Specific Axioms Research Articles

Composing group action to permutation representation

A mapping that satisfies two specific axioms provides a common notion of group action. A homomorphism translating from a group to a symmetric group of a certain set can also be used to describe group action. Therefore, any example of the group actions can be stated based on the second equivalent definition, such as the regular action, natural matrix action, coset action, and Z^2 acting on R^2, etc. It is necessary to examine the concepts of the orbit and stabilizer of a group in order to reveal the orbit-stabilizer theorem. After the preparatory work, the orbit-stabilizer theorem can be proved by defining a mapping from the orbit to the stabilizer and then checking that the mapping is well-defined and bijective. To derive Burnsides lemma, it needs to introduce the set of fixed points which is related to the concept of the stabilizer. Through the orbit-stabilizer theorem along with the fact that a set is a disjoint union of orbits, Burnside's lemma can be confirmed. Moreover, it is natural to compose a group action with a linear representation, and then a representation would be obtained, which is permutation representation. Further, one must calculate the character of the permutation representation, the dimension of the fixed subspace, and the dimension of CX^G. Then it can show Burnsides lemma in another way by permutation representation.

Functions of criminal law axioms in law-making activities

The article analyzes the functions of criminal law axioms in law-making activity, on the basis of which such functions as: social, regulatory, evaluative, of stimulating law-making activity, compensatory, epistemological and aesthetic are distinguished functions. Each of them are revealed and justified. In particular, it is concluded that the social function of criminal law axioms underlies the idea of social justice, equality of all before the law, responsibility and humanism; the regulatory function manifests itself both in achieving the necessary law-making result in the form of the adoption of legal acts, and in the establishment of certain prohibitions, duties and regulations; with the help of the evaluation function, a study of a specific legal phenomenon, an event that may be embedded in a certain norm; the function of stimulating law-making activity motivates the legislator to create and change legal norms; the compensation function allows you to fill in the gap by applying an analogy; the epistemological function is manifested in the fact that with its help a certain doubt is allowed, without which the process of cognition is impossible; the aesthetic function is to give the axiom a logically constructed, meaningful and complete form. Examples of the implementation of the functions of specific criminal law axioms are given and the opinions of various authors on this issue are analyzed. At the end of the article, relevant conclusions are drawn based on the results of the study.

Relative Symmetric Reductions under Multi-Choice Non-Transferable-Utility Situations

In many game-theoretical results, the reduction axiom and its converse have been regarded as important requirements under axiomatic processes for solutions. However, it is shown that the replicated core counters a specific (inferior) converse reduction axiom under multi-choice non-transferable-utility situations. Thus, two modified reductions and relative properties of the reduction axiom and its converse are proposed to characterize the replicated core in this article.The main methods and relative results are as follows. First, two different types of reductions are proposed by focusing on both participants and participation levels under relative symmetric reducing behavior. Further, relative reduction axioms and their converse are adopted to characterize the replicated core.

BOUNDED-ANALYTIC SEQUENT CALCULI AND EMBEDDINGS FOR HYPERSEQUENT LOGICS

AbstractA sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory.

Retracted] A Formal Verification Method of Compilation Based on C Safety Subset

With the rapid increase in the number of wireless terminals and the openness of wireless networks, the security of wireless communication is facing serious challenges. The safety and security of computer communication have always been a research hotspot, especially the wireless communication that still has a more complex architecture which leads to more safety problems in the communication system development. In recent years, more and more wireless communication systems are applied in the safety‐critical field which tends to need high safety guarantees. A compiler is an important tool for system development, and its safety and reliability have an important impact on the development of safety‐critical software. As the strictest method, formal verification methods have been widely paid attention to in compiler verification, but the current formal verification methods have some problems, such as high proof complexity, weak verification ability, and low algorithm efficiency. In this paper, a compiler formal verification method based on safety C subsets is proposed. By abstracting the concept of C grammar units from safety C subsets, the formal verification of the compiler is transformed into the verification of limited C grammar units. In this paper, an axiom system of first‐order logic and special axioms are introduced. On this axiom system, the semantic consistency verification of C grammar unit and target code pattern is completed by means of theorem proving, and the formal verification of the compiler is completed.

