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Outline 2 LP Correctness DD Summary On logic programming and locating errors in programs W lodzimierz Drabent Institute of Computer Science, Polish Academy of Sciences (IPI PAN); IDA, Link opings universitet, Sweden SaS seminar

  1. Outline 2 LP Correctness DD Summary On logic programming and locating errors in programs W� lodzimierz Drabent Institute of Computer Science, Polish Academy of Sciences (IPI PAN); IDA, Link¨ opings universitet, Sweden SaS seminar 2019-11-08 Version 1.0, compiled November 15, 2019 1 / 24

  2. Outline 2 LP Correctness DD Summary Outline ◮ Introduction to Logic Programming (LP) ◮ On proving program correctness (and completeness), i.e. how to reason about our programs ◮ Approximate specifications ◮ Declarative Diagnosis (DD) Why abandoned; a cure Inadequacy of Prolog debuggers ◮ Summary 2 / 24

  3. Outline 2 LP Correctness DD Summary Outline Logic Programming (LP) is declarative We can do declarative programming in Prolog Debugging should be declarative too Methods exist: Declarative Diagnosis (DD), a.k.a. algorithmic debugging [Shapiro’83,Pereira’86,Naish,...] ·· Tools do not ⌢ We discuss the (possibly) main reason for non-acceptance of DD 3 / 24

  4. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control Declarative programming WHAT to compute Program – a description of the problem not a description of computer actions Logic Programming Program – a set of axioms Results – its logical consequences Computation – proof construction Main programming language – Prolog 4 / 24

  5. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control Logic Programming (LP). The core part Program – a set of axioms ( of the form A 0 ← A 1 , . . . , A n A i – atoms (atomic formulae) ). Computation – search for logical consequences of the program. Query Q ( of the form A 1 , . . . , A n ). Answers Qθ such that P | = Qθ ( P – the program, θ – substitution). Any answer Q ′ computed for P is a logical consequence of P , P | = Q ′ . And conversely (if P | = Qθ then Qθ is an instance of a computed answer for Q ). Note: untyped logic 5 / 24

  6. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP, example, puzzle Build a sequence out of three 1’s, three 2’s, . . . , three 9’s, so that between each consecutive occurrences of i there are exactly i elements. [1,9,1,2,1,8,2,4,6,2,7,9,4,5,8,6,3,4,7,5,3,9,6,8,3,5,7] [1,8,1,9,1,5,2,6,7,2,8,5,2,9,6,4,7,5,3,8,4,6,3,9,7,4,3] [1,9,1,6,1,8,2,5,7,2,6,9,2,5,8,4,7,6,3,5,4,9,3,8,7,4,3] [3,4,7,8,3,9,4,5,3,6,7,4,8,5,2,9,6,2,7,5,2,8,1,6,1,9,1] [3,4,7,9,3,6,4,8,3,5,7,4,6,9,2,5,8,2,7,6,2,5,1,9,1,8,1] [7,5,3,8,6,9,3,5,7,4,3,6,8,5,4,9,7,2,6,4,2,8,1,2,1,9,1] 6 / 24

  7. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control Notation Variables in programs – begin with upper case – anonymous variable (each occurrence of – a distinct variable) [ a 1 , . . . , a n ] – list, its elements a 1 , . . . , a n ( n ≥ 0 ) [ ] – empty list [ h | t ] – the list with head h and tail t [ h 1 , h 2 | t ] – the list with head h 1 and tail [ h 2 | t ] , i.e. [ h 1 | [ h 2 | t ]] 7 / 24

  8. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control Notation Variables in programs – begin with upper case – anonymous variable (each occurrence of – a distinct variable) [ a 1 , . . . , a n ] – list, its elements a 1 , . . . , a n ( n ≥ 0 ) [ ] – empty list [ h | t ] – the list with head h and tail t [ h 1 , h 2 | t ] – the list with head h 1 and tail [ h 2 | t ] , i.e. [ h 1 | [ h 2 | t ]] 7 / 24

  9. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP, example, puzzle solution ( S ) ← sequence 27( S ) , sublist ( [1 , , 1 , , 1] , S ) , sublist ( [2 , , , 2 , , , 2] , S ) , sublist ( [3 , , , , 3 , , , , 3] , S ) , sublist ( [4 , , , , , 4 , , , , , 4] , S ) , sublist ( [5 , , , , , , 5 , , , , , , 5] , S ) , sublist ( [6 , , , , , , , 6 , , , , , , , 6] , S ) , sublist ( [7 , , , , , , , , 7 , , , , , , , , 7] , S ) , sublist ( [8 , , , , , , , , , 8 , , , , , , , , , 8] , S ) , sublist ( [9 , , , , , , , , , , 9 , , , , , , , , , , 9] , S ) . sublist ( Y , XYZ ) ← app ( , YZ , XYZ ) , app ( Y , , YZ ) . sequence 27([ , , , , , , , , , , , , , , , , , , , , , , , , , , ]) . app ( [ ] , L, L ) . app ( [ H | K ] , L, [ H | M ] ) ← app ( K, L, M ) . 8 / 24

  10. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control Y Z LP, example, puzzle � �� � X Y Z solution ( S ) ← � �� � XY Z sequence 27( S ) , sublist ( [1 , , 1 , , 1] , S ) , sublist ( [2 , , , 2 , , , 2] , S ) , sublist ( [3 , , , , 3 , , , , 3] , S ) , sublist ( [4 , , , , , 4 , , , , , 4] , S ) , sublist ( [5 , , , , , , 5 , , , , , , 5] , S ) , sublist ( [6 , , , , , , , 6 , , , , , , , 6] , S ) , sublist ( [7 , , , , , , , , 7 , , , , , , , , 7] , S ) , sublist ( [8 , , , , , , , , , 8 , , , , , , , , , 8] , S ) , sublist ( [9 , , , , , , , , , , 9 , , , , , , , , , , 9] , S ) . sublist ( Y , XYZ ) ← app ( , YZ , XYZ ) , app ( Y , , YZ ) . sequence 27([ , , , , , , , , , , , , , , , , , , , , , , , , , , ]) . app ( [ ] , L, L ) . app ( [ H | K ] , L, [ H | M ] ) ← app ( K, L, M ) . 8 / 24

  11. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP, example, puzzle solution ( S ) ← sequence 27( S ) , H K L sublist ( [1 , , 1 , , 1] , S ) , � �� � sublist ( [2 , , , 2 , , , 2] , S ) , M sublist ( [3 , , , , 3 , , , , 3] , S ) , sublist ( [4 , , , , , 4 , , , , , 4] , S ) , sublist ( [5 , , , , , , 5 , , , , , , 5] , S ) , sublist ( [6 , , , , , , , 6 , , , , , , , 6] , S ) , sublist ( [7 , , , , , , , , 7 , , , , , , , , 7] , S ) , sublist ( [8 , , , , , , , , , 8 , , , , , , , , , 8] , S ) , sublist ( [9 , , , , , , , , , , 9 , , , , , , , , , , 9] , S ) . sublist ( Y , XYZ ) ← app ( , YZ , XYZ ) , app ( Y , , YZ ) . sequence 27([ , , , , , , , , , , , , , , , , , , , , , , , , , , ]) . app ( [ ] , L, L ) . app ( [ H | K ] , L, [ H | M ] ) ← app ( K, L, M ) . 8 / 24

  12. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP. Two levels of reading a program declarative – a set of axioms, operational – a description of computations. ALGORITHM = LOGIC + CONTROL [Robert Kowalski, 1974] Operational level (prog. lang. Prolog): control information (the ordering within the program, some special constructs). Important: The two levels can be considered separately. ☞ Program correctness is a property of the declarative level. 9 / 24

  13. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP. Two levels of reading a program declarative – a set of axioms, operational – a description of computations. ALGORITHM = LOGIC + CONTROL [Robert Kowalski, 1974] Operational level (prog. lang. Prolog): control information (the ordering within the program, some special constructs). Important: The two levels can be considered separately. ☞ Program correctness is a property of the declarative level. We do not need to reason in terms of von Neumann machine. J.Backus, Can programming be liberated from the von Neumann style? CACM, 1978 (One may also program operationally, neglecting the 1st level.) 9 / 24

  14. Outline 2 LP Correctness DD Summary Ex. ex.program Logic+control LP. Two levels of reading a program declarative – a set of axioms, operational – a description of computations. ALGORITHM = LOGIC + CONTROL [Robert Kowalski, 1974] Operational level (prog. lang. Prolog): control information (the ordering within the program, some special constructs). Important, often neglected: The two levels can be considered separately. ☞ Program correctness is a property of the declarative level. We do not need to reason in terms of von Neumann machine. J.Backus, Can programming be liberated from the von Neumann style? CACM, 1978 (One may also program operationally, neglecting the 1st level.) 9 / 24

  15. Outline 2 LP Correctness DD Summary method 1 method 2 Approximate... Program correctness How to reason about program results ? Imperative programming: partial correctness + termination 10 / 24

  16. Outline 2 LP Correctness DD Summary method 1 method 2 Approximate... Program correctness How to reason about program results ? Imperative programming: partial correctness + termination Correctness – the program answers compatible with the specification Completeness – all the required answers will be produced (by the specification) 10 / 24

  17. Outline 2 LP Correctness DD Summary method 1 method 2 Approximate... Program correctness How to reason about program results ? Imperative programming: partial correctness + termination ւ ց LP : correctness completeness Correctness – the program answers compatible with the specification Completeness – all the required answers will be produced (by the specification) 10 / 24

  18. Outline 2 LP Correctness DD Summary method 1 method 2 Approximate... Program correctness How to reason about program results ? Imperative programming: partial correctness + termination ւ ց LP : correctness completeness full correctness (?) Correctness – the program answers compatible with the specification Completeness – all the required answers will be produced (by the specification) 10 / 24

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