Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The Decision Problem and the
Model Theory of First-order Logic
Richard Zach
University of Calgary
www.ucalgary.ca/rzach/
CSHPM
May 29, 2015
Richard Zach
The Decision Problem
May 29, 2015
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Outline
1
Algebraic Approaches to Decidability
2
Decidability by Semantic Methods
3
Development of Semantic Reasoning
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Introduction
Decision Problem: find a procedure which decides after
finitely many steps if a given formula is derivable/valid or
not
One of the fundamental problems for the development of
modern logic in the 1920s and 1930s
Connected to:
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Hilbert’s philosophical project
development of notion of decision procedure/computation
development of model-theoretic semantics for first-order
logic
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Early Work on Propositional Logic
Attempt at a decidability proof
for propositional logic in
Hilbert’s 1905 lectures on logic.
Hilbert’s 1917/18 lecture course
Principles of Mathematics
contain a completeness proof
for propositional logic.
Proof by algebraic
manipulation; uses normal
forms in essential way.
Decidability follows and
semantic completeness follows
(Bernays 1918).
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The General Decision Problem
“[We require] not only the individual operations but also the
path of calculation as a whole should be specified by rules, in
other words, an elimination of thinking in favor of mechanical
calculation.
If a logical or mathematical assertion is given, the required
procedure should give complete instructions for determining
whether the assertion is correct or false by a determinate
calculation after finitely many steps.
The problem thus formulated I want to call the general decision
problem.”
Behmann, “Entscheidungsproblem und Algebra der Logik”, Göttingen, May
10, 1921
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Behmann’s Algebraic-Syntactic Proof
Result: decidability of monadic
second-order logic
Procedure modelled on
algebraic elimination problem
Successively removes
quantifiers from a given (closed)
formula
Reduces truth of formula to
condition on the size of the
domain
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Hilbert and Bernays on the Decision Problem
Lectures Logische Grundlagen der Mathematik, co-taught by
Hilbert and Bernays, Winter 1922/23
Statement of decision problem
Discussion of importance, examples from geometry
New proof of decidability of first-order monadic logic
(without =)
Proof is not algebraic, but semi-semantical:
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If not valid, not valid in some finite individual domain with
at most 2n elements
Validity in a finite domain reducible to propositional
calculus
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Schönfinkel’s Original Proof
Talk given by Bernays and
Schönfinkel in December 1921
Manuscript (in Bernays
Nachlass) from Winter term
1922/23
Treats case of validity of
first-order formulas of the form
(∃x)(∀y)A where A only
contains one binary predicate
symbol
Uses infinite sums and products
(like in
Schröder/Löwenheim/Skolem)
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Bernays and Schönfinkel 1927
“Zum Entscheidungsproblem der mathematischen Logik”, Math.
Ann. 99 (1928), submitted March 1927.
Discussion of decision problem in general, cardinality
questions in particular
Duality of satisfiability and validity made explicit
Discussion of satisfiability and finite satisfiability
(Löwenheim and Skolem)
Interpretation of predicates extensional
Proof of finite controllability of monadic class with bound
(from 1922/23 lectures)
Discussion of prefix classes
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The Bernays-Schönfinkel Class
Class ∃∗ ∀∗ A now called
“Bernays-Schönfinkel class”
The class ∀∗ ∃∗ A is shown to be
decidable for validity in Bernays
and Schönfinkel 1928.
It is the preliminary “trivial”
case before Bernays goes on to
the actual contribution of the
paper.
The Bernays-Schönfinkel class is
not dealt with in Schönfinkel’s
manuscript.
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The Schönfinkel Class
(∃x)(∀y)A with A quantifier-free and containing n binary
predicate symbols decidable for validity
Proof is by giving bound on countermodel: 2n
Schönfinkel’s manuscript gives algebraic argument
Semantic proof thus due to Bernays
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Semantic Methods
Development of Semantic Reasoning
The Ackermann Class
“Über die Erfüllbarkeit gewisser
Zählausdrücke”, Math. Ann. 100
(1929), sumitted February 1928
Deals with formulas of the form
∃∗ ∀∃∗ A and satisfiability
(i.e., ∀∗ ∃∀∗ A and validity)
Emphasizes satisfiability and
upper bound on model
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The Ramsey Class
“On a problem of formal logic”,
Proc. LMS, ser. 2, vol. 30 (1929),
submitted November 1928
Deals with satisfiability of
formulas of the form ∀∗ A and
=; sketch for ∃∗ ∀∗ A.
Requires finite Ramsey theorem
(and thus sparked Ramsey
theory)
Like Behmann, focuses on
spectra: sizes of domain where
formula satisfiable
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
The Development of Semantics
No clear semantics in H 1917/18
Issues:
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Domain of quantification
Interpretation of predicates and predicate variables
Range of quantified variables
Open formulas
“correct [richtig]” vs. “valid [allgemeingültig]”
validity vs. satisfiability
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Behmann on Semantics: Correctness
Distinguishes predicate constants and variables
Formulas with (free) predicate variables are not
propositions [Aussagen]
Basic notion: correct (applies to 2nd order sentences only)
Russellian notion, but domain is variable
Decision problem: decide if a 2nd order sentence is correct
for every domain of individuals
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The Decision Problem
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Bernays on Semantics: Notions
Formulas are schemata for propositions
They contain two variable elements:
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domain of individuals (values of x’s)
predicates (values of P ’s—considered as functions from
objects to truth values)
Validity defined for a specific domain
Decision problem: Is A valid for every domain?
Dual notion: satisfiability (taken most likely from Skolem
1920)
Empty domain excluded: trivial for decision problem
(∀ always valid, ∃ always unsatisfiable)
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Algebraic Manipulation
Semantic Methods
Development of Semantic Reasoning
Bernays on Semantics: Results
Validity and satisfiability are dual
If A satisfiable in domain X, it is satisfiable in any domain Y
where Y and X have same cardinality (X finite)
(Essentially a proof that isomorphism implies elementary
equivalence)
If A is satisfiable in X, then A satisfiable in any Y ⊇ X.
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