Showing papers on "Modal operator published in 2011"

First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees

Abstract: We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S.

Modal restriction semigroups: towards an algebra of functions

Abstract: Restriction semigroups model algebras of partial maps under composition and domain. Here we consider restriction semigroups for which the usual Boolean operations on domains are modeled. Such algebras are capable of modeling the usual modal operators considered in dynamic logic. Indeed adding a natural functional variant of union to the signature gives a deterministic version of the modal semirings of Moller and Struth, but also a monoidal version of the classical restriction categories of Cockett and Manes. Other operations modeled are intersection and (in the finite case) functional iteration. In each case, axiomatizations of the concrete functional examples are given, leading to algebraic models of partial maps incorporating all the domain-related and set-theoretic operations previously considered. Our algebras furnish natural algebraic semantics for the logics of deterministic computer programs, leading to new results for some variants of propositional dynamic logic.

Computing Distances between Probabilistic Automata

Abstract: We present relaxed notions of simulation and bisimulation on Probabilistic Automata (PA), that allow some error . When = 0 we retrieve the usual notions of bisimulation and simulation on PAs. We give logical characterisations of these notions by choosing suitable logics which differ from the elementary ones, L and L¬, by the modal operator. Using flow networks, we show how to compute the relations in PTIME. This allows the definition of an efficiently computable non discounted distance between the states of a PA. A natural modification of this distance is introduced, to obtain a discounted distance, which weakens the influence of long term transitions. We compare our notions of distance to others previously defined and illustrate our approach on various examples. We also show that our distance is not expansive with respect to process algebra operators. Although L(¬) is a suitable logic to characterise -(bi)simulation on deterministic PAs, it is not for general PAs; interestingly, we prove that it does characterise weaker notions, called a priori -(bi)simulation, which we prove to be NP-difficult to decide.

Proof-Carrying code in a session-typed process calculus

Abstract: Dependent session types allow us to describe not only properties of the I/O behavior of processes but also of the exchanged data. In this paper we show how to exploit dependent session types to express proof-carrying communication. We further introduce two modal operators into the type theory to provide detailed control about how much information is communicated: one based on traditional proof irrelevance and one integrating digital signatures.

A new four-valued approach to modal logic

Abstract: In this paper several systems of modal logic based on four-valued matrices are presented. We start with pure modal logics, i.e. modal logics with modal operators as the only operators, using the Polish framework of structural consequence relation. We show that with a four-valued matrix we can define modal operators which have the same behavior as in pure S5 (S5 with only modal operators). We then present modal logics with conjunction and disjunction based on four-valued matrices. We show that if we use partial order instead of linear order, we are avoiding Lukasiewicz’s paradox. We then introduce classical negation and we show than defining implication in the usual way using negation and disjunction Kripke law is valid using either linear or partial order. On the other hand we show that the difference between linear and partial order appears at the level of the excluded middle and the replacement theorem.

Finite-valued Lukasiewicz modal logic Is PSPACE-complete

Abstract: It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.

Complexity of Model Checking for Modal Dependence Logic

Abstract: Modal dependence logic (MDL) was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n the atomic formula =(p_1,...,p_(n-1),p_n) intuitively states that the value of p_n is determined solely by those of p_1,...,p_(n-1). We show that model checking for MDL formulae over Kripke structures is NP-complete and further consider fragments of MDL obtained by restricting the set of allowed propositional and modal connectives. It turns out that several fragments, e.g., the one without modalities or the one without propositional connectives, remain NP-complete. We also consider the restriction of MDL where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the model checking problem for this bounded MDL is still NP-complete. We additionally extend MDL by allowing classical disjunction - introduced by Sevenster - besides dependence disjunction and show that classical disjunction is always at least as computationally bad as bounded arity dependence atoms and in some cases even worse, e.g., the fragment with nothing but the two disjunctions is NP-complete. Furthermore we almost completely classifiy the computational complexity of the model checking problem for all restrictions of propositional and modal operators for both unbounded as well as bounded MDL.

Expressiveness of the interval logics of Allen's relations on the class of all linear orders: complete classification

Abstract: We compare the expressiveness of the fragments of Halpern and Shoham's interval logic (HS), i.e., of all interval logics with modal operators associated with Allen's relations between intervals in linear orders. We establish a complete set of inter-definability equations between these modal operators, and thus obtain a complete classification of the family of 212 fragments of HS with respect to their expressiveness. Using that result and a computer program, we have found that there are 1347 expressively different such interval logics over the class of all linear orders.

