CS156: The Calculus of Computation

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1 Page 1 of 31 CS156: The Calculus of Computation Zohar Manna Winter 2010 Chapter 3: First-Order Theories

2 Page 2 of 31 First-Order Theories I First-order theory T consists of Signature Σ T - set of constant, function, and predicate symbols Set of axioms A T - set of closed (no free variables) Σ T -formulae A Σ T -formula is a formula constructed of constants, functions, and predicate symbols from Σ T, and variables, logical connectives, and quantifiers. The symbols of Σ T are just symbols without prior meaning the axioms of T provide their meaning.

3 Page 3 of 31 First-Order Theories II A Σ T -formula F is valid in theory T (T-valid, also T = F), iff every interpretation I that satisfies the axioms of T, i.e. I = A for every A A T (T-interpretation) also satisfies F, i.e. I = F A Σ T -formula F is satisfiable in T (T-satisfiable), if there is a T-interpretation (i.e. satisfies all the axioms of T) that satisfies F Two formulae F 1 and F 2 are equivalent in T (T-equivalent), iff T = F 1 F 2, i.e. if for every T-interpretation I, I = F 1 iff I = F 2 Note: I = F stands for F true under interpretation I T = F stands for F is valid in theory T

4 Page 4 of 31 Fragments of Theories A fragment of theory T is a syntactically-restricted subset of formulae of the theory. Example: a quantifier-free fragment of theory T is the set of quantifier-free formulae in T. A theory T is decidable if T = F (T-validity) is decidable for every Σ T -formula F; i.e., there is an algorithm that always terminate with yes, if F is T-valid, and no, if F is T-invalid. A fragment of T is decidable if T = F is decidable for every Σ T -formula F obeying the syntactic restriction.

5 Page 5 of 31 Theory of Equality T E I Signature: consists of Σ = : {=, a, b, c,, f, g, h,, p, q, r, } =, a binary predicate, interpreted with meaning provided by axioms all constant, function, and predicate symbols Axioms of T E 1. x. x = x (reflexivity) 2. x, y. x = y y = x (symmetry) 3. x, y, z. x = y y = z x = z (transitivity) 4. for each positive integer n and n-ary function symbol f, x 1,...,x n, y 1,...,y n. i x i = y i f (x 1,...,x n ) = f (y 1,...,y n ) (function congruence)

6 Page 6 of 31 Theory of Equality T E II 5. for each positive integer n and n-ary predicate symbol p, x 1,...,x n, y 1,...,y n. i x i = y i (p(x 1,...,x n ) p(y 1,...,y n )) (predicate congruence) (function) and (predicate) are axiom schemata. Example: (function) for binary function f for n = 2: x 1, x 2, y 1, y 2. x 1 = y 1 x 2 = y 2 f (x 1, x 2 ) = f (y 1, y 2 ) (predicate) for unary predicate p for n = 1: x, y. x = y (p(x) p(y)) Note: we omit congruence for brevity.

7 Page 7 of 31 Decidability of T E I T E is undecidable. The quantifier-free fragment of T E is decidable. Very efficient algorithm. Semantic argument method can be used for T E Example: Prove F : a = b b = c g(f (a), b) = g(f (c), a) is T E -valid.

8 Page 8 of 31 Decidability of T E II Suppose not; then there exists a T E -interpretation I such that I = F. Then, 1. I = F assumption 2. I = a = b b = c 1, 3. I = g(f (a), b) = g(f (c), a) 1, 4. I = a = b 2, 5. I = b = c 2, 6. I = a = c 4, 5, (transitivity) 7. I = f (a) = f (c) 6, (function) 8. I = b = a 4, (symmetry) 9. I = g(f (a), b) = g(f (c), a) 7, 8, (function) 10. I = 3, 9 contradictory F is T E -valid.

9 Page 9 of 31 Natural Numbers and Integers Natural numbers N = {0, 1, 2, } Integers Z = {, 2, 1, 0, 1, 2, } Three variations: Peano arithmetic T PA : natural numbers with addition, multiplication, = Presburger arithmetic T N : natural numbers with addition, = Theory of integers T Z : integers with +,, >,=, multiplication by constants

10 Page 10 of Peano Arithmetic T PA (first-order arithmetic) Σ PA : {0, 1, +,, =} Equality Axioms: (reflexivity), (symmetry), (transitivity), (function) for +, (function) for. And the axioms: 1. x. (x + 1 = 0) (zero) 2. x, y. x + 1 = y + 1 x = y (successor) 3. F[0] ( x. F[x] F[x + 1]) x. F[x] (induction) 4. x. x + 0 = x (plus zero) 5. x, y. x + (y + 1) = (x + y) + 1 (plus successor) 6. x. x 0 = 0 (times zero) 7. x, y. x (y + 1) = x y + x (times successor) Line 3 is an axiom schema.

