Diaspora of a Mathematics of Argument

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Diaspora of a Mathematics of Argument

• R. LouiDept of Computer Science
• Washington University
• St. Louis

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Outline

• I. Intellectual History of Process-Based Models
  of Reasoning
• II. Some Technical Issues regarding Argument
• III. Foundations
• A. Probability
• B. Decision
• C. Legal Reasoning
• D. Belief Revision/Deontic Logic
• E. Negotiation
• F. Rhetoric
• IV. Future Work
• A. Fairness
• B. Computation

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CS TALKScope of My Current CS Work

• cgi in gawk
• book with S. Sachs
• independent co-malloc for localizing dynamically
  allocated objects
• optimal average hash chain length for gawk
• malloc with a vmstat time series estimator for
  elective memory expansion
• gnu release(s) with M. Waldvogel, M. Pachos, K.
  Krouse
• something with FPGA's
• patent license with J. Lockwood, J. Moscola,
  M. Pachos

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CS TALKScope of My Current CS Work (cont.)

• purely probabilistic negotiating agents
• model, simulations
• real-time object recognition for aerial targets
• hiding among non-combatants
• half-baked ideas, proposal with R. Pless
• AI and Law service
• journal, ICAIL, JURIX, special issues,
  workshops, JD-PhD's, no

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WHAT IS LOGIC?What do Computer Scientists think
is Logic?

• Roughly Hilbert-Russell-Whitehead tradition
• 1. there is one correct logic
• it is either the predicate logic or the
  propositional logic or both
• 2. entailment (syntactic or semantic?) has
  something to do with mathematical proof
• 3. logic codifies correct ways of reasoning
• 4. logic has something to do with the success
  of hardware

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WHAT IS LOGIC?What do Computer Scientists think
is Logic?(cont.)

• Some more advanced members of our species
• Knowledge Representation
• 1. logics are like programming languages can
  be chosen or designed
• w/o metaphysical consequence
• 2. some logics are more expressive than others
• 3. some logics license more inferences than
  others
• 4. inferential license and expressiveness are
  complementary

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WHAT'S NEW/DIFFERENTWhat is the Difference
between Argument and Deduction?

• Diagram of an argument
• ltp, lt a,b, p gt,
• ltc, agt,
• ltd, bgt
• gt
• Diagram of a proof
• ltp1, p2, , pgt

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Obvious

• Argument Deduction
• nondemonstrative demonstrative
• if p then defeasibly q if p then materially q
• nonmonotonic monotonic
• argument proof
• subargument subproof
• counterargument (counterproof?)
• defeat (fallibility? corrigibility?)
• inconsistency-tolerating inconsistency-degenerati
  ng

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Less Obvious

• Argument Deduction
• focus is on metalanguage focus is on object
  language
• conflict, rebuttal, warrant and, or, not
• anytime ideal
• warrant w.r.t. arguments commitment at all t
• produced in time t
• Warranted(S,t) Thm(S), Proved(S,t)
• constructive nonconstructive
• p's warrant underdetermined
• ampliative nondeterminism nonampliative
  conclusions
• of process in the meanings of words

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Less Obvious (continued)

• Argument Deduction
• strategy-based constraint-based
• choices proof constraint-propagation
• dialectical unilateral
• sits between objectivist and invites principle
  of charity
• relativist conceptions of truth
• formally, 10-20 years old formally, 100-150
  years old
• 20thC seminal in social sciences 20thC dominant
  in logic

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Variations of Logic

• Modal Logic opaque contexts
• gtgt notation for beliefs about beliefs
• Fuzzy Logic predication weakened
• gtgt smoother control, washing machines
• Multivalued Logic truth weakened
• gtgt semantic curiosities, reductions
• Relevance Logic implication weakened
• gtgt model of limited inferential capacity

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Variations of Logic (cont.)

• Intuitionist Logic weak negation added
• gtgt first step toward elevation of process
• Counterfactual Logic second implication added
• gtgt plausible alternative conditional
• Paraconsistent Logic meaning from inconsistency
• gtgt proof-theory for consistent subsets
• Belief Revision recovery from inconsistency
• gtgt model of premise adoption/retraction
• Argument ties logic to computation in a
  fundamental way
• gtgt rewrite foundations of other fields

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INTELLECUTAL HISTORY

• 1. where did the idea of defeasibility come
  from?
• 2. where did the idea of procedural rationality
  come from?
• 3. where did the idea of argument come from?

