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First order theories

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Satisfiability

The classic SAT problem: given a propositional formula , is satisfiable ?

Example:

Let x1,x2 be propositional variables

Let := (x1 → x2) Æ (x1 Æ :: x2)

Is satisfiable ?

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Satisfiability Modulo Theories

Now let the predicates be equalities

Example:

Let x1,x2,x3 2 Z

Let := ((x1 = x2) → (x2 = x3)) Æ x1 x3

Is satisfiable ?

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Satisfiability Modulo Theories

Now let the predicates be from linear arithmetic

Example:

Let x1,x2,x3 2 R

Let := ((x1 + 2x3 < 5) Ç :(x3 · 1) Æ (x1 ¸ 3))

Is satisfiable ?

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Satisfiability Modulo Theories

Now let the predicates represent arrays

Example:

Let i,j,a[] 2 Z

Let := (i = j Æ a[j] = 1) Æ a[i] 1

Is satisfiable ?

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Generalization

Equalities, linear predicates, arrays, ... Is there a general framework to define them ?

Yes ! It is called first-order logic

Each of the above examples is a first-order theory.

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Some useful first-order theories

The SMT web page: http://combination.cs.uiowa.edu/smtlib/

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What lies ahead...

As in the case of propositional logic, we need to define Syntax Semantics

But it is more difficult, because: First-order logic is a template for defining theories Semantics cannot be defined with truth-tables

The domains can be infinite !

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First-order logic: Terms & predicates

Let x,y be variables f,g be function symbols (unary & binary, respectively) a,b,c be propositional variables p be a binary predicate symbol

Example of terms: x f(f(x)) g(f(x),g(y,z)))

Example of predicates: a p(f(x),g(x,y))

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First-order logic – Grammar

A FOL formula is a predicate, or An application of a propositional connective over FOL

formula(s) An application of a quantifier (9, 8) over a FOL formula

Examples:

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First-order Logic / scopes

Each quantifier has a scope

If a variable is not quantified, it is called free

A formula without free variables is called a sentence

Here, b is free

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First order logic

We distinguish between Logical symbols: Æ Ç : 8 9 ( ) Non-logical Symbols :

function symbols (incl. constants) predicate symbols

is the ‘signature’ of the theory restricts the syntax The theory also restricts the semantics of the symbols in

We will see how...

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Example 1

= {0,1,f,p} ‘0’,’1’ are constant symbols ‘f’ is a binary function symbol ‘p’ is a binary predicate

An example of a -formula:

9y 8x. p(x, y)

Is it true ? depends on the interpretation we give to p. Does there exists a satisfying interpretation?

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Example 2

= {1,p1,p2, p3} ‘1’ is a constant symbol p1, p2 are binary predicates p3 is a unary predicate

An example -formula:

8n 9k. p2(n,1) ! p3(k) Æ p1(n,k) Æ p1(k, 2n).

is this formula satisfiable ? i.e., is there an interpretation that makes it true ?

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Example 2, interpreted

set p1 = ‘>’, p2 = ‘<‘, p3 = isPrime.

So now the formula is:

8n 9k. n > 1 ! isprime(k) Æ n < k < 2n.

This is Bertrand’s postulate Proven separately by Chhebyshev, Ramanujan, Erdos.

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Without interpretations...

Technically, we should also assign interpretation to symbols that are already ‘taken’: ‘+’,’*’, ‘<‘, ... In practice no one in their right mind gives them non-

standard interpretation.

Example: Let = {0,1,+,=} where 0,1 are

constants, + is a binary function and = a predicate.

Let = 9x. x + 0 = 1 Q: Is true in N0 ?

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An interpretation is given by a structure A structure is given by:

1. A domain to each variable

2. An interpretation of the nonlogical symbols: i.e., Maps each function symbol to a function of the same arity Maps each constant symbol to a domain element Maps each predicate symbol to a predicate of the same arity

3. An assignment of a domain element to each free (unquantified) variable

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Structures

= 9x. f(x, 0) = 1 Consider the structure S:

Domain: N0

Interpretation: ‘f’ * (multiplication) ‘0’ and ‘1’ are mapped to 0 and 1 in N0

‘=‘ = (equality)

Now, is true in S ?

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Satisfying structures

Definition: A formula is satisfiable if there exists a structure that satisfies it

Example: = 9x. f(x, 0) = 1 is satisfiable Consider the structure S’:

Domain: N0

Interpretation: ‘0’ and ‘1’ are mapped to 0 and 1 in N0

‘=’ = (equality) ‘f’ + (addition)

is satisfied by S’. S’ is a model of .

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Two ways to define a theory

We can define a theory

1. The easy, “direct” way, Define the grammar, Assume standard interpretation of the symbols.

2. The hard, “formal” way. Define the signature Restrict possible interpretations with sentences/axioms

Very useful for defining algorithms and proving

general theorems

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1. Defining a theory: the easy way

Example: The quantifier-free theory of equalities

Grammar: A Boolean combination of predicates of the form “x = y”, where x,y 2 R.

Example: x = y Ç :(y = z ! x = z)

Note that here... we implicitly say that ‘=‘ is the equality predicate, and we predefine the domain of the variables

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2. Defining a theory: the hard way

Let be a signature

A theory defined over is called a -theory

A -theory restricts the interpretations that we can give the symbols in . How ?

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2. Definitions...

For a signature :

-formula: a formula over symbols + logical symbols.

A -theory T is defined by a set of -sentences.

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2. Definitions…

Let T be a -theory

A -formula is T-satisfiable if there exists a structure

that satisfies both and the sentences defining T.

A -formula is T-valid if all structures that satisfy the

sentences defining T also satisfy .

