Concerning Measures in a First-order Logic with Actions and Meta-beliefs

In this paper we investigate agents that have incomplete information and make decisions based on their beliefs, expressed as situation calculus bounded action theories. Such theories have an infinite object domain, but the number of objects that belong to fluents at each time point is bounded by a given constant. Recently it has been shown that verifying temporal properties over such theories is decidable. Here, we first show that we can actually check whether an arbitrary action theory maintains boundedness. Secondly, we examine progression. Progression can be thought of as capturing the notion of belief states resulting from actions in the situation calculus. In the general case, such belief states can be expressed only in second-order logic. Here, we show that for bounded action theories, progression, and hence belief states, can always be represented in first-order logic. Based on this result, we further prove decidability of temporal verification over online executions, i.e., those executions resulting from agents performing only actions that are feasible according to their beliefs.

Dynamic Complexity (as introduced by Patnaik and Immerman [14]) tries to express how hard it is to update the solution to a problem when the input is changed slightly. It considers the changes required to some stored data structure (possibly a massive database) as small quantities of data (or a tuple) are inserted or deleted from the database (or a structure over some vocabulary). The main difference from previous notions of dynamic complexity is that instead of treating the update quantitatively by finding the the time/space trade-offs, it tries to consider the update qualitatively, by finding the complexity class in which the update can be expressed (or made). In this setting, DynFO, or Dynamic First-Order, is one of the smallest and the most natural complexity class (since SQL queries can be expressed in First-Order Logic), and contains those problems whose solutions (or the stored data structure from which the solution can be found) can be updated in First-Order Logic when the data structure undergoes small changes.Etessami [7] considered the problem of isomorphism in the dynamic setting, and showed that Tree Isomorphism can be decided in DynFO. In this work, we show that isomorphism of Planar 3-connected graphs can be decided in DynFO + (which is DynFO with some polynomial precomputation). We maintain a canonical description of 3-connected Planar graphs by maintaining a database which is accessed and modified by First-Order queries when edges are added to or deleted from the graph. We specifically exploit the ideas of Breadth-First Search and Canonical Breadth-First Search to prove the results. We also introduce a novel method for canonizing a 3-connected planar graph in First-Order Logic from Canonical Breadth-First Search Trees.KeywordsShort PathPlanar GraphUndirected GraphDynamic ComplexityComplexity ClassThese 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.

Reports an error in "Bridging the gap between subjective probability and probability judgments: The quantum sequential sampler" by Jiaqi Huang, Jerome R. Busemeyer, Zo Ebelt and Emmanuel M. Pothos (Psychological Review, 2025[Jul], Vol 132[4], 916-955; see record 2025-25219-001). The authors noticed a typo in Equation 17. In the matrix shown, the entry in the fourth row should be "(0 0 β+ … . .)" instead of "(0 β+ … . .)". A zero is missing from the fourth row. The online version of this article has been corrected. (The following abstract of the original article appeared in record 2025-25219-001.) One of the most important challenges in decision theory has been how to reconcile the normative expectations from Bayesian theory with the apparent fallacies that are common in probabilistic reasoning. Recently, Bayesian models have been driven by the insight that apparent fallacies are due to sampling errors or biases in estimating (Bayesian) probabilities. An alternative way to explain apparent fallacies is by invoking different probability rules, specifically the probability rules from quantum theory. Arguably, quantum cognitive models offer a more unified explanation for a large body of findings, problematic from a baseline classical perspective. This work addresses two major corresponding theoretical challenges: first, a framework is needed which incorporates both Bayesian and quantum influences, recognizing the fact that there is evidence for both in human behavior. Second, there is empirical evidence which goes beyond any current Bayesian and quantum model. We develop a model for probabilistic reasoning, seamlessly integrating both Bayesian and quantum models of reasoning and augmented by a sequential sampling process, which maps subjective probabilistic estimates to observable responses. Our model, called the Quantum Sequential Sampler, is compared to the currently leading Bayesian model, the Bayesian Sampler (J. Zhu et al., 2020) using a new experiment, producing one of the largest data sets in probabilistic reasoning to this day. The Quantum Sequential Sampler embodies several new components, which we argue offer a more theoretically accurate approach to probabilistic reasoning. Moreover, our empirical tests revealed a new, surprising systematic overestimation of probabilities. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

A new first-order logic, functional logic, was proposed recently by Staples, Robinson and Hazel. The logic provides a formal means of describing and reasoning about dependence on an implicit parameter, a prime motivation being the unification of the Hoare logic used to reason about procedural programs with the powerful and well established techniques of classical logic. Viewed more abstractly, independently of possible applications, functional logic may be described as a logic with primitives, axioms and inference rules appropriate for reasoning about the properties of mathematical functions. In this paper, the completeness of functional logic is proved; that is, it is shown that any term of a theory in the logic which is true in all models is a theorem.

Chapter 3 - Plan Recognition Using Statistical–Relational Models

Probabilistic stit logic and its decomposition

We consider a modified version of the situation calculus built using a two-variable fragment of the first-order logic extended with counting quantifiers. We mention several additional groups of axioms that can be introduced to capture taxonomic reasoning. We show that the regression operator in this framework can be defined similarly to regression in Reiter’s version of the situation calculus. Using this new regression operator, we show that the projection and executability problems (the important reasoning tasks in the situation calculus) are decidable in the modified version even if an initial knowledge base is incomplete. We also discuss the complexity of solving the projection problem in this modified language in general. Furthermore, we define description logic based sub-languages of our modified situation calculus. They are based on the description logics ALCO(U) (or ALCQO(U), respectively). We show that in these sub-languages solving the projection problem has better computational complexity than in the general modified situation calculus. We mention possible applications to formalization of Semantic Web services and some connections with reasoning about actions based on description logics.

