OUTLINE 1. The Discipline of Computer Science 2. Semantics 3.
Logic and Computation 4. Computability 5. The Ontology of Computer
Science 6. Computational Knowledge 7. Computational Thinking 8.
Legal and Ethical Questions 9. Further Issues
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I THE DISCIPLINE of COMPUTER SCIENCE
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What is Computer Science? More than programming The methodology
of programming and system design The design and exploration of
programming and specification Languages The invention and
exploration of new computational paradigms The semantics of
programming languages The design of interpreters and compilers The
relationship between deduction and Computation The design of
embedded and distributed systems The design and exploration of
architectural description languages Software engineering AI,
robotics, Computational science: computational biology,
computational physics, Computational finance
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Computer Science, Mathematics, Science and Engineering
Mathematics used to build mathematical models of computational
artefacts e.g., analysis of algorithms, semantics of programming
languages. Mathematics used in the activity of programming:
correctness. Experimentation: empirical testing to see if a system
meets its specification (which acts as a theory) Engineering :
design methodologies CS involves all these aspects: but is this
different to engineering in general?
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Lessons and Overlaps Philosophy of mathematics: the nature of
mathematical objects and mathematical knowledge Philosophy of
language: theories of meaning and semantics Philosophy of science
and the nature of experimentation e.g. computational experiments in
evolutionary programming. Specification and testing in software
development. Metaphysics: the ontological status of computational
artefacts Ethics: computer ethics Philosophy of engineering: the
nature of practical knowledge
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Some Questions What kinds of things are programs? Are they
abstract or concrete? Do programs have a dual existence do they
exist in both physical and symbolic forms? What are their equality
criteria: when are two programs taken to be the same (legal
aspect)? What is the relationship between programs and algorithms?
Are the former linguistic and the latter mathematical objects? How
do the they differ? Many computer scientists will claim its syntax
all the way down. What kind of things are types/data types? Are
they best seen/modelled as sets or categories (in both senses) or
something else? What distinguishes hardware from software? What
kinds of things are digital objects? What is their identity
criteria? Do we need a new ontological category to house them? What
are the objectives of the various semantic theories of programming
languages? What is the difference between operational and
denotational semantics? What does it mean for a program to be
correct? What do correctness proofs establish? What is abstraction
in computer science? How is it related to abstraction in
mathematics? What are formal methods? What is the difference
between a formal method and informal one? How is mathematics used
in computer science? Are mathematical models used in a descriptive
or normative way? Does the Church-Turing thesis apply to physical
machines? Is there such a thing as computational thinking?
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II SEMANTICS
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Syntax and Semantics A programming language is not completely
determined by its grammar; it only pins down what the legal strings
of the language are. We also need to supply a semantic account that
provides the meaning of the constructs of the language. In the
first instance a semantics is normative. But it is also needed for
users of the language. It must facilitate and support 1. The design
and verification of correct programs 2. The construction of
compilers 3. Proofs of properties of programs and systems 4.
Etc..
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Approaches to Semantics 1. Physical Machine 2. Natural language
interpretation 3. A translation into a (known) Programming Language
4. A mathematical account
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Machine Semantics One popular view is that the semantic account
is supplied by an implementation on physical machine i.e., the
intended meaning is to be given by the actual effect on the state
of a physical machine. Consider a device found in nature with red
and yellow buttons. In order to discover what the yellow button
does we push it. We observe that the yellow button moves the head.
But designed machines are not like that. They are designed with a
purpose that is independently specified. The meaning for the
instruction push the yellow button must be given independently of
the physical machine. Without this there can be no notion of the
machine malfunctioning (not meeting the designers intentions): it
just does what it does. But how is such an independent
specification to be given?
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Natural Language Consider the NL specification push the yellow
button means move the head forward Here the semantics of the
instruction given in a natural language. For example, we might
specify that If B then C else D means Test B. If true then run C.
If B is false then run D But what do test and run mean? These words
are vague and so cannot really pin down the meaning of an
artificial language.
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Metacircular Interpreters Defined via an interpretation into
another programming language (or a subset of the source one) L L
Here the second language is intended to provide the meaning of the
first. There are two objections: 1. Infinite regress: Kripkes
Wittgenstein (later) L L L L........... 2. Logical Requirement
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Logical Requirements Such accounts do not easily support
argumentation and exploration of the properties of the language and
programs written in it. We need to be able to reason about programs
and articulate and establish their properties. There is no
mathematics here.
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Mathematical Accounts
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Operational Semantics Abstract machine semantics: given by
rules of state change e.g., the reduction rules of the lambda
calculus or the SECD machine. Some argue that operational accounts
are not mathematical We can apparently get quite a long way
expounding the properties of a language with purely syntactic rules
and transformations........One such language is the Lambda Calculus
and, as we shall see, it can be presented solely as a formal system
with syntactic conversion rules..........But we must remember that
when working like this all we are doing is manipulating symbols-we
have no idea at all of what we are talking about. To solve any real
problem, we must give some semantic interpretation. We must say,
for example, "these symbols represent the integers". Stoy. Peter
Landin ISWIM/Dana Scott OWHY.