Algebraic Notions of Termination

Abstract: Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence Kleene algebras and omega algebras. The resulting notions and techniques are used in calculational proofs of classical theorems of rewriting theory. These applications show that modal semirings are powerful tools for reasoning algebraically about the finite and infinite dynamics of programs and transition systems.

Two-dimensional description logics for context-based semantic interoperability

Abstract: Description Logics (DLs) provide a clear and broadly accepted paradigm for modeling and reasoning about terminological knowledge. However, it has been often noted, that although DLs are well-suited for representing a single, global viewpoint on an application domain, they offer no formal grounding for dealing with knowledge pertaining to multiple heterogeneous viewpoints — a scenario ever more often approached in practical applications, e.g. concerned with reasoning over distributed knowledge sources on the Semantic Web. In this paper, we study a natural extension of DLs, in the style of two-dimensional modal logics, which supports declarative modeling of viewpoints as contexts, in the sense of McCarthy, and their semantic interoperability. The formalism is based on two-dimensional semantics, where one dimension represents a usual object domain and the other a (possibly infinite) domain of viewpoints, addressed by additional modal operators and a metalanguage, on the syntactic level. We systematically introduce a number of expressive fragments of the proposed logic, study their computational complexity and connections to related formalisms.

Modal operators on commutative residuated lattices

Abstract: We prove some fundamental properties of monotone modal operators on bounded commutative integral residuated lattices (CRL). Moreover we give a positive answer to the problem left open in [RACHŮNEK, J.—SALOUNOV A, D.: Modal operators on bounded commutative residuated Rl-monoids, Math. Slovaca 57 (2007), 321–332].

Logics for information systems and their dynamic extensions

Abstract: The article proposes logics for information systems, which provide information about a set of objects regarding a set of attributes. Both “complete” and “incomplete” information systems are dealt with. The language of these logics contains modal operators, and constants corresponding to attributes and attribute values. Sound and complete deductive systems for these logics are presented, and the problem of decidability is addressed. Furthermore, notions of information and information update are defined, and dynamic extensions of the above logics are presented to accommodate these notions. A set of reduction axioms enables us to obtain a complete axiomatization of the dynamic logics.

Conceptual modeling in full computation-tree logic with sequence modal operator

Abstract: In this paper, we propose a method for modeling concepts in full computation-tree logic with sequence modal operators. An extended full computation-tree logic, CTLS*, is introduced as a Kripke semantics with a sequence modal operator. This logic can appropriately represent hierarchical tree structures in cases where sequence modal operators in CTLS* are applied to tree structures. We prove a theorem for embedding CTLS* into CTL*. The validity, satisfiability, and model-checking problems of CTLS* are shown to be decidable. An illustrative example of biological taxonomy is presented using CTLS* formulas. © 2011 Wiley Periodicals, Inc. (This paper is an extended version of Kamide and Kaneiwa.)

Model checking epistemic and probabilistic properties of multi-agent systems

Abstract: Model checking, a formal automatic verification method, has been widely used in multi-agent systems to verify specifications that contain qualitative properties (e.g safety and liveliness) and quantitative properties. Decision making processes based on inherent knowledge are necessary for agents to act appropriately, particularly in uncertain settings. In order to check epistemic (i.e knowledge) and measurable properties in multi-agent systems, we propose a new logic PCTLK, which uses probabilistic, epistemic, and temporal modal operators. We exploit Discrete-Time Markov Chains (DTMC), in which we are able to represent measurable properties with probability, to model uncertainty in multi-agent systems. We extend the formalism of interpreted systems by adding probabilistic features to suit DTMC models and to present the model checking algorithm for our logic. At the end of this paper, we simulate our algorithm using an example of online shopping.

Epistemic modals in the past

Abstract: The aim of this paper is to provide additional arguments against the view that on the epistemic reading of modal verbs, the time of the modal is always the utterance time. The hypothesis defended, also adopted by Eide (2002, 2003) and von Fintel and Gillies (2008) is that epistemic modals can be in the scope of Tense/Aspect. Three possible translations of might have been in French (with a passe compose or an imparfait on the modal and a simple infinitival, or with a present on the modal and a perfect infinitival) are semantically differentiated. The analysis describes the distribution of past tenses on epistemic modality and explains the differences in their interpretation. Possibilities are the sort of thing that comes into and goes out of existence, that can be ‘dated’ (Mondadori 1978, p. 246) It is obvious that we don’t have a good understanding of what happens when a modal is combined with temporal operators. (Portner 2009, p. 230)

A new deduction system for deciding validity in modal logic K

Abstract: A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this paper. Modal logics could be very helpful in modeling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently it has been presented the Connectionist Modal Logics which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness, and the fact that the system itself is a decision procedure are proved in this paper. The main advantages of this approach are: first, the system is deterministic, that is, it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula (if such exists); third, the language of deduction and the language of a logic coincide. Some of these advantages are compared to other classical approaches.