In the domain of ontology design as well as in Conceptual Modeling, representing part-whole relations is a long-standing challenging problem. However, in most papers the focus has been on properties of the part-whole relation, rather than on its semantics. In the last decades, most approaches which have addressed the formal specification of the part-whole relation (i) rely on First Order Logic (FOL) which is unable to address multiple levels of granularity and (ii) do not support any typing mechanism useful for the extensional side of concepts and then, many difficulties remain especially about expressiveness. In mathematical logic and program checking, type theories have proved to be appealing but so far, they have not been applied in the formalization of ontological relations. To bridge this gap, we present an axiomatization of the part-whole relation which hold between typed terms. Relation structures in the dependently-typed framework rely on a constructive logic. We define in a precise way what relation structures and their meta-properties, are in term of type classes using the Coq language.

On Expressive Completeness of Duration and Mean Value Calculi (Extended Abstract)

Authors like Keynes, H. Jeffreys and Carnap advocated using a concept of "logical probability". Logical probability had the following properties: (a) it was representable as a function from potential states of full belief (or "evidence") to states of subjective or credal probability judgment. (b) Such functions were alleged to be constrained by principles of probability logic. (c) All rational agents were supposed to be obliged to adopt the standard function that probability logic prescribed. In this essay, it is argued that these three requirements could be satisfied only if probability logic prescribed that credal probability should be numerically determinate. Keynes denied that it should numerically determinate and Carnap abandoned the idea that probability logic could supply a determinate function from states of full belief to numerically determinate credal states that all rational agents ought to adopt. The paper explains that once this is conceded, logical probability ought to be interpreted rather differently than it is customarily is.

We provide a new link between deductive and probabilistic reasoning fallacies. Illusory inferences from disjunction are a broad class of deductive fallacies traditionally explained by recourse to a matching procedure that looks for content overlap between premises. In two behavioral experiments, we show that this phenomenon is instead sensitive to real-world causal dependencies and not to exact content overlap. A group of participants rated the strength of the causal dependence between pairs of sentences. This measure is a near perfect predictor of fallacious reasoning by an independent group of participants in illusory inference tasks with the same materials. In light of these results, we argue that all extant accounts of these deductive fallacies require non-trivial adjustments. Crucially, these novel indirect illusory inferences from disjunction bear a structural similarity to seemingly unrelated probabilistic reasoning problems, in particular the conjunction fallacy from the heuristics and biases literature. This structural connection was entirely obscure in previous work on these deductive problems, due to the theoretical and empirical focus on content overlap. We argue that this structural parallelism provides arguments against the need for rich descriptions and individuating information in the conjunction fallacy, and we outline a unified theory of deductive illusory inferences from disjunction and the conjunction fallacy, in terms of Bayesian confirmation theory.

Action is one of the most important concepts in computer science, and situation calculus is the standard formalism for representing and reasoning about actions and their effects. Situation calculus essentially could be presented in a logic framework. In this paper, based on Fangzhen and Reiter's first-order logic framework LR (1994), we propose such a logic framework and study it from the point view of model theory. By theorems of mathematical logic, we research minimal action theory and progression.

Bayesian belief networks (also known asbelief networks,causal probabilistic networks,causal nets,graphical probability networks,probabilistic cause–effect models, andprobabilistic influence diagrams) provide decision‐support for a wide range of problems involving uncertainty and probabilistic reasoning. The underlying theory of BBNs is Bayesian probability theory and the notion of propagation. Although this has been around for a long time it is only since the late 1980s that efficient algorithms and tools to implement them have been developed to enable propagation in networks with a reasonable number of variables. The recent explosion of interest in BBNs is due to these developments, which mean that for the first time realistic size problems can be solved. These developments make BBNs an excellent method for reasoning about uncertainty. BBNs have proved useful in applications such as medical diagnosis and diagnosis of mechanical failures. Within the software engineering community the interest in BBNs is motivated by the need to integrate diverse types of evidence to provide rigorous quantitative dependability/safety arguments. BBNs enable us to incorporate not just the results from testing but also other evidence about the process and product; this evidence can be both objective and subjective. The BBN forces the assessor to expose all assumptions about the impact of different forms of evidence and hence provides a visible and auditable dependability or safety argument. This article provides a very brief introduction to BBNs and explains how they can be used for assessing critical systems.

In this paper we will present a graph-transformation based method for the verification of heterogeneous first order logic (FOL) and Euler/Venn proofs. In previous work, it has been shown that a special collection of directed acyclic graphs (DAGs) can be used interchangeably with Euler/Venn diagrams in reasoning processes. Thus, proofs which include Euler/Venn diagrams can be thought of as proofs with DAGs where steps involving only Euler/Venn diagrams can be treated as particular DAG transformations. Here we will show how the characterization of these manipulations can be used to verify Euler/Venn proofs. Also, a method for verifying the use of heterogeneous Euler/Venn and FOL reasoning rules will be presented that is also based upon DAG transformations .

Starting from John MacFarlane's recent survey of answers to the question ‘What is assertion?’, I defend an account of assertion that draws on elements of MacFarlane's and Robert Brandom's commitment accounts, Timothy Williamson's knowledge norm account, and my own previous work on the normative status of logic. I defend the knowledge norm from recent attacks. Indicative conditionals, however, pose a problem when read along the lines of Ernest Adams' account, an account supported by much work in the psychology of reasoning. Furthermore, there seems to be no place for degrees of belief in the accounts of belief and assertion given here. Degrees of belief do have a role in decision-making, but, again, there is much evidence that the orthodox theory of subjective utility maximization is not a good description of what we do in decision-making and, arguably, neither is it a good normative guide to how we ought to make decisions.