The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia Logica,doi:10.1007/s11225-018-9815-

Axiomatic systems and topological semantics for intuitionistic temporal logic

We propose four axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the four logics obtained from the axiomatic systems are distinct. Finally, we show that when the language is restricted to the `henceforth'-free fragment, the set of valid formulas for the relational and topological semantics coincide

Propositional Logics Complexity and the Sub-Formula Property

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle for a deductive system for a logic L states that whenever F1,...,Fk proves A, there is a proof in which each formula occurrence is either a sub-formula of A or of some of Fi. In this work we extend Statman result and show that any propositional (possibly modal) structural logic satisfying a particular formulation of the sub-formula principle is in PSPACE. If the logic includes the minimal purely implicational logic then it is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our technique can be used to prove that any finitely many-valued logic has the set of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192

A Logical Foundation for Environment Classifiers

Taha and Nielsen have developed a multi-stage calculus {\lambda}{\alpha} with a sound type system using the notion of environment classifiers. They are special identifiers, with which code fragments and variable declarations are annotated, and their scoping mechanism is used to ensure statically that certain code fragments are closed and safely runnable. In this paper, we investigate the Curry-Howard isomorphism for environment classifiers by developing a typed {\lambda}-calculus {\lambda}|>. It corresponds to multi-modal logic that allows quantification by transition variables---a counterpart of classifiers---which range over (possibly empty) sequences of labeled transitions between possible worlds. This interpretation will reduce the "run" construct---which has a special typing rule in {\lambda}{\alpha}---and embedding of closed code into other code fragments of different stages---which would be only realized by the cross-stage persistence operator in {\lambda}{\alpha}---to merely a special case of classifier application. {\lambda}|> enjoys not only basic properties including subject reduction, confluence, and strong normalization but also an important property as a multi-stage calculus: time-ordered normalization of full reduction. Then, we develop a big-step evaluation semantics for an ML-like language based on {\lambda}|> with its type system and prove that the evaluation of a well-typed {\lambda}|> program is properly staged. We also identify a fragment of the language, where erasure evaluation is possible. Finally, we show that the proof system augmented with a classical axiom is sound and complete with respect to a Kripke semantics of the logic

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later