11 Page 11 of 31 Example: 3x + 5 = 2y can be written using Σ PA as x + x + x = y + y Note: we have > and since 3x + 5 > 2y write as z. z 0 3x + 5 = 2y + z 3x + 5 2y write as z. 3x + 5 = 2y + z Example: Existence of pythagorean triples (F is T PA -valid): F : x, y, z. x 0 y 0 z 0 x x + y y = z z

12 Page 12 of 31 Decidability of Peano Arithmetic T PA is undecidable. (Gödel, Turing, Post, Church) The quantifier-free fragment of T PA is undecidable. (Matiyasevich, 1970) Remark: Gödel s first incompleteness theorem Peano arithmetic T PA does not capture true arithmetic: There exist closed Σ PA -formulae representing valid propositions of number theory that are not T PA -valid. The reason: T PA actually admits nonstandard interpretations. For decidability: no multiplication

13 Page 13 of Presburger Arithmetic T N Signature Σ N : {0, 1, +, =} no multiplication! Axioms of T N (equality axioms, with 1-5): 1. x. (x + 1 = 0) (zero) 2. x, y. x + 1 = y + 1 x = y (successor) 3. F[0] ( x. F[x] F[x + 1]) x. F[x] (induction) 4. x. x + 0 = x (plus zero) 5. x, y. x + (y + 1) = (x + y) + 1 (plus successor) Line 3 is an axiom schema. T N -satisfiability (and thus T N -validity) is decidable (Presburger, 1929)

14 Page 14 of Theory of Integers T Z Signature: Σ Z : {..., 2, 1, 0, 1, 2,..., 3, 2, 2, 3,..., +,, >, =} where..., 2, 1, 0, 1, 2,... are constants..., 3, 2, 2, 3,... are unary functions (intended meaning: 2 x is x + x, 3 x is x x x) +,, >,= have the usual meanings. Relation between T Z and T N : T Z and T N have the same expressiveness: For every ΣZ -formula there is an equisatisfiable Σ N -formula. For every ΣN -formula there is an equisatisfiable Σ Z -formula. Σ Z -formula F and Σ N -formula G are equisatisfiable iff: F is T Z -satisfiable iff G is T N -satisfiable

15 Page 15 of 31 Σ Z -formula to Σ N -formula I Example: consider the Σ Z -formula F 0 : w, x. y, z. x + 2y z 7 > 3w + 4. Introduce two variables, v p and v n (range over the nonnegative integers) for each variable v (range over the integers) of F 0 : F 1 : w p, w n, x p, x n. y p, y n, z p, z n. (x p x n ) + 2(y p y n ) (z p z n ) 7 > 3(w p w n ) + 4 Eliminate by moving to the other side of >: F 2 : w p, w n, x p, x n. y p, y n, z p, z n. x p + 2y p + z n + 3w p > x n + 2y n + z p w n + 4

16 Page 16 of 31 Σ Z -formula to Σ N -formula II Eliminate > and numbers: w p, w n, x p, x n. y p, y n, z p, z n. u. F 3 : (u = 0) x p + y p + y p + z n + w p + w p + w p = x n + y n + y n + z p + w n + w n + w n + u which is a Σ N -formula equisatisfiable to F 0. To decide T Z -validity for a Σ Z -formula F: transform F to an equisatisfiable Σ N -formula G, decide T N -validity of G.

17 Page 17 of 31 Σ Z -formula to Σ N -formula III Example: The Σ N -formula x. y. x = y + 1 is equisatisfiable to the Σ Z -formula: x. x > 1 y. y > 1 x = y + 1.

18 Rationals and Reals Signatures: Σ Q = {0, 1, +,, =, } Σ R = Σ Q { } Theory of Reals T R (with multiplication) x x = 2 x = ± 2 Theory of Rationals T Q (no multiplication) }{{} 2x = 7 x = 7 2 x+x Note: strict inequality okay; simply rewrite as follows: x + y > z (x + y = z) x + y z Page 18 of 31

19 Page 19 of Theory of Reals T R Signature: Σ R : {0, 1, +,,, =, } with multiplication. Axioms in text. Example: is T R -valid. a, b, c. b 2 4ac 0 x. ax 2 + bx + c = 0 T R is decidable (Tarski, 1930) High time complexity

20 Page 20 of Theory of Rationals T Q Signature: Σ Q : {0, 1, +,, =, } without multiplication. Axioms in text. Rational coefficients are simple to express in T Q. Example: Rewrite as the Σ Q -formula 1 2 x y 4 3x + 4y 24 T Q is decidable Quantifier-free fragment of T Q is efficiently decidable

21 Page 21 of 31 Recursive Data Structures (RDS) I Tuples of variables where the elements can be instances of the same structure: e.g., linked lists or trees. 1. Theory T cons (LISP-like lists) Signature: where Σ cons : {cons, car, cdr, atom, =} cons(a, b) list constructed by concatenating a and b car(x) left projector of x: car(cons(a, b)) = a cdr(x) right projector of x: cdr(cons(a, b)) = b atom(x) true iff x is a single-element list Note: an atom is simply something that is not a cons. In this formulation, there is no NIL value.

