stone coalgebras we argue that the category of stone spaces forms an interesting base category for coalgebras in particular if one considers the vietoris functor as an analogue to the power set functor on the category of sets we prove that the so called descriptive general frames which play a fundamental role in the semantics of modal logics can be seen as stone coalgebras in a natural way this yields a duality between modal algebras and coalgebras for the vietoris functor building on this idea we introduce the notion of a vietoris polynomial functor over the category of stone spaces for each such functor t we provide an adjunction between t sorted boolean algebras with operators and the stone coalgebras for t we also identify the subcategory of algebras on which the adjunction restricts to an equivalence and show that the final t coalgebra is the dual of the initial t bao

calculation of brs cohomology with spectral sequences a method for finding the general form of the brs cohomology spaceh for the various gauge and supersymmetry theories is presented the method is adapted for use in the space of integrated local polynomials of the gauge fields and ghosts with arbitrary numbers of fields and dervivatives the technique uses the hodge decomposition in a fock space with a euclidean inner product and combines this with spectral sequences to generate simple and soluble equations whose solutions span a simple spacee isomorphic to the complicated spaceh the technique is illustrated for pedagogic purposes by the detailed calculation of the ghost charge zero and one sectors ofh for yang mills theory with gauge groupso 32 in ten dimensions the method is appropriate for supersymmetric theories gravity supergravity and superstrings where higher order terms with many derivatives occur naturally in the effective action

logical vs behavioural specifications abstract there are two fundamentally different approaches for specifying and verifying properties of systems the logical approach makes use of specifications given as formulae of temporal or modal logics and relies on efficient model checking algorithms the behavioural approach exploits various equivalence or refinement checking methods provided the specifications are given in the same formalism as implementations in this paper we provide translations between the logical formalism of calculus and the behavioural formalism of disjunctive modal transition systems the translations preserve structural properties of the input specification and allow us to perform logical operations on the behavioural specifications as well as behavioural compositions on logical formulae the unification of both approaches provides additional methods for component based stepwise design

the seiberg witten map for the 4d noncommutative bf theory we describe the seiberg witten map taking the 4d noncommutative bf theory ncbf into its pure commutative version the existence of this map is in agreement with the hypothesis that such maps are available for any noncommutative theory with schwarz type topological sectors and represents a strong indication for the renormalizability of these theories in general

algebraic characterization of the wess zumino consistency conditions in gauge theories a new way of solving the descent equations corresponding to the wess zumino consistency conditions is presented the method relies on the introduction of an operator which allows to decompose the exterior space time derivatived as a brs commutator the case of the yang mills theories is treated in detail

probabilistic bisimilarity as testing equivalence abstract larsen and skou initiated the study of probabilistic bisimilarity and its characterisation in terms of tests later on van breugel et al showed that for labelled markov processes with continuous state spaces probabilistic bisimilarity nicely coincides with a simple notion of testing equivalence their proof employs advanced machinery from topology in the discrete case of finite state reactive probabilistic processes we prove that coincidence result with an elementary and more accessible proof

w gauge structures and their anomalies an algebraic approach starting from flat two dimensional gauge potentials we propose the notion of w gauge structure in terms of a nilpotent becchi rouet stora brs differential algebra the decomposition of the underlying lie algebra with respect to an sl 2 subalgebra is crucial for the discussion of conformal covariance in particular the appearance of a projective connection different sl 2 embeddings lead to different w gauge structures we present a general soldering procedure which allows one to express zero curvature conditions for the w currents in terms of conformally covariant differential operators acting on the w gauge fields and to obtain at the same time the complete nilpotent brs differential algebra generated by w currents gauge fields and the ghost fields corresponding to w diffeomorphisms as illustrations we treat the cases of sl 2 itself and the two different sl 2 embeddings in sl 3 viz the w 1 3 and w 2 3 gauge structures in some detail in these cases we determine algebraically w anomalie

