experiments prove the promising capacity of the proposed MINES from various aspects, especially for the superiority, effectiveness, and transfer ability.
Foundation models have vast potential to enable diverse AI applications. The powerful yet incomplete nature of these models has spurred a wide range of mechanisms to augment them with capabilities such as in-context learning, information retrieval, and code interpreting. We propose Vieira, a declarative framework that unifies these mechanisms in a general solution for programming with foundation models. Vieira follows a probabilistic relational paradigm and treats foundation models as stateless functions with relational inputs and outputs. It supports neuro-symbolic applications by enabling the seamless combination of such models with logic programs, as well as complex, multi-modal applications by streamlining the composition of diverse sub-models. We implement Vieira by extending the Scallop compiler with a foreign interface that supports foundation models as plugins. We implement plugins for 12 foundation models including GPT, CLIP, and SAM. We evaluate Vieira on 9 challenging tasks that span language, vision, and structured and vector databases. Our evaluation shows that programs in Vieira are concise, can incorporate modern foundation models, and have comparable or better accuracy than competitive baselines.
We present a general framework for abstracting agent behavior in multi-agent synchronous games in the situation calculus, which provides a first-order representation of the state and allows us to model how plays depend on the data and objects involved. We represent such games as action theories of a special form called situation calculus synchronous game structures (SCSGSs), in which we have a single action "tick" whose effects depend on the combination of moves selected by the players. In our framework, one specifies both an abstract SCSGS and a concrete SCSGS, as well as a refinement mapping that specifies how each abstract move is implemented by a Golog program defined over the concrete SCSGS. We define notions of sound and complete abstraction with respect to a mapping over such SCSGS. To express strategic properties on the abstract and concrete games we adopt a first-order variant of alternating-time mu-calculus mu-ATL-FO. We show that we can exploit abstraction in verifying mu-ATL-FO properties of SCSGSs under the assumption that agents can always execute abstract moves to completion even if not fully controlling their outcomes.
Conflict detection is relevant in various application scenarios, ranging from interactive decision-making to the diagnosis of faulty knowledge bases. Conflicts can be regarded as sets of constraints that cause an inconsistency. In many scenarios (e.g., constraint-based configuration), conflicts are repeatedly determined for the same or similar sets of constraints. This misses out on the valuable opportunity for leveraging knowledge reuse and related potential performance improvements, which are extremely important, specifically interactive constraint-based applications. In this paper, we show how to integrate knowledge reuse concepts into non-instructive conflict detection. We introduce the InformedQX algorithm, which is a reuse-aware variant of QuickXPlain. The results of a related performance analysis with the Linux-2.6.3.33 configuration knowledge base show significant improvements in terms of runtime performance compared to QuickXPlain.
Artificial Intelligence for Theorem Proving (AITP) has given rise to a plethora of benchmarks and methodologies, particularly in Interactive Theorem Proving (ITP). Research in the area is fragmented, with a diverse set of approaches being spread across several ITP systems. This presents a significant challenge to the comparison of methods, which are often complex and difficult to replicate. Addressing this, we present BAIT, a framework for the fair and streamlined comparison of learning approaches in ITP. We demonstrate BAIT’s capabilities with an in-depth comparison, across several ITP benchmarks, of state-of-the-art architectures applied to the problem of formula embedding. We find that Structure Aware Transformers perform particularly well, improving on techniques associated with the original problem sets. BAIT also allows us to assess the end-to-end proving performance of systems built on interactive environments. This unified perspective reveals a novel end-to-end system that improves on prior work. We also provide a qualitative analysis, illustrating that improved performance is associated with more semantically-aware embeddings. By streamlining the implementation and comparison of Machine Learning algorithms in the ITP context, we anticipate BAIT will be a springboard for future research.
We present an FCA-based axiomatization method that produces a complete OWL 2 EL TBox (the terminological part of an OWL 2 EL ontology) from a graph dataset in at most exponential time. We describe technical details that allow for efficient implementation as well as variations that dispense with the computation of extremely large axioms, thereby rendering the approach applicable albeit some completeness is lost. Moreover, we evaluate the prototype on real-world datasets.
Recurrent Neural Cascades (RNCs) are the recurrent neural networks with no cyclic dependencies among recurrent neurons. This class of recurrent networks has received a lot of attention in practice. Besides training methods for a fixed architecture such as backpropagation, the cascade architecture naturally allows for constructive learning methods, where recurrent nodes are added incrementally one at a time, often yielding smaller networks. Furthermore, acyclicity amounts to a structural prior that even for the same number of neurons yields a more favourable sample complexity compared to a fully-connected architecture. A central question is whether the advantages of the cascade architecture come at the cost of a reduced expressivity. We provide new insights into this question. We show that the regular languages captured by RNCs with sign and tanh activation with positive recurrent weights are the star-free regular languages. In order to establish our results we developed a novel framework where capabilities of RNCs are assessed by analysing which semigroups and groups a single neuron is able to implement. A notable implication of our framework is that RNCs can achieve the expressivity of all regular languages by introducing neurons that can implement groups.
Minimal Macro-Based Rewritings of Formal Languages: Theory and Applications in Ontology Engineering (and Beyond)
PDF ↗In this paper, we introduce the problem of rewriting finite formal languages using syntactic macros such that the rewriting is minimal in size. We present polynomial-time algorithms to solve variants of this problem and show their correctness. To demonstrate the practical relevance of the proposed problems and the feasibility and effectiveness of our algorithms in practice, we apply these to biomedical ontologies authored in OWL. We find that such rewritings can significantly reduce the size of ontologies by capturing repeated expressions with macros. This approach not only offers valuable assistance in enhancing ontology quality and comprehension but can also be seen as a general methodology for evaluating features of rewriting systems (including syntactic macros, templates, or other forms of rewriting rules), which can be analyzed in terms of their influence on computational problems.
