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Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the global trace condition (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston’s approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce activation algebras which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete.

The influence of Yelp reviews on independent music venues is acknowledged, though the exact nature of this impact remains unclear. As negative reviews increase, they can shape public perception, leading even those with neutral or positive experiences to avoid the venue. This behavior, known as the bandwagon effect or herd behavior, reinforces the venue’s negative reputation and further influences potential patrons. The study examines 1,423 independent venues and their Yelp reviews, dating back to the venues’ opening or as far as 2004, the year Yelp was founded. Subsequently, the study leverages a meticulous extraction methodology involving 30,486 keywords derived from an extensive dataset of 2,305,734 mostly negative or neutral Yelp reviews that were rated 1, 2, or 3 out of 5 stars. These keywords have been systematically categorized into seven codified groups: (1) Noise/Music, (2) Food/Drink, (3) Service, (4) Place/Atmosphere, (5) Transactions, (6) Positive words, and (7) Negative words. Moreover, a devised sentence extraction mechanism has been coded and implemented to capture contextually relevant information for each distinct word category. Subsequently, a sentiment analysis was performed on the extracted sentences to evaluate the overall impressions associated with music/noise-related topics. The significance of this study is underscored by its potential to augment our comprehension of the impact of Yelp reviews on independent music venues, offering actionable insights for venue management, and contributing substantively to the broader discourse surrounding the intersection of online reviews, big data, live music, noise studies, and hospitality.

The topics of structural proof theory and logic programming have influenced each other for more than three decades. Proof theory has contributed the notion of sequent calculus, linear logic, and higher-order quantification. Logic programming has introduced new normal forms of proofs and forced the examination of logic-based approaches to the treatment of bindings. As a result, proof theory has responded by developing an approach to proof search based on focused proof systems in which introduction rules are organized into two alternating phases of rule application. Since the logic programming community can generate many examples and many design goals (e.g., modularity of specifications and higher-order programming), the close connections with proof theory have helped to keep proof theory relevant to the general topic of computational logic.

The aim of this article is to offer a method for determining the ontological commitments of a formalized theory. The second section shows that determining the consequence relation of a language model-theoretically entails that the ontology of a theory is tied very closely to the variables that feature in that theory. The third section develops an alternative way of determining the ontological commitments of a theory given a proof-theoretic account of the consequence relation for the language that theory is in. It is shown that the proof-theoretic account of ontological commitment does not entail that the ontological commitments of a theory depend on the variables of that theory. The last section of the article discusses how this account of ontological commitment can be used in other philosophical projects such as Wright’s (Frege’s conception of numbers as objects, Aberdeen University Press, Aberdeen, 1983 ) abstractionism. The article concludes with a discussion of the upshots of adopting the proof-theoretic account of ontological commitment for ontology generally.

An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No prerequisites are needed beyond high-school algebra. New material is presented in small chunks that are easy for beginners to digest. The author offers a friendly style without sacrificing mathematical rigor. Ideas are developed through motivating examples, precise definitions, carefully stated theorems, clear proofs, and a continual review of preceding topics. Features Study aids including section summaries and over 1100 exercises Careful coverage of individual proof-writing skills Proof annotations and structural outlines clarify tricky steps in proofs Thorough treatment of multiple quantifiers and their role in proofs Unified explanation of recursive definitions and induction proofs, with applications to greatest common divisors and prime factorizations About the Author: Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra. Logic Propositions; Logical Connectives; Truth Tables Logical Equivalence; IF-Statements IF, IFF, Tautologies, and Contradictions Tautologies; Quantifiers; Universes Properties of Quantifiers: Useful Denials Denial Practice; Uniqueness Proofs Definitions, Axioms, Theorems, and Proofs Proving Existence Statements and IF Statements Contrapositive Proofs; IFF Proofs Proofs by Contradiction; OR Proofs Proof by Cases; Disproofs Proving Universal Statements; Multiple Quantifiers More Quantifier Properties and Proofs (Optional) Sets Set Operations; Subset Proofs More Subset Proofs; Set Equality Proofs More Set Quality Proofs; Circle Proofs; Chain Proofs Small Sets; Power Sets; Contrasting ∈ and ⊆ Ordered Pairs; Product Sets General Unions and Intersections Axiomatic Set Theory (Optional) Integers Recursive Definitions; Proofs by Induction Induction Starting Anywhere: Backwards Induction Strong Induction Prime Numbers; Division with Remainder Greatest Common Divisors; Euclid’s GCD Algorithm More on GCDs; Uniqueness of Prime Factorizations Consequences of Prime Factorization (Optional) Relations and Functions Relations; Images of Sets under Relations Inverses, Identity, and Composition of Relations Properties of Relations Definition of Functions Examples of Functions; Proving Equality of Functions Composition, Restriction, and Gluing Direct Images and Preimages Injective, Surjective, and Bijective Functions Inverse Functions Equivalence Relations and Partial Orders Reflexive, Symmetric, and Transitive Relations Equivalence Relations Equivalence Classes Set Partitions Partially Ordered Sets Equivalence Relations and Algebraic Structures (Optional) Cardinality Finite Sets Countably Infinite Sets Countable Sets Uncountable Sets Real Numbers (Optional) Axioms for R; Properties of Addition Algebraic Properties of Real Numbers Natural Numbers, Integers, and Rational Numbers Ordering, Absolute Value, and Distance Greatest Elements, Least Upper Bounds, and Completeness Suggestions for Further Reading Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra.