This article is intended to be an int roduction (for real novices) to the area of proof systems and type systems. We provide very basic notions and their fundamental properties together with some inforrnal ideas behind them.

We start by describing hov 'logical reasoning' can be formulated into a proof system, and how structures of proofs can be studied based on this system. We then proceed to a concise exposition of the notion of computable functions, which emerged from proof theory in the early 1930's$ (during the development of the incompleteness theorems by Göodel). Next, after a short account of pure (type-free) $\lambda$-calculus in connection with computable functions, we discuss three typed $\lambda$-calculi (the simple type system, the second-order type system, and the calculus of constructions) as representatives of type svstems, with emphasis on their relation to the corresponding intuitionistic proof systems (tlle implicational fragment, the second-order propositional logic, and the higher-order predicate logic, respectively).

This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of $\lambda$-calculus. We start by describing the well-known results showing the deep connection between intersection type systems and normalization properties, i.e., their power of naturally characterizing solvable, normalizing, and strongly normal- izing pure $\lambda$-terms. We then explain the importance of intersection types for the semantics of $\lambda$-calculus, through the construction of filter models and the representation of algebraic lattices. We end with an original result that shows how intersection types also allow to naturally characterize tree representations of unfoldings of $\lambda$-terms (Böohm trees).

This paper gives a survey on syntax and semantics for type assignment systems, with a special attention to semantic completeness of the systems. Starting with the most basic system with function types only, it introduces polymorphic types, intersection types, union types, and existential type quantifier in a step-by-step manner. It also provides several sequent-style formulations of type assignment systems. With the sequent calculi, it shows the properties of type assignment systems concerning the completeness and the conservativity of various systems.

We present a simple framework for "non-standard type inference based" analyses of functional programs and show how to apply it to dead-code and strictness analyses. A key feature of this framework is that is based directly on operational semantics.

We first point out some nature of linear logic, in comparison with tra- ditional logics, in Introduction (§ 1), then give the syntax and the intuitive meaning of the syntax in § 2. We give phase semantics of linear logic and a phase semantic proof for the completeness and cut-elimination theorems (at once) in § 3. We formulate a concurrent process calculus as a conservative subsystem of linear logic and give some examples of concurrent process specification by linear logic in § 4. In § 5 we give a simplified phase semantics and the completeness for a fragment of linear logic, in which some concurrent process models such as Petri nets can be formulated.