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AI"tweety is a bird, but tweety does not fly"

• McCarthy-Hayes -- Modal Belief
• 1973
• Reiter --Closed World Databases
• 1978
• Doyle -- TMS
• 1978
• Kowalski -- PROLOG
• 1974
• Clark -- Negation as Failure
• 1978
• Argument systems
• 1987...
• Pollock -- Defeasible Reasoning
• 1968-1974-1986-1987
• Nute -- Defeasible PROLOG
• 1985
• Kyburg -- system for probability based on defeat
• 1961-1974

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Epistemology"it seems red therefore it is red"
is defeasible

• Belzer -- Defeasible Reasoning
• 1986
• Swain -- Epistemic Defeasibility
• 1978
• Pollock -- Knowledge Justification
• 1974
• Sosa -- Conceptions of Knowledge
• 1970
• Lehrer-Paxson -- Knowlege
• 1969
• Firth -- Coherence
• 1964
• Chisholm -- Perceiving
• 1957-1964
• Ladd -- Structure of a Moral Code
• 1957

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Reasoning (Qualitative Decision Theory)"doing a
achieves the goal, therefore do a" is defeasible

• Nozick -- Practical Reason/Explanations
• 1981
• Searle -- Prima Facie Obligations/Practical
  Reason
• 1978-1985
• Raz -- Practical Reason/Norms
• 1970
• Gauthier -- Practical Reasoning
• 1963

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Ethical Reasoning"a person has a prima facie
obligation or responsibility"

• Glover -- Responsibility
• 1970
• Nozick -- Moral Structures
• 1968
• Feinberg -- Action and Responsibility
• 1965
• Wellman -- Language of Ethics
• 1961
• Brandt -- Blameworthiness and obligation
• 1958
• Melden -- Action/Rights
• 1956-1959
• Mackie -- Responsibility and Language
• 1955
• Hare -- Language of Morals
• 1952
• (note Stevenson 1938 and Ross 1930)

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Political Justification

• Barry -- Political Argument
• 1965
• Rawls -- Pure Procedural Justice
• 1958-1974

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Dialectics/Rhetoric

• Rescher -- Dialectics
• 1977
• Perelman -- Justice and Argument
• 1963
• Toulmin -- Uses of Argument
• 1958

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Origins

• Ladd
• Raz
• Gauthier
• Wellman
• Brandt/Melden/Hare
• Barry
• Rawls
• Perelman/Toulmin
• Hart

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Origins (continued)

• Hart -- Ascription of Responsibility
• 1948
• Wisdom -- Gods
• 1945
• Waismann -- Verifiability
• 1951
• Austin -- Speech Acts
• 1947?
• Wittgenstein -- Remarks on Foundations of
  Mathematics
• 1935?
• Keynes -- Treatise on Probability
• 1908 thesis begins Part of our knowledge we
  obtain direct and part by argument.
• Bentham -- Principles of Morals and Legislation
• 185?

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Defeasibility

• When the student has learnt that in English law
  here are positive conditions required for the
  existence of a valid contract, - his
  understanding of the legal concept of a contract
  is still incomplete, ... For ... he has still to
  learn what can defeat a claim that there is a
  valid contract, even though all these conditions
  are satisfied. The student has still to learn
  what can follow on the word "unless", which
  should accompany the statement of these
  conditions. This characteristic of legal
  concepts is one for which no word exists in
  ordinary English ... but the law has a word which
  with some hesitation I borrow and extend this
  is the word defeasible...
• (Hart vs. Aristoteliean Society, 1951, p. 152)

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Process

• ... Principia Mathematica gives rise to
  questions about the relation in which ordinary
  reasoning stands to this ordered system, and in
  particular, as to the precise connection between
  the process of inference, in which the older
  logicians were principally interested, but which
  Russell ignores, and the relation of
  implication, on which his scheme depends.
• The gradual perfection of the formal treatment
  ... had been to empty logic of content and to
  reduce it more and more to mere dry bones, until
  finally it seemed to exclude ... most of the
  principles, usually deemed logical, of reasonable
  thought.
• (Keynes vs. Russell, Whitehead, Ramsey,
  1908/1921/1973, p. 118, 1972, p. 243)