Dynamical Phenomena and Their Models: Truth and Empirical Correctness

In the epistemological tradition, there are two main interpretations of the semantic relation that an empirical theory may bear to the real world. According to realism, the theory-world relationship should be conceived as truth; according to instrumentalism, instead, it should be limited to empirical adequacy. Then, depending on how empirical theories are conceived, either syntactically as a class of sentences, or semantically as a class of models, the concepts of truth and empirical adequacy assume different and specific forms. In this paper, we review two main conceptions of truth (one sentence-based and one model-based) and two of empirical adequacy (one sentence-based and one model-based), we point out their respective difficulties, and we give a first formulation of a new general view of the theory-world relationship, which we call Methodological Constructive Realism (MCR). We then show how the content of MCR can be further specified and expressed in a definite and precise form. The bulk of the paper shows in detail how it is possible to accomplish this goal for the special case of deterministic dynamical phenomena and their correlated deterministic models. This special version of MCR is formulated as an axiomatic extension of set theory, whose specific axioms constitute a formal ontology that provides an adequate framework for analyzing the two semantic relations of truth and empirical correctness, as well as their connections.

The Dynamics of Epistemic Attitudes in Resource-Bounded Agents

The paper presents a new logic for reasoning about the formation of beliefs through perception or through inference in non-omniscient resource-bounded agents. The logic distinguishes the concept of explicit belief from the concept of background knowledge. This distinction is reflected in its formal semantics and axiomatics: (i) we use a non-standard semantics putting together a neighborhood semantics for explicit beliefs and relational semantics for background knowledge, and (ii) we have specific axioms in the logic highlighting the relationship between the two concepts. Mental operations of perceptive type and inferential type, having effects on epistemic states of agents, are primitives in the object language of the logic. At the semantic level, they are modelled as special kinds of model-update operations, in the style of dynamic epistemic logic. Results about axiomatization, decidability and complexity for the logic are given in the paper.

JaVerT: JavaScript verification toolchain

The dynamic nature of JavaScript and its complex semantics make it a difficult target for logic-based verification. We introduce JaVerT, a semi-automatic JavaScript Verification Toolchain, based on separation logic and aimed at the specialist developer wanting rich, mechanically verified specifications of critical JavaScript code. To specify JavaScript programs, we design abstractions that capture its key heap structures (for example, prototype chains and function closures), allowing the developer to write clear and succinct specifications with minimal knowledge of the JavaScript internals. To verify JavaScript programs, we develop JaVerT, a verification pipeline consisting of: JS-2-JSIL, a well-tested compiler from JavaScript to JSIL, an intermediate goto language capturing the fundamental dynamic features of JavaScript; JSIL Verify, a semi-automatic verification tool based on a sound JSIL separation logic; and verified axiomatic specifications of the JavaScript internal functions. Using JaVerT, we verify functional correctness properties of: data-structure libraries (key-value map, priority queue) written in an object-oriented style; operations on data structures such as binary search trees (BSTs) and lists; examples illustrating function closures; and test cases from the official ECMAScript test suite. The verification times suggest that reasoning about larger, more complex code using JaVerT is feasible.

The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol c, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of c satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.

