Finite Set Of Formulas Research Articles

Finite Based Contraction and Expansion via Models

We propose a new paradigm for Belief Change in which the new information is represented as sets of models, while the agent's body of knowledge is represented as a finite set of formulae, that is, a finite base. The focus on finiteness is crucial when we consider limited agents and reasoning algorithms. Moreover, having the input as arbitrary set of models is more general than the usual treatment of formulas as input. In this setting, we define new Belief Change operations akin to traditional expansion and contraction, and we identify the rationality postulates that emerge due to the finite representability requirement. We also analyse different logics concerning compatibility with our framework.

Measuring Inconsistency in Some Logics with Modal Operators

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

用于修正命题理论的可靠和完备的R演算

The AGM postulates are for the belief revision (revision by a single belief), and the DP postulates for the iterated revision (revision by a finite sequence of beliefs). Li gave an R-calculus for R-configurations Δ|Γ, where Δ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. In this paper, two deduction systems for the revision of a theory by another theory are given such that the systems are sound and complete, that is, if Δ|Γ ⇒ Γ′ is provable then Γ′ ⊇ Δ is consistent and Γ′ - Δ is a maximal subset of Γ such that (Γ′ - Δ) ∪ Δ is consistent; and for any finite theories Δ and Γ, there is a finite theory Γ ′ such that Γ′ - Δ is a maximal subset of G such that (Γ′ - Δ) ∪ Δ is consistent, and Δ|Γ ⇒ Γ′ is provable. Moreover, if Δ|Γ ⇒ Γ′ is provable then Γ satisfies the AGM and the DP postulates.

Non-finite axiomatizability of dynamic topological logic

Dynamic topological logic (DTL) is a polymodal logic designed for reasoning about dynamic topological systems . These are pairs 〈 X , f 〉, where X is a topological space and f : X → X is continuous. DTL uses a language L which combines the topological S4 modality □ with temporal operators from linear temporal logic. Recently, we gave a sound and complete axiomatization DTL * for an extension of the logic to the language L * , where ◊ is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known in the language L, although one proof system, which we shall call KM, was conjectured to be complete by Kremer and Mints. In this article, we show that given any language L' such that L ⊆ L' ⊆ L * , the set of valid formulas of L' is not finitely axiomatizable. It follows, in particular, that KM is incomplete.

A sound and complete R-calculi with respect to contraction and minimal change

AGM postulates are for belief revision (revision by a single belief), and DP postulates are for iterated revision (revision by a finite sequence of beliefs). R-calculus is given for R-configurations Δ|Γ, where Δ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. We shall give two R-calculi C and M (sets of deduction rules) such that for any finite consistent sets Γ, Δ of formulas in the propositional logic, there is a consistent set Θ ⊆ Γ of formulas such that Δ|Γ ⟹ Δ, Θ is provable and Θ is a contraction of Γ by Δ or a minimal change of Γ by Δ; and prove that C and M are sound and complete with respect to the contraction and the minimal change, respectively.

One Sound and Complete <i>R</i>-Calculus with Pseudo-Subtheory Minimal Change Property

The AGM axiom system is for the belief revision (revision by a single belief), and the DP axiom system is for the iterated revision (revision by a finite sequence of beliefs). Li [1] gave an R-calculus for R-configurations Δ|Γ, where Δ is a set of atomic formulas or the negations of atomic formulas, and Γ is a finite set of formulas. In propositional logic programs, one R-calculus N will be given in this paper, such that N is sound and complete with respect to operator s(Δ,t), where s(Δ,t)is a pseudo-theory minimal change of t by Δ.

The Sound and Complete <i>R</i>-Calculi with Respect to Pseudo-Revision and Pre-Revision

The AGM postulates ([1]) are for the belief revision (revision by a single belief), and the DP postulates ([2]) are for the iterated revision (revision by a finite sequence of beliefs). Li [3] gave an R-calculus for R-configurations △|Γ, where Δ is a set of literals, and Γ is a finite set of formulas. We shall give two R-calculi such that for any consistent set Γ and finite consistent set △ of formulas in the propositional logic, in one calculus, there is a pseudo-revision Θ of Γ by Δ such that is provable and and in another calculus, there is a pre-revision Ξ of Γ by Δ such that is provable, and for some pseudo-revision Θ; and prove that the deduction systems for both the R-calculi are sound and complete with the pseudo-revision and the pre-revision, respectively.

Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for orderof-magnitude reasoning and its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object.