a partial ordering semantics for ccs a new operational semantics for pure ccs is proposed that considers the parallel operator as a first class one and permits a description of the calculus in terms of partial orderings the new semantics also for unguarded agents is given in the sos style via the partial ordering derivation relation ccs agents are decomposed into sets of sequential subagents the new derivations relate sets of subagents and describe their actions and the casual dependencies among them the computations obtained by composing partial ordering derivations are observed either as interleaving or partial orderings of events interleavings coincide with milner s many step derivations and linearizations of partial orderings are all and only interleavings abstract semantics are obtained by introducing two relations of observational equivalence and congruence that preserve concurrency these relations are finer than milner s in that they distinguish interleaving of sequential nondeterministic agents from their concurrent execution

completeness of kozen s axiomatisation of the propositional m calculus propositional calculus is an extension of the propositional modal logic with the least fixpoint operator in the paper introducing the logic kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the model system k it is shown that this axiomatisation is complete

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

nested semantics over finite trees are equationally hard this paper studies nested simulation and nested trace semantics over the language bccsp a basic formalism to express finite process behaviour it is shown that none of these semantics affords finite in equational axiomatizations over bccsp in particular for each of the nested semantics studied in this paper the collection of sound closed in equations over a singleton action set is not finitely based

becchi rouet stora cohomology of zero curvature systems ii the noncomplete ladder case the yang mills type theories and their brs cohomologies are analyzed within the zero curvature formalism

an analytical investigation of shape control of large space structures by applied temperatures an analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure the temperatures at the control elements are defined so as to minimize the overall thermal distortion of the structure from its ideal shape and a matrix equation is obtained which can be solved for the set of optimum control temperatures a formulation of the procedure for continuous structures governed by differential equations and a formulation for discrete finite element modeled structures governed by matrix equations are presented the equations from the continuous formulation are employed for the shape control of a simple beam distorted by nonuniform heating and the discrete formulation is applied in a general purpose finite element structural analysis computer program for the shape control of a 750 m radiometer antenna reflector dish subjected to orbital heating a reduction in thermal distortion by a factor of nearly 50 was obtained with the use of only seven control elements results for four different sets of control locations for the antenna are presented in which reductions in distortion of up to a factor of four were obtained

divide and congruence ii from decomposition of modal formulas to preservation of delay and weak bisimilarity abstract earlier we presented a method to decompose modal formulas for processes with the internal action and congruence formats for branching and bisimilarity were derived on the basis of this decomposition method the idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation in this follow up paper the decomposition method is enhanced to deal with modal characterisations that contain a modality a to derive congruence formats for delay and weak bisimilarity

algebraic characterization of vector supersymmetry in topological field theories a cohomological characterization of a class of linearly broken ward identities is provided the examples of the topological vector supersymmetry and of the landau ghost equation are discussed the existence of these linearly broken ward identities lies in the brst exactness of suitable antifield dependent cocycles with negative ghost number this algebraic set up turns out to be related to the cohomological reformulation of the noether theorem given by g barnich f brandt and m henneaux commun math phys 174 57 93 1995 m henneaux j pure appl algebra 100 3 1995

three logics for branching bisimulation three temporal logics are introduced that induce on labeled transition systems the same identifications as branching bisimulation a behavioral equivalence that aims at ignoring invisible transitions while preserving the branching structure of systems the first logic is an extension of hennessy milner logic with an until operator the second one is another extension of hennessy milner logic which exploits the power of backward modalities the third logic is ctl without the next time operator a relevant side effect of the last characterization is that it sets a bridge between the state and action based approaches to the semantics of concurrent systems

deciding observational congruence of finite state ccs expressions by rewriting abstract we propose a term rewriting approach to verify observational congruence between guarded recursive finite state ccs expressions starting from the complete axiomatization of observational congruence for this subset of ccs a non terminating rewriting relation has been defined this rewriting relation is canonical over a subclass of infinite derivations structured fair derivations which compute all the normal forms the rewriting relation is shown to be complete with respect to the axiomatization by proving that every structured fair derivation computes a term that denotes an r normal process graph the existence of a finite representation for normal forms allows the definition of a rewriting strategy that in a finite number of rewriting steps decides observational congruence of guarded recursive finite state ccs expressions

