Consistency of a Programming Logic for a Version of PCF Using Domain Theory

Transcription

1 Consistency of a Programming Logic for a Version of PCF Using Domain Theory Andrés Sicard-Ramírez EAFIT University Logic and Computation Seminar EAFIT University 5 April, 3 May 2013

2 A Core Functional Programming Language Plotkin s PCF language

3 PCF Features (Plotkin 1977) Typed λ-calculus Basic data types: Natural numbers and Booleans

4 A Programming Logic Logical Theory of Constructions (LTC) (Bove, Dybjer and Sicard-Ramírez 2009) 1 LTC = type-free version of PCF (terms, conversion and discrimination rules) + first-order logic + inductive predicates (not considered in this talk) 1 A Bove, P Dybjer and A Sicard-Ramírez (2009) Embedding a Logical Theory of Constructions in Agda In: Proceedings of the 3rd workshop on Programming languages meets program verification (PLPV 09), pp 59 66

5 LTC-Terms Terms t = x t t λxt fix xt true false if 0 succ pred iszero variable application λ-abstraction fixed-point operator Boolean constants natural number constants Convention Ṭhe binary application function symbol is left-associative

6 LTC-Formulae Formulae A = A A A A A A xa xa t = t P(t,, t) truth, falsehood binary logical connectives quantifiers equality predicate Abbreviations A = A, t t = (t = t )

7 Conversion and Discrimination Rules of LTC Conversion rules t t if true t t = t, t t if false t t = t, pred 0 = 0, t pred (succ t) = t, iszero 0 = true, t iszero (succ t) = false, t t (λxt) t = t[x = t ], t fix xt = t[x = fix xt], where t[x = t ] is the capture-free substitution of x for t in t Discrimination rules true false, t 0 succ t

8 LTC Consistency How we know that LTC is a consistent theory? 2 C C Chang and H J Keisler (1992) Model Theory 3rd Vol 73 Studies in Logic and the Foundations of Mathematics 3rd impression North-Holland

9 LTC Consistency How we know that LTC is a consistent theory? Standard answer: To build a model for LTC (Chang and Keisler 1992, theorem 1321) 2 2 C C Chang and H J Keisler (1992) Model Theory 3rd Vol 73 Studies in Logic and the Foundations of Mathematics 3rd impression North-Holland

10 LTC Consistency How we know that LTC is a consistent theory? Standard answer: To build a model for LTC (Chang and Keisler 1992, theorem 1321) 2 domain model for LTC 2 C C Chang and H J Keisler (1992) Model Theory 3rd Vol 73 Studies in Logic and the Foundations of Mathematics 3rd impression North-Holland

11 Introduction to Domain Theory Motivation: Does λ-calculus have models? Historically my first model for the λ-calculus was discovered in 1969 and details were provided in Scott (1972) 1 (written in 1971) (Scott 1980, p 226) 2 1 D Scott (1972) Continuous lattices In: Lecture Notes in Mathematics 274, pp D Scott (1980) Lambda Calculus: Some Models, Some Philosophy In: The Kleene Symposium Ed by J Barwise, H J Keisler and K Kunen Vol 101 Studies in Logic and the Foundations of Mathematics North-Holland Publishing Company, pp

12 Introduction to Domain Theory Non-standard definitions pre-domain, domain, complete partial order (cpo), ω-cpo, bottomless ω-cpo, Scott s domain, Convention ḍomain ω-complete partial order

13 Introduction to Domain Theory Some bibliographic references G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press

14 Introduction to Domain Theory Some bibliographic references G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press J C Mitchell (1996) Foundations for Programming Languages The MIT Press

15 Introduction to Domain Theory Some bibliographic references G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press J C Mitchell (1996) Foundations for Programming Languages The MIT Press T Streicher (2006) Domain-Theoretic Foundations of Functional Programming World Scientific Publishing Co Pte Ltd

16 Introduction to Domain Theory Some bibliographic references G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press J C Mitchell (1996) Foundations for Programming Languages The MIT Press T Streicher (2006) Domain-Theoretic Foundations of Functional Programming World Scientific Publishing Co Pte Ltd G Plotkin (1992) Post-graduate Lecture Notes in Advance Domain Theory (incorporating the Pisa Notes ) Electronic edition prepared by Yugo Kashiwagi and Hidetaka Kondoh

17 Partially Ordered Sets Definition (Partially ordered set) A partially ordered set (poset) (D, ) is a set D on which the binary relation satisfies the following properties: x x x x y z x y y z x z x y x y y x x = y (reflexive) (transitive) (antisymmetry)