Answer Set Programming (ASP) has emerged as a promising paradigm in knowledge representation and automated reason- ing owing to its ability to model hard combinatorial problems from diverse domains in a natural way. Building on advances in propositional SAT solving, the past two decades have wit- nessed the emergence of well-engineered systems for solv- ing the answer set satisfiability problem, i.e., finding mod- els or answer sets for a given answer set program. In re- cent years, there has been growing interest in problems be- yond satisfiability, such as model counting, in the context of ASP. Akin to the early days of propositional model count- ing, state-of-the-art exact answer set counters do not scale well beyond small instances. Exact ASP counters struggle with handling larger input formulas. The primary contribu- tion of this paper is a new ASP counting framework, called sharpASP, which counts answer sets avoiding larger input formulas. This relies on an alternative way of defining answer sets that allows lifting of key techniques developed in the con- text of propositional model counting. Our extensive empirical analysis over 1470 benchmarks demonstrates significant per- formance gain over current state-of-the-art exact answer set counters. Specifically, by using sharpASP, we were able to solve 1062 benchmarks with PAR2 score of 3082 whereas using prior state-of-the-art, we could only solve 895 bench- marks with PAR2 score of 4205, all other experimental con- ditions being the same.
This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.
Many inductive logic programming approaches struggle to learn programs from noisy data. To overcome this limitation, we introduce an approach that learns minimal description length programs from noisy data, including recursive programs. Our experiments on several domains, including drug design, game playing, and program synthesis, show that our approach can outperform existing approaches in terms of predictive accuracies and scale to moderate amounts of noise.
Recent research on predicting the binding affinity between drug molecules and proteins use representations learned, through unsupervised learning techniques, from large databases of molecule SMILES and protein sequences. While these representations have significantly enhanced the predictions, they are usually based on a limited set of modalities, and they do not exploit available knowledge about existing relations among molecules and proteins. Our study reveals that enhanced representations, derived from multimodal knowledge graphs describing relations among molecules and proteins, lead to state-of-the-art results in well-established benchmarks (first place in the leaderboard for Therapeutics Data Commons benchmark ``Drug-Target Interaction Domain Generalization Benchmark", with an improvement of 8 points with respect to previous best result). Moreover, our results significantly surpass those achieved in standard benchmarks by using conventional pre-trained representations that rely only on sequence or SMILES data. We release our multimodal knowledge graphs, integrating data from seven public data sources, and which contain over 30 million triples. Pretrained models from our proposed graphs and benchmark task source code are also released.
On the Structural Hardness of Answer Set Programming: Can Structure Efficiently Confine the Power of Disjunctions?
PDF ↗Answer Set Programming (ASP) is a generic problem modeling and solving framework with a strong focus on knowledge representation and a rapid growth of industrial applications. So far, the study of complexity resulted in characterizing hardness and determining their sources, fine-grained insights in the form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Unfortunately, for the well-known parameter treewidth disjunctive programs require double-exponential runtime under reasonable complexity assumptions. This quickly becomes out of reach. We deal with the classification of structural parameters for disjunctive ASP on the program's rule structure (incidence graph). First, we provide a polynomial kernel to obtain single-exponential runtime in terms of vertex cover size, despite subset-minimization being not represented in the program’s structure. Then we turn our attention to strictly better structural parameters between vertex cover size and treewidth. Here, we provide double-exponential lower bounds for the most prominent parameters in that range: treedepth, feedback vertex size, and cliquewidth. Based on this, we argue that unfortunately our options beyond vertex cover size are limited. Our results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.
The need to model and analyse dynamic systems operating over complex data is ubiquitous in AI and neighboring areas, in particular business process management. Analysing such data-aware systems is a notoriously difficult problem, as they are intrinsically infinite-state. Existing approaches work for specific datatypes, and/or limit themselves to the verification of safety properties. In this paper, we lift both such limitations, studying for the first time linear-time verification for so-called data-aware processes modulo theories (DMTs), from the foundational and practical point of view. The DMT model is very general, as it supports processes operating over variables that can store arbitrary types of data, ranging over infinite domains and equipped with domain-specific predicates. Specifically, we provide four contributions. First, we devise a semi-decision procedure for linear-time verification of DMTs, which works for a very large class of datatypes obeying to mild model-theoretic assumptions. The procedure relies on a unique combination of automata-theoretic and cover computation techniques to respectively deal with linear-time properties and datatypes. Second, we identify an abstract, semantic property that guarantees the existence of a faithful finite-state abstraction of the original system, and show that our method becomes a decision procedure in this case. Third, we identify concrete, checkable classes of systems that satisfy this property, generalising several results in the literature. Finally, we present an implementation and an experimental evaluation over a benchmark of real-world data-aware business processes.
Expressing system specifications using Computation Tree Logic (CTL) formulas, formalising programs using Kripke structures, and then model checking the system is an established workflow in program verification and has wide applications in AI. In this paper, we consider the task of model enumeration, which asks for a uniform stream of output systems that satisfy the given specification. We show that, given a CTL formula and a system (potentially falsified by the formula), enumerating satisfying submodels is always hard for CTL--regardless of which subset of CTL-operators is considered. As a silver lining on the horizon, we present fragments via restrictions on the allowed Boolean functions that still allow for fast enumeration.