22 Page 22 of 31 Recursive Data Structures (RDS) II Axioms: 1. The axioms of reflexivity, symmetry, and transitivity of = 2. Function Congruence axioms x 1, x 2, y 1, y 2. x 1 = x 2 y 1 = y 2 cons(x 1, y 1 ) = cons(x 2, y 2 ) x, y. x = y car(x) = car(y) x, y. x = y cdr(x) = cdr(y)

23 Page 23 of Predicate Congruence axiom x, y. x = y (atom(x) atom(y)) 4. x, y. car(cons(x, y)) = x (left projection) 5. x, y. cdr(cons(x, y)) = y (right projection) 6. x. atom(x) cons(car(x), cdr(x)) = x (construction) 7. x, y. atom(cons(x, y)) (atom) Note: the behavior of car and cons on atoms is not specified. T cons is undecidable Quantifier-free fragment of T cons is efficiently decidable

24 Lists with equality 2. Theory T E cons (lists with equality theory) Signature: T E cons = T E T cons Σ E Σ cons (this includes uninterpreted constants, functions, and predicates) Axioms: union of the axioms of T E and T cons T E cons is undecidable Quantifier-free fragment of T E cons is efficiently decidable Example: The Σ E cons-formula F : car(x) = car(y) cdr(x) = cdr(y) atom(x) atom(y) f (x) = f (y) is T E cons-valid. Page 24 of 31

25 Suppose not; then there exists a T E cons-interpretation I such that I = F. Then, 1. I = F assumption 2. I = car(x) = car(y) 1,, 3. I = cdr(x) = cdr(y) 1,, 4. I = atom(x) 1,, 5. I = atom(y) 1,, 6. I = f (x) = f (y) 1, 7. I = cons(car(x), cdr(x)) = cons(car(y), cdr(y)) 2, 3, (function) 8. I = cons(car(x), cdr(x)) = x 4, (construction) 9. I = cons(car(y), cdr(y)) = y 5, (construction) 10. I = x = y 7, 8, 9, (transitivity) 11. I = f (x) = f (y) 10, (function) Lines 6 and 11 are contradictory, so our assumption that I = F must be wrong. Therefore, F is T E cons-valid. Page 25 of 31

26 Page 26 of 31 Theory of Arrays T A Signature: Σ A : { [ ],, =} where a[i] binary function read array a at index i ( read(a,i) ) a i v ternary function write value v to index i of array a ( write(a,i,v) ) Axioms 1. the axioms of (reflexivity), (symmetry), and (transitivity) of T E 2. a, i, j. i = j a[i] = a[j] (array congruence) 3. a, v, i, j. i = j a i v [j] = v (read-over-write 1) 4. a, v, i, j. i j a i v [j] = a[j] (read-over-write 2)

27 Page 27 of 31 Note: = is only defined for array elements not T A -valid, but is T A -valid. Also F : a[i] = e a i e = a F : a[i] = e j. a i e [j] = a[j], a = b a[i] = b[i] is not T A -valid: We have only axiomatized a restricted congruence. T A is undecidable Quantifier-free fragment of T A is decidable

28 Page 28 of Theory of Arrays T = A (with extensionality) Signature and axioms of T = A are the same as T A, with one additional axiom a, b. ( i. a[i] = b[i]) a = b (extensionality) Example: is T = A -valid. F : a[i] = e a i e = a T A = is undecidable Quantifier-free fragment of T A = is decidable

29 Page 29 of 31 First-Order Theories Quantifiers QFF Theory Decidable Decidable T E Equality T PA Peano Arithmetic T N Presburger Arithmetic T Z Linear Integer Arithmetic T R Real Arithmetic T Q Linear Rationals T cons Lists Tcons E Lists with Equality

30 Page 30 of 31 Combination of Theories How do we show that 1 x x 2 f (x) f (1) f (x) f (2) is (T E T Z )-satisfiable? Or how do we prove properties about an array of integers, or a list of reals...? Given theories T 1 and T 2 such that Σ 1 Σ 2 = {=} The combined theory T 1 T 2 has signature Σ 1 Σ 2 axioms A 1 A 2

31 Page 31 of 31 Nelson & Oppen showed that, if satisfiability of the quantifier-free fragment (qff) of T 1 is decidable, satisfiability of qff of T 2 is decidable, and certain technical simple requirements are met, then satisfiability of qff of T 1 T 2 is decidable.