Equilibrium semantics of languages of imperfect information

In this paper, we introduce a new approach to independent quantifiers, as originally introduced in Informational independence as a semantic phenomenon by Hintikka and Sandu (1989) [9] under the header of independence-friendly (IF) languages. Unlike other approaches, which rely heavily on compositional methods, we shall analyze independent quantifiers via equilibriums in strategic games. In this approach, coined equilibrium semantics, the value of an IF sentence on a particular structure is determined by the expected utility of the existential player in any of the game’s equilibriums. This approach was suggested in Henkin quantifiers and complete problems by Blass and Gurevich (1986) [2] but has not been taken up before. We prove that each rational number can be realized by an IF sentence. We also give a lower and upper bound on the expressive power of IF logic under equilibrium semantics.     

Dependence of variables construed as an atomic formula

We define a logic D capable of expressing dependence of a variable on designated variables only. Thus D has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic D achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our D avoids some difficulties arising in the original independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of D is definable inside D. We give such a definition for D in the spirit of [11] and [2] and [1].     

Note on witnessed Gödel logics with Delta

Witnessed Gödel logics are based on the interpretation of ∀ (∃) by minimum (maximum) instead of supremum (infimum). Witnessed Gödel logics appear for many practical purposes more suited than usual Gödel logics as the occurrence of proper infima/suprema is practically irrelevant. In this note we characterize witnessed Gödel logics with absoluteness operator △ w.r.t. witnessed Gödel logics using a uniform translation.     

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(～) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.     

The expressive power of k-ary exclusion logic

In this paper we study the expressive power of k-ary exclusion logic, EXC[k], that is obtained by extending first order logic with k-ary exclusion atoms. It is known that without arity bounds exclusion logic is equivalent with dependence logic. By observing the translations, we see that the expressive power of EXC[k] lies in between k-ary and ($k+1$)-ary dependence logics. We will show that, at least in the case when $k=1$, both of these inclusions are proper.In a recent work by the author it was shown that k-ary inclusion-exclusion logic is equivalent with k-ary existential second order logic, ESO[k]. We will show that, on the level of sentences, it is possible to simulate inclusion atoms with exclusion atoms, and in this way express ESO[k]-sentences by using only k-ary exclusion atoms. For this translation we also need to introduce a novel method for “unifying” the values of certain variables in a team. As a consequence, EXC[k] captures ESO[k] on the level of sentences, and we obtain a strict arity hierarchy for exclusion logic. It also follows that k-ary inclusion logic is strictly weaker than EXC[k].Finally we use similar techniques to formulate a translation from ESO[k] to k-ary inclusion logic with an alternative strict semantics. Consequently, for any arity fragment of inclusion logic, strict semantics is strictly more expressive than lax semantics.     

Arithmetical complexity of fuzzy predicate logics — A survey II

Results on arithmetical complexity of important sets of formulas of several fuzzy predicate logics (tautologies, satisfiable formulas, …) are surveyed and some new results are proven.     

On the interpolation property of some intuitionistic modal logics

LetL be one of the intuitionistic modal logics considered in [7] (or one of its extensions) and letM _L be the algebraic semantics ofL. In this paper we will extend toL the equivalence, proved in the classical case (see [6]), among he weak Craig interpolation theorem, the Robinson theorem and the amalgamation property of varietyM _L. We will also prove the equivalence between the Craig interpolation theorem and the super-amalgamation property of varietyM _L. Then we obtain the Craig interpolation theorem and Robinson theorem for two intuitionistic modal logics, one ofS ₄-type and the other one ofS ₅-type, showing the super-amalgamation property of the corresponding algebraic semantics.     

Axiomatizing first-order consequences in dependence logic

Dependence logic, introduced in Väänänen (2007) [11], cannot be axiomatized. However, first-order consequences of dependence logic sentences can be axiomatized, and this is what we shall do in this paper. We give an explicit axiomatization and prove the respective Completeness Theorem.     

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants.     

On the rules of intermediate logics

If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser rule are derivable, admissible but nonderivable, or not admissible. Supported by the Austrian Science Fund FWF under projects P16264 and P16539.     

Possible world semantics for first-order logic of proofs

In the tech report Artemov and Yavorskaya (Sidon) (2011) [4] an elegant formulation of the first-order logic of proofs was given, FOLP. This logic plays a fundamental role in providing an arithmetic semantics for first-order intuitionistic logic, as was shown. In particular, the tech report proved an arithmetic completeness theorem, and a realization theorem for FOLP. In this paper we provide a possible-world semantics for FOLP, based on the propositional semantics of Fitting (2005) [5]. We also give an Mkrtychev semantics. Motivation and intuition for FOLP can be found in Artemov and Yavorskaya (Sidon) (2011) [4], and are not fully discussed here.     

An optimal lower bound on the number of variables for graph identification

In this paper we show that (n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k–1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.Research supported by NSF grant CCR-8709818.Research supported by NSF grant CCR-8805978 and Pennsylvania State University Research Initiation grant 428-45.Research supported by NSF grants DCR-8603346 and CCR-8806308.     

Symmetry in information flow

The article investigates information flow properties of symmetric multi-party protocols. It gives a sound and complete axiomatic system for properties of the functional dependence predicate that are common to all protocols with the same group of symmetries.     

Special transformations in algebraically closed valued fields

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan (2006) [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV F_S(0,0), that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (V F,Γ)-generated substructure S in the language L_RV. The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski-Kazhdan theory.     

Weakened Semantics and the Traditional Square of Opposition

In this paper we present a proposal that (i) could validate more relations in the square than those allowed by classical logic (ii) without a modification of canonical notation neither of current symbolization of categorical statements though (iii) with a different but reliable semantics.      

Capturing k-ary existential second order logic with k-ary inclusion–exclusion logic

In this paper we analyze k-ary inclusion–exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k].We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.     

Vectorization hierarchies of some graph quantifiers

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier in , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two -equivalent models with certain additional structure, yields a pair of -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity. Received: 15 October 1996     

Note on Deduction Theorems in contraction‐free logics

This paper provides a finer analysis of the well‐known form of the Local Deduction Theorem in contraction‐free logics (understood as Tarski‐style finitary structural consequence relations). An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.