Synthesizing memory models from framework sketches and Litmus tests

A memory consistency model specifies which writes to shared memory a given read may see. Ambiguities or errors in these specifications can lead to bugs in both compilers and applications. Yet architectures usually define their memory models with prose and litmus tests—small concurrent programs that demonstrate allowed and forbidden outcomes. Recent work has formalized the memory models of common architectures through substantial manual effort, but as new architectures emerge, there is a growing need for tools to aid these efforts. This paper presents MemSynth, a synthesis-aided system for reasoning about axiomatic specifications of memory models. MemSynth takes as input a set of litmus tests and a framework sketch that defines a class of memory models. The sketch comprises a set of axioms with missing expressions (or holes). Given these inputs, MemSynth synthesizes a completion of the axioms—i.e., a memory model—that gives the desired outcome on all tests. The MemSynth engine employs a novel embedding of bounded relational logic in a solver-aided programming language, which enables it to tackle complex synthesis queries intractable to existing relational solvers. This design also enables it to solve new kinds of queries, such as checking if a set of litmus tests unambiguously defines a memory model within a framework sketch. We show that MemSynth can synthesize specifications for x86 in under two seconds, and for PowerPC in 12 seconds from 768 litmus tests. Our ambiguity check identifies missing tests from both the Intel x86 documentation and the validation suite of a previous PowerPC formalization. We also used MemSynth to reproduce, debug, and automatically repair a paper on comparing memory models in just two days.

Regret theory: State dominance and expected utility

Choices made according to regret theory (RT) may violate the expected utility (EU) model. We propose a stochastic dominance (SD) method for comparing RT and EU paradigms holistically, without focusing on a specific axiom or on a specific numerical example. We show that in some important cases, including the two-state case, e.g., war or peace, Republican or Democratic Party winning the Presidential election etc., RT does not violate both EU and Cumulative Prospect Theory (CPT). Obviously, when EU is not violated by RT the economic results derived under EU are intact also under RT.

Pitowsky\u2019s Kolmogorovian Models and Super-determinism

In an attempt to demonstrate that local hidden variables are mathematically possible, Pitowsky constructed “spin- functions” and later “Kolmogorovian models”, which employs a nonstandard notion of probability. We describe Pitowsky’s analysis and argue (with the benefit of hindsight) that his notion of hidden variables is in fact just super-determinism (and accordingly physically not relevant). Pitowsky’s first construction uses the Continuum Hypothesis. Farah and Magidor took this as an indication that at some stage physics might give arguments for or against adopting specific new axioms of set theory. We would rather argue that it supports the opposing view, i.e., the widespread intuition “if you need a non-measurable function, it is physically irrelevant”.

A comparison between monoidal and substructural logics

Monoidal logics were introduced as a foundational framework to analyse the proof theory of deontic logic. Building on Lambek’s work in categorical logic, logical systems are defined as deductive systems, that is, as collections of equivalence classes of proofs satisfying specific rules and axiom schemata. This approach enables the classification of deductive systems with respect to their categorical structure. When looking at their proof theory, however, one can see that there are similarities between monoidal and substructural logics. The purpose of the present paper is to address this issue and highlight the differences between these two approaches. We argue that monoidal logics provide a more flexible foundational framework that enables a finer analysis of the relationship between negation(s) and other logical connectives. We show that the elimination of double negation(s) is independent from the de Morgan dualities, that monoidal deductive systems are not necessarily weakly distributive and that deductive systems satisfying the elimination of double negation(s) and the law of excluded middle are not necessarily classical.

