Rigid First-Order Hybrid Logic

We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators , thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K , a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

14 Hybrid logics

Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

The usual Skolemization procedure, which removes strong quantifiers by introducing new function symbols, is in general unsound for first-order substructural logics defined based on classes of complete residuated lattices. However, it is shown here (following similar ideas of Baaz and Iemhoff for first-order intermediate logics in [1]) that first-order substructural logics with a semantics satisfying certain witnessing conditions admit a “parallel” Skolemization procedure where a strong quantifier is removed by introducing a finite disjunction or conjunction (as appropriate) of formulas with multiple new function symbols. These logics typically lack equivalent prenex forms. Also, semantic consequence does not in general reduce to satisfiability. The Skolemization theorems presented here therefore take various forms, applying to the left or right of the consequence relation, and to all formulas or only prenex formulas.KeywordsFunction SymbolResiduated LatticeIntuitionistic LogicPredicate SymbolSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hybrid logics are an extension of modal logics where it is possible to refer to a specific state, thus allowing the description of what happens at specific states, equalities and transitions between them. This makes hybrid logics very desirable to work with relational structures.

The question of axiomatizability of first-order temporal logics is studied w.r.t. different semantics and several restrictions on the language. The validity problem for logics admitting flexible interpretations of the predicate symbols or allowing at least binary predicate symbols is shown to be Π1 1-complete. In contrast, it is decidable for temporal logics with rigid monadic predicate symbols but without function symbols and identity.

In this paper, we integrate Separation Logic with propositional Projection Temporal Logic (PPTL) to obtain a two-dimensional logic, named $$\text {PPTL}^{\tiny \text{ SL }}$$ PPTL SL . The spatial dimension is realized by a decidable fragment of separation logic which can be used to describe linked lists, and the temporal dimension is expressed by PPTL. Furthermore, we prove that any $$\text {PPTL}^{\tiny \text{ SL }}$$ PPTL SL formula can be transformed into its normal form. Example are given to show how to specify temporal heap properties by this hybrid logic.

A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable GTRS, the subject reduction property does not hold. Using the principal type for the left-hand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

We generalize the notion of symmetries of propositional formulas in\nconjunctive normal form to modal formulas. Our framework uses the coinductive\nmodels and, hence, the results apply to a wide class of modal logics including,\nfor example, hybrid logics. Our main result shows that the symmetries of a\nmodal formula preserve entailment.\n

Information flow control is a long-studied approach for establishing non-interference properties of programs. For instance, it can be used to prove that a secret does not interfere with some computation, thereby establishing that the former does not leak through the latter. Despite their potential as a holy grail for security reasoning and their maturity within the literature, information flow type systems have seen limited adoption. In practice, information flow specifications tend to be excessively complex and can easily spiral out of control even for simple programs. Additionally, while non-interference is well-behaved in an idealized setting where information leakage never occurs, most practical programs must violate non-interference in order to fulfill their purpose. Useful information flow type systems in prior work must therefore contend with a definition of non-interference extended with declassification , which often offers weaker modular reasoning properties. We introduce structural information flow , which both illuminates and addresses these issues from a logical viewpoint. In particular, we draw on established insights from the modal logic literature to argue that information flow reasoning arises from hybrid logic , rather than conventional modal logic as previously imagined. We show with a range of examples that structural information flow specifications are straightforward to write and easy to visually parse. Uniquely in the structural setting, we demonstrate that declassification emerges not as an aberration to non-interference, but as a natural and unavoidable consequence of sufficiently general machinery for information flow. This flavor of declassification features excellent local reasoning and enables our approach to account for real-world information flow needs without compromising its theoretical elegance. Finally, we establish non-interference via a logical relations approach, showing off its simplicity in the face of the expressive power captured.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

On the interpretability of arithmetic in temporal logic

By a theory is meant an applied first-order predicate calculus with at least one predicate symbol and perhaps some individual constants and function symbols and a specified set of axioms. In addition to the terms defined by means of the individual variables, constants, and function symbols a theory may also include among its terms those constructed by means of operators such as the epsilon or iota operators; that is, expressions like (εχΡ) or (οχΡ), where P is a well formed formula (wff) of the theory, may also be terms. A constant term of a theory F is then a term in which no variable occurs free. We are interested only in theories which have at least one constant term so that if a theory doesn't have any individual constants it must necessarily admit as terms expressions constructed by means of operators. A sentence of a theory F is a closed wff.

Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because first-order logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program. Higher-order quantification can be use to hide predicates and function symbols. If such higher-order quantification is restricted so that operationally, only hiding is specified, then the cost of such higher-order quantifiers within proof search can be small: one only needs to deal with adding new eigenvariables and clauses involving such eigenvariables. On the other hand, the specification of hiding via quantification can allow for novel and interesting proofs of logical equivalence between programs. This paper will present several example of how reasoning directly on a logic program can benefit significantly if higher-order quantification is used to provide abstractions.KeywordsLogic ProgramLogic ProgrammingSecurity ProtocolLinear LogicHorn ClauseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.