Atomic Formulas Research Articles

Atomic saturation of reduced powers

AbstractOur aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any which is SOP3”, (where Δ consists of atomic or conjunction of atomic formulas). However, the theorem on “it is enough to deal with symmetric pre‐cuts” (so the theorem) cannot be generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context.

АЛГОРИТМЫ РЕШЕНИЯ СИСТЕМ УРАВНЕНИЙ НАД РАЗЛИЧНЫМИ КЛАССАМИ КОНЕЧНЫХ ГРАФОВ

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n > 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.

SYMBOLIC LOGIC: RETURN TO THE ORIGINS. PAPER II. BASIC CATEGORIES

Problem setting. The paper is the Part II of the large research, dedicated to both revision of the system of basic logical categories and generalization of the modern predicate logic to functional logic. Basic categories of functional logic are the following: an individual, a function, representation, and a sequence. Paper objective. T he main task of the paper is to describe every one of the categories in question. The more expansive task of all the paper series is to expose the whole system of functional logic and to prove its advantagies. Recent research and publications analysis. Functional logic was discovered by the author in 1997, and there is no investigation in this field up today except of papers of the autor. Paper main body. An individual is any theoretical object regarded as atomic. Functions are regarded as ambiguous in general case, maybe nullary, and are treated as methods to represent some (any) objects. Most important types of functions are individuals’ functions (for example, arithmetic operations), functionals, and operators. Representation is the ultimate generalization of equality; we treat representation as any specification of an object, as a way to make it present. Fundamentals of the theory of representation are following: any individual can be represented either by itself as a function, or by a nullary function, or via other objects by a function with argument places. The corrected form (with respect to the Paper I) of representation formulas, i. e., atomic formulas of functional logic are involved. The form of terms that represents general names in formal languages of functional logic is present. Both these corrections use the machinery of choice functions, i. e., functions that “elect” one value of any ambiguous function in any case of its use. A sequence treated as a basic nondefinable logical notion. The nondefinability of the notion of a sequence will be proved in another paper. When we operate with individuals, functions, and representation, we structure them in sequences such that every latter has the first element and is discrete (and many of them also have the last element). Conclusions of the research. An individual and a function are relative categories because, first, when we start to talk about a function, it became an individual with the name “such and such a function” and, secondly, when an individual is represented by itself, it became a function. Also, categories of an individual, a function, representation, and a sequence are basic but not universal for logic, i. e., not every logical object is either an individual or a function or representation or a sequence. Say, semantical categories of an expression, a sense, and a denotation are independent of four described above. This means that we described above the field of logic, independent from semantics. The author calls this field (and its theory) logistics.

The KK principle and rotational symmetry

The KK principle and rotational symmetry

A Novel Categorical Approach to Semantics of Relational First-Order Logic

We present a categorical formalization of a variant of first-order logic. Unlike other texts on this topic, the goal of this paper is to give a very transparent and self-contained account without requiring more background than basic logic and set theory. Our focus is to show how the semantics of first-order formulas can be derived from their usual deduction rules. For understanding the core ideas, it is not necessary to investigate the internal term structure of atomic formulas, thus we abstract atomic formulas to (syntactically opaque) relations; in this sense, our variant of first-order logic is “relational”. While the derived semantics is based on categorical principles (even the duality that arises from a symmetry between two ways of looking at something where there is no reason to choose one over the other), it is nevertheless “constructive” in that it describes explicit computations of the truth values of formulas. We demonstrate this by modeling the categorical semantics in the RISCAL (RISC Algorithm Language) system which allows us to validate the core propositions by automatically checking them in finite models.

Funnel control for fully actuated systems under a fragment of signal temporal logic specifications

Temporal logics have lately proven to be a valuable tool for various control applications by providing a rich specification language. Existing temporal logic-based control strategies discretize the underlying dynamical system in space and/or time. We will not use such an abstraction and consider continuous-time systems under a fragment of signal temporal logic specifications by using the associated robust semantics. In particular, this paper provides computationally-efficient funnel-based feedback control laws for a class of systems that are, in a sense, feedback equivalent to single integrator systems, but where the dynamics are partially unknown for the control design so that some degree of robustness is obtained. We first leverage the transient properties of a funnel-based feedback control strategy to maximize the robust semantics of some atomic temporal logic formulas. We then guarantee the satisfaction for specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals by a suitable switched control system. The result is a framework that satisfies temporal logic specifications with a user-defined robustness when the specification is satisfiable. When the specification is not satisfiable, a least violating solution can be found. The theoretical findings are demonstrated in simulations of the nonlinear Lotka–Volterra equations for predator–prey models.

