Bridging the gap between LLMs and structured program vulnerability analysis: An agent reasoning approach with first-order logic modeling

From First-Order Self-Extensional Paradefinite Four-Valued Logic to First-Order Classical Logic

This paper discusses a multimodal AI system applied to legal reasoning for tax law. The results given here are very general and apply to systems developed for other areas besides tax law. A central goal of this work is to gain a better understanding of the relationships between LLMs (Large Language Models) and automated theorem-proving methodologies. To do this, we suppose (1) two cases for the theorem-proving system: one where it has a countable number of total meanings for its countable number of atoms and the other is where it has an uncountable number of total meanings for its countable number of atoms, and (2) LLMs can have an uncountable number of token meanings. With this in mind, the results given in this paper use the downward and upward Löwenheim–Skolem theorems and logical model theory to contrast these two AI modalities. One modality focuses on syntactic proofs and the other focuses on logical semantics based on LLMs. Particularly, one modality uses a rule-based first-order logic theorem-proving system to perform legal reasoning. The objective of this theorem-proving system is to provide proofs as evidence of valid legal reasoning when enacted laws are applied to particular situations. These proofs are syntactic structures that can be presented in the form of narrative explanations of how the answer to the legal question was determined. The second modality uses LLMs to analyze and transform a user’s tax query so this query can be sent to a first-order logic theorem-proving system to perform its legal reasoning function. The main goal of our application of LLMs is to enhance and simplify user input and output for the theorem-proving system. Using logical model theory, we show how there can exist an equivalence between laws represented in logic of the theorem-proving system, fixed in time when the theorem-proving system was set up, and new semantics given by LLMs. These results are based on logical model theory and Löwenheim–Skolem theorems.

We introduce Qiana, a logic framework for reasoning on formulas that are true only in specific contexts. In Qiana, it is possible to quantify over both formulas and contexts to express, e.g., that “everyone knows everything Alice says”. Qiana also permits paraconsistent logics within contexts, so that contexts can contain contradictions. Furthermore, Qiana is based on first-order logic, and is finitely axiomatizable, so that Qiana theories are compatible with pre-existing first-order logic theorem provers. We show how Qiana can be used to represent temporality, event calculus, and modal logic. We also discuss different design alternatives of Qiana.

A canonical result in model theory is the homomorphism preservation theorem (h.p.t.) which states that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula, standardly proved via a compactness argument. Rossman (2008) established that the h.p.t. remains valid when restricted to finite structures. This is a significant result in the field of finite model theory. It stands in contrast to the other preservation theorems proved via compactness where the failure of the latter also results in the failure of the former [ 2 ], [ 27 ]. Moreover, almost all results from traditional model theory that do survive to the finite are those whose proofs work just as well when considering finite structures. Rossman’s result is interesting as an example of a result which remains true in the finite but whose proof uses entirely different methods. It is also of importance to the field of constraint satisfaction due to the equivalence of existential-positive formulas and unions of conjunctive queries [ 7 ]. Adjacently, Dellunde and Vidal (2019) established a version of the h.p.t. holds for a collection of first-order many-valued logics, namely those whose structures (finite and infinite) are defined over a fixed finite MTL-chain. In this paper we unite these two strands. We show how one can extend Rossman’s proof of a finite h.p.t. to a very wide collection of many-valued predicate logics. In doing so, we establish a finite variant to Dellunde and Vidal’s result, one which not only applies to structures defined over algebras more general than MTL-chains but also where we allow for those algebra to vary between models. We identify the fairly minimal critical features of classical logic that enable Rossman’s proof from a model-theoretic point of view, and demonstrate how any non-classical logic satisfying them will inherit an appropriate finite h.p.t. This investigation provides a starting point in a wider development of finite model theory for many-valued logics and, just as the classical finite h.p.t. has implications for constraint satisfaction, the many-valued finite h.p.t. has implications for valued constraint satisfaction problems.

Modern socio-technical systems demand management initiatives that acknowledge work as dynamic and negotiated, rather than fixedly prescribed. This paper discusses the benefits of pattern finding (inductive detection of weak signals) and pattern priming (deductive use of a Work-as-X archetype catalogue) to achieve a larger understanding of work via dedicated patterns. The notion of work system pattern is first formalised with first-order logic and then positioned within the Structured Exploration of Complex Adaptations (SECA) method, designed as a dual-loop engine for its operationalisation towards weak signals identification. This pattern-based analytical inquiry is meant to accelerate sense-making, support organisational learning, and sharpen resilience potentials, especially in volatile high-risk socio-technical system.

