A
Survey of
Computability
Logic
გამოთვლადობის ლოგიკა
Логика
вычислимости
可计算性逻辑
(CoL)
Under the approach of CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their “truth” is seen as algorithmic solvability. In turn, computational problems --- understood in their most general, interactive sense --- are defined as games played by a machine against its environment, with “algorithmic solvability” meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question “what can be computed?”, just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases “what can be computed” always allows itself to be replaced by “how can be computed”, which makes CoL of potential interest in not only theoretical computer science, but many applied areas as well, including constructive applied theories, interactive knowledgebase systems, resource oriented systems for planning and action, or declarative programming languages.
Currently
CoL is still at
an early stage of development, with open problems prevailing over
answered
questions. For this reason it offers plenty of research opportunities,
with
good chances of interesting findings, for those with interests in logic
and its
applications in computer science.
This article
presents a survey of the subject: its philosophy and motivations, main
concepts and most significant results obtained so far. No proofs of
those results are included.
Contents
1.5
CoL vs.
intuitionistic logic
1.6
CoL vs.
independence-friendly logic
1.7 The ‘from
semantics
to syntax’ paradigm
2
Games
2.1
The two players
2.3
Constant games
2.4
Not-necessarily-constant
games
2.5
Static games
3
The CoL zoo of game
operations
3.1 Preview
3.2
Prefixation
3.3 Negation
3.6
Reduction
3.7
Blind operations
3.10
Toggling operations
3.11
Cirquents
5
The
language
of CoL and its semantics
5.1
Formulas
5.2
Interpretations
5.3
Validity
6.1
Outline
6.2
The Gentzen-style
system CL7
6.3
The Gentzen-style
system Int+
6.4
The cirquent
calculus system CL15
6.5
The brute force
system CL13
6.6
The brute force
system CL4
6.7
The brute force
system CL12
7
Clarithmetic (CoL-based
arithmetic)
7.1
Introduction
7.2
Clarithmetic versus
bounded arithmetic
7.3
Motivations
7.4
Common preliminaries
for all our theories of
clarithmetic
7.6
Clarithmetics for
provable computability
8
CoL-based knowledgebase
and resourcebase systems
9.1
Selected papers on
CoL by Japaridze
9.2
Selected papers on
CoL by other authors
9.3
PhD theses, MS
theses and externally funded
projects on CoL