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Formal Inconsistency

• WITTGENSTEIN Think of the case of the Liar. It
  is very queer in a way that this should have
  puzzled anyone-- Because the thing works like
  this if a man says 'I am lying' we say that it
  follows that he is not lying, from which it
  follows that he is lying and so on. Well, so
  what? ... It does not matter. ... it is just a
  useless language-game, and why should anyone be
  excited?
• TURING ... one usually uses a contradiction as
  a criterion for having done something wrong. But
  in this case one cannot find anything done wrong.
• WITTGENSTEIN Yes -- and more nothing has been
  done wrong. ... where will the harm come?
• (Wittgenstein vs. Turing, 1939, Hodges, p. 154)

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RECENT TECHNICAL QUESTIONS

• 0. knee-jerk reaction (deductivists)
• Q. isn't
• p gt r,
• p q gt r
• reducible to
• p q --gt r
• p q --gt r ?
• A. no.
• what can be concluded with just p? r.
• does that imply q? no.

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TECHNICAL

• I. old issues (Touretzky, Horty, Thomason, 1987)
• Q. skeptical vs. credulous
• A. skeptical
• Q. ambiguity-propagating vs. blocking
• A. depends whether undercut or rebut
• Q. syntactic specificity
• 1. (strict specificity)
• p gt r p q gt r
• 2. (shortcut specificity)
• p gt q q gt r p gt r
• 3. (defeasible specificity)
• p gt r q gt r
• p gt q

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TECHNICAL (continued)

• A. keep it simple (Nute, 1990)
• A. appeal to convention (Simari-Loui, 1992)
• A. give explicit ordering (Lin-Shoham, 1987,
  Vreeswijk, 1991)
• A. provide for meta-argument about defeat
• e.g., context-dependent defeat
  (Prakken-Sartor, 1995)
• r1 p gt q
• r2 r gt q
• s gt r1gtr2

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TECHNICAL (continued)

• II. principles versus rules (Loui-Norman, Hage,
  Verheij, 1995-2001)
• Q. Is there a formal difference between
• 1. no vehicles in the park
• 2. parks should be peaceful
• A. Rationales can recall rationales during
  dispute
• A. Principles can be weighed free speech
  versus privacy

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TECHNICAL (continued)

• III. rules for fair dialectic (Loui, Gordon,
  Vreeswijk, Lodder, 1992-2001)
• Q. What is the exact procedural burden?
• 1. pro argument 1 for p
• 2. con argument 2 for p, and
• argument 2 defeats argument 1.
• a. should con have the burden of showing
• argument 2 defeats argument 1?
• or
• b. should the claim be presumable and subject
  to dispute?

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TECHNICAL (continued)

• Q. are normal default rules fair?
• p q / q
• 1. pro argument 1 for p
• 2. con a. proof of q
• b. (it does not suffice to argue q?)
• Q. what is the penalty of failed attempts to
  rebut?
• 1. pro argument 1 for p based on b and c,
  etc.
• 2. con b and c.
• 3. pro why b?
• 4-15. con ... pro ... con loses the
  subdispute over b.
• 16. con nevertheless, c.
• A. rhetorical costs HIGH. logical costs
  NONE?

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TECHNICAL (continued)

• IV. rules extracted from cases
• Q. What is the structure of a precedent case?
• A. (Raz, 1970)
• a b c d e f / q
• A. (Rissland-Ashley, 1985-1990)
• a b c d- e- f- / q
• A. (Loui-Norman, 1992)
• argument 1(a,(b,c) q)
• argument 2(a,d,e q)
• argument 3(e,f d)
• argument 4(c f)
• _______________
• q

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TECHNICAL (continued)

• Q. What is the rule of the case?
• A. (Loui-Norman, 1995)
• sufficient premises of arguments in
  dialectical subtree with leafs that are pro
  arguments
• a b c d e f gt q
• but no false specificity
• a b c f gt q
• A. (Prakken-Sartor, Bench-Capon, 1992-2001)
• Argument1, Argument3 gt Argument2, Argument4

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TECHNICAL (continued)

• V. criteria for theory-formation when theories
  are defeasible? (Peczenik, McCarty, 1997-2001)
• Q. Given a set of cases
• case 1 a b d e f j q
• case 2 a b d q
• ...
• case n b d r
• what is the "best-fitting" set of defeasible
  rules?
• 1. all cases predicted by rules
• 2. no error
• (so far this is a learning problem with no
  simplicity measure)
• 3. sets of rules restricted or justified by
  principles?