18 Partially Ordered Sets Definition (Partially ordered set) A partially ordered set (poset) (D, ) is a set D on which the binary relation satisfies the following properties: x x x x y z x y y z x z x y x y y x x = y (reflexive) (transitive) (antisymmetry) Examples (Z, ) is a poset Let a, b Z with a 0 The divisibility relation is defined by a b = c (ac = b) Then (Z +, ) is a poset (P(A), ) is a poset

19 Partially Ordered Sets Example Hasse diagram for the poset ({1, 2, 3, 4, 6, 8, 12}, )

20 Partially Ordered Sets Example Hasse diagram for the poset ({a, b, c}, ) {a, b, c} {a, c} {a, b} {b, c} {a} {c} {b}

21 Monotone Functions Definition (Monotone function) Let (D, ) and (D, ) be two posets A function f D D is monotone if x y x y f(x) f(y)

22 Notable Elements Definition (Upper bound) Let (D, ) be a poset and let A D Let u D be an element such that a u for all elements a A, then u is an upper bound of A Examples A = {a, b, c} Upper bounds: {e, f, j, h} A = {j, h} No upper bounds A = {a, c, d, f} Upper bounds: {f, h, j} g d b h a j f e c

23 Notable Elements Definition (Supremum or least upper bound) An element x is the supremum or the least upper bound of the subset A, denoted by A, if x is an upper bound that is less than every other upper bound of A Example A = {b, d, g} Upper bounds: {g, h} A = g g d b h a j f e c

24 ω-complete Partial Orders Definition (ω-chain) Let D = (D, ) be a poset A ω-chain of D is an increasing chain d d d, where d D

25 ω-complete Partial Orders Definition (ω-complete partial order) Let D = (D, ) be a poset The poset D is a ω-complete partial order (ω-cpo) if (Plotkin 1992) 3 1 There is a least element D, that is, x x The element is called bottom 2 For every ω-chain d d d, the least upper bound d D exists 3 G Plotkin (1992) Post-graduate Lecture Notes in Advance Domain Theory (incorporating the Pisa Notes ) Electronic edition prepared by Yugo Kashiwagi and Hidetaka Kondoh

26 ω-complete Partial Orders Definition (Lifted set) Let A be a set The symbol A denotes the ω-cpo whose elements A { } are ordered by x y, if and only if, x = or x = y (Mitchell 1996) 4 The ω-cpo A is called A lifted 4 J C Mitchell (1996) Foundations for Programming Languages The MIT Press

27 ω-complete Partial Orders Definition (Lifted set) Let A be a set The symbol A denotes the ω-cpo whose elements A { } are ordered by x y, if and only if, x = or x = y (Mitchell 1996) 4 The ω-cpo A is called A lifted Examples The lifted unit type and the lifted Booleans B () false true data () = () data Bool = True False 4 J C Mitchell (1996) Foundations for Programming Languages The MIT Press

28 ω-complete Partial Orders Example The lifted natural numbers N n

29 ω-complete Partial Orders Example The ω-cpo LN of lazy natural numbers arises from a non-strict successor function, that is, S( ) (Escardó 1993) n 2 S(S( )) 1 S( ) data Nat = Z S Nat 0 5 M H Escardó (1993) On lazy natural numbers with applications to computability theory and functional programming In: SIGACT News 24 (1), pp 61 67

30 ω-complete Partial Orders Definition (Continuous function) Let (D, ) and (D, ) be two ω-cpos A function f D D is continuous if (Plotkin 1992) 6 1 The function is monotone 2 The function preserves the least upper bounds of the ω-chains, that is, f(d ) = f( d ), for all ω-chains d d d 6 G Plotkin (1992) Post-graduate Lecture Notes in Advance Domain Theory (incorporating the Pisa Notes ) Electronic edition prepared by Yugo Kashiwagi and Hidetaka Kondoh

31 ω-complete Partial Orders Definition (Function space of continuous functions) Let (D, ) and (D, ) be two ω-cpos The function space of continuous functions is the set (Winskel 1994) 7 [D D ] = {f D D f is continous } 7 G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press

32 ω-complete Partial Orders Definition (Function space of continuous functions) Let (D, ) and (D, ) be two ω-cpos The function space of continuous functions is the set (Winskel 1994) 7 [D D ] = {f D D f is continous } Theorem The function space [D D ] is an ω-cpo [D D ] can be partially ordered point-wise by f g d D f(d) g(d) The bottom element is λx 7 G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press

33 ω-complete Partial Orders Definition Let f D D be a function, then f (d) = d, f + (d) = f(f (d)) Theorem (The Fixed-Point Theorem) Let (D, ) be an ω-cpo Given f [D D], then Fix(f) = f ( ), is the least fixed-point of f (Winskel 1994) 8, that is, d f(d) d Fix(f) d, f(fix(f)) = Fix(f) 8 G Winskel (1994) The Formal Semantics of Programming Languages An Introduction 2nd printing The MIT Press