b r s algebras analysis of the consistency equations in gauge theory we compute all possible anomalous terms in quantum gauge theory in the natural class of polynomials of differential forms by using the appropriate cohomological and algebraic methods we do it for all dimensions of spacetime and all structure groups with reductive lie algebras

generic weakest precondition semantics from monads enriched with order we devise a generic framework where a weakest precondition semantics in the form of indexed posets is derived from a monad whose kleisli category is enriched by posets it is inspired by jacobs recent identification of a categorical structure that is common in various predicate transformers but adds generality in the following aspects 1 different notions of modality such as may vs must are captured by eilenberg moore algebras 2 nested alternating branching like in games and in probabilistic systems with nondeterministic environments is modularly modeled by a monad on the eilenberg moore category of another

reasoning about knowledge and probability we provide a model for reasoning about knowledge and probability together we allow explicit mention of probabilities in formulas so that our language has formulas that essentially say according to agent i formula holds with probability at least b the language is powerful enough to allow reasoning about higher order probabilities as well as allowing explicit comparisons of the probabilities an agent places on distinct events we present a general framework for interpreting such formulas and consider various properties that might hold of the interrelationship between agents probability assignments at different states we provide a complete axiomatization for reasoning about knowledge and probability prove a small model property and obtain decision procedures

a new realization for the d 2 topological algebra we derive topological algebras for two dimensional models admitting a d b decomposition with b the brst operator in general the topological algebra we obtain is not derived from the twisting of an n 2 supersymmetry algebra we show for the particular case of an abelian model described by a ladder of matter how to derive n 2 supersymmetry generators from the odd generators of the topological algebra

a fully abstract denotational semantics for the calculus of higher order communicating systems abstract in this paper we study the calculus of higher order communicating systems chocs thomsen proc of popl 89 acm 1989 pp 143 154 inform comput 116 1 1995 38 57 in a denotational setting we present a construction of a denotational semantics for chocs which resides in a domain constructed using the standard constructions of separated sum cartesian product the plotkin power domain constructor and recursively defined domains we show under mild restrictions that the denotational semantics and the operational semantics of chocs are fully abstract we have previously proved using bisimulation arguments that processes as first class objects are powerful enough to simulate recursion however the proof is very long and tedious to demonstrate the power of the denotational approach we use it to obtain a very simple proof of the simulation of recursion result

coalgebraic modal logic soundness completeness and decidability of local consequence this paper studies finitary modal logics interpreted over coalgebras for an endofunctor and establishes soundness completeness and decidability results the logics are studied within the abstract framework of coalgebraic modal logic which can be instantiated with arbitrary endofunctors on the category of sets this is achieved through the use of predicate liftings which generalise atomic propositions and modal operators from kripke models to arbitrary coalgebras predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle this induction principle is systematically exploited to establish soundness completeness and decidability of the logics we believe that this induction principle also opens new ways for reasoning about modal logics our proof of completeness does not rely on a canonical model construction and the proof of the finite model property does not use filtrations

cut elimination in coalgebraic logics we give two generic proofs for cut elimination in propositional modal logics interpreted over coalgebras we first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cut rule is admissible in the ensuing sequent calculus we then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination apart from the fact that cut elimination holds our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics as applications we present a new proof of the already known interpolation property for coalition logic and newly establish the interpolation property for the conditional logics lck and lckid

zero curvature conditions and conformal covariance two dimensional zero curvature conditions with special emphasis on conformal properties are investigated in detail and the appearance of covariant higher order differential operators constructed in terms of a projective connection is elucidated the analysis is based on the kostant decomposition of simple lie algebras in terms of representations with respect to their principal sl 2 subalgebra

a modal proof theory for final polynomial coalgebras an infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets the canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor the states of this final coalgebra are certain maximal sets of formulas that have natural syntactic closure properties the syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the termination of certain functions induced by transition paths a completeness theorem is proven for the logic of functors which have the lindenbaum property that every consistent set of formulas has a maximal extension this property is shown to hold if the deducibility relation is generated by countably many inference rules a counter example to completeness is also given this is a polynomial functor that is not lindenbaum it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable even though all of its countable subsets are satisfiable