Formula-inaccessible cardinals and a characterization of all natural models of Zermelo-Fraenkel set theory

E. Zermelo (1930) and J. C. Sheperdson (1952) proved that a cumulative set is a standard model of von Neumann-Bernays-Gödel set theory if and only if for some inaccessible cardinal number . The problem of a canonical form for all natural models of ZF theory turned out to be more complicated. Since the notion of a model of ZF theory cannot be defined by a finite set of formulae, we introduce a new notion of (strongly) formula-inaccessible cardinal number using a schema of formulae and its relativization on the set , and prove a formula-analogue of the Zermelo-Sheperdson theorem giving a canonical form for all natural models of ZF theory.

Definable fixed points in modal and temporal logics — a survey

The paper presents a survey of author's results on definable fixed points in modal, temporal, and intuitionistic propositional logics. The well-known Fixed Point Theorem considers the modalized case, but here we investigate the positive case. We give a classification of fixed point theorems, describe some classes of models with definable least fixed points of positive operators, special positive operators, and give some examples of undefinable least fixed points. Some other interesting phenomena are discovered – definability by formulas that do not preserve positivity of parameters and definability by finite sets of formulas. We also consider negative operators, graded modalities, construct undefinable inflationary fixed points, and put some problems.

A polynomial translation of S4 into intuitionistic logic

It is known that both S4 and the Intuitionistic prepositional calculus Int are P-SPACE complete. This guarantees that there is a polynomial translation from each system into the other.However, no sound and faithful polynomial translation from S4 into Int is commonly known. The problem of finding one was suggested by Dana Scott during a very informal gathering of logicians in February 2005 at UCLA. Grigori Mints then brought it to my attention, and in this paper I present a solution. It is based on Kripke semantics and describes model-checking for S4 using formulas of Int.A simple translation from Int into S4, the Gödel-Tarski translation, is wellknown; given a formula φ of Int, one obtains by prefixing □ to every subformula. For example,.That the translation is sound and faithful can be seen by considering topological semantics, which assign open sets both to □-formulas of S4 and arbitrary formulas of Int; the interpretation of φ and turn out to be identical. See Tarski's paper [6] for details. Gödel's original paper can be found in [3].In [2], Friedman and Flagg present a kind of inverse to Gödel-Tarski. Given a formula φ of S4 and a finite set of formulas Γ of Int, for each ε ∈ Γ one gets an intuitionistic formula given recursively by

Bookmaking over infinite-valued events

We extend De Finetti’s coherence criterion to the infinite-valued propositional logic of Łukasiewicz. Given a finite set of formulas ψ i and corresponding real numbers β i ∈ [0, 1], we prove that the β i ’s arise from a finitely additive measure on formulas if, and only if, there is no possible choice of “stakes” σ i ∈ R such that, for every valuation V the quantity ∑ i = 1 n σ i ( β i - V ( ψ i ) ) is <0. This solves a problem of Jeff Paris, and generalizes previous work on Dutch Books in finite-valued logics, by B. Gerla and others. We also extend our result to infinitely many formulas, and to the case when the formulas ψ i are logically related. In a final section we deal with the problem of deciding if a book is Dutch.

The generation of formulas held in common knowledge

Common knowledge of a finite set of formulas implies a special relationship between syntactic and semantic common knowledge. If S, a set of formulas held in common knowledge, is implied by the common knowledge of some finite subset of S, and A is a non-redundant semantic model where exactly S is held in common knowledge, then the following are equivalent: (a) S is maximal among the sets of formulas that can be held in common knowledge, (b) A is finite, and (c) the set S determines A uniquely; otherwise there are uncountably many such A. Even if the knowledge of the agents are defined by their knowledge of formulas, 1) there is a continuum of distinct semantic models where only the tautologies are held in common knowledge and, 2) not assuming that S is finitely generated (a) does not imply (c), (c) does not imply (a), and (a) and (c) together do not imply (b).

Uniform and non uniform strategies for tableaux calculi for modal logics

ABSTRACT The major emphasis of this paper is on the definition of complete strategies in tableaux calculi for propositional modal logics. The strategies use a non exhaustive backtracking mechanism, a selective periodicity test and a uniform or a non uniform priority on the order of application of the tableaux rules. The propositional modal logics treated herein are those having a tableaux calculus with finite sets of formulas possibly occurring in the tableaux. Experimental results with the ATINF modal prover are presented.

Implicational formulas in intuitionistic logic

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.