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

Natural language processing (NLP) is one of many AI applications that use logic-based techniques. A promising development in natural language processing is the development of controlled natural languages (CNLs). Using controlled natural languages, humans can communicate with intelligent machines naturally and simply. Currently, logic-based controlled Arabic (LBCA) is behind those for other natural languages in terms of quality. A lot of research is thus needed to improve them. This paper provides all the theoretical justifications needed to develop a logic-based controlled Arabic for knowledge representation, where Arabic texts can be translated to first-order logic (FOL). In this paper, we introduce a formalism and translation of Arabic expressions to first-order logic and discuss various exemplary applications. Such formalism and translation are presented here as a core design that would be useful for any individual or organization implementing a logic-based controlled Arabic. The theoretical justification we have indicated is that a good range of Arabic expressions is successfully translated to first-order logic as a pre-step towards automatic translation and ultimately developing a logic-based controlled Arabic for knowledge representation. In the future, logic-based controlled Arabic, in combination with appropriate controlled forms of rational argumentation, may serve as a communication layer between Arabic language users, especially Arab domain experts, and intelligent machines.

Operational semantics on arbitrary first-order fuzzy logic programming

Due to the importance of linear algebra and matrix operations in data analytics, there is significant interest in using relational query optimization and processing techniques for evaluating (sparse) linear algebra programs. In particular, in recent years close connections have been established between linear algebra programs and relational algebra that allow transferring optimization techniques of the latter to the former. In this paper, we ask ourselves which linear algebra programs in MATLANG correspond to the free-connex and q-hierarchical fragments of conjunctive first-order logic. Both fragments have desirable query processing properties: free-connex conjunctive queries support constant-delay enumeration after a linear-time preprocessing phase, and q-hierarchical conjunctive queries further allow constant-time updates. By characterizing the corresponding fragments of MATLANG, we hence identify the fragments of linear algebra programs that one can evaluate with constant-delay enumeration after linear-time preprocessing and with constant-time updates. To derive our results, we improve and generalize previous correspondences between MATLANG and relational algebra evaluated over semiring-annotated relations. In addition, we identify properties on semirings that allow to generalize the complexity bounds for free-connex and q-hierarchical conjunctive queries from Boolean annotations to general semirings.

This paper presents the development of a software tool that enables the translation of first-order predicate logic with at most three variables into relation algebra. The tool was developed using the Z3 theorem prover, leveraging its capabilities to enhance reliability, generate code, and expedite development. The resulting standalone Python program allows users to translate first-order logic formulas into relation algebra, eliminating the need to work with relation algebra explicitly. This paper outlines the theoretical background of first-order logic, relation algebra, and the translation process. It also describes the implementation details, including validation of the software tool using Z3 for testing correctness. By demonstrating the feasibility of utilizing first-order logic as an alternative language for expressing relation algebra, this tool paves the way for integrating first-order logic into tools traditionally relying on relation algebra as input. 15 pages, 2 figures, 2 tables, to be published in Fundamenta Informaticae

In this note, we give a linear-size translation from formulas of first-order logic into equations of the calculus of relations preserving validity and finite validity. Our translation also gives a linear-size conservative reduction from formulas of first-order logic into formulas of the three-variable fragment of first-order logic.

Text-to-image retrieval in remote sensing (RS) has advanced rapidly with the rise of large vision-language models (LVLMs) tailored for aerial and satellite imagery, culminating in remote sensing large vision-language models (RS-LVLMS). However, limited explainability and poor handling of complex spatial relations remain key challenges for real-world use. To address these issues, we introduce RUNE (Reasoning Using Neurosymbolic Entities), an approach that combines Large Language Models (LLMs) with neurosymbolic AI to retrieve images by reasoning over the compatibility between detected entities and First-Order Logic (FOL) expressions derived from text queries. Unlike RS-LVLMs that rely on implicit joint embeddings, RUNE performs explicit reasoning, enhancing performance and interpretability. For scalability, we propose a logic decomposition strategy that operates on conditioned subsets of detected entities, guaranteeing shorter execution time compared to neural approaches. Rather than using foundation models for end-to-end retrieval, we leverage them only to generate FOL expressions, delegating reasoning to a neurosymbolic inference module. For evaluation, we repurpose the DOTA dataset—originally designed for object detection—by augmenting it with more complex queries than in existing benchmarks. We show the LLM’s effectiveness in text-to-logic translation and compare RUNE with state-of-the-art RS-LVLMs, demonstrating superior performance. We introduce two metrics, Retrieval Robustness to Query Complexity (RRQC) and Retrieval Robustness to Image Uncertainty (RRIU), which evaluate performance relative to query complexity and image uncertainty. RUNE outperforms joint-embedding models in complex RS retrieval tasks, offering gains in performance, robustness, and explainability. We demonstrate RUNE’s potential for real-world RS applications through a use case on post-flood satellite image retrieval.

We consider a class of formula equations in first-order logic, Horn formula equations, which are defined by a syntactic restriction on the occurrences of predicate variables. Horn formula equations play an important role in many applications in computer science. We state and prove a fixed-point theorem for Horn formula equations in first-order logic with a least fixed-point operator. Our fixed-point theorem is abstract in the sense that it applies to an abstract semantics which generalises standard semantics. We describe several corollaries of this fixed-point theorem in various areas of computational logic, ranging from the logical foundations of program verification to inductive theorem proving.

