Inductive First-Order Formula Synthesis by ASP: A Case Study in Invariant Inference

In this paper, we present a system that translates first order logic (FOL) formulas into natural language (NL) sentences. The motivation comes from an intelligent tutoring system teaching logic as a knowledge representation language, where it is used as a means for feedback to the users. FOL to NL conversion is achieved by using a rule-based approach, where we exploit the pattern matching capabilities of rules. So, the system consists of a rule-based component and a lexicon. The rule-based unit implements the conversion process, which is based on a linguistic analysis of a FOL sentence, and the lexicon provides lexical and grammatical information that helps in producing the NL sentences. The whole system is implemented in Jess, a java-based expert system shell. The conversion process currently covers a restricted set of FOL formulas.

Link mining problems are characterized by high complexity (since linked objects are not statistically independent) and uncertainty (since data is noisy and incomplete). Thus they necessitate a modeling language that is both probabilistic and relational. Markov logic provides this by attaching weights to formulas in first-order logic and viewing them as templates for features of Markov networks. Many link mining problems can be elegantly formulated and efficiently solved using Markov logic. Inference algorithms for Markov logic draw on ideas from satisfiability testing, Markov chain Monte Carlo, belief propagation, and resolution. Learning algorithms are based on convex optimization, pseudo-likelihood, and inductive logic programming. Markov logic has been used successfully in a wide variety of link mining applications and is the basis of the open-source Alchemy system.

Robots executing tasks following human instructions in domestic or industrial environments essentially require both adaptability and reliability. Behavior Tree (BT) emerges as an appropriate control architecture for these scenarios due to its modularity and reactivity. Existing BT generation methods, however, either do not involve interpreting natural language or cannot theoretically guarantee the BTs' success. This paper proposes a two-stage framework for BT generation, which first employs large language models (LLMs) to interpret goals from high-level instructions, then constructs an efficient goal-specific BT through the Optimal Behavior Tree Expansion Algorithm (OBTEA). We represent goals as well-formed formulas in first-order logic, effectively bridging intent understanding and optimal behavior planning. Experiments in the service robot validate the proficiency of LLMs in producing grammatically correct and accurately interpreted goals, demonstrate OBTEA's superiority over the baseline BT Expansion algorithm in various metrics, and finally confirm the practical deployability of our framework. The project website is https://dids-ei.github.io/Project/LLM-OBTEA.

While a number of weak consistency mechanisms have been developed in recent years to improve performance and ensure availability in distributed, replicated systems, ensuring the correctness of transactional applications running on top of such systems remains a difficult and important problem. Serializability is a well-understood correctness criterion for transactional programs; understanding whether applications are serializable when executed in a weakly-consistent environment, however remains a challenging exercise. In this work, we combine a dependency graph-based characterization of serializability and leverage the framework of abstract executions to develop a fully-automated approach for statically finding bounded serializability violations under any weak consistency model. We reduce the problem of serializability to satisfiability of a formula in First-Order Logic (FOL), which allows us to harness the power of existing SMT solvers. We provide rules to automatically construct the FOL encoding from programs written in SQL (allowing loops and conditionals) and express consistency specifications as FOL formula. In addition to detecting bounded serializability violations, we also provide two orthogonal schemes to reason about unbounded executions by providing sufficient conditions (again, in the form of FOL formulae) whose satisfiability implies the absence of anomalies in any arbitrary execution. We have applied the proposed technique on TPC-C, a real-world database program with complex application logic, and were able to discover anomalies under Parallel Snapshot Isolation (PSI), and verify serializability for unbounded executions under Snapshot Isolation (SI), two consistency mechanisms substantially weaker than serializability.

