Related Version: A compact, peer-reviewed version of this paper will be available in the conference proceedings of Formal Methods of Computer-Aided Design (FMCAD) 2020
We propose trace logic, an instance of many-sorted first-order logic, to automate the partial correctness verification of programs containing loops. Trace logic generalizes semantics of program locations and captures loop semantics by encoding properties at arbitrary timepoints and loop iterations. We guide and automate inductive loop reasoning in trace logic by using generic trace lemmas capturing inductive loop invariants. Our work is implemented in the RAPID framework, by extending and integrating superposition-based first-order reasoning within RAPID. We successfully used RAPID to prove correctness of many programs whose functional behavior are best summarized in the first-order theories of linear integer arithmetic, arrays and inductive data types.
We consider one problem that was largely left open by Rudolf Carnap in his work on inductive logic, the problem of analogical inference. After discussing some previous attempts to solve this problem, we propose a new solution that is based on the ideas of Bruno de Finetti on probabilistic symmetries. We explain how our new inductive logic can be developed within the Carnapian paradigm of inductive logic-deriving an inductive rule from a set of simple postulates about the observational process-and discuss some of its properties.
The entailment between separation logic formulae with inductive predicates, also known as symbolic heaps, has been shown to be decidable for a large class of inductive definitions. Recently, a 2-EXPTIME algorithm was proposed and an EXPTIME-hard bound was established; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines.
We develop a doubly-exponential decision procedure for the satisfiability problem of guarded separation logic -- a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the above connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly-ex…
Recent work on neuro-symbolic inductive logic programming has led to promising approaches that can learn explanatory rules from noisy, real-world data. While some proposals approximate logical operators with differentiable operators from fuzzy or real-valued logic that are parameter-free thus diminishing their capacity to fit the data, other approaches are only loosely based on logic making it difficult to interpret the learned "rules". In this paper, we propose learning rules with the recently proposed logical neural networks (LNN). Compared to others, LNNs offer strong connection to classica…
The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many elements. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this pr…
Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-ord…
We identify difference-bound set constraints (DBS), an analogy of difference-bound arithmetic constraints for sets. DBS can express not only set constraints but also arithmetic constraints over set elements. We integrate DBS into separation logic with linearly compositional inductive predicates, obtaining a logic thereof where set data constraints of linear data structures can be specified. We show that the satisfiability of this logic is decidable. A crucial step of the decision procedure is to compute the transitive closure of DBS-definable set relations, to capture which we propose an exten…
A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii)…
In this paper, we present an abstract framework of many-valued modal logic with the interpretation of atomic propositions and modal operators as predicate lifting over coalgebras for an endofunctor on the category of sets. It generalizes Pattinson's stratification method for colagebraic modal logic to the many-valued setting. In contrast to standard techniques of canonical model construction and filtration, this method employs an induction principle to prove the soundness, completeness, and finite model property of the logics. As a consequence, we can lift a restriction on the previous approac…
This paper presents an approach to lemma synthesis to support advanced inductive entailment procedures based on separation logic. We first propose a mechanism where lemmas are automatically proven and systematically applied. The lemmas may include universal guard and/or unknown predicate. While the former is critical for expressivity, the latter is essential for supporting relationships between multiple predicates. We further introduce lemma synthesis to support (i) automated inductive reasoning together with frame inference and (ii) theorem exploration. For (i) we automatically discover and p…
The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification languages and systems. This paper is a formal study of some of the most common forms of inductive definitions found in scientific text: monotone inductive definition, definition by induction over a well-founded order and iterated inductive definitions. We define a logic of definitions offering a uniform formal syntax to express definitions of the different sorts…
We propose a novel paradigm for solving Inductive Logic Programming (ILP) problems via deep recurrent neural networks. This proposed ILP solver is designed based on differentiable implementation of the deduction via forward chaining. In contrast to the majority of past methods, instead of searching through the space of possible first-order logic rules by using some restrictive rule templates, we directly learn the symbolic logical predicate rules by introducing a novel differentiable Neural Logic (dNL) network. The proposed dNL network is able to learn and represent Boolean functions efficient…
Distributed Autoepistemic Logic with Inductive Definitions (dAEL(ID)) is a recently proposed non-monotonic logic for says-based access control. We define a query-driven decision procedure for dAEL(ID) that is implemented in the knowledge-base system IDP. The decision procedure is designed in such a way that it allows one to determine access rights while avoiding redundant information flow between principals in order to enhance security and reduce privacy concerns. Given that the decision procedure has in the worst case an exponential runtime, it is to be regarded as a proof of concept that inc…
