Cooperation,Knowledge, and Time: Alternating-time Temporal Epistemic Logic and its Applications

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal Logic (ATL). The key insight in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems.     

A finite analog to the löwenheim-skolem theorem

The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. With each (consistent) derivation there is associated a Buridan-Volpin (orBV) structure [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas has a Tarski modelM, then from any normal derivationD inLK* of can be constructed aBV modelM _D=[M, {r(x)}] of where each ranger(x) is finite.Presented byMelvin Fitting;     

Models for stronger normal intuitionistic modal logics

This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated.This paper presents results of an investigation of intuitionistic modal logic conducted in collaboration with Dr Milan Boi.     

Superintuitionistic Companions of Classical Modal Logics

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic , the set L[] of L-companions of . Here L[] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize if the law of excluded middle p V p is added. Questions addressed are, for instance, whether there exist logics with the disjunction property in L[], whether L[] contains a smallest element, and whether L[] contains lower covers of . Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and ap plies duality theory from [34].     

Four-valued semantics for relevant logics (and some of their rivals)

This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the Australian Plan semantics, which uses a unary operator for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it provides a semantics for the contraction-free relevant logicC (orRW). Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof.     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

Derivation of likelihood-ratio tests for Guttman quasi-simplex covariance structures

In this paper, maximum-likelihood estimates have been obtained for covariance matrices which have the Guttman quasi-simplex structure under each of the following null hypotheses: (a) The covariance matrix , can be written asTT + where and are both diagonal matrices with unknown elements andT is a known lower triangular matrix, and (b) the covariance matrix *, is expressible asT*T + I where is an unknown scalar. The linear models from which these covariance structures arise are also stated along with the underlying assumptions. Two likelihood-ratio tests have been constructed, one each for the above null hypotheses, against the alternative hypothesis that the population covariance matrix is simply positive definite and has no particular pattern. A numerical example is provided to illustrate the test procedure. Possible applications of the proposed test are also suggested.Adapted from portions of the author's dissertation under the same title submitted to the Department of Psychology, University of North Carolina, in partial fulfillment of the requirements for the Ph.D. degree. The author wishes to express his gratitude to his thesis chairman Dr. R. Darrell Bock and to his committee members Professors Samarendra Nath Roy, Lyle V. Jones, Thelma G. Thurstone, and Dorothy Adkins. Indebtedness is also acknowledged to Dr. Somesh Das Gupta who was quite helpful during the initial stage of the study.Formerly at the Department of Psychology, Indiana University. The author is grateful both to Indiana University and University of North Carolina for the support extended to him during his doctoral studies.     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.     

The Classification of Propositional Calculi

We discuss Smirnovs problem of finding a common background for classifying implicational logics. We formulate and solve the problem of extending, in an appropriate way, an implicational fragment H of the intuitionistic propositional logic to an implicational fragment TV of the classical propositional logic. As a result we obtain logical constructions having the form of Boolean lattices whose elements are implicational logics. In this way, whole classes of new logics can be obtained. We also consider the transition from implicational logics to full logics. On the base of the lattices constructed, we formulate the main classification principles for propositional logics.     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

On reduced matrices

It is shown that the class of reduced matrices of a logic is a 1 ^st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 ^st order.     

On two problems of Harvey Friedman

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).The second part includes a proof of the theorem ensuring existence of a greatest structurally complete intermediate logic having the disjunction property. This theorem confirms H. Friedman's conjecture 41 (cf. [2], problem 41).In the third part the reader will find a criterion which allows us to obtain sets satisfying the conditions of Friedman's problem 42, on the basis of intermediate logics satisfying the conditions of problem 41.Finally, the fourth part contains a proof of a criterion which allows us to obtain modal logics endowed with Hallden's property on the basis of structurally complete intermediate logics having the disjunction property.Dedicated to Professor Roman SuszkoThe author would like to thank professors J. Perzanowski and A. Wroski for valuable suggestions.     

Mathematical quantum theory I: Random ultrafilters as hidden variables

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables.More precisely, these constructions permit us to write down a category-theoretically natural correlation between ideal outcomes of quantum measurements u of a universal wave function, and possible worlds of an Everett-Wheeler-like many-worlds-theory.The universal wave function, first, is simply a pure state of the Hilbert space (L ₂([0, 1]) ^M in a model M an appropriate mathematical-physical theory T, where T includes enough set-theory to derive all the analysis needed for von Neumann-algebraic formulations of quantum theory.The worlds of this framework can then be given a genuine model-theoretic construal: they are random models M(u) determined by M-random elements u of the unit interval [0, 1], where M is again a fixed model of T.Each choice of a fixed basis for a Hilbert space H in a model of M of T then assigns ideal spectral values for observables A on H (random ultrafilters on the range of A regarded as a projection-valued measure) to such M-random reals u. If is the universal Lebesgue measure-algebra on [0, 1], these assignments are interrelated by the spectral functional calculus with value 1 in the boolean extension (V( )) ^M , and therefore in each M(u).Finally, each such M-random u also generates a corresponding extension M(u) of M, in which ideal outcomes of measurements of all observables A in states are determined by the assignments just mentioned from the random spectral values u for the universal position-observable on L ₂([0, 1]) in M.At the suggestion of the essay's referee, I plan to draw on its ideas in the projected sequel to examine more recent modal and decoherence-interpretations of quantum theory, as well as Schrödinger's traditional construal of time-evolution. A preliminary account of the latter — an obvious prerequisite for any serious many-worlds-theory, given that Everett's original intention was to integrate time-evolution and wave-function collapse — is sketched briefly in Section 5.3. The basic idea is to apply results from the theory of iterated measure-algebras to reinterpret time-ordered processes of measurements (determined, for example, by a given Hamiltonian observable H in M) as individual measurements in somewhat more complexly defined extensions M(u) of M.In plainer English: if one takes a little care to distinguish boolean- from measure-algebraic tensor-products of the universal measure-algebra L, one can reinterpret formal time-evolution so that it becomes internal to the universal random models M(u).     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

On Intermediate Predicate Logics of some Finite Kripke Frames,I. Levelwise Uniform Trees

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.     

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure).     

On Extensions of Intermediate Logics by Strong Negation

In this paper we will study the properties of the least extension n() of a given intermediate logic by a strong negation. It is shown that the mapping from to n() is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure of the lattice of extensions of n(LC).     

Set theory as modal logic

A logical systemBM ⁺ is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, and These modalities are submitted to a finite number of axioms. is the usual sign of necessity, corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbedding of the usual theory of classesM intoBM ⁺ is constructed, such that a formulaA is provable inM if and only if(A) is provable inBM ⁺. There is also an inverse imbedding with an analogous property.     

Many-Valued Points And Equality

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we present some conservative translations involving classical logic, Lukasiewicz three-valued system L ₃, the intuitionistic system I ¹ and several paraconsistent logics, as for instance Sette's system P ¹, the D'Ottaviano and da Costa system J ₃ and da Costa's systems C _n, 1 n.     

Simplified semantics for relevant logics (and some of their rivals)

This paper continues the work of Priest and Sylvan inSimplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such asB ⁺ andB. We show that the simplified semantics can also be used for a large number of extensions of the positive base logicB ⁺, and then add the dualising^* operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.