1 First order theories

2 Satisfiability The classic SAT problem: given a propositional formula , is  satisfiable ? Example:  Let x 1,x 2 be propositional variables  Let  := (x 1 → x 2 ) Æ (x 1 Æ :: x 2 )  Is  satisfiable ?

3 Satisfiability Modulo Theories Now let the predicates be equalities Example:  Let x 1,x 2,x 3 2 Z  Let  := ((x 1 = x 2 ) → (x 2 = x 3 )) Æ x 1  x 3  Is  satisfiable ?

4 Satisfiability Modulo Theories Now let the predicates be from linear arithmetic Example:  Let x 1,x 2,x 3 2 R  Let  := ((x 1 + 2x 3 < 5) Ç : (x 3 · 1) Æ (x 1 ¸ 3))  Is  satisfiable ?

5 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 ?

6 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.

7 Some useful first-order theories The SMT web page:

8 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 !

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

10 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:

11 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

12 First-order theories First-order logic is a framework. It gives us a generic syntax and building blocks for constructing restrictions thereof. Each such restriction is called a first-order theory.

13 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...

14 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: 9 y 8 x. p(x, y) Is it true ?  depends on the interpretation we give to p.  Does there exists a satisfying interpretation?

15 Example 2  = {1,p 1,p 2, p 3 }  ‘1’ is a constant symbol  p 1, p 2 are binary predicates  p 3 is a unary predicate An example  -formula: 8 n 9 k. p 2 (n,1 ) ! p 3 ( k ) Æ p 1 (n,k ) Æ p 1 (k, 2n). is this formula satisfiable ?  i.e., is there an interpretation that makes it true ?

16 Example 2, interpreted set p 1 = ‘>’, p 2 = ‘<‘, p 3 = isPrime. So now the formula is: 8 n 9 k. n > 1 ! isprime(k) Æ n < k < 2n. This is Bertrand’s postulate  Proven separately by Chhebyshev, Ramanujan, Erdos.

17 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  = 9 x. x + 0 = 1  Q: Is  true in N 0 ?

18 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

19 Structures  = 9 x. f(x, 0) = 1 Consider the structure S:  Domain: N 0  Interpretation: ‘f’  * (multiplication) ‘0’ and ‘1’ are mapped to 0 and 1 in N 0 ‘=‘  = (equality) Now, is  true in S ?

20 Satisfying structures Definition: A formula is satisfiable if there exists a structure that satisfies it Example:  = 9 x. f(x, 0) = 1 is satisfiable Consider the structure S’:  Domain: N 0  Interpretation: ‘0’ and ‘1’ are mapped to 0 and 1 in N 0 ‘=’  = (equality) ‘f’  + (addition)  is satisfied by S’. S’ is a model of .

21 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

22 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

23 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 ?

24 2. Definitions... For a signature  :  - formula: a formula over  symbols + logical symbols. A  -theory T is defined by a set of  -sentences.

25 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 .

26 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.

27 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)

28 2. Axioms for some ‘famous’ predicates Let ‘T’ be: (defining ‘<‘) Example: Á : 8 x. 9 y. y < x Is Á T-satisfiable? We will show a model for it.  Domain: Z  ‘< ‘  < 8 x,y,z. x < y Æ y < z → x < z(transitivity) 8 x,y. x < y → : (y < x)(anti-symmetry)

29 2. Axioms for some ‘famous’ predicates Let ‘T’ be: (defining ‘<‘) Example: Á : 8 x. 9 y. y < x Is Á T-valid? 8 x,y,z. x < y Æ y < z → x < z(transitivity) 8 x,y. x < y → : (y < x)(anti-symmetry) We will show a structure to the contrary  Domain: N 0  ‘<‘  <

30 2. Axioms for some ‘famous’ predicates Let  = {‘<‘} Consider the  -formula Á : 8 x. 9 y. y < x Consider the theory T: Is Á T-valid? We will show a structure to the contrary  Domain: N 0  ‘<‘  < 8 x,y,z. x < y Æ y < z → x < z(transitivity) 8 x,y. x < y → : (y < x)(anti-symmetry)

31 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}

32 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.

33 Theories Let  = {}  (T must be empty: no nonlogical symbols to interpret) Q: What is T ? A: Quantified Boolean Formulas (QBF) Example:  8 x 1 9 x 2 8 x 3. x 1 → (x 2 Ç x 3 )

34 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 …

35 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 formula literal: Boolean-variable | : Boolean-variable ( x 1 Ç : x 2 ) Æ ( : x 3 Ç x 2 )

36 Expressiveness of a theory Now consider a Propositional Logic formula  ( x 1 Ç x 2 Ç x 3 ). Q: Can we express this language with 2-CNF? A: No. Proof:  The language accepted by  has 7 words: all assignments other than x 1 = x 2 = x 3 = F.  The first 2-CNF clause removes ¼ of the assignments, which leaves us with 6 accepted words. Additional clauses only remove more assignments.

37 Expressiveness of a theory Claim: 2-CNF Á Propositional Logic Generally there is only a partial order between theories. Languages defined by L 2 Languages defined by L 1 L 2 is more expressive than L 1. Denote: L 1 Á L 2

38 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.

39 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. Induction 4. 8 x. x + 0 = x 5. 8 x, y. (x + y) + 1 = x + (y + 1) 6. 8 x. x * 0 = x, y. x * (y + 1) = x * y + x + * Undecidable! These axioms define the semantics of ‘+’

40 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. Induction 4. 8 x. x + 0 = x 5. 8 x, y. (x + y) + 1 = x + (y + 1) 6. 8 x. x * 0 = x, y. x * (y + 1) = x * y + x + * decidable! These axioms define the semantics of ‘+’

41 Tradeoff: expressiveness/computational hardness. Assume we are given theories L 1 Á … Á L n More expressive Easier to decide UndecidableDecidable Intractable (exponential) Tractable (polynomial) Computational Challenge! LnLn L1L1 Our course

42 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.

43 Expressiveness and complexity Q1: Let L 1 and L 2 be two theories whose satisfiability problem is decidable and in the same complexity class. Is the satisfiability problem of an L 1 formula reducible to a satisfiability problem of an L 2 formula?

44 Expressiveness and complexity Q2: Let L 1 and L 2 be two theories whose satisfiability problems are reducible to one another. Are L 1 and L 2 in the same complexity class ?