A More Unified Approach to Free Logics

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (namely self-identity) are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (i), are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics (free and classical). The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework.

THE EXACT STRENGTH OF THE CLASS FORCING THEOREM

AbstractThe class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$ , that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$ . It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal {L}_{\text {Ord},\omega }(\in ,A)$ , allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}(\in ,A)$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega ,\omega }(\in ,A)$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$ . Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$ .

The Russell-Prawitz embedding and the atomization of universal instantiation

Abstract Given the recent interest in the fragment of system $\mathbf{F}$ where universal instantiation is restricted to atomic formulas, a fragment nowadays named system ${\mathbf{F}}_{\textbf{at}}$, we study directly in system $\mathbf{F}$ new conversions whose purpose is to enforce that restriction. We show some benefits of these new atomization conversions: (i) they help achieving strict simulation of proof reduction by means of the Russell–Prawitz embedding of $\textbf{IPC}$ into system $\mathbf{F}$, (ii) they are not stronger than a certain ‘dinaturality’ conversion known to generate a consistent equality of proofs, (iii) they provide the bridge between the Russell–Prawitz embedding and another translation, due to the authors, of $\textbf{IPC}$ directly into system ${\mathbf{F}}_{\textbf{at}}$ and (iv) they give means for explaining why the Russell–Prawitz translation achieves strict simulation whereas the translation into ${\mathbf{F}}_{\textbf{at}}$ does not.

Decision Analysis Methods Combining Quantitative Logic and Fuzzy Soft Sets

In this paper we propose a new decision analysis method combining quantitative logic and fuzzy soft set theory. Firstly, we transform a fuzzy information system into a fuzzy soft set, and then establish a formal language based on the fuzzy soft set, in which the parameters of fuzzy soft set are regarded as atomic formulas, some atomic formulas are connected by the logical connectives and then a logical formula is formed, and a implicative type of formula is interpreted as a soft decision rule (SDR). Secondly, various types of measures to evaluate the SDR are introduced and then the soft metric between two logical formulas is established. Thirdly, we apply the soft metric to the soft decision analysis, a SDR extraction algorithm for fuzzy decision information system and a corresponding recommendation algorithm are proposed. Finally, some attribute analysis examples, including the example as shown in rough sets and the practical credit card application example, are given to illustrate the newly proposed method and related concepts.

Bioavailability and Bioequivalence of Allopurinol in Two Tablet Formulations

Introduction Description The chemical structure of allopurinol is 1, 5-dihydro-4H-pyrazolo [3, 4-d] pyrimidin-4-one. It is a xanthine oxidase inhibitor that’s administered orally and intravenously. It may be a white undefined mass with an atomic weight of 158.09 and atomic formula C5H3N4NaO.

Uniformly Bounded Arrays and Mutually Algebraic Structures

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

Linking Focusing and Resolution with Selection

Focusing and selection are techniques that shrink the proof-search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atomic formula can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory and the related framework called superdeduction; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof-search space.

The Interaction Potential between an Atom and a Conductive Wall

In this work, the interaction potential between an atom (with definite Static Polarizability) and a perfect neutral conductive wall is evaluated in two approaches. First, using, the explicit form of the singlet and doublet parts of the vacuum vector potential operator in atomic quadratic stark shift formula and taking the expectation value in vacuum state. Second, writing, the quadratic stark shift formula in terms of electric field correlation function, relating the correlation function to the imaginary parts of the vector potential Green function tensor components (via fluctuation dissipation theorem in statistical mechanics) and the evaluation of the required vector potential Green function tensor components. The consistency of the two methods is shown.