34 ω-complete Partial Orders Definition (Coalesced sum) Let D = (D, ),, D = (D, ) be ω-cpos The coalesced sum, that is, disjoint union with bottom elements identified D D is the ω-cpo (Plotkin 1992) 9 with the order {(i, d) d D d } x y x = or i n d, d D d d x = (i, d) y = (i, d ) 9 G Plotkin (1992) Post-graduate Lecture Notes in Advance Domain Theory (incorporating the Pisa Notes ) Electronic edition prepared by Yugo Kashiwagi and Hidetaka Kondoh

35 ω-complete Partial Orders Associated with the coalesced sum are the injection functions in D D D if d =, in (d) = (i, d) otherwise

36 Domain Model for LTC Terms: The term language of LTC From domain theory it is known that a domain model for Terms, where self-application is allowed and where the terms will have values in the Booleans or the lazy natural numbers is a solution to the recursive domain equation (Plotkin 1992) 10 D B LN (D D) 10 G Plotkin (1992) Post-graduate Lecture Notes in Advance Domain Theory (incorporating the Pisa Notes ) Electronic edition prepared by Yugo Kashiwagi and Hidetaka Kondoh

37 Domain Model for LTC Notation Let D be a domain and let ρ be a valuation on D (a function from the set of variables to D) ρ(x d): the valuation which maps x to d and otherwise acts like ρ λxe: λ-abstraction on D

38 Domain Model for LTC Convention D: A solution to the recursive domain equation for LTC From terms to functions and viceverse The domain D comes equipped with the continuous functions (Barendregt 2004) 11 F D [D D], G [D D] D 11 H Barendregt (2004) The Lambda Calculus Its Syntax and Semantics 2nd Vol 103 Studies in Logic and the Foundations of Mathematics 6th impression Elsevier

39 Domain Model for LTC Interpretation function Terms D: (Based on Pitts (1994) 12 ) x = ρ(x), λxt = G(λd t ( ) ), t t = f( t ) if t = G(f), otherwise, fix xt = Fix(λd t ( ) ), true = true, false = false, if = G(if), 0 = 0, succ = G(succ), pred = G(pred), iszero = G(iszero), where 12 A M Pitts (1994) Computational Adequacy via Mixed Inductive Definitions In: Mathematical Foundations of Programming Semantics Ed by S Brookes, M Main, A Melton, M Mislove and D Schmidt Vol 802 LNCS Springer, pp 72 82

40 Domain Model for LTC we omit the use of the injection functions in, and the continuous functions if, succ, pred and iszero from D to D are defined by λxyx if(d) = λxyy if d = true, if d = false, otherwise, n + 1 succ(d) = n + 1 if d = n LN, if d = n LN, otherwise,

41 Domain Model for LTC 0 if d = 0, pred(d) = d if d = succ(d ), otherwise, true if d = 0, iszero(d) = false if d = succ(d ), otherwise

42 Domain Model for LTC If the LTC equality is interpreted as the equality in D, it is possible verify that the conversion and discrimination rules of LTC are satisfied in D

43 Bonus Slides

44 Monotone Functions Example (Counter-example of monotone function) halt N B true if n, halt(n) = false if n = Let n N Since n, and not necessarily halt( ) halt(n), that is, false true, the halt function is non-monotone (Schmidt 1986) D A Schmidt (1986) Denotational Semantics A Methodology for Language Development Allyn and Bacon, Inc

45 Continuous Functions Example (Monotone but non-continuous function 14 ) f [Bool] Bool f(xs) = False True if xs is finite, that is, the function f is non-continuous if xs = [False, False, ], otherwise Given d = [], d = [False], d = [False, False], we have f(d ) = False = f([false, False, ]) = f( d ), 14 behind_continuity_in_winskels/

46 Complete Partial Orders Definition (Directed set) Let D = (D, ) be a poset A subset X D is directed if X and x y X z X x z y z Definition (Complete partial order) Let D = (D, ) be a poset The poset D is a complete partial order (cpo) if (Barendregt 2004) 15 1 There is a least element D, that is, x x The element is called bottom 2 For every directed X D, the least upper bound X D exists 15 H Barendregt (2004) The Lambda Calculus Its Syntax and Semantics 2nd Vol 103 Studies in Logic and the Foundations of Mathematics 6th impression Elsevier

47 Complete Partial Orders Note The Scott domains are built from complete partial orders See, for example, Gunter and Scott (1990) C A Gunter and D S Scott (1990) Semantics Domains In: Handbook of Theoretical Computer Science Ed by J van Leeuwen Vol B Formal Models and Semantics The MIT Press Chap 12