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FOUNDATIONS I. Probability

• A probability calculation is an argument.
• A statistical argument is an argument.
• Reference Class
• Reichenbach (1949) "use the narrowest
  reference class for which there are adequate
  statistics"
• Kyburg (1961,1974,1983) maximum in a partial
  order? dominance defeat. Each "inference
  structure" permits an argument from a different
  sample class.
• Prob(A B C D)?
• Sample from BCD 5 A's/9
• Sample from BC 14 A's/20
• what is the logic of combinatorial significance
  tests?

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FOUNDATIONS I. Probability (cont.)

• ltS1,p1,q1gt and ltS2,p2,q2gt disagree
• iff
• not(p1,q1 in p2,q2) and
• not(p2,q2 in p1,q1)
• ltS1,p1,q1gt defeats ltS2,p2,q2gt
• iff
• they disagree and
• S1 strict subset of S2
• also Pollock (1985, 1990) who uses defeat
  explicitly

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FOUNDATIONS I. Probability (cont.)

• Protocols
• Shafer on Monte-Hall type probability
  "paradoxes" (1985)
• the probability argument
• is improved through knowledge of the protocol
• Neyman-Pearson tradition of crucial tests
• two crucial tests
• produce two competing statistical arguments?

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FOUNDATIONS II. Decision

• Problem of Small Worlds
• Savage (1954, 1967)
• considering fresh and stale
• should not change the calculation
• based on good and rotten.
• But of course it does.
• So a grand world which contains all detail.
• pseudomicrocosm vs. real microcosm.
• Shafer-Tversky (1988)
• framing problems
• constructive decision theory

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FOUNDATIONS II. Decision (cont.)

• Loui (1990)
• u(s) given T(P1,s) m(P1) 5
• u(s) given T(P2,s) m(P2) -4
• u(s) given T(P1 P2, s) m(P1 P2) 6 u(s)
  5-46
• ("defeasibility" of linearity)
• Holds(P1P2, s) gt u(s) 7
• defeats
• Holds(P1,s) gt u(s) 5
• (defeasibility of valuation)

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FOUNDATIONS II. Decision (cont.)

• But s is a lottery s r/p t(1-p)
• Prob(p) .5 u(r) 10 u(t) 0 so expected
  u(s) .5(10) .5(0) 5
• (defeasibility of outcome)
• Simon (1955-1967)
• substantive vs. procedural rationality, yes,
• but more importantly
• decision is more like chess than
  constraint-propagation
• heuristic valuation changes as
  search/computation proceeds
• a defeasible independence/substitution axiom?
• paradoxes of certainty, menu-dependence, framing
  effects based on description

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FOUNDATIONS III. Legal Reasoning

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FOUNDATIONS IV. Belief Revision/Deontic Logic

• contrary-to-duty imperatives (Chisholm, 1963,
  Nute etc., 1996)
• 1. O(p--gtq), O(p--gtr) and O(p) are
  consistent.
• (there can be two expiations)
• 2. O(p) entails O(p --gt x) for any x.
• (all expiations are obliged)
• von Wright (1982)
• "It only means that, if the prohibition is
  violated, the coordinated Contrary-to-Duty
  imperatives require, for their satisfaction, that
  both q and that r come true. ... If ... the
  conjunction of the two ... is a logical
  impossibility ..., the legislator would
  presumably take steps to remove the conflict."
• "deontic obligation" is different from
  "technical obligation"

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FOUNDATIONS IV. Belief Revision/Deontic Logic

• 3. O(AB), O(AC) entails (BC)
• Alchourron (1993)
• "a set of conditional general norms entails ...
  a non-tautological sentence ... iff it follows in
  the logic for normative propositions that the
  authority has inconsistently normed some action
  for some circumstance."
• If norms are defeasible rules, no such problems
• 1. two different arguments for expiation.
• 2. material conditionals are NOT deontic
  conditionals.
• 3. the entailment is not a result for
  defeasible conditionals

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FOUNDATIONS IV. Belief Revision/Deontic Logic

• AGM belief revision choice and refinement,
  ampliativity
• if p then defeasibly q
• if p and r then defeasibly q
• p --gt q in K0
• p r --gt q in K0pr--gtqr
• Alchourron (1993)
• "It seems to me unquestionable that the main
  conditions are the formal representation of the
  revisions effectively performed by an agent and
  of his dispositions to revise."
• "The particular details of the revisions (and
  the choice functions) are never analyzed by a
  logician (as a logician)..." Yes, defeasible
  conditionals would inform choice functions, but
  invoke "possible confusion of logic and
  revision," hiding "conceptually weaker
  conclusions" in "quiet darkness."

