ABSTRACT. We give an overview of the syntax and semantics of Clocked Type Theory (CloTT), a dependent type theory for guarded recursion with many clocks, in which one can encode coinductive types and capture the notion of productivity in types. The main novelty of CloTT is the notion of ticks, which allows one to open the delay type modality, and, e.g., define a dependent form of applicative functor action, which can be used for reasoning about coinductive data. In the talk we will give examples of programming and reasoning about guarded recursive and coinductive data in CloTT, and we will present the main syntactic results: Strong normalisation, canonicity and decidability of type checking. If time permits, we will also sketch the main ideas of the denotational semantics for CloTT.

ABSTRACT. Higher-Order Fixpoint Logic (HFL) is an extension of the modal mu-calculus by a typed lambda calculus. As in the mu-calculus, whether the nesting of least and greatest fixpoints increases expressive power is an important question. It is known that at low type theoretic levels, the fixpoint alternation hierarchy is strict. We present classes of structures over which the alternation hierarchy of HFL-formulas at low type level collapses into the alternation-free fragment, albeit at increase in type level by one.

ABSTRACT. In recent years \emph{schematic representations} of proofs by induction have been studied for there interesting proof theoretic properties, i.e.\ allowing extensions of Herbrand's theorem to certain types of inductive proofs. Most of the work concerning these proof theoretic properties presented schematic proofs as sets of proofs connected by \emph{links} together with a global soundness condition. Recently, the $\mathcal{S}\mathit{i}\mathbf{LK}$-calculus was introduced which provides inferences for expanding the sets of proofs within a schematic proof as well as introducing links without violating the soundness condition. In this work we discuss a simplification of the $\mathcal{S}\mathit{i}\mathbf{LK}$-calculus which isolates the essential mechanisms and provides a path towards the automated construction of schematic proofs.

ABSTRACT. We study the logical complexity of proofs in cyclic arithmetic (CA), as introduced in Simpson '17, in terms of quantifier alternations of formulae occurring. Writing CΣ_n for (the logical consequences of) cyclic proofs containing only Σ_n formulae, our main result is that IΣ_{n+1} and CΣ_n prove the same Π_{n+1} theorems, for n > 0. Furthermore, due to the 'uniformity' of our method, we also show that CA and PA proofs differ only elementarily in size.

The inclusion IΣ_{n+1} ⊆ CΣ_n is obtained by proof theoretic techniques, relying on normal forms and structural manipulations of Peano arithmetic (PA) proofs. It improves upon the natural result that IΣ_n ⊆ CΣ_n.

The inclusion CΣ_n ⊆ IΣ_{n+1} is obtained by calibrating the approach of Simpson '17 with recent results on the reverse mathematics of Büchi’s theorem (Kolodziejczyk, Michalewski, Pradic & Skrzypczak '16), and specialising to the case of cyclic proofs.

These results improve upon the bounds on proof complexity and logical complexity implicit in Simpson '17 and Berardi & Tatsuta '17.

This talk will be based on the following work: http://www.anupamdas.com/wp/log-comp-cyc-arith/

ABSTRACT. We propose a coinductive extension of Miller et. al.'s framework of uniform proof as a machinery for formulating and proving coinductive invariants arising from first-order Horn clause logic programming. It helps the study of coinductive logic programming.

ABSTRACT. Circular (ie. non-wellfounded but regular) proofs received increasing interest in recent years with the simultaneous development of their applications and meta-theory: infinitary proof theory is now well-established in several proof-theoretical frameworks such as Martin Löf's inductive predicates, linear logic with fixed points, etc.

In the setting of non-wellfounded proofs, a validity criterion is necessary to distinguish, among all infinite derivation trees (aka. pre-proofs), those which are logically valid proofs. A standard approach is to consider a pre-proof to be valid if every infinite branch is supported by a progressing thread.

This work focuses on circular proofs for MALL with fixed points. Among all representations of valid circular proofs, a new fragment is described, based on a stronger validity criterion.

This new criterion is based on a labelling of formulas and proofs, whose validity is purely local.

This allows this fragment to be easily handled, while being expressive enough to still contain all circular embeddings of Baelde's muMALL finite proofs with (co)inductive invariants: in particular deciding validity and computing a certifying labelling can be done efficiently. Moreover the Brotherston-Simpson conjecture holds for this fragment: every labelled representation of a circular proof in the fragment is translated into a standard finitary proof.

Finally we explore how to extend these results to a bigger fragment, by relaxing the labelling discipline while retaining (i) the ability to locally certify the validity and (ii) to some extent, the ability to finitize circular proofs.

ABSTRACT. We propose LMSO, a proof system inspired from Linear Logic, as a proof-theoretical framework to extract finite-state stream transducers from linear-constructive proofs of omega-regular specifications. We advocate LMSO as a stepping stone toward semi-automatic approaches to Church’s synthesis com- bining computer assisted proofs with automatic decisions procedures. LMSO is correct in the sense that it comes with an automata-based realizability model in which proofs are interpreted as finite-state stream transducers. It is moreover complete, in the sense that every solvable instance of Church’s synthesis problem leads to a linear-constructive proof of the formula specifying the synthesis problem.

ABSTRACT. Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. We present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria.

ABSTRACT. In the setting of classical first-order logic with inductive predicates, we give as example a theory for which the conjectures that cannot be proved without induction reasoning are still not provable by adding non-trivial explicit induction reasoning.