on the final sequence of a finitary set functor a standard construction of the final coalgebra of an endofunctor involves defining a chain of iterates starting at the final object of the underlying category and successively applying the functor in this paper we show that for a finitary set functor this construction always yields a final coalgebra in 2 steps

a hierarchy of probabilistic system types we arrange various classes of probabilistic systems studied in the literature in an expressiveness hierarchy our expressiveness criterion is the existence of a system translation from the less expressive type into the more expressive type that preserves and reflects probabilistic bisimilarity we model the different system types as coalgebras of suitable behaviour functors and argue that the corresponding coalgebraic bisimilarity coincides with probabilistic bisimilarity for the classes for which the latter notion has been proposed in the literature the theory of coalgebras provides a unified framework for the presentation of the different classes and the system translations we needed to establish the hierarchy all these translations arise in a standard way from natural transformations between the two behaviour functors involved such a translation generally preserves coalgebraic bisimilarity we exploit a new result that under mild assumptions on the behaviour functors a system translation induced by a natural transformation with injective components also reflects bisimilarity

automata and fixed point logic a coalgebraic perspective this paper generalizes existing connections between automata and logic to a coalgebraic abstraction level let f set to set be a standard functor that preserves weak pullbacks we introduce various notions of f automata devices that operate on pointed f coalgebras the criterion under which such an automaton accepts or rejects a pointed coalgebra is formulated in terms of an infinite two player graph game we also introduce a language of coalgebraic fixed point logic for f coalgebras and we provide a game semantics for this language finally we show that the two approaches are equivalent in expressive power we prove that any coalgebraic fixed point formula can be transformed into an f automaton that accepts precisely those pointed f coalgebras in which the formula holds and conversely we prove that any f automaton can be converted into an equivalent fixed point formula that characterizes the pointed f coalgebras accepted by the automaton

modular construction of complete coalgebraic logics we present a modular approach to defining logics for a wide variety of state based systems the systems are modelled as coalgebras and we use modal logics to specify their observable properties we show that the syntax semantics and proof systems associated with such logics can all be derived in a modular fashion moreover we show that the logics thus obtained inherit soundness completeness and expressiveness properties from their building blocks we apply these techniques to derive sound complete and expressive logics for a wide variety of probabilistic systems for which no complete axiomatisation has been obtained so far

testing probabilistic equivalence through reinforcement learning checking if a given system implementation respects its specification is often done by proving that the two are equivalent the equivalence is chosen in particular for its computability and of course for its meaning that is for its adequacy with what is observable from the two systems implementation and specification trace equivalence is easily testable decidable from interaction but often considered too weak in contrast bisimulation is accepted as the canonical equivalence for interaction but it is not testable richer than an equivalence is a form of distance it is zero between equivalent systems and it provides an estimation of their difference if the systems are not equivalent our main contribution is to define such a distance in a context where 1 the two systems to be compared have a stochastic behavior 2 the model of one of them e g the implementation is unknown hence our only knowledge is obtained by interacting with it 3 consequently the target equivalence observed when distance is zero must be testable to overcome the problem that the model is unknown we use a reinforcement learning approach that provides powerful stochastic algorithms that only need to interact with the model our second main contribution is a new family of testable equivalences called k moment the weakest of them 1 moment equivalence is trace equivalence as k grows k moment equivalences become finer all remaining as well as their limit weaker than bisimulation we propose a framework to define and test a bigger class of testable equivalences test observation equivalences toes and we show how they can be made coarser or not by tuning some parameters

a compositional approach to defining logics for coalgebras we present a compositional approach to defining expressive logics for coalgebras of endofunctors on set this approach uses a notion of language constructor and an associated notion of semantics to capture one inductive step in the definition of a language for coalgebras and of its semantics we show that suitable choices for the language constructors and for their associated semantics yield logics which are both adequate and expressive w r t behavioural equivalence moreover we show that type building operations give rise to corresponding operations both on language constructors and on their associated semantics thus allowing the derivation of expressive logics for increasingly complex coalgebraic types our framework subsumes several existing approaches to defining logics for coalgebras and at the same time allows the derivation of new logics with logics for probabilistic systems being the prime example