AGGREGATION METHODS FOR FUZZY JUDGMENTS

Keywords: judgment aggregation, fuzzy, collective decisionJEL Code: C02, C60, D711.INTRODUCTION AND LITERATURE REVIEWAggregation theories have been of considerable importance for the social scientists since the 18th century, with significant research work done in many disciplines. The theory shed light on how judgments are collected into a set of consistent group judgment and how democratic majority is unable to give one. This problem arises as one of the most fundamental barrier of collective decision making in various situations, ranging from the most basic collective decision making parliaments to small one case based committees and from economic expert panel to court judges' committee. Judgment aggregation is also explained as a tool that looks at each of the interdisciplinary problems that they have in common.The latest attention to judgment aggregation was triggered by the so called Paradox or Discursive Dilemma which was explained by Kornhauser (1992). This problem was primarily faced when the situation was multifaceted. In literature it was first discussed by Kornhauser and Sager (1986), where they explicated a three member court. This court had to arrive at a judgment in a breach of contract case; comprising of three propositions which had to be true in order for the defendant to be held liable. Further explaining this case, a propositions means a statement on which decision is to be made. Here propositions are denoted by a, and 'c'. Where 'a' states that the defendant was not liable to do an act, and 'b' states that the defendant did that act. The conclusion premise, c, states that whether a person is liable to pay compensation or not. Judges are then required to state whether those premises are true or false called judgment, together the decision on all the premises written individual or collective is called as judgment set. Important to know is that 'c' is true if both a and b are true, which means that if a person was liable not to do an act and still did it hence he is liable to pay compensation. Together (a λ b) hold a person guilty of breach of contract. The following example portrays the Doctrinal paradox through Table 1.Analyzing Table 1, it can be assessed that first judge holds only first premise to be true. Second judge find both the premises to be true, however the third apprehends first to be false but second to be true. Judgment set of majority voting holds two premises to be true but the collection of third turns out to be false, resulting in logically inconsistent judgment set {TTF}, inconsistency here signifies the lack of rationality. Likewise Table 1 majority outcome means that the person was liable not to do an act and did that act but is not guilty of the breach of contract. Pettit (2003) explains this paradox as 'discursive dilemma'. But the question was how general this doctrinal paradox, is it limited to some specific case of majority outcomes or is it not possible for any majority to reach a consistent outcome. In response to this List and Pettit (2002) presented a model of judgment aggregation, which was inspired by Arrowian Social Choice Theory, explained in Arrow (1963) and illustrated a generic form proving the impossibility theorem and inability of the majority outcome to reach a consistent judgment set. After List and Pettit (2004) discussed model of judgment aggregation, much debate started assessing if judgment aggregation is familiar with the preference aggregation. Preference aggregation is endorsed with problem of cyclical preferences, judgment aggregation similarly bear the consequences of inconsistent majority output. The concept of discursive dilemma resembles Condorcet paradox. Dietrich and List (2007) explained the analogue of Arrow's theorem in theory of judgment aggregation.Dietrich (2016) later spoke about agenda sensitivity issues and proved new impossibility theorem based on the previous one.Arrow (1963) started working in the field of social welfare functions and analyzed how many of them satisfy specific axioms. …

THREE FORMS OF PHYSICAL MEASUREMENT AND THEIR COMPUTABILITY

AbstractWe have begun a theory of measurement in which an experimenter and his or her experimental procedure are modeled by algorithms that interact with physical equipment through a simple abstract interface. The theory is based upon using models of physical equipment as oracles to Turing machines. This allows us to investigate the computability and computational complexity of measurement processes. We examine eight different experiments that make measurements and, by introducing the idea of an observable indicator, we identify three distinct forms of measurement process and three types of measurement algorithm. We give axiomatic specifications of three forms of interfaces that enable the three types of experiment to be used as oracles to Turing machines, and lemmas that help certify an experiment satisfies the axiomatic specifications. For experiments that satisfy our axiomatic specifications, we give lower bounds on the computational power of Turing machines in polynomial time using nonuniform complexity classes. These lower bounds break the barrier defined by the Church-Turing Thesis.

Scoring rules and social choice properties: some characterizations

In many voting systems, voters’ preferences on a set of candidates are represented by linear orderings. In this context, scoring rules are well-known procedures to aggregate the preferences of the voters. Under these rules, each candidate obtains a fixed number of points, \(s_k\), each time he/she is ranked \(k\)th by one voter and the candidates are ordered according to the total number of points they receive. In order to identify the best scoring rule to use in each situation, we need to know which properties are met by each of these procedures. Although some properties have been analyzed extensively, there are other properties that have not been studied for all scoring rules. In this paper, we consider two desirable social choice properties, the Pareto-optimality and the immunity to the absolute loser paradox, and establish characterizations of the scoring rules that satisfy each of these specific axioms. Moreover, we also provide a proof of a result given by Saari and Barney (The Mathematical Intelligencer 25:17–31, 2003), where the scoring rules meeting reversal symmetry are characterized. From the results of characterization, we establish some relationships among these properties. Finally, we give a characterization of the scoring rules satisfying the three properties.