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FOUNDATIONS V. Negotiation

• Acceptability can be argued
• Fisher-Ury-Patton (1981) Principled
  negotiation gives arguments for proposals.
• why not open the window? Im cold I have a
  sweater
• Sycara (1988-1995), Loui-Moore (1993-1997),
  Parsons-Jennings etc. (1998-2001) case-based
  arguments from precedent settlements
• that raise was acceptable to you last year
• Utility can be argued
• search can lift utilities at the proposed
  settlements

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FOUNDATIONS V. Negotiation

• Utility can be argued
• Instead of strategic form a1, , a4 X b1, ,
  b5 with
• utilities Ua and Ub,
• suppose
• OPTa(x,y) and OPTb(x,y), a hard optimization
  problem for
• each agent with parameters determined by the
  agreement
• Ua(x,y) is as current best solution for
  OPTa(x,y)
• Ua and Ub are lifted at ltp,qgt which is the focus
  of dialogue
• or when joint-problem-solving
• Sunk cost-of-search arguments lead to settlement

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FOUNDATIONS VI. Rhetoric/"Informal Logic"

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Future Work Fairness (Procedural)

• Claim 1. Fairness depends on the computational
  abilities of agents (the known subspace S x T of
  the possible strategies S x T
• thus, rules are changed when a strategy s is
  discovered for which all known responses t are
  inadequate.
• Claim 2. To be fair (just), the procedure must
  construct its output upon the right inputs, with
  adequate monotonicity and invariance properties
• thus, the justification of social procedure
  resembles the proof of program correctness.
• (concatenation) ex-post asymmetry of position
  should be the result of fair (just) ex-ante
  asymmetry adjusted only by the procedures
  effects on elective inputs (strategic choices).

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Future Work Fairness (cont.)

• Obsv (political scientists).
• the purposes of the procedure can limit the
  degree of stochastics, the maximum variation of
  outcome, and the permissible input types.
• Claim 3.
• procedures should be non-dictatorial (for every
  important different kind of outcome, e.g.,
  victory/defeat, there is a strategy pair that
  would reach this outcome). (dominance is more
  interesting)
• Obsv
• rule symmetry and equivalent initial position
  are prima facie fair(but sometimes there are
  good reasons for bias, e.g. plaintiff)
• Claim 4.
• Fairness can be inherited from class
  relationships among procedural types.

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Future Work Computation

• Observations. Social procedures which
  regulate/distribute/construct distributions are
  games. Social programming is like distributed
  programming (quantify over strategy tuples).
  Building societies is like inventing algorithms
  for distributed decision-making. Argument games.
  Welfare distributions. Elections. Tournaments.
• Objective. I want computer science to be at the
  foundation of the study of social procedures.
• Obstacle. Game-playing is not considered
  computation (yet).
• Claim. Two people playing chess compute the
  outcome of the game.
• Why do you have trouble with this claim? But
  not modems or chess tournaments for charities.

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Future Work Computation

• Paradigmatic computation
• on a machine but, long division by hand?
• causally connected but, two people doing long
  division?
• deterministic but, probabilistic algorithms?
• locus of control but, distributed algorithms?
• fully specified but, pseudo-code? uncompiled
  code?
• algorithmic but, protocols? interactions?
• (control systems?)
• non-elective but, frequently arbitrary
  choices
• (e.g., search algorithms)
• Protocol-design is a kind of programming.

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Future Work Computation

• Broad computation intentional and teleological
  rule-following upon symbols.
• The existence of a program is the test for
  computation, not the existence of an algorithm.
• Not just any rule-following is computation, since
  the objects must be symbolic and the
  rule-following purposive and non-accidental.