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

Knowledge and reasoning in design systems

Induction is a successful approach for verification of hardware and software systems. A common practice is to model a system using logical formulas, and then use a decision procedure to verify that some logical formula is an inductive safety invariant for the system. A key ingredient in this approach is coming up with the inductive invariant, which is known as invariant inference. This is a major difficulty, and it is often left for humans or addressed by sound but incomplete abstract interpretation. This paper is motivated by the problem of inductive invariants in shape analysis and in distributed protocols. This paper approaches the general problem of inferring first-order inductive invariants by restricting the language L of candidate invariants. Notice that the problem of invariant inference in a restricted language L differs from the safety problem, since a system may be safe and still not have any inductive invariant in L that proves safety. Clearly, if L is finite (and if testing an inductive invariant is decidable), then inferring invariants in L is decidable. This paper presents some interesting cases when inferring inductive invariants in L is decidable even when L is an infinite language of universal formulas. Decidability is obtained by restricting L and defining a suitable well-quasi-order on the state space. We also present some undecidability results that show that our restrictions are necessary. We further present a framework for systematically constructing infinite languages while keeping the invariant inference problem decidable. We illustrate our approach by showing the decidability of inferring invariants for programs manipulating linked-lists, and for distributed protocols.

In latest paper “Commonsense visual sensemaking for autonomous driving: on generalized neuro-symbolic online abduction integrating vision and semantics”, which appeared in journal “Artificial intelligence’’(Vol.299,2021), the authors claimed that It incorporates the latest technologies in visual computing and has been developed as a modular framework that can be generally used in hybrid architectures for real-time perception and control. It has been developed as a modular framework that can be generally used in hybrid architectures for real-time perception and control. This advancement represents a useful side of symbolism.Research on the visual computation has received a great deal of attention The demands on human-based cognitive computing are very high. for example, the implementation of autonomous driving functions depends on the effective cooperation of three components: perception, decision making and execution. The perception layer identifies people, objects, and signs on the road by simulating the human eye through multi-dimensional sensors; the decision layer evaluates and makes decisions through pre-processing such as algorithm fusion and feature extraction; the data is fused and output to each control unit in the execution layer; and finally, the hardware mechanism in the execution layer makes feedback actions to realize the full set of autonomous driving operations.In this work, we try to show that this kind of symbolism framework will encounter the so-called “the framing problem” as earlier attempts to formalize the changes of knowledge in event flows. The framing problem is the problem of finding a sufficient set of axioms for a feasible description for environments of the robot or automatic driving. The framing problem is naturally not a problem for connectionism, because the mathematical basis behind it is not formal logic. But among old-school AI, formal logic is still dominant, and implicit in the framing problem is also the flaw in old-school AI's use of symbolic computation as a model of the mind and perception. While end-to-end vision and control based on deep learning has (arguably) been successful for self-driving cars, integrating hybrid vision and semantic solutions at each step, there are clear needs in fulfilling the fundamental legal and ethical responsibilities involving interpretable, human-centered AI and industrialization. However, other paradigms may be worth adopting, for example, the abstract neuro-symbolic approach to online visual perception using answer set programming (ASP) is systematically formalized and fully implemented.In fact, the framing problem describes the problem of expressing facts about the world using first-order logic (FOL). Representing the state of a robot or an automatic driving in traditional FOL requires the use of several axioms that simply imply that things in the environment do not change arbitrarily. For example, we describe a 'block world' in terms of rules about stacking blocks on top of each other. In the FOL system, other axioms need to be used to infer the environment (for example, blocks cannot change position unless they are moved).

Invariant inference algorithms such as interpolation-based inference and IC3/PDR show that it is feasible, in practice, to find inductive invariants for many interesting systems, but non-trivial upper bounds on the computational complexity of such algorithms are scarce, and limited to simple syntactic forms of invariants. In this paper we achieve invariant inference algorithms, in the domain of propositional transition systems, with provable upper bounds on the number of SAT calls. We do this by building on the monotone theory, developed by Bshouty for exact learning Boolean formulas. We prove results for two invariant inference frameworks: (i) model-based interpolation, where we show an algorithm that, under certain conditions about reachability, efficiently infers invariants when they have both short CNF and DNF representations (transcending previous results about monotone invariants); and (ii) abstract interpretation in a domain based on the monotone theory that was previously studied in relation to property-directed reachability, where we propose an efficient implementation of the best abstract transformer, leading to overall complexity bounds on the number of SAT calls. These results build on a novel procedure for computing least monotone overapproximations.