exact three dimensional analysis for static torsion of piezoelectric rods based on the theory of elasticity exact analytical and numerical solutions of piezoelectric rods under static torsion are studied in this paper direct solution method is used the main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading it is shown that previous assumptions are valid only in some types of piezoelectric materials while in other types these assumptions lead to considerable deviations from accurate modeling the present analytical solutions are compared with three dimensional finite element analysis results and absolute agreements are found at the end of this article torsional rigidity shape effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

bialgebraic methods and modal logic in structural operational semantics bialgebraic semantics invented a decade ago by turi and plotkin is an approach to formal reasoning about well behaved structural operational semantics sos an extension of algebraic and coalgebraic methods it abstracts from concrete notions of syntax and system behaviour thus treating various kinds of operational descriptions in a uniform fashion in this paper bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel general approach to proving the compositionality of process equivalences for languages defined by structural operational semantics to prove compositionality one provides a notion of behaviour for logical formulas and defines an sos like specification of modal operators which reflects the original sos specification of the language this approach can be used to define sos congruence formats as well as to prove compositionality for specific languages and equivalences

diffeomorphism cohomology in beltrami parametrization considering local conformal field theories on a riemann surface by coupling conformal matter fields with a complex structure parametrized by a beltrami differential the local diffeomorphism cohomology modulo d of the becchi rouet stora operator is computed directly by means of the spectral sequences method then thanks to both power counting and locality principles of the feynman algorithm the local theory is analyzed on the one hand in the ghost number one sector consistent anomalies turn out to be exactly after elimination of a trace anomaly involving matter fields the well known holomorphically split diffeomorphism anomaly due to the vacuum on the other hand in the ghost number zero sector local observables of the theory namely the vertex operators are generically calculated as well as all possible classical actions for lagrangian conformal models

a quantum field theoretic description of linking numbers and their generalization it is shown that the linking number of two surfaces of dimensionp andn p 1 in ann dimensional manifold has a natural description in terms of the two point funciton of a simple topological field theory the two point function of a more general theory whose partition function yields the ray singer torsion provides a definition for a generalized linking number

a note on local brst cohomology of yang mills type theories with free abelian factors we extend previous work on antifield dependent local becchi rouet stora tyutin brst cohomology for matter coupled gauge theories of yang mills type to the case of gauge groups that involve free abelian factors more precisely we first investigate in a model independent way how the dynamics enters the computation of the cohomology for a general class of lagrangians in general spacetime dimensions we then discuss explicit solutions in the case of specific models our analysis has implications for the structure of characteristic cohomology and for consistent deformations of the classical models as well as for divergences counterterms and for gauge anomalies that may appear during perturbative quantization

the quantitative linear time branching time spectrum we present a distance agnostic approach to quantitative verification taking as input an unspecified distance on system traces or executions we develop a game based framework which allows us to define a spectrum of different interesting system distances corresponding to the given trace distance thus we extend the classic linear time branching time spectrum to a quantitative setting parametrized by trace distance we also prove a general transfer principle which allows us to transfer counterexamples from the qualitative to the quantitative setting showing that all system distances are mutually topologically inequivalent

presenting functors on many sorted varieties and applications this paper studies several applications of the notion of a presentation of a functor by operations and equations we show that the technically straightforward generalisation of this notion from the one sorted to the many sorted case has several interesting consequences first it can be applied to give equational logic for the binding algebras modelling abstract syntax second it provides a categorical approach to algebraic semantics of first order logic third this notion links the uniform treatment of logics for coalgebras of an arbitrary type t with concrete syntax and proof systems analysing the many sorted case is essential for modular completeness proofs of coalgebraic logics