Generating Descriptions of Incomplete City-Traffic States with Agents

Abstract: Multiple approaches aim at describing numerical data with words. In this paper we give a brief overview of a distributed agent-based system providing summaries of city traffic in a textual form. The system deals locally with a problem of incomplete data adopting a method for grounding of modal statements in artificial cognitive agents. The internal uncertainty of an agent about a current state of the traffic is expressed with autoepistemic modal operators of possibility, belief, and knowledge.

Philosophical Logic: An Introduction to Advanced Topics

Abstract: 1. Introduction / Sentences / Truth and Falsity / Defense and Refutation / Inference, Form and Implication / Formally Valid Inference / Conjunctions / Inference with Conjunctions / Negation / Inference with Negation / Truth-Functionality and Negation / Grouping / 2. Sentential Logic / Simple Sentences / Sentences / Derivations: A First Look / A Note on Sets / Lines / Derivations Again / Theorems / Truth Sets / Soundness / Completeness / Extensions of SL / Conditionalization / Model Sets / Syntax and Semantics / 3. Quantificational Logic / Singular Terms / Predicates / Some Symbolic Conventions / Some / The Language QL / Derivations / Truth Sets / All / Further Extensions of QL / Model Sets / Identity / Model Sets for QL / 4. Sentential Modal Logic / Non-Truth-Functional Sentential Operators / Sentential Modal Operators / Derivations / S5, S4, T, and B / Possible Worlds / At a World and In a World / Model Sets and Model Systems / Deontic Logic and Model Sets / 5. Quantification and Modality / Some Derivations / Model Sets and Systems / An Alternative / 6. Set Theory / The Axiom of Extensionality / Axioms of Separation / Pairing Axiom and Rule U / The Restriction on the A2 Axiom / The Null Set / An Interpretation / More Axioms / General Intersection Operation / Order and Relations / Functions / Sizes of Sets / The Power Set Axiom / A Basic Theorem / 7. Incompleteness / The Language of Arithmetic / Three Key Concepts / Three Key Theorems / The Core Argument / Concluding Observations / 8. An Introduction to Term Logic / Syllogistic / The Limits of Syllogistic / Term Functor Logic / Singular Terms and Identity in TFL / Relationals in TFL / The Logic of Sentences in TFL / Rules of Inference for Derivations in TFL / Derivation in TFL / The Bridge to TFL / 9. Modal Term Logic / Modal Operators on Terms / Modal Operators on Sentences / Rules of Derivation for Modal TFL / Modal Inference in TFL / Rules, Axioms and Principles / List of Symbols / Glossary / Index.

Contingency-based equilibrium logic

Abstract: We investigate an alternative language for equilibrium logic that is based on the concept of positive and negative contingency. Beyond these two concepts our language has the modal operators of necessity and impossibility and the Boolean operators of conjunction and disjunction. Neither negation nor implication are available. Our language is just as expressive as the standard language of equilibrium logic (that is based on conjunction and intuitionistic implication).

Reasoning with models of probabilistic knowledge over probabilistic knowledge

Abstract: In multi-agent systems, the knowledge of agents about other agents’ knowledge often plays a pivotal role in their decisions. In many applications, this knowledge involves uncertainty. This uncertainty may be about the state of the world or about the other agents’ knowledge. In this thesis, we answer the question of how to model this probabilistic knowledge and reason about it efficiently. Modal logics enable representation of knowledge and belief by explicit reference to classical logical formulas in addition to references to those formulas’ truth values. Traditional modal logics (see e.g. [Fitting, 1993; Blackburn et al., 2007]) cannot easily represent scenarios involving degrees of belief. Works that combine modal logics and probabilities apply the representation power of modal operators for representing beliefs over beliefs, and the representation power of probability for modeling graded beliefs. Most tractable approaches apply a single model that is either engineered or learned, and reasoning is done within that model. Present model-based approaches of this kind are limited in that either their semantics is restricted to have all agents with a common prior on world states, or are resolving to reasoning algorithms that do not scale to large models. In this thesis we provide the first sampling-based algorithms for model-based reasoning in such combinations of modal logics and probability. We examine a different point than examined before in the expressivity-tractability tradeoff for that combination, and examine both general models and also models which use Bayesian Networks to represent subjective probabilistic beliefs of agents. We provide exact inference algorithms for the two representations, together with correctness results, and show that they are faster than comparable previous ones when some structural conditions hold. We also present sampling-based algorithms, show that those converge under relaxed conditions and that they may not converge otherwise, demonstrate the methods on some examples, and examine the performance of our algorithms experimentally.