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2. Theories through axioms

The number of sentences that are necessary for defining a theory might be large or infinite.

Instead, it is common to define a theory through a set of axioms.

The theory is defined by these axioms, and everything that can be inferred from them by a sound

inference system.

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2. Axioms for some ‘famous’ predicates Let ‘T’ be: (defining ‘=‘):

Example: = ((x = y) Æ :(y = z)) ! :(x = z) Is Á T-valid ?

Every structure that satisfies these axioms, satisfies . Hence is T-valid.

8 x. x = x(reflexivity)

8 x,y. x = y ! y = x (symmetry)

8 x,y,z. x = y Æ y = z ! x = z (transitivity)

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2. Axioms for some ‘famous’ predicates Let ‘T’ be: (defining ‘<‘)

Example: Á: 8x. 9y. y < x Is Á T-satisfiable?

We will show a model for it. Domain: Z ‘< ‘ <

8x,y,z. x < y Æ y < z → x < z (transitivity)

8x,y. x < y → :(y < x) (anti-symmetry)

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2. Axioms for some ‘famous’ predicates Let ‘T’ be: (defining ‘<‘)

Example: Á: 8x. 9y. y < x Is Á T-valid?

8x,y,z. x < y Æ y < z → x < z (transitivity)

8x,y. x < y → :(y < x) (anti-symmetry)

We will show a structure to the contrary

Domain: N0

‘<‘ <

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2. Axioms for some ‘famous’ predicates Let = {‘<‘} Consider the -formula Á: 8x. 9y. y < x Consider the theory T:

Is Á T-valid?

We will show a structure to the contrary Domain: N0

‘<‘ <

8x,y,z. x < y Æ y < z → x < z (transitivity)

8x,y. x < y → :(y < x) (anti-symmetry)

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Fragments

So far we only restricted the nonlogical symbols. Sometimes we want to

restrict the grammar restrict the allowed logical symbols.

These are called logic fragments.

Examples: The quantifier-free fragment over = {=,+,0,1} The conjunctive fragment over = {=,+,0,1}

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Fragments

Let = {} (T must be empty: no nonlogical symbols to interpret)

Q: What is the quantifier-free fragment of T ? A: propositional logic

Thus, propositional logic is also a first-order theory. A very degenerate one.

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Theories

Let = {} (T must be empty: no nonlogical symbols to interpret)

Q: What is T ? A: Quantified Boolean Formulas (QBF)

Example: 8x1 9x2 8x3. x1 → (x2 Ç x3)

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Some famous theories

Presburger arithmetic: = {0,1,+,=} Peano arithmetic: = {0,1,+,*,=} Theory of reals Theory of integers Theory of arrays Theory of pointers Theory of sets Theory of recursive data structures …

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Expressiveness of a theory

Each formula defines a language:the set of satisfying assignments (‘models’) are the words accepted by this language.

Consider the fragment ‘2-CNF’formula : ( literal Ç literal ) | formula Æ formulaliteral: Boolean-variable | :Boolean-variable

(x1 Ç :x2) Æ (:x3 Ç x2)

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Expressiveness of a theory

Now consider a Propositional Logic formula (x1 Ç x2 Ç x3).

Q: Can we express this language with 2-CNF? A: No.

Proof: The language accepted by has 7 words: all assignments

other than x1 = x2 = x3 = F. The first 2-CNF clause removes ¼ of the assignments,

which leaves us with 6 accepted words. Additional clauses only remove more assignments.

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Expressiveness of a theory

Claim: 2-CNF Á Propositional Logic Generally there is only a partial order between

theories.

Languages defined by L2

Languages defined by L1

L2 is more expressive than L1.

Denote: L1 Á L2

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The tradeoff

So we see that theories can have different expressive power

Q: why would we want to restrict ourselves to a theory or a fragment ? why not take some ‘maximal theory’…

A: Adding axioms to the theory may make it harder to decide or even undecidable.

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Example: First Order Peano Arithmetic = {0,1,‘+’, ‘*’, ‘=’} Domain: Natural numbers

Axioms (“semantics”):1. 8 x. (0 x + 1) 2. 8 x, y. (x y) ! (x + 1 y + 1) 3. Induction4. 8 x. x + 0 = x 5. 8 x, y. (x + y) + 1 = x + (y + 1) 6. 8 x. x * 0 = 0 7. 8 x, y. x * (y + 1) = x * y + x

+

*

Undecidable!

These axioms define the semantics of ‘+’

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Example: First Order Presburger Arithmetic = {0,1,‘+’, ‘*’, ‘=’} Domain: Natural numbers

Axioms (“semantics”):1. 8 x. (0 x + 1) 2. 8 x, y. (x y) ! (x + 1 y + 1) 3. Induction4. 8 x. x + 0 = x 5. 8 x, y. (x + y) + 1 = x + (y + 1) 6. 8 x. x * 0 = 0 7. 8 x, y. x * (y + 1) = x * y + x

+

*

decidable!

These axioms define the semantics of ‘+’

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Tradeoff: expressiveness/computational hardness. Assume we are given theories L1 Á … Á Ln

More expressiveEasier to decide

UndecidableDecidable

Intractable(exponential)

Tractable(polynomial)

Computational Challenge! LnL1

Our course

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When is a specific theory useful?

1. Expressible enough to state something interesting.

2. Decidable (or semi-decidable) and more efficiently solvable than richer theories.

3. More expressive, or more natural for expressing some models in comparison to ‘leaner’ theories.

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Expressiveness and complexity

Q1: Let L1 and L2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L1 formula reducible to a satisfiability problem of an L2 formula?

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Expressiveness and complexity

Q2: Let L1 and L2 be two theories whose satisfiability problems are reducible to one another. Are L1 and L2 in the same complexity class ?