A Formal Approach to Problem Solving.* - Universiteit Gent

A Formal Approach to Problem Solving.? Diderik Batens Centre for Logic and Philosophy of Science Universiteit Gent, Belgium [email protected]

1

Aim of this paper

For more than forty years, problem solving is recognized as central for the understanding of reasoning in the sciences. In this paper a formal approach to problem solving is spelled out. The essential aim is to devise a formal procedure that can function as an explication of a problem-solving process—Sects. 2–5. Touchy topics in this connection are (i) the relation to the relevant philosophy of science literature and (ii) the role of erotetic logic in the approach. For (i) I shall rely on the approach developed by Thomas Nickles ([19], [20], [21], [22]), on the model developed by myself ([1], [2] and [3]) and on the work of Joke Meheus ([14], [15], [16], [17], [18]). A particular difficulty concerns the function of constraints in a formal approach to problem solving. The relation can be shown to be sensible but cannot be discussed here. As for (ii), previous work, for example [10], suggests that the required erotetic logic differs in central respects from the ‘standard approach’ ([24], [25], [26]). However, it can be shown that all derivations of (sets of) questions in terms of instruction DP (Sect. 2) are derivable (possibly in several steps) according to the standard approach. A problem-solving process is clearly different from a proof—for one thing, problem-solving processes may be unsuccessful. As “proof” is used to denote both the explicanda in natural language and their explicata in a formal language, I shall use “problem-solving process”—psp for short—to denote the explicata as well as the explicanda. Two features seem essential to psps. First, some of their parts may be justified at a stage of the process, but may turn out dead ends at a later stage. The formal approach to this feature is familiar from the dynamic proofs of adaptive logics—see for example [4] or [5]. Next, realistic psps require ‘prospective’ steps: if A1 , . . . , An are true, the solution of problem P is B. Such considerations enable one to proceed in a goal-directed way: to tackle the problem P by first solving the sequence of problems “Is A1 true?”, . . . , “Is An true?”. The formal approach to this feature is provided by the prospective proofs studied in [11], [6] and other ?

This abstract is based on technical report [7] on which I received interesting comments from Kristof De Clercq, Andrzej Wi´sniewski and others. The research was supported by the Fund for Scientific Research – Flanders, by the Royal Flemish Academy of Belgium for Science and the Arts, and indirectly by the Flemish Minister responsible for Science and Technology (contract BIL01/80).

papers. Such prospective steps are sometimes deemed “proof heuristics”, for example by Hintikka in [13], and thus expelled from the realm of logic proper. In my view this is a serious mistake, but I cannot argue for this view in the present context. Given the generality of the notion of problem solving, a psp should be able to incorporate most logics, and especially most adaptive logics developed so far. I can only provide some hints in this respect for the reader unfamiliar with adaptive logics. There is no positive test1 for many actual reasoning processes—see [5] for a host of examples. Hence, those reasoning processes cannot be explicated by proofs of the traditional sort—(of which any initial fragment is itself a proof). Adaptive logics circumvent this limitation by having dynamic proofs in which a formula (a line in an annotated proof) may be introduced at some stage of a proof but may be marked (and hence considered as not derived) at a later stage. By doing so, dynamic proofs enable one to explicate reasoning processes that range from interpreting a set of premises as consistently as possible to deriving inductive generalizations or inductive predictions. To keep this paper within bounds, I shall restrict psps to a special case: solving the problem whether A or ∼A is derivable by CL (Classical Logic) from (a consistent subset of) the premises (Sects. 2–3), possibly extended with the answers to questions that may be directly answered by standard means (Sect. 4). This allows, for example, for the explication of predictive problems from data and theories—a problem falling beyond the scope of the present paper but easily solved in view of the sections on background knowledge from [9]. The way in which the restriction may be removed is suggested in Sect. 5. My approach is computational in that it defines a problem-solving procedure, which is intended to explicate (a specific form of) human reasoning.2 I shall not present a completely deterministic version of it. To do so would take away the attention from the basic mechanisms. Moreover, the efficiency of the possible specifications of the procedure has not been sufficiently studied to obtain a final judgement on the matter.

2

The Elements of a Psp

A stage of a psp is a sequence of lines; a psp is a chain of stages. The first stage consists of a single line. Where Si and Si+1 are subsequent stages, Si+1 is obtained by adding one line to Si and possibly changing the marks—see below— of the lines of Si+1 . The instructions determine which lines may be added to Si in view of Si , viz. in view of the formulas that occur in unmarked lines of Si . The instructions obviously depend on the underlying logic or logics.3 1 2 3

See [12] on decidability, positive test, and other computational matters. Hence results on logic programming are at best indirectly relevant. To save some space, I describe the rules as part of the instructions. A more general and more systematic outlook considers the rules as given and sees the instructions as specifying whether a rule may or should be applied in view of the stage of the psp.

Two kinds of lines will occur in a psp: problem lines and declarative lines. In the narrow sense, a problem is a non-empty set of questions. For the wider sense I refer to the aforementioned philosophy of science literature. In this paper, the main problem will be a single yes–no question ?{M, ∼M }, and all derived problems will be sets of yes–no questions. A and ∼A will be called the direct answers of the question ?{A, ∼A}. Γ will always be the premise set. The formula of a declarative line may be a member of Γ or a formula derived from members of Γ . It may also have the form [B1 , . . . , Bn ] A which indicates that Γ ∪ {B1 , . . . , Bn } ` A. {B1 , . . . , Bn } will be called the condition of A. If A is the formula of a declarative line, this formula is said to be conditionless—the condition is then also said to be empty. Given a problem P, the problem solver has to chose a direct answer of a question of the problem as a target (from P), which is noted on a target line.4 A target is a formula the problem solver tries to establish. There is no objection against introducing several targets at once. A target A will be written as [A] A which is logically redundant but guides the procedure, as we shall see. As the psp proceeds, some (declarative lines as well as question) lines may become useless for solving the main problem. Such lines will be marked. I now turn to the formal machinery. First some preparatory definitions. Let us distinguish α-formulas from β-formulas (varying on a theme from [23]) and assign to each formula two other formulas according to the following table: α α1 α2 β β1 β2 A∧B A B ∼(A ∧ B) ∼A ∼B A ≡ B A ⊃ B B ⊃ A ∼(A ≡ B) ∼(A ⊃ B) ∼(B ⊃ A) ∼(A ∨ B) ∼A ∼B A∨B A B ∼(A ⊃ B) A ∼B A⊃B ∼A B ∼∼A A A The positive part relation is defined recursively by the following three clauses: 1. pp(A, A). 2. pp(A, α) if pp(A, α1 ) or pp(A, α2 ). 3. pp(A, β) if pp(A, β1 ) or pp(A, β2 ). The instructions These rely on the prospective rules from [11]. Γ is the premise set. A psp starts with an application of Main, which introduces the main problem ?{M, ∼M }. In all instructions, k denotes a suitable line number. 4

The formal machinery may be defined without target lines, but the latter greatly clarify the goal-directed character of psps.

Main

Start a psp with the line: 1

{?{M, ∼M }}

Main

Target If P is the problem of an unmarked problem line, and A is a direct answer of a member of P, then one may add: k Prem

[A] A

Target

If A is an unmarked target,5 B ∈ Γ , and pp(A, B), then one may add: k

B

Prem

Let ∗A denote the ‘complement’ of A, viz. B if A has the form ∼B and ∼A otherwise. The formula analysing rules of CL (see [11]) may be summarized as follows:6 [∆] α [∆] β [∆] α1 [∆] α2 [∆, ∗β2 ] β1 [∆, ∗β1 ] β2 The general form of the rules is [∆] A / [∆ ∪ ∆0 ] B. The following instruction handles their application: FAR

If C is an unmarked target, [∆] A is the formula of an unmarked line i, [∆] A / [∆ ∪ ∆0 ] B is a formula analysing rule, and pp(C, B), then one may add: k

[∆ ∪ ∆0 ] B

i; R

in which R is the name of the formula analysing rule. The condition analysing rules of CL are summarized by: [∆ ∪ {β}] A [∆ ∪ {α}] A [∆ ∪ {α1 , α2 }] A [∆ ∪ {β1 }] A [∆ ∪ {β2 }] A They all have the form [∆ ∪ {B}] A / [∆ ∪ ∆0 ] A. The following instruction refers to them: CAR

If A is an unmarked target, [∆ ∪ {B}] A is the formula of an unmarked line i, and [∆ ∪ {B}] A / [∆ ∪ ∆0 ] A is a condition analysing rule, then one may add: k

[∆ ∪ ∆0 ] A

i; R

in which R is the name of the condition analysing rule. 5 6

Phrased more precisely: A is the formula of an unmarked target line. The left rule states that both [∆] α1 and [∆] α2 may be derived (separately) from [∆] α (notational abuse here and in the text).

The instructions EM (excluded middle) and EM0 allow one to eliminate certain problems without answering them. Line i (either line i or line j) will be R-marked after an application of EM0 (EM)—see below for marking. EM0

If [∆ ∪ {∗A}] A is the formula of a line i that is neither R-marked nor I-marked, then one may add: k

EM

[∆] A

i; EM0

If A is an unmarked target, [∆ ∪ {B}] A and [∆0 ∪ {∼B}] A are the respective formulas of the unmarked or only D-marked lines i and j, and ∆ ⊆ ∆0 or ∆0 ⊆ ∆, then one may add: k

[∆ ∪ ∆0 ] A

i, j; EM

Transitivity is essential for eliminating solved questions from an (at least implicit) problem as well as for summarizing the remaining problems (and paths) in a problem-solving process. If ∆0 ⊆ ∆, line i is R-marked after the application of Trans. Trans

If A is an unmarked target, and [∆∪{B}] A and [∆0 ] B are the respective formulas of the at most S-marked7 lines i and j, then one may add: k

[∆ ∪ ∆0 ] A

i, j; Trans

The last instruction handles derived problems: DP

If A is an unmarked target from problem line i and [B1 , . . . , Bn ] A is the formula of an unmarked line j, then one may add: k

{?{B1 , ∼B1 }, . . . , ?{Bn , ∼Bn }}

i, j; DP

In view of the intended applications (deriving predictions, explanations, etc.) the procedural system has no instruction for applying EFQ—the paraconsistent procedural variant logic of CL is studied in [8]. Marking It is necessary to introduce several kinds of marks, which have distinct effects on the procedure. Each kind is governed by a definition, which applies to stages of the psp—marks may come and go with each new stage. The fist kind of mark indicates that a line is redundant. Definition 1. An at most S-marked declarative line i that has [∆] A as its formula is R-marked at a stage iff, at that stage, [Θ] A is the formula of a line for some Θ ⊂ ∆.8 Definition 2. An unmarked problem line i is R-marked at a stage iff, at that stage, a direct answer A of a question of line i is the formula of a line. 7 8

A line is at most S-marked iff it is not R-marked, not I-marked and not D-marked. A formula A is identified with [∅] A.

The next marking definitions require some terminology. Remember that a target is a target from a problem line (and from its problem). If A is a target, any line in which [∆] A is derived for some ∆ 6= {A} will be called a resolution line (for target A). The line called j in instruction DP will be said to generate the problem line introduced by DP. A is a direct target from [∆] B iff [∆] B is the formula of the resolution line that generates problem P, A is a target from P, and A ∈ ∆—remark that some targets from P are not members of ∆. A target sequence is a sequence h[∆1 ] A1 , . . . , [∆n ] An i in which every Ai+1 (1 ≤ i < n) is a direct target from [∆i ] Ai . A target sequence h[∆1 ] A1 , . . . , [∆n ] An i is grounded iff A1 is not a direct target from any unmarked [Θ] B derived in the psp. A set is flatly inconsistent iff it contains A as well as ∼A for some A. I-marked lines are inoperative: acting on the line is not useful for solving any extant problem. Definition 3. An at most S-marked target line that has [A] A as its formula is I-marked at a stage iff every problem line from which A is a target is marked at that stage. Definition 4. An at most S-marked resolution line of which [∆1 ] A1 is the formula for some ∆1 6= ∅ is I-marked at a stage iff, at that stage, for every grounded target sequence h[∆n ] An , . . . , [∆1 ] A1 i, (i) some target [Ai ] Ai (1 ≤ i ≤ n) is marked, or (ii) {An , . . . , A1 } ∩ ∆1 6= ∅, or (iii) ∆1 ∪ . . . ∪ ∆n ∪ Γs◦ is flatly inconsistent. Definition 5. An unmarked problem line is I-marked iff no unmarked resolution line generates it. Dead end marks indicate that no further action can be taken in view of a line. A is a dead end iff A is (in the present propositional context) a literal and A is not a positive part of a premise. If A is not a literal, then CAR leads from the unmarked [∆ ∪ {A}] B to one or more [∆ ∪ ∆0 ] B. The latter is called a CAR-descendant of [∆ ∪ {A}] B. Definition 6. An at most S-marked resolution line with formula [∆] A is Dmarked at a stage iff some B ∈ ∆ is a dead end or, at that stage, all CARdescendants of [∆] A occur in the psp and are D-marked. Definition 7. An at most S-marked target line with formula [A] A is D-marked at a stage iff A is a dead end or no further action can be taken in view of target A. Whether an action that can be taken in view of target A is obvious in view of the instructions. If no further action can be taken in view of target A, all instructions that could be applied in view of that target were taken and resulted in lines that are R-marked, I-marked or D-marked. The proofs from [11] may be easily adapted to show that, for all consistent Γ , if Γ ` A, then the procedure applied to Γ and {?{A, ∼A}} results in the

answer A, and if Γ 0 A, then the procedure applied to Γ and {?{A, ∼A}} stops without the main problem being answered or results in the answer ∼A. The procedure may be sped up (in a way that derives from a straightforward insight in the psp) by S-marks. Let Γs◦ be the union of the set of premises and the set of conditionless formulas that occur at stage s of the psp. Definition 8. A R-unmarked resolution line in which [∆1 ] A1 is derived is Smarked iff (i) ∆1 ∩ Γs◦ 6= ∅, or (ii) for some target sequence h[∆n ] An , . . . , [∆1 ] A1 i, {An } ∪ ∆1 is flatly inconsistent whereas ∆1 is not flatly inconsistent, or (iii) ∆1 ⊂ ∆n ∪ . . . ∪ ∆2 for some target sequence h[∆n ] An , . . . , [∆1 ] A1 i.

3

An example

I can only offer an extremely simple example, but the reader can easily explore other examples. Let ?{p ∨ q, ∼(p ∨ q)} be the main problem and let {∼s, ∼u ∨ r, (r ∧ t) ∨ s, (q ∨ u) ⊃ (∼t ∨ q), t ⊃ u} be the premise set. The superscripts of the marks name the stage at which the line is (thus) marked—this convention saves rewriting. 1 2 3

{?{p ∨ q, ∼(p ∨ q)}} [∼(p ∨ q)] ∼(p ∨ q) [∼p, ∼q)] ∼(p ∨ q)

Main Target 2; C∼∨E

D3 D3

As ∼p is not a positive part of any premise, line 3 is D-marked, and hence so is line 2. 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

[p ∨ q] p ∨ q [p] p ∨ q [q] p ∨ q {?{q, ∼q}} [q] q (q ∨ u) ⊃ (∼t ∨ q) [q ∨ u] ∼t ∨ q [q] ∼t ∨ q [q, t] q [u] ∼t ∨ q [u, t] q {?{u, ∼u}, ?{t, ∼t}} [t] t (r ∧ t) ∨ s [∼s] r ∧ t [∼s] t {?{s, ∼s}} [∼s] ∼s ∼s

Target 4; C∨E 4; C∨E 4, 6; DP Target Prem 9; ⊃E 10; C∨E 11; ∨E 10; C∨E 13; ∨E 8, 14; DP Target Prem 17; ∨E 18; ∧E 16, 19; DP Target Prem

D5

I12

S22 R22 R22

The S-mark of line 19 indicates that Trans can be applied. I repeat part of the psp: 14 15 16 17 18 19 20 21 22 23

[u, t] q {?{u, ∼u}, ?{t, ∼t}} [t] t (r ∧ t) ∨ s [∼s] r ∧ t [∼s] t {?{s, ∼s}} [∼s] ∼s ∼s t

13; ∨E 8, 14; DP Target Prem 17; ∨E 18; ∧E 16, 19; DP Target Prem 19, 22; Trans

S23 R23 R23 S22 R23 R22 R22

The S-mark of line 14 again indicates how to proceed. Once line 24 is added to the psp, line 14 is R-marked (“R24 ” is added to it on the present convention). 24 25 26 27 28 29 30

[u] q {?{u, ∼u}} [u] u t⊃u [t] u u q

14, 23; Trans 8, 24; DP Target Prem 27; ⊃E 23, 28; Trans 24, 29; Trans

S29 R30 R29 R29 S28 R29

Lines 14 and 15 are R-marked at stage 24. The S-mark of line 24 triggers Trans (line 30). Several marks are added at stage 30, most importantly, line 6 is Smarked, which leads to 31

p∨q

6, 30; Trans

whence the problem is solved (and line 1 is R-marked). This is not the shortest way to derive an answer to ?{p ∨ q, ∼(p ∨ q)} from the premise set. However, (most of) the heuristics is pushed into the psp. Moreover, each step of the psp is sensible in view of the previous stage—remark, for example, ∼u ∨ r, [∼s] r, etc. are not the formula of a line in any psp for this main problem and premises.

4

Answerable Questions

A straightforward extension of the described procedure introduces, next to the main problem and the premises, a set A of questions that can be directly answered by standard means, for example questions that are answered (obviously at a cost) by observational or experimental means. The procedure should decide when such a question may be ‘launched’, and hence when its answer surfaces in the psp as a new premise. This simple extension drastically enhances the realistic

character of psps and is a first step away from psps that reduce to proof search devices for CL. Whether a question ?{A, ∼A} ∈ A is launched seems to depend on pragmatic considerations. If the expense of directly answering the main problem, ?{M, ∼M }, is near to nil, to do so may be preferred over engaging in logical deduction. If the expense is extremely high, one may renounce from seeking a direct answer, even where this entails that the main problem is left unsolved. However, such considerations do also affect logical deduction, and hence will be neglected in the present context. It should not be required that A be an unmarked target in order for ?{A, ∼A} ∈ A to be launched. Thus a target p may justify that ?{p ∧ q, ∼(p ∧ q)} is launched. If the answer is p ∧ q, the target is reached. A target p may also justify launching ?{p ∨ q, ∼(p ∨ q)}. The answer p ∨ q leads to [∼q] p, from which p is obtained if ∼q is reached by deduction or by launching other members of A. Where p is the target and ?{p ∧ q, ∼(p ∧ q)} is launched, the answer ∼(p ∧ q) is useless for reaching p, but may be useful for reaching ∼p, which forcibly is a possible target. All this suggest that a member of A may be launched iff an unmarked target B is a positive part of A or of ∼A. A launched question is answered ‘outside of’ the psp, but its answer enters the psp. It does so with the status of a new premise. As any instruction handling this would be awkward from a logical point of view, it is advisable to redefine A as a set of couples: their first element is a question, their second element a direct answer to this question—in other words, each couple has the form h?{A, ∼A}, Ai or the form h?{A, ∼A}, ∼Ai.9 That the second element is unknown to the problem solver until the question ?{A, ∼A} is launched, is captured by the fact that the instruction than handles the introduction of new premises (second elements of members of A) depends only on the stage of the psp and the first element of the member of A. This suggests the following instruction: New

If A is the target of the unmarked target line i, pp(A, B) or pp(A, ∼B), and h?{B, ∼B}, Ci ∈ A (where C ∈ {B, ∼B}), then one may add: k

C

i; New

It is obvious from a line justified by New which member of A was launched and in view of which target it was launched. The present extension requires that one redefines: A is a dead end iff A is (in the present propositional context) a literal and A is not a positive part of a premise or of a direct answer to the first element of a member of A. This redefinition has direct effects on I-marking. Let us consider an example. The main problem is {?{p, ∼p}}, the premise set contains (q ∧ r) ⊃ p, ∼s ∨ q, s, and h?{q ⊃ r, ∼(q ⊃ r)}, q ⊃ ri ∈ A. 9

I presuppose consistency in the present paper (up to this section). So the second elements of the members of A should form a consistent set. I do however use a formulation that is sufficiently general to be adjusted to the inconsistent case. Moreover, even if the union of the set of premises and the set of new premises is inconsistent, the present psps do not lead to triviality, as may be seen from [8].

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

{?{p, ∼p}} [p] p (q ∧ r) ⊃ p [q ∧ r] p [q, r] p {?{q, ∼q}, ?{r, ∼r}} [r] r q⊃r [q] r [q] p {?{q, ∼q}} [q] q ∼s ∨ q [s] q {?{s, ∼s}} [s] s s q p

Main Target Prem 3; ⊃E 4; C∧E 2, 5; DP Target 7; New 8; C⊃E 5, 9; Trans 2, 10; DP Target Prem 13; ∨E 12, 14; DP Target Prem 14, 17; Trans 10, 18; Trans

R19 R19 R19 S9 R10 I10 I10 I10 R19 R18 R18 S14 R18 R17 R17

The application of Trans in line 10 is triggered by the S-mark of line 5 and has a rather dramatic effect: five lines are marked at stage 10.

5

Variants and Extensions

Instead of deriving sets of questions by DP, one might allow for, or even restrict the procedure to, the derivation of singleton problems. Problem lines are then more easily unmarked—problems are more easily ‘reused’ at later stages.10 Many other variants of the specific instructions are possible. More interesting variants are obtained by modifying the underlying logic. They are not discussed here because all realistic applications require the predicative level. Moreover, many different (combinations of) logics may be required, depending on the context. Thus CL may be replaced by an inconsistencyadaptive logic in order to explicate psps (as explicanda) which presuppose that the premises are interpreted as consistently as possible. Similarly, if CL is replaced by an adaptive logic of inductive generalization (see for example [9]), the psps will be able to answer problems of the form ?{(∀α)A, ∼(∀α)A} in view of data and (possibly) background knowledge.11 The procedure from Sect. 2 may also be transformed to a proof-search method for full CL by introducing instructions for the predicative matters together with an instruction handling the EFQ rule from [11]. 10

11

Compare this to: according to Definition 3, a target line may be unmarked at stage s although it was introduced in view of a question line that is marked at stage s. It is obviously determined by the underlying logic (or logics) whether a set of premises together with the set of directly answerable questions and their answers is sufficient to answer a specific problem.

6

Some Comments

It can be shown (i) that the procedure leads from a premise set Γ to a solution of {?{M, ∼M }} (for contingent M ) iff there is a consistent Γ 0 ⊆ Γ such that Γ 0 `CL M or Γ 0 `CL ∼M and (ii) that the procedure stops iff this condition is not fulfilled. This is exactly what one needs with respect to the intended domain of application. The extension to the predicate level is as expected. It can also be shown that there is a specific sense in which each line in a psp (as defined by the present procedure) is (at the stage at which it is introduced) sensible with respect to the solution of the main problem. The procedure is not maximally efficient. The intention was to define a procedure that (i) can explicate actual problem-solving processes, (ii) avoids steps that are useless with respect to the main problem, the premises and the directly answerable questions (see the last paragraph of Sect. 3), and (iii) can be easily generalized to other logics. Especially adaptive logics (maximal consistent interpretations, inductive generalization, abduction, . . . ) are interesting in this respect because they arguably are the underlying logics of actual problem-solving processes. The adjustment of the procedure to such logics is straightforward, but fully beyond the scope of the present paper. Apparently any logic can be defined by a set of prospective rules of inference. These rules are then handled by instructions that are determined by the sensible relations between premise sets and conclusions. Finally, the ideas underlying each kind of mark are independent of CL and can be adjusted to other logics. I do not think that the described procedure is even maximally efficient in its kind. It may possibly be made more efficient by modifying the marking definitions or by introducing further kinds of marks. The subject clearly requires for further research. Moreover, some problem-solving processes (explicanda) may require logics that are very different from CL. However, such logics are within reach, as adaptive logics and their dynamic proofs illustrate. In other words, what is required are different logics. The requirement does not undermine the present plot, which boils down to this: turning the suitable logics into adequate proof-search procedures provides the formal road to problem solving.

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