Completions of µ-Algebras

Abstract: A µ-algebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f,µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of µ-algebras contains a µ-algebra that has no embedding into a complete µ-algebra. We focus then on modal µ-algebras, i.e. algebraic models of the propositional modal µ-calculus. We prove that free modal µ-algebras satisfy a condition – reminiscent of Whitman’s condition for free lattices – which allows us to prove that (i) modal operators are adjoints on free modal µalgebras, (ii) least prefixed points of �1-operations satisfy the constructive relation µx.f = W n�0 f n (?). These properties imply the following statement: the MacNeille-Dedekind completion of a free modal µ-algebra is a complete modal µ-algebra and moreover the canonical embedding preserves all the operations in the class Comp(�1,�1) of the fixed point alternation hierarchy.

A uniform approach to three-valued semantics for μ -calculus on abstractions of hybrid automata

Abstract: In this paper, we consider the definition of a three-valued semantics for a μ-calculus on abstractions of hybrid automata. To this end, we first develop a framework that is general in the sense that it provides a preservation result for several possible semantics of the modal operators. In a second step, we instantiate our framework to two particular abstractions. To this end, a key issue is the consideration of both over- and underapproximated reachability, while classic simulation-based abstractions rely only on overapproximations, and therefore limit the preservation to the universal (μ-calculus’) fragment. To specialize our general result, we consider (1) modal abstractions, where the notions of ‘may’ and ‘must’ transitions are extended from the purely discrete to the hybrid time framework, and (2) so-called discrete bounded bisimulation abstractions.

Modelling relationship between antecedent and consequent in modal conditional statements

Abstract: We consider a cognitive agent that is able to utter natural language conditional statements with modal operators of possibility and belief. Our long term aim is to simulate humans' ability to choose a statement within an agent. We are developing formal criteria on when a conditional statement can be said based on agent's knowledge. In previous works we defined a formal model for choosing a proper conditional statement. Here we focus more on a relation between antecedent and consequent. The relation between two phrases that is required to utter a conditional statement. We generalise our previous model to make it more flexible according to requirements and constraints imposed on an agent.

The γ-admissibility of Relevant Modal Logics II -- The Method using Metavaluations

Abstract: The γ/em>-admissibility is one of the most important problems in the realm of relevant logics. To prove the γ-admissibility, either the method of normal models or the method using metavaluations may be employed. The γ-admissibility of a wide class of relevant modal logics has been discussed in Part I based on a former method, but the γ-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist axioms are well known as a means of expressing generalized forms of formulas with modal operators. This paper shows that γ is admissible for relevant modal logics with restricted Sahlqvist axioms in terms of the method using metavaluations.

Organizational decision-makers bias supports market wave formation : evidence with logical formalization

Abstract: Imitation of first-mover firms in opting for a merger or acquisition (M&A) facilitates merger-wave formation. Empirical evidence suggests that, under uncertainty of outcomes, firms regret more not following their rivals’ strategy than possibly failing jointly by copying it. We explore the outcomes and look for underlying behavioral assumptions in that decision-making framework by modal logic. Biased expectations, represented by the B (belief) modal operator, filter out relevant scenarios from managerial consideration. The theorems highlight the drive to imitate first-mover M&As. Our approach goes against the view that human behavior, being non-logical in many respects, defies logic-based rendering. Logic is a flexible representation tool that can model even faulty behavior in a transparent way, also exploring the consequences of the cognitive mistakes made. Our findings suggest that threats to wealth creation may not necessarily find their origins in morally dubious organizational behavior, but rather in modalities of decision making under uncertainty.

Model checking EGF on basic parallel processes

Abstract: In this paper we study the problem of model checking the logic EGF over Basic Parallel Processes (BPP). The logic EGF is obtained by extending the logic EF with the CTL* notation EGF, which means that there exists an infinite path on which there are infinitely many entries satisfying certain property. We prove that this problem is PSPACE-complete, and Σdp-complete for certain classes of fixed formula with the nesting depth d of modal operators.

Modality for Free: Notes on Adding the Tarskian M\

Abstract: We briefly examine the modal formulae that can be derived in Multiplicative Additive Linear Logic (MALL) and some extensions by using Tarksi's extensional modal operators. We also breifly compare this with a substructural form of the modal logic K.