The increasing interest in automated code conversion and transcompilation—driven by the need to support multiple platforms efficiently—has raised new challenges in verifying that translated codes preserve the intended behaviors of the originals. Although it has not yet been widely adopted, transcompilation offers promising applications in software reuse and cross-platform migration. With the growing use of Large Language Models (LLMs) in code translation, where internal reasoning remains inaccessible, verifying the equivalence of their generated outputs has become increasingly essential. However, existing evaluation metrics—such as BLEU and CodeBLEU, which are commonly used as baselines in transcompiler evaluation—primarily measure syntactic similarity, even though this does not guarantee semantic correctness. This syntactic bias often leads to misleading evaluations where structurally different but semantically equivalent code is penalized. This syntactic bias often leads to misleading evaluations, where structurally different but semantically equivalent code is penalized. To address this limitation, we propose a formal verification framework based on equivalence checking using First-Order Logic (FOL). The approach models core programming constructs—such as loops, conditionals, and function calls—that function as logical axioms, enabling equivalence to be assessed at the behavioral level rather than simply by their textual similarity. We initially used the Z3 solver to manually encode Swift and Java code into FOL. To improve scalability and automation, we later integrated ANTLR to parse and translate both the source and transcompiled codes into logical representations. Although the framework is language-agnostic, we demonstrate its effectiveness through a case study of Swift-to-Java transcompilation. The experimental results demonstrated that our method effectively identifies semantic equivalence, even when syntax differs significantly. Our method achieves an average semantic accuracy of 86.1%, compared to BLEU’s syntactic accuracy of 64.45%. This framework bridges the gap between code translation and formal semantic verification. These results highlight the potential for formal equivalence checking to serve as a more reliable validation method in code translation tasks, enabling more trustworthy cross-language code conversion.

Abstract Cyclic-proof systems are sequent-calculus-style proof systems that allow circular structures that represent induction. Cyclic-proof systems are considered suitable for automated inductive reasoning, and the cut-elimination property is desirable because finding cut formulas often requires heuristics. However, the cut-elimination property does not hold in some cyclic-proof systems, such as the sequent calculus for first-order logic and the entailment system for symbolic-heap separation logic. This paper proves that the cyclic-proof system for the logic of bunched implications does not satisfy the cut-elimination property, even if the system is restricted to the positive fragment with inductively defined propositions. To prove this, we use a new proof technique called proof unrolling. To demonstrate that proof unrolling is a general technique, this paper adapts proof unrolling to another cyclic-proof system for multiplicative additive linear logic with fixed-point operators.

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes. In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem’s parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube. Given that several such First Order Logic (FO) definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

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.

Infinite probabilistic databases (PDBs) are a framework of probability distributions over infinitely many different database instances. We systematically study the representability problem for such PDBs by means of tuple-independence and first-order views. Although first-order views over tuple-independent PDBs are not a complete representation system for infinite PDBs, they form a fairly robust class: Adding first-order constraints does not give them additional expressive power, and they cover many relevant special cases such as block-independent disjoint PDBs, and PDBs of bounded instance size. We identify criteria for representability (or non-representability) in this class and explore their limits. In addition, we study the expressive power of fragments of first-order logic over tuple-independent PDBs. In general, for infinite PDBs, the landscape of relative expressive power for various classes of views over PDBs with independence assumptions is much more complex than in the finite setting.

Neurosymbolic AI is an emerging paradigm that combines neural network learning capabilities with the structured reasoning capacity of symbolic systems. Although machine learning has achieved cutting-edge outcomes in diverse fields, including healthcare, agriculture, and environmental science, it has potential limitations. Machine learning and neural models excel at identifying intricate data patterns, yet they often lack transparency, depend on large labelled datasets, and face challenges with logical reasoning and tasks that require explainability. These challenges reduce their reliability in high-stakes applications such as healthcare. To address these limitations, we propose a hybrid framework that integrates symbolic knowledge expressed in First-Order Logic into neural learning via a Logic Tensor Network (LTN). In this framework, expert-defined medical rules are embedded as logical axioms with learnable thresholds. As a result, the model gains predictive power, interpretability, and explainability through reasoning over the logical rules. We have utilized this neurosymbolic method for predicting diabetes by employing the Pima Indians Diabetes Dataset. Our experimental setup evaluates the LTN-based model against several conventional methods, including Support Vector Machines (SVM), Logistic Regression (LR), K-Nearest Neighbors (K-NN), Random Forest Classifiers (RF), Naive Bayes (NB), and a Standalone Neural Network (NN). The findings demonstrate that the neurosymbolic framework not only surpasses traditional models in predictive accuracy but also offers improved explainability and robustness. Notably, the LTN-based neurosymbolic framework achieves an excellent balance between recall and precision, along with a higher AUC-ROC score. These results underscore its potential for trustworthy medical diagnostics. This work highlights how integrating symbolic reasoning with data-driven models can bridge the gap between explainability, interpretability, and performance, offering a promising direction for AI systems in domains where both accuracy and explainability are critical.