Yoshihiko Kakutani

Duality between Call-by-Name Recursion and Call-by-Value Iteration

We investigate the duality between call-by-name recursion and call-by-value iteration in the ?µ-calculi and their models. Semantically, we consider that iteration is the dual notion of recursion. Syntactically, we extend the call-by-name ?µ-calculus and the call-by-value one with a fixed-point operator and an iteration operator, respectively. This paper shows that the dual translations between the call-byname ?µ-calculus and the call-by-value one, which is constructed by Selinger, can be expanded to our extended ?µ-calculi. Another result of this study provides uniformity principles for those operators.

Axioms for Recursion in Call-by-Value

We propose an axiomatization of fixpoint operators in typed call-by-value programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform T-fixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinskis fixpoint operator derived from an iterator (infinite loop constructor) in the presence of first-class continuations, provided that we define the uniformity principle on such an iterator via a notion of effect-freeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.

A logic for formal verification of quantum programs

This paper provides a Hoare-style logic for quantum computation. While the usual Hoare logic helps us to verify classical deterministic programs, our logic supports quantum probabilistic programs. Our target programming language is QPL defined by Selinger, and our logic is an extension of the probabilistic Hoare-style logic defined by den Hartog. In this paper, we demonstrate how the quantum Hoare-style logic proves properties of well-known algorithms.

Call-by-name and call-by-value in normal modal logic

This paper provides a call-by-name and a call-by-value calculus, both of which have a Curry-Howard correspondence to the minimal normal logic K. The calculi are extensions of the λµ-calculi, and their semantics are given by CPS transformations into a calculus corresponding to the intuitionistic fragment of K. The duality between call-byname and call-by-value with modalities is investigated in our calculi.

Parameterizations and Fixed-Point Operators on Control Categories

The λμ-calculus features both variables and names together with their binding mechanisms. This means that constructions on open terms are necessarily parameterized in two different ways for both variables and names. Semantically, such a construction must be modeled by a biparameterized family of operators. In this paper, we study these biparameterized operators on Selingers categorical models of the λμ-calculus called control categories. The overall development is analogous to that of Lambeks functional completeness of cartesian closed categories via polynomial categories. As a particular and important application of such consideration, we study the parameterizations of uniform fixed-point operators on control categories. We show a bijective correspondence between biparameterized fixed-point operators and nonparameterized ones under the uniformity conditions.

Semi-automated verification of security proofs of quantum cryptographic protocols

This paper presents a formal framework for semi-automated verification of security proofs of quantum cryptographic protocols. We simplify the syntax and operational semantics of quantum process calculus qCCS so that verification of weak bisimilarity of configurations becomes easier. In addition, we generalize qCCS to handle security parameters and quantum states symbolically. We then prove the soundness of the proposed framework. A software tool, named the verifier, is implemented and applied to the verification of Shor and Preskills unconditional security proof of BB84. As a result, we succeed in verifying the main part in Shor and Preskills unconditional security proof of BB84 against an unlimited adversarys attack semi-automatically, i.e., it is automatic except for giving user-defined equations.

Observational Equivalence Using Schedulers for Quantum Processes

In the study of quantum process algebras, researchers have introduced different notions of equivalence between quantum processes like bisimulation or barbed congruence. However, there are intuitively equivalent quantum processes that these notions do not regard as equivalent. In this paper, we introduce a notion of equivalence named observational equivalence into qCCS. Since quantum processes have both probabilistic and nondeterministic transitions, we introduce schedulers that solve nondeterministic choices and obtain probability distribution of quantum processes. By definition, the restrictions of schedulers change observational equivalence. We propose some definitions of schedulers, and investigate the relation between the restrictions of schedulers and observational equivalence.

Induction by Coinduction and Control Operators in Call-by-Name.

This paper studies emulation of induction by coinduction in a call-by-name language with control operators. Since it is known that call-by-name programming languages with control operators cannot have general initial algebras, interaction of induction and control operators is often restricted to effect-free functions. We show that some class of such restricted inductive types can be derived from full coinductive types by the power of control operators. As a typical example of our results, the type of natural numbers is represented by the type of streams. The underlying idea is a counterpart of the fact that some coinductive types can be expressed by inductive types in call-by-name pure language without side-effects.

A formal approach to unconditional security proofs for quantum key distribution

We present an approach to automate Shor-Preskill style unconditional security proof of QKDs. In Shor-Preskills proof, the target QKD, BB84, is transformed into another QKD based on an entanglement distillation protocol (EDP), which is more feasible for direct analysis. We formalized heir method as program transformation in a quantum programming language, QPL. The transform is defined as rewriting rules which are sound with respect to the security in the semantics of QPL. We proved that rewriting always terminates for any program and that the normal form is unique under appropriate conditions. By applying the rewriting rules to the program representing BB84, we can obtain the corresponding EDP-based protocol automatically. We finally proved the security of the obtained EDP-based protocol formally in the quantum Hoare logic, which is a system for formal verification of quantum programs. We show also that this method can be applied to B92 by a simple modification.

Classical Natural Deduction for S4 Modal Logic

This paper proposes a natural deduction system CNDS4 for classical S4 modal logic with necessity and possibility modalities. This new system is an extension of Parigot’s Classical Natural Deduction with dualcontext to formulate S4 modal logic. The modal λμ-calculus is also introduced as a computational extraction of CNDS4. It is an extension of both the λμ-calculus and the modal λ-calculus. Subject reduction, confluency, and strong normalization of the modal λμ-calculus are shown. Finally, the computational interpretation of the modal λμ-calculus, especially the computational meaning of the modal possibility operator, is discussed.