Proximity-Based Unification: An Efficient Implementation Method

Unification is a central concept in logic systems based on the resolution principle. As well, in knowledge representation, proximity relations (i.e., reflexive, symmetric, fuzzy binary relations) are useful for introducing semantics into a syntactic level by modeling the semantic closeness of different syntactic objects and managing vague or imprecise information. Proximity relations, in combination with the unification algorithm, make possible expressing certain forms of approximate reasoning in a logic programming framework. In this article, we use proximity relations in the context of a (fuzzy) logic programming system, called Bousi ~ Prolog, as a way of solving the limitations introduced by similarity relations (i.e., transitive proximity relations) to correctly represent fuzzy information. Recently, we introduced an accurate definition of proximity between expressions (terms or atomic formulas) and a new unification algorithm able to manage proximity relations properly. However, the so-called weak unification algorithm, which is an extension of Martelli and Montanari's unification algorithm supported by the new notion of proximity, does not have an efficient implementation. In this article, we present a method that facilitates such an efficient implementation, including an adaptation of the weak SLD resolution rule based on the new unification algorithm, and its integration and implementation into the fuzzy logic programming system Bousi ~ Prolog. A performance analysis to show its efficiency is also presented.

On the initialization of clocks in timed formalisms

Constraint LTL over clocks (CLTLoc) is an extension of LTL allowing for atomic formulae of the form x<c or x=c, which constrain the time delay measured by clock x with respect to constant value c. In a previous work, we showed that CLTLoc is equivalent to Timed Automata. The result was proven by considering a clock semantics that conforms to the original definition by Alur and Dill, i.e., when clocks are reset (i.e., equal to 0) in the origin, both CLTLoc and Timed Automata define the class of Timed ω-Regular languages. In this paper, we show that if we allow the clocks to have any value in the origin, the power of the formalism to express timed languages does not change, as long as non-Zeno languages are considered. If Zeno languages are allowed, then CLTLoc is strictly more powerful than TA. As a consequence of these results, we also show that non-Zeno Timed ω-Regular languages are closed with respect to the left quotient operation with timed regular languages over finite words.

Efficient Decentralized LTL Monitoring Framework Using Tableau Technique

This paper presents a novel framework for decentralized monitoring of Linear Temporal Logic (LTL) formulas, under the situation where processes are synchronous and the formula is represented as a tableau. The tableau technique allows one to construct a semantic tree for the input LTL formula, which can be used to optimize the decentralized monitoring of LTL in various ways. Given a system P and an LTL formula φ, we construct a tableau T φ . The tableau T φ is used for two purposes: (a) to synthesize an efficient round-robin communication policy for processes, and (b) to find the minimal ways to decompose the formula and communicate observations of processes in an efficient way. In our framework, processes can propagate truth values of both atomic and compound formulas (non-atomic formulas) depending on the syntactic structure of the input LTL formula and the observation power of processes. We demonstrate that this approach of decentralized monitoring based on tableau construction is more straightforward, more flexible, and more likely to yield efficient solutions than alternative approaches.

Determinate logic and the Axiom of Choice

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice.We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions.A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals.

Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

Team Semantics generalizes Tarski's Semantics for First Order Logic by allowing formulas to be satisfied or not satisfied by sets of assignments rather than by single assignments. Because of this, in Team Semantics it is possible to extend the language of First Order Logic via new types of atomic formulas that express dependencies between different assignments. Some of these extensions are much more expressive than First Order Logic proper; but the problem of which atoms can instead be added to First Order Logic without increasing its expressive power is still unsolved. In this work, I provide an answer to this question under the additional assumptions (true of most atoms studied so far) that the dependency atoms are relativizable and non-jumping. Furthermore, I show that the global (or Boolean) disjunction connective can be added to any strongly first order family of dependencies without increasing the expressive power, but that the same is not true in general for non strongly first order dependencies.

Another plan for negation

The paper presents a plan for negation, proposing a paradigm shift from the Australian plan for negation, &#x0D; leading to a family of contra-classical logics. The two main ideas are the following: &#x0D; &#x0D; Instead of shifting points of evaluation (in a frame), shift the evaluated formula. &#x0D; Introduce an incompatibility set for every atomic formula, extended to any compound formula, and impose the condition on valuations that a formula evaluates to true iff all the formulas in its incompatibility set evaluate to false. Thus, atomic sentences are not independent in their truth-values. &#x0D; The resulting negation, in addition to excluding the negated formula, provides a positive alternative&#x0D; to the negated formula.&#x0D; &#x0D; I also present a sound and complete natural deduction proof systems for those logics.&#x0D; In addition, the kind of negation considered in this paper is shown to provide an innovative notion of grounding negation. &#x0D;