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Inference system interpretation

The inferential interpretation of a quadruple is as follows. Think of a system of first order inference steps, in which elements of the set A have the role of judgements, which are the premises and conclusions of first-order inference steps. (A first-order inference step is one without local constants, and in which no hypotheses are discharged. In general, the premise of a rule in which subdeductions may use local constants and hypotheses are not basic judgements.)

First-order inference steps are the simplest kind, having as a special case steps with no premises (a particularly severe cardinality restriction). However, to make up for this deficiency, there are no cardinality restrictions on how many premises an inference step requires, so that one may have infinitary inference steps.

The inference steps of such a rule system can be depicted as follows.


\begin{picture}(45,30)(0,0)
\put(10,25){\makebox(0,0){\ldots}}
\put(20,15){\line...
...(41,10){\makebox(0,0)[l]{$b$ }}
\put(20,6){\makebox(0,0)[b]{$a$ }}
\end{picture}

A map between rule systems is a way of translating formulas of the domain to formulas of the codomain, in such a way that a proof (partial or total) in the codomain of the translation of a formula can be systematically translated back, in a top-down fashion, to a proof of the formula in the domain. It is a `conservative extension' of the domain.

There is a correspondence between the inference-system and the state-machine interpretations. Given an inference system, there is a `sceptical' state-machine in which the states are judgements, the inputs are (codes for) inference steps, and each output is a premise location within the immediately preceding rule, the point at which further sceptical enquiry is to continue. You tell the machine you have a proof of the initial judgement, and it calls your bluff. Here is a log of the machine's operations, starting in state a1.

state input output
a1 b1 : B(a1) c1 : C(a1, b1)
a2 = d(a1, b1, c1) b2 : B(a2) c2 : C(a2, b2)
a3 = ... ... ...
an = ... bn : B(an) $ \mbox{(awaited)} $
The machine asks you to describe a path in your so-called proof, at each stage asking a question of the form `How is the cn-th premise of step bn proved?', and getting an answer of the form `By step bn+1'. It may happen that at some stage there are no further questions to be asked, as the last step required no premises. If you really have a proof, this will always happen. If not, well you might get away with it, but you've only yourself to blame if you're found out, or the skeptic keeps asking you questions forever. Given a state-machine, and a state a, there is a set of sets of states $Bar(a) = \{ \{ d^\ast(a,b,c) \vert c : C^\ast(a,b) \} \vert b : B^\ast(a) \}$ which `bar' a, in the sense that if we start in state a and follow strategy b, the machine's escape is barred by states in the set $\{d^\ast(a,b,c) \vert c : C^\ast(a,b) \}$. (Road blocks can be set there) In the corresponding inference system, the judgements can be thought of as saying that such and such a state is barred by any set, which is equivalent to saying that it is barred by the empty set.

belongs with stuff at end of accessibility


next up previous
Next: Topological interpretation Up: Partial constructions Previous: State machine interpretation
Peter Hancock
1998-04-08