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Denotational Semantics Meaning via denotation into mathematical
structures. Set-theoretic semantics. Often generalised to category
theory. Programs and types associated with abstract mathematical
objects. But why are such accounts philosophically different to
ones given by abstract machines/reduction rules? These issues
reflect the realism/antirealism debates in the philosophy of
mathematics.
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Infinite Regress Challenge The infinite regress generated by
replacing one language by another? Are mathematical accounts
different to translational ones? Even set theory is written in a
language and rules or axioms need to be grasped or interpreted. All
end in axiomatic theories Kripke, Wittgenstein and Rule
following
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Axiomatic Axiomatic theory of language constructs e.g. Direct
axiomatisation of state change: Hoare logic provide a relation
between states induced by programming constructs. Axiomatic
accounts can be viewed as the postulation of a new ontology: an
ontology of computational objects.
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Criteria For An Adequate Semantic Theory 1. It must deliver the
correct truth conditions e.g. while b do S b S; while b do S (x t
)s t[s/x] 2. It must provide a plausible explanation of what it is
that a users understanding of the language consists in 3. And it
must be comprehensible how someone might acquire that understanding
of the language 4. Which of the accounts best satisfies these
requirements? 5. Different roles for different approaches
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III LOGIC and COMPUTATION
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Logical Questions Many issues here. Many new logics: Hoare
Logic, Dynamic logic, sub-structural logics, partial and three-
valued logics, temporal and modal logics,.... What is the
appropriate logic with which to reason about program correctness
and termination? Programs may not terminate. So should we use a
logic with truth value gaps? Or should we employ partial logic?
Does it need to be typed? Should the underlying logic be
constructive or classical? How do we decide? Are the criteria
practical or conceptual?
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The Development of Programs and Systems The core of the subject
and the home of many conceptual questions Specification
Implementation Program construction Proofs of correctness
Testing
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Functional Specifications A so-called functional specification
lays down the relationship that should hold between the input and
the output. Two parts 1. Declaration of variables and their types;
some indicated as inputs and some as outputs 2. A statement of the
relationship between the input and output 3. It can be represented
as: [x:T,y:S | R(x,y)]
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Correctness A program p with input type T and output type S is
taken to be correct relative to a specification [x:T,y:S | R(x,y)]
if, for every input x:T, p yields an output y that together with x
satisfies R i.e., For all x:T, R(x,p(x))
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Proofs of Correctness Usually carried out in some version of
(typed) predicate logic etc. There have to be rules that govern the
relationship between programs and specifications e.g. program
sequencing x:T R ( x, p ( x )) y : S G ( y, q ( y )) x:T RG ( x,
p;q ( x )) The proofs are carried out inside such a system
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Two Philosophical Concerns 1. The Epistemological challenge 2.
The Empirical challenge
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The Epistemological Challenge Correctness proofs are usually
long, shallow and tedious to carry out. Often they are constructed
with the aid of theorem provers. What is the epistemological status
of correctness proofs? Generally, are they too long to be
graspable? What is it to be graspable? Do we have apriori knowledge
of the results? The role of new mathematical concepts and
abstraction in creating graspable proofs. Should these be seen as
making a proof graspable? Or are we constructing a new proof or
even putting in place a proof that was not there before?
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The Empirical Challenge Mathematical correctness is established
up to the abstract specification But there is always an empirical
residue that pertains to the target physical machine: this can
still go wrong Hence, proofs of correctness cannot establish that
the program will work on a physical machine Whatever we prove
formally, the machine may malfunction
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Physical Correctness We cannot prove correctness of a machine
code relative to an actual physical machine. Indeed, how is the
machine code fixed? Not by the machine itself (recall semantic
discussion). To meaningfully talk about the correctness of the code
we require an independent semantic specification (presumably of the
designers intentions) Whether the physical machine behaves
according to this specification of its machine code is an empirical
matter. The question is not whether code is correct but whether the
physical machine correctly (in terms of the specification of the
machine code ) implements the code. It is the machine that is right
or wrong not the software.
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IV COMPUTABILITY
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Turing Machines Turing machines were intended to capture the
computational ability of human calculators One interpretation of
the Church-Turing thesis is that Turing machines do this and only
this (Copeland) But do Turing machines also provide the limit to
what can be physically computed?
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Gandy Machines Gandy stated four principles that define a class
of discrete deterministic mechanical devices (Gandy machines).
Gandy machines are intended to reflect what can be computed in the
physical world. Gandy proved that Gandy machines are equivalent to
Turing machines. But are there examples of (ideal) discrete
deterministic mechanical devices that contravene Gandys principles
but are consistent with the physics of the real world?