On axiomatization of Łukasiewicz's four-valued modal logic

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £.

The Shapley value for shortest path games: a non-graph-based approach

The shortest path games are considered in this paper. The transportation of a good in a network has costs and benefits. The problem is to divide the profit of the transportation among the players. Fragnelli et al. (Math Methods Oper Res 52: 251–264, 2000) introduce the class of shortest path games and show it coincides with the class of monotone games. They also give a characterization of the Shapley value on this class of games. In this paper we consider further five characterizations of the Shapley value (Hart and Mas-Colell’s in Econometrica 57:589–614, 1989; Shapley’s in Contributions to the theory of games II, annals of mathematics studies, vol 28. Princeton University Press, Princeton, pp 307–317, 1953; Young’s in Int J Game Theory 14:65–72, 1985, Chun’s in Games Econ Behav 45:119–130, 1989; van den Brink’s in Int J Game Theory 30:309–319, 2001 axiomatizations), and conclude that all the mentioned axiomatizations are valid for the shortest path games. Fragnelli et al. (Math Methods Oper Res 52:251–264, 2000)’s axioms are based on the graph behind the problem, in this paper we do not consider graph specific axioms, we take \(TU\) axioms only, that is we consider all shortest path problems and we take the viewpoint of an abstract decision maker who focuses rather on the abstract problem than on the concrete situations.

Constructive Boolean circuits and the exactness of timed ternary simulation

We classify gate level circuits with cycles based on their stabilization behavior. We define a formal class of combinational circuits, the constructive circuits, for which signals settle to a unique value in bounded time, for any input, under a simple conservative delay model, called the up-bounded non-inertial (UN) delay. Since circuits with combinational cycles can exhibit asynchronous behavior, such as non-determinism or metastability, it is crucial to ground their analysis in a formal delay model, which previous work in this area did not do. We prove that ternary simulation, such as the practical algorithm proposed by Malik, decides the class of constructive circuits. We prove that three-valued algebra is able to maintain correct and exact stabilization information under the UN-delay model, and thus provides an adequate electrical interpretation of Malik's algorithm, which has been missing in the literature. Previous work on combinational circuits used the upbounded inertial (UI) delay to justify ternary simulation. We show that the match is not exact and that stabilization under the UI-model, in general, cannot be decided by ternary simulation. We argue for the superiority of the UN-model for reasons of complexity, compositionality and electrical adequacy. The UN-model, in contrast to the UI-model, is consistent with the hypothesis that physical mechanisms cannot implement non-deterministic choice in bounded time. As the corner-stone of our main results we introduce UN-Logic, an axiomatic specification language for UN-delay circuits that mediates between the real-time behavior and its abstract simulation in the ternary domain. We present a symbolic simulation calculus for circuit theories expressed in UN-logic and prove it sound and complete for the UN-model. This provides, for the first time, a correctness and exactness result for the timing analysis of cyclic circuits. Our algorithm is a timed extension of Malik's pure ternary algorithm and closely related to the timed algorithm proposed by Riedel and Bruck, which however was not formally linked with real-time execution models.

Contraction-free calculi for modal logics S5 and KD45

It is known that termination and backtracking are among the most important problems in constructing derivations in non-classical logics. In this paper contractionfree and backtracking-free sequent calculi for modal logics S5 and KD45 are presented and founded. These are index-style sequent calculi with some specific indexed axioms. The main new ideas in considered calculi are to use metavariables (along with natural numbers) as elements of indexes and to numerate by different natural number all the positive occurrences of modality 2.