Recursively defined linked data structures embedded in a pointer-based heap and their properties are naturally expressed in pure first-order logic with least fixpoint definitions (FO+lfp) with background theories. Such logics, unlike pure first-order logic, do not admit even complete procedures. In this paper, we undertake a novel approach for synthesizing inductive hypotheses to prove validity in this logic. The idea is to utilize several kinds of finite first-order models as counterexamples that capture the non-provability and invalidity of formulas to guide the search for inductive hypothes…
Inductive logic programming is a type of machine learning in which logic programs are learned from examples. This learning typically occurs relative to some background knowledge provided as a logic program. This dissertation introduces bottom preprocessing, a method for generating initial constraints on the programs an ILP system must consider. Bottom preprocessing applies ideas from inverse entailment to modern ILP systems. Inverse entailment is an influential early ILP approach introduced with Progol. This dissertation also presents $\bot$-Popper, an implementation of bottom preprocessing fo…
Transformer architectures have achieved great success in solving natural language tasks, which learn strong language representations from large-scale unlabeled texts. In this paper, we seek to go further beyond and explore a new logical inductive bias for better language representation learning. Logic reasoning is known as a formal methodology to reach answers from given knowledge and facts. Inspired by such a view, we develop a novel neural architecture named FOLNet (First-Order Logic Network), to encode this new inductive bias. We construct a set of neural logic operators as learnable Horn c…
In inductive learning of a broad concept, an algorithm should be able to distinguish concept examples from exceptions and noisy data. An approach through recursively finding patterns in exceptions turns out to correspond to the problem of learning default theories. Default logic is what humans employ in common-sense reasoning. Therefore, learned default theories are better understood by humans. In this paper, we present new algorithms to learn default theories in the form of non-monotonic logic programs. Experiments reported in this paper show that our algorithms are a significant improvement …
The entailment problem $φ\models ψ$ in Separation Logic \cite{IshtiaqOHearn01,Reynolds02}, between separated conjunctions of equational ($x \iseq y$ and $x \not\iseq y$), spatial ($x \mapsto (y_1,\ldots,y_\rank)$) and predicate ($p(x_1,\ldots,x_n)$) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called \emph{establishment} \cite{IosifRogalewiczSimacek13,KatelaanMathejaZuleger19,P…
Reachability Logic is a formalism that can be used, among others, for expressing partial-correctness properties of transition systems. In this paper we present three proof systems for this formalism, all of which are sound and complete and inherit the coinductive nature of the logic. The proof systems differ, however, in several aspects. First, they use induction and coinduction in different proportions. The second aspect regards compositionality, broadly meaning their ability to prove simpler formulas on smaller systems, and to reuse those formulas as lemmas for more complex formulas on large…
A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that of the classical system of Martin-Lof's inductive definitions. This paper studies the conjecture for intuitionistic logic. This paper first points out that the countermodel of FOSSACS 2017 paper by the same authors shows the conjecture for intuitionistic logic is false in general. Then this paper shows the conjecture for intuitionistic logic is true under ari…
Inductive Logic Programming in Databases: from Datalog to DL+log
Università degli Studi di Bari（巴里大学）
在本文中，我们解决了一个问题，已经引起了数据库社区的注意，随着语义Web的出现，即本体论（和它们所传达的语义）如何帮助解决典型的数据库问题的问题，通过更好地了解KR方面相关的数据库。特别是，我们调查这个问题，从ILP的角度考虑两个数据库的问题，（i）的定义的意见和（ii）的定义的约束，为数据库的模式表示也通过本体。两者都可以重新表述为ILP问题，并可以受益于KR框架DL+log的表达和演绎能力。我们通过实例说明了应用场景。关键词：归纳逻辑程序设计，关系数据库，本体，描述逻辑，混合知识表示和推理系统。逻辑程序设计理论与实践（Theory and Practice of Logic Programming，TPLP）
Hexoses are simple sugars that play a key role in many cellular pathways, and in the regulation of development and disease mechanisms. Current protein-sugar computational models are based, at least partially, on prior biochemical findings and knowledge. They incorporate different parts of these findings in predictive black-box models. We investigate the empirical support for biochemical findings by comparing Inductive Logic Programming (ILP) induced rules to actual biochemical results. We mine the Protein Data Bank for a representative data set of hexose binding sites, non-hexose binding sites…
We define a class of Separation Logic formulae, whose entailment problem: given formulae $ϕ, ψ_1, \ldots, ψ_n$, is every model of $ϕ$ a model of some $ψ_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated locatio…
University of Klagenfurt（克拉根福大学）；Graz University of Technology（格拉茨工业大学）
Efficient omission of symmetric solution candidates is essential for combinatorial problem-solving. Most of the existing approaches are instance-specific and focus on the automatic computation of Symmetry Breaking Constraints (SBCs) for each given problem instance. However, the application of such approaches to large-scale instances or advanced problem encodings might be problematic since the computed SBCs are propositional and, therefore, can neither be meaningfully interpreted nor transferred to other instances. As a result, a time-consuming recomputation of SBCs must be done before every in…
From Circuit Evidence to Mechanistic Theory: An Inductive Logic Approach
Department of Computer Science, University of Manchester（曼彻斯特大学计算机科学系）；Idiap Research Institute（Idiap研究机构）；CRUK National Biomarker Centre, University of Manchester（曼彻斯特大学癌症研究联盟国家生物标志物中心）
ANDRE: An Attention-based Neuro-symbolic Differentiable Rule Extractor for Inductive Logic Programming
Dept. of Mechanical and Aerospace Engineering, George Washington University, USA（机械与航空航天工程系，乔治华盛顿大学）；Dept. of Mechanical Engineering Sciences, University of Surrey, UK（机械工程科学系，萨里大学）
Spatial Reasoning（空间推理）；The DesignSpace Group（设计空间集团）；University of Bremen（不来梅大学）；University of Münster（明斯特大学）
We present an inductive spatio-temporal learning framework rooted in inductive logic programming. With an emphasis on visuo-spatial language, logic, and cognition, the framework supports learning with relational spatio-temporal features identifiable in a range of domains involving the processing and interpretation of dynamic visuo-spatial imagery. We present a prototypical system, and an example application in the domain of computing for visual arts and computational cognitive science.