saint venant end effects in piezoceramic materials the stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated however those for piezoelectric materials have not been studied in this paper we examine the saint venant end effects of piezoceramic materials by considering the problem of a semi infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces it is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction thus the governing equations in terms of the airy stress function and electric potential function can be uncoupled the governing equation in terms of the airy stress function involves only two non dimensional parameters after non dimensionalization finally the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically and their variation with the two non dimensional parameters is presented

algebraic laws for nondeterminism and concurrency since a nondeterministic and concurrent program may in general communicate repeatedly with its environment its meaning cannot be presented naturally as an input output function as is often done in the denotational approach to semantics in this paper an alternative is put forth first a definition is given of what it is for two programs or program parts to be equivalent for all observers then two program parts are said to be observation congruent if they are in all program contexts equivalent the behavior of a program part that is its meaning is defined to be its observation congruence class the paper demonstrates for a sequence of simple languages expressing finite terminating behaviors that in each case observation congruence can be axiomatized algebraically moreover with the addition of recursion and another simple extension the algebraic language described here becomes a calculus for writing and specifying concurrent programs and for proving their properties

coalgebraic semantics of modal logics coalgebras can be seen as a natural abstraction of kripke frames in the same sense coalgebraic logics are generalised modal logics in this paper we give an overview of the basic tools techniques and results that connect coalgebras and modal logic we argue that coalgebras unify the semantics of a large range of different modal logics such as probabilistic graded relational conditional and discuss unifying approaches to reasoning at this level of generality we review languages defined in terms of the so called cover modality languages induced by predicate liftings as well as their common categorical abstraction and present abstract results on completeness expressiveness and complexity in these settings both for basic languages as well as a number of extensions such as hybrid languages and fixpoints

coequational logic for accessible functors covarieties of coalgebras are those classes of coalgebras for an endofunctor h on the category of sets that are closed under coproducts subcoalgebras and quotients equivalently covarieties are classes of h coalgebras that can be presented by coequations adamek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets here this result is extended to accessible functors given a presentation of an accessible functor h simple deduction systems for coequations are formulated and it is shown that regularity of the presentation implies soundness and completeness of these deduction systems the converse is true whenever h has a non trivial terminal coalgebra also a method is found to obtain concrete descriptions of cofree and thus terminal coalgebras of accessible functors and is applied to the finite and countable powerset functor as well as to the finite distribution functor

what is the coalgebraic analogue of birkhoff s variety theorem logical definability is investigated for certain classes of coalgebras related to state transition systems hidden algebras and kripke models the filter enlargement of a coalgebra a is introduced as a new coalgebra a whose states are special observationally rich filters on the state set of a the ultra filter enlargement is the subcoalgebra a of a whose states are ultrafilters boolean combinations of equations between terms of observable or output type are identified as a natural class of formulas for specifying properties of coalgebras these observable formulas are permitted to have a single state variable and form a language in which modalities describing the effects of state transitions are implicitly present a and a validate the same observable formulas it is shown that a class of coalgebras is definable by observable formulas iff the class is closed under disjoint unions images of bisimulations and ultra filter enlargements closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms moreover every set of observable formulas has the same models as some set of conditional equations examples are constructed to show that the use of enlargements is essential in these characterisations and that there are classes of coalgebras definable by conditional observable equations but not by equations alone the main conclusion of the paper is that to structurally characterise classes of coalgebras that are logically definable by modal languages requires a new construction of stone space type in addition to the coalgebraic duals of the three constructions homomorphisms subalgebras direct products that occur in birkhoff s original variety theorem for algebras copyright 2001 elsevier science b v

final coalgebras for functors on measurable spaces we prove that every functor on the category meas of measurable spaces built from the identity and constant functors using products coproducts and the probability measure functor d has a final coalgebra our work builds on the construction of the universal harsanyi type spaces by heifetz and samet and papers by rosziger and jacobs on coalgebraic modal logic we construct logical languages probabilistic logics of transition systems and interpret them on coalgebras the final coalgebra is carried by the set of descriptions of all points in all coalgebras for the category set we work with the functor d of discrete probability measures we prove that every functor on set built from d and the expected functors has a final coalgebra the work for set differs from the work for meas negation in needed for final coalgebras on set but not for meas