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A Infinite Speed-up machine Imagine a machine that speeds-up at
every step i.e., it performs the same number of computation steps
in half the time of the previous step. This will perform 4 steps in
1+1/2+1/4+1/8 seconds In the limit it will perform an infinite
number of steps in 2 seconds But is this physically possible? Is
this consistent with the physics of the real world?
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V ONTOLOGY
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The Objects of Computer Science Programs Algorithms
Specifications Data types Software systems Processes Digital
objects Programming languages Specification languages.....
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Some Questions What kinds of things are programs? Are they
abstract or concrete? Are programs (literally ) strings of symbols
that are well-formed according to some grammar i.e., linguistic
objects. Are they different to algorithms? Are the former
linguistic and the latter mathematical objects? What is the
difference? What is the relationship between programs and
algorithms? What are the equality criteria for programs? Are two
programs taken to be the same when they encode the same algorithm?
But what is the equality criteria for algorithms? When are two
types taken to be the same? Is it when they are extensionally the
same. Can these issues can only be sensibly addressed in the
context of a programming language?
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Semantics and Ontological Questions A tradition that has its
modern origins in Frege and Dummett has it that ontology is
determined by semantics. Each semantic theory for a programming
language generates an ontology and a notion of program equality.
The criteria for a correct semantics is clearer to articulate (than
metaphysical ones)
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Ontological Frameworks Are types abstract mathematical objects?
If so which ones: sets or categories? Most textbooks implicitly
treat them as sets. However, the average user does not know or
employ these frameworks. In practice, types (e.g. numbers, lists,
trees, operations and abstract and polymorphic types) play a sortal
role and this is articulated as type-inference rules e.g. x:T t: S
s: T S a:T ------------------ x:T t :T S sa:S Here type membership
is decidable.
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Computational Ontology Do these remarks argue for a direct
axiomatisation of programming languages? This would argue for a new
ontological framework in which programs and types are taken as
ontologically primitive notions And this framework does not
supervene on sets or categories of anything else.
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VI COMPUTATIONAL KNOWLEDGE
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Kinds of Knowledge Theoretical Knowledge is traditionally
characterised as knowing that something is the case e.g. 2+2=4
Practical Knowledge is traditionally characterised as knowing how
to do something: I know how to ride a bicycle Are these distinct
and exhaustive?
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Knowledge of a Third Kind? Can you do the tango? Silly answer:
I dont know; I have never tried it. Better: No. I do not even know
exactly what is involved. In learning the tango, you are not only
learning how to do it but also what it is to do it. Of course, you
may not be able to articulate that knowledge; the best you may be
able to do is to demonstrate it. Your knowledge lies midway between
theoretical and practical knowledge. Dummett argues that knowledge
of natural language is of this kind
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Knowing How to Program Is knowledge of programming and
programming languages also of this kind? Can you articulate your
knowledge of your ability to program in Miranda, C or Java? Is it
possible to express the knowledge involved in being able program
other than by demonstration?
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VII COMPUTATIONAL THINKING
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Thinking like a Computer Scientist Approach to problem solving.
Think procedurally. Use abstraction. Use type checking. Go for
implementable solutions... Provides a general set of skills that
transcend the ability to write software Gaining credibility. E.g.
UK RAE and benchmarks But is it distinctive? Is it different to
mathematical thinking? How?
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Two Styles of Problem Solving Computational Mathematical Think
procedurally to solve problems Specification with implementation as
the solution Tractability and efficiency as desiderata for solution
Abstraction, both data and procedural Use problem decomposition
etc. Type-checking, as a generalization of dimensional analysis.
Solve Problems any way you can The solution can take non-
constructive forms Solution does not even have to be computable
Abstraction is at the heart of mathematics Use problem
decomposition etc. Type-checking not necessarily a part of
mathematical problem solving
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Abstraction in Maths and CS Is there one kind of abstraction?
Is this captured by Freges principle? The direction of A=The
direction of B iff A is parallel to B How does CS abstraction
differ from abstraction in mathematics? Is abstraction in CS
distinctive because the implementation into lower levels must be
(eventually) present? In mathematics the examples of the
abstraction may involve proof to check that they are instances.
However, they are not necessarily algorithmically checkable.
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Types in Maths and CS How are abstract types and polymorphism
used in CS? Type-checking as a semantic extension to syntax. How
are types used in mathematics? Originally to avoid paradox. But
functions are typed.
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VIII LEGAL AND ETHICAL QUESTIONS
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Legal Questions Identity of programs: semantics and identity
revisited. The law, algorithms and the ideas behind/underneath
programs Copyright and patents
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Ethical Questions Obligations and correctness: Safety critical
software. Any different to the civil engineer? The size and
complexity of software and correctness proofs: does it mitigate
against the obligation to carry out correctness proofs not feasible
in practice
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IX FURTHER ISSUES Issues in the philosophy of mind and
computationalism Hypercomputation The design of programming
languages Computational paradigms..............