In the Nuclear Power Plant (NPP) control room, the operators’ performance in emergencies is impacted by the need to monitor many indicators on the control room boards, the limited time to interact with dynamic events, and the incompleteness of the operator’s knowledge. Recent research has been directed toward increasing the level of automation in the NPP system by employing modern AI techniques that support the operator’s decisions. In previous work, the authors have employed a novel AI-guided declarative approach (namely, Answer Set Programming (ASP)) to represent and reason with human qualitative knowledge. This represented knowledge is structured to form a reasoning-based operator support system that assists the operator and compensates for any knowledge incompleteness by performing reasoning to diagnose failures and recommend executing actions in real time. A general ASP code structure has been proposed and tested against simple scenarios, e.g., diagnosis of pump failures that result in loss of flow transients and generating the needed plans for resolving the issue of stuck valves in the secondary loop. In this work, we investigate the potential of the previously proposed ASP structure by applying ASP to a realistic case study of the Three Mile Island, Unit 2 (TMI-2) accident event sequence (in particular, the first 142 minutes). The TMI scenario presents many challenges for a reasoning system, including a large number of variables, the complexity of the scenario, and the misleading readings. The capability of the ASP-based reasoning system is tested for diagnosis and recommending actions throughout the scenario. This paper is the first work to test and demonstrate the capability of an automated reasoning system by applying it to a realistic nuclear accident scenario, such as the TMI-2 accident.

In the last decade, advances in satisfiability-modulo-theories (SMT) solvers have powered a new generation of software tools for verification and testing. These tools transform various program analysis problems into the problem of satisfiability of formulas in propositional or first-order logic, where they are discharged by SMT solvers, such as Z3 from Microsoft Research. This paper briefly summarizes four initiatives from Microsoft Research that build upon Z3 and move beyond first-order satisfaction: Fixed points—μZ is a scalable, efficient engine for discharging fixed point queries over recursive predicates with logical constraints, integrated in Z3; Interpolants—Interpolating Z3 uses Z3’s proof generation capability to generate Craig interpolants in the first-order theory of uninterpreted functions, arrays and linear arithmetic; Automata—The symbolic automata toolkit lifts classical automata analyses to work modulo symbolic constraints on alphabets; Polynomials—a new decision procedure for the existential theory of the reals allows efficient solving of systems of non-linear arithmetic constraints.KeywordsPoint QueryAbstract DomainSymbolic Model CheckLinear ArithmeticUninterpreted FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT (“satisfiability modulo theories”) combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains.

We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn formula equations which is based on expressing the fixed-point computation of a minimal model of a set of Horn clauses on the object level as a formula in first-order logic with a least fixed point operator. We describe several corollaries of this fixed-point theorem, in particular concerning the logical foundations of program verification, and sketch how to generalise it to incorporate abstract interpretations.

This paper proposes a method to automatically generate composite services. The function of a service is specified by its Inputs, Output, Preconditions, and Result (IOPR). These functional descriptions are translated into a first-order logic (FOL) formula. An Automatic Theorem Prover (ATP) is used to generate a proof from known services (as axioms) to the composite service (as an object formula). Based on the deductive program synthesis theory, the implementation of the composite service is extracted from the proof. The “proof to program” method used here guarantees the completeness and correctness of the result. An brief introduction of the prototype system is given.KeywordsSemantic Web ServicesAutomatic Service CompositionAutomatic Theorem ProofDeductive Program Synthesis

Logic as a knowledge representation and reasoning language is a fundamental topic of an Artificial Intelligence (AI) course and includes a number of sub-topics. One of them, which brings difficulties to students to deal with, is converting natural language (NL) sentences into first-order logic (FOL) formulas. To assist students to overcome those difficulties, we developed the NLtoFOL system and equipped it with a strong assistance and feedback mechanism. In this work, first, we present that feedback mechanism. The mechanism can provide assistance before an answer is submitted, if requested, but mainly it provides assistance after an answer is submitted. To that end, it characterizes the answer in terms of completeness and accuracy to determine the level of incorrectness, based on an answer categorization scheme, introduced in this paper. The automatically generated natural language feedback sequences grow from general to specific and can include statements on a student’s metacognitive state. Feedback is provided as natural language sentences automatically generated through a template-based natural language generation mechanism. Second, we present an extensive evaluation of the effectiveness of the assistance and feedback mechanism on students’ learning. The evaluation of feedback with students showed that full feedback sequences lead to greater learning gains than sequences consisting of only flag feedback and bottom-out hints (n = 226), and that generic, template-based feedback sequences are comparable to the utility of problem-specific hints generated by human tutors (n = 120).