A temporal logic for reasoning about partially ordered computations (Extended Abstract)

This work presents three increasingly expressive Dynamic Logics in which the programs are described in a language based on CCS. Our goal is to build dynamic logics that are suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. In order to accomplish that, CCS’s operators and constructions are added to a basic modal logic. Doing this, the semantics of CCS’s parallel operator allows us to build dynamic logics that support communicating and concurrent programs. We build a simple Kripke semantics for these logics, provide complete axiomatizations for them and show that they have the finite model property. This contrasts with other dynamic logics with parallel operators presented in the literature, such as Peleg’s Concurrent PDL with Channels, where either the parallel programs cannot communicate, or at least one of the properties mentioned above (simple Kripke semantics, complete axiomatization and finite model property) is missing.

LetL be any modal or tense logic with the finite model property. For eachm, definer L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr L (m) for some linear modal and tense logicsL.

We present an axiomatisation for an extension of a temporal epistemic logic with an epistemic reset operator defined on the intersection between epistemic and temporal relations. Additionally we show the logic has the finite model property, hence it is decidable.

Tense logics in the bimodal propositional language are investigated with respect to the Finite Model Property. In order to prove positive results techniques from investigations of modal logics above K4 are extended to tense logic. General negative results show the limits of the transfer.

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

Abstract—This paper shows that a classic metalogical frame-work, including all Boolean operators, can be used to support the development of a metric behavioural theory for Markov pro-cesses. Previously, only intuitionistic frameworks or frameworks without negation and logical implication have been developed to fulfill this task. The focus of this paper is on continuous Markovian logic (CML), a logic that characterizes stochastic bisimulation of Markov processes with arbitrary measurable state space and continuous-time transitions. For a parameter ε> 0 interpreted as observational error, we introduce an ε-parametrized metatheory for CML: we define the concepts of ε-satisfiability and ε-provability related by a sound and complete axiomatization and prove a series of ”parametrized” metatheorems including decidability, weak completeness and fi-nite model property. We also prove results regarding the relations between metalogical concepts defined for different parameters. Using this framework, we can characterize both the stochastic bisimulation relation and various observational preorders based on behavioural pseudometrics. The main contribution of this paper is proving that all these analyses can actually be done using a unified complete Boolean framework. And this extends the state of the art in this field, since the related works only propose intuitionistic contexts that limit, for instance, the use of the Boolean logical implication. I.

We consider an extension of the unary negation fragment of first-order logic in which arbitrarily many binary symbols may be required to be interpreted as equivalence relations. We show that this extension has the finite model property. More specifically, we show that every satisfiable formula has a model of at most doubly exponential size. We argue that the satisfiability (= finite satisfiability) problem for this logic is 2-ExpTime-complete. We also transfer our results to a restricted variant of the guarded negation fragment with equivalence relations.

The fluted fragment with transitive relations

We consider a generalised Chisholm set of contrary to duty obligations (CTD) of the form $$ O q_0 $$ and for i = 0,…, n we have the CTD is $$ \begin{array} {l} q_i\to O q_{i+1}\\ \neg q_i \to O \neg q_{i+1} \end{array} $$ and the facts ±q j for some j ∈ J ⊆ {0,1,…, n + 1}. Note that for the case of n = 1 and fact ¬q 0 we have the Chisholm paradox.We also allow for temporal sequencing of the q i in the form that q i + 1 may come temporally before or after q i .We offer a representation of this problem in a variation of standard deontic logic that we call TSDL, with the standard temporal operator \(\lozenge\), the deontic obligation operator O, and the past operator Y for “yesterday”. This formalism is free of the above paradoxes. We provide an axiomatization and show it to be complete. The logic formalism enjoys the finite tree model property and hence is decidable.

Recently, Baltag and van Benthem introduced a decidable logic of functional\ndependence (LFD) that extends the logic of Cylindrical Relativized Set Algebras\n(CRS) with atomic local dependence statements. Its semantics can be given in\nterms of generalised assignment models or their modal counterparts, hence the\nlogic is both a first-order and a modal logic. We show that LFD has the finite\nmodel property (FMP) using Herwig's theorem on extending partial isomorphisms,\nand prove a bisimulation invariance theorem characterizing LFD as a fragment of\nfirst-order logic.\n

Kleene Algebra (KA) is a useful tool for proving that two programs are equivalent. Because KA's equational theory is decidable, it integrates well with interactive theorem provers. This raises the question: which equations can we (not) prove using the laws of KA? Moreover, which models of KA are complete, in the sense that they satisfy exactly the provable equations? Kozen (1994) answered these questions by characterizing KA in terms of its language model. Concretely, equivalences provable in KA are exactly those that hold for regular expressions. Pratt (1980) observed that KA is complete w.r.t. relational models, i.e., that its provable equations are those that hold for any relational interpretation. A less known result due to Palka (2005) says that finite models are complete for KA, i.e., that provable equivalences coincide with equations satisfied by all finite KAs. Phrased contrapositively, the latter is a finite model property (FMP): any unprovable equation is falsified by a finite KA. Both results can be argued using Kozen's theorem, but the implication is mutual: given that KA is complete w.r.t. finite (resp. relational) models, Palka's (resp. Pratt's) arguments show that it is complete w.r.t. the language model. We embark on a study of the different complete models of KA, and the connections between them. This yields a novel result subsuming those of Palka and Pratt, namely that KA is complete w.r.t. finite relational models. Next, we put an algebraic spin on Palka's techniques, which yield a new elementary proof of the finite model property, and by extension, of Kozen's and Pratt's theorems. In contrast with earlier approaches, this proof relies not on minimality or bisimilarity of automata, but rather on representing the regular expressions involved in terms of transformation automata.

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

We propose a validity preserving translation from a subset of epistemic Alternating-time Temporal Logic (ATL) to epistemic Computation Tree Logic (CTL). The considered subset of epistemic ATL is known to have the finite model property and decidable model-checking. This entails the decidability of validity but the implied algorithm is unfeasible. Reducing the validity problem to that in a corresponding system of CTL makes the techniques for automated deduction for that logic available for the handling of the apparently more complex system of ATL.

The guarded fragment of first-order logic, GF, enjoys the finite model property, so the satisfiability and the finite satisfiability problems coincide. We are concerned with two extensions of the two-variable guarded fragment that do not possess the finite model property, namely, GF2 with equivalence and GF2 with transitive guards. We prove that in both cases every finitely satisfiable formula has a model of at most double exponential size w.r.t. its length. To obtain the result we invent a strategy of building finite models that are formed from a number of multidimensional grids placed over a cylindrical surface. The construction yields a 2NEXPTIME-upper bound on the complexity of the finite satisfiability problem for these fragments. For the case with equivalence guards we improve the bound to 2EXPTIME.

Introduction. For the definitions of propositional calculi, models and the finite model property (f.m.p.) we refer to [I] (we use 'model' for 'strong model'). In this note we describe two equivalent propositional calculi (equivalent in that they have identical sets of theorems) which differ in that one has the f.m.p. and the other does not, and in fact has no nontrivial finite model at all. This shows the f.m.p. is an attribute of a propositional calculus and not of the equivalence class of the calculus, unlike, for instance, decidability. I should like to thank R. Harrop for conversations on the topics of this paper, and also thank the referee for suggesting considerable improvements.