It is no exaggeration to say that the synthesis of computing and mathematical logic has become one of the most active research areas in the general field of theoretical computer science. The reasons for this development may be manifold, but the following factors certainly deserve some special emphasis: Mathematical logic is a (the) science of symbol manipulation and symbolic computation. Logic is a framework for knowledge representation and knowledge engineering. Logic concepts form a link between computing, artificial intelligence, expert systems, decision support systems and the like. One should also mention the raise of P R O L O G and the Japanese 5 th generation computing and observe that during the last few years generations of new computers and computer architectures have been developed which are so powerful that now many theoretical concepts can be realized \"on the machine\" which had been considered unfeasible until recently. In a way, constructive mathematics at the top and P R O L O G at the bottom form the borderlines of the world o f logical programming one of the most fascinating areas of research where logic meets computer science. A lot is known about both extremes, and serious efforts are made to bridge the gap between the theoretical analysis of constructive mathematics and the practical experience one has gained in many PROLOG like applications. In the following we give some examples of the interplay between logic and computer science where choice of topics is necessarily influenced by our personal scientific background. The red thread is the idea to survey several formalism for knowledge representation in a very wide sense. We emphasize the notions of proof and provability and try to convey principal concepts rather than precise technical details. However, this article is too brief to deal with these topics in detail; other topics cannot be discussed at all.

Proof Theory

Between constructive mathematics and PROLOG

We consider an extension of propositional Godel logic by an unary operator that enables the addition of a positive real to truth values. We provide a suitable calculus of relations and show completeness and cut elimination.

background

Cut Elimination for Gödel Logic with an Operator Adding a Constant

We define an ordinal notation system simultaneously with its ordering. Our simultaneous definitions generate only the ordinal terms in Cantor normal form which are in one-to-one correspondence with the ordinals below $\varepsilon_0$. We also define the ordinal arithmetic operations on our ordinal notations and show that they coincide with those on set-theoretic ordinals. We implement the ordinal notation system in the Agda proof assistant as simultaneous inductive-inductive-recursive definitions.

Ordinal notations via simultaneous definitions

The course is focussed on the articulation of deductions and computations in informal and formal mathematical proofs. The axiomatic method put the emphasis on deduction steps, but since the seventies several notions of proofs have given an important role to both notions and recent mathematical proofs such as proof of the four color theorem, the double bubble theorem or Hale’s theorem have shown that computation seems to be unavoidable. The rst part of the course will be dedicated to the use the articulation of deductions and computations in proof theory, in particular in the study of cut elimination processes. We will show that computation rules allow to avoid axiom in many theories and that a general cut elimination theorem can be proved for all these theories. The second part of the course will be dedicated to the use of the articulation of deductions and computations in proof processing systems such as Coq. We will show that some theorems cannot be proved in practice when we do not have computation rules and how computation rules give a new status to proofs of programs.

https://asimod.informatik.tu-muenchen.de/2005/abstracts/Dowek_05.pdf

Deduction and Computation with Coq

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.

Proof Theory

Citations

Between constructive mathematics and PROLOG

Cut Elimination for Gödel Logic with an Operator Adding a Constant

Ordinal notations via simultaneous definitions